diff --git a/Code/Makefile b/Code/Makefile
index be2aafb9376d8cf9bacf06f9f534ccdd5790507d..b0ae417a58b95716995e0b6a4fef39e92501d812 100644
--- a/Code/Makefile
+++ b/Code/Makefile
@@ -41,7 +41,7 @@ mpi-omp : export EXEC_SKY = sky3d.mpi-omp
#
seq omp mpi debug seq_debug mpi_debug omp_debug mpi-omp ifort ifort_seq ifort_omp:
@$(MAKE) -f $(firstword $(MAKEFILE_LIST)) makeit
-
+
clean:
-@rm -f *.o *.mod 2>/dev/null || true
@@ -100,7 +100,7 @@ grids.o : grids.f90 params.o forces.o Makefile
densities.o : densities.f90 params.o grids.o levels.o trivial.o Makefile
-fragments.o : fragments.f90 params.o grids.o forces.o levels.o twobody.o \
+fragments.o : fragments.f90 params.o grids.o forces.o levels.o \
$(PARALLEL_SKY) Makefile
twobody.o : twobody.f90 params.o grids.o densities.o moment.o forces.o Makefile
diff --git a/Code/coulomb.f90 b/Code/coulomb.f90
index 101e0dfc0cdf44a39aa8db1b32c1e51865172268..5c111cc069012840e0761aafcd19a6ceaffa177b 100644
--- a/Code/coulomb.f90
+++ b/Code/coulomb.f90
@@ -148,12 +148,13 @@ CONTAINS
!!transform of \f$1/r\f$, but to compute it on the actual grid the same way
!!as the densities and potentials are transformed
!---------------------------------------------------------------------------
- SUBROUTINE poisson
+ SUBROUTINE poisson(rhocharge)
+ REAL(db),INTENT(IN) :: rhocharge(nx,ny,nz)
COMPLEX(db),ALLOCATABLE :: rho2(:,:,:)
ALLOCATE(rho2(nx2,ny2,nz2))
! put proton density into array of same or double size, zeroing rest
IF(.NOT.periodic) rho2=(0.D0,0.D0)
- rho2(1:nx,1:ny,1:nz)=rho(:,:,:,2)
+ rho2(1:nx,1:ny,1:nz)=rhocharge(:,:,:)
! transform into momentum space
CALL dfftw_execute_dft(coulplan1,rho2,rho2)
! add charge factor and geometric factors
@@ -217,7 +218,7 @@ CONTAINS
q(1,1,1)=(0.D0,0.D0)
ELSE
q=1.D0/SQRT(REAL(q))
- q(1,1,1)=2.84D0/(dx*dy*dz)**(1.D0/3.D0)
+ q(1,1,1)=2.84D0*3.D0/(dx+dy+dz)
CALL dfftw_execute_dft(coulplan1,q,q)
END IF
DEALLOCATE(iqx,iqy,iqz)
diff --git a/Code/dynamic.f90 b/Code/dynamic.f90
index ad313d3b25accb508a5a2069aa61730e4ff1f88c..57865b35e7cfdcb9d363de13ece82dc4a351b62c 100644
--- a/Code/dynamic.f90
+++ b/Code/dynamic.f90
@@ -15,7 +15,7 @@ MODULE DYNAMIC
USE Levels
USE Energies
USE Moment
- USE Twobody, ONLY: twobody_case,istwobody,roft,rdot, twobody_print
+ USE Twobody, ONLY: twobody_analysis,istwobody,roft,roft_old
USE Parallel
USE Meanfield, ONLY: skyrme, hpsi, spot
USE Trivial, ONLY: overlap
@@ -371,7 +371,8 @@ CONTAINS
!! a full printout (determined by \c mprint).
!! - <b> Step 2: twobody analysis </b> the twobody analysis is
!! performed, but only if the calculation started as a twobody
-!! scenario.
+!! scenario. This is the simplified version which essentially only
+!! determines the fragment in case of separation.
!! - <b> Step 3: moments: </b> the moments of the distribution are
!! calculated using subroutine \c moments. This includes total mass,
!! momenta, and angular momenta. They are printed out if indicated by
@@ -408,25 +409,16 @@ CONTAINS
!! \c plot_densities and the binary densities are written onto <tt> *.tdd </tt>
!! files using \c write_densities.
!! - <b> Step 7: other output: </b> in the proper \c mprint interval
-!! the two-body analysis results, the single-particle state
+!! the single-particle state
!! information, and the moments are printed on standard output, using
-!! also the routines \c twobody_print and \c moment_print.
-!!
-!! It is important to note that when \c tinfo is called before the
-!! time step iteration starts, the
-!! two-body analysis cannot work because the fragment centers of mass
-!! from the previous time step are either not known yet (restart) or
-!! identical to the present ones (initialization). The subroutine \c twobody_case
-!! is still called to find the fragment properties,
-!! but the derived kinetic energy etc. will be incorrect. Therefore
-!! the call to \c twobody_print is suppressed in this case. The
-!! logical variable \c initialcall is used to recognize this case.
+!! also the routine \c moment_print.
!!
!! - <b> Step 8: check for final separation: </b> for the twobody case
!! it is checked whether the separation found between the two fragments
!! is larger than the input quantity \c rsep with positive time
!! derivative \c rdot of the separation distance, in which case the
-!! program is terminated with an appropriate message.
+!! program complete twobody analysis is done and the wave functions
+!! are saved; then the job is terminated with an appropriate message.
!---------------------------------------------------------------------------
SUBROUTINE tinfo
REAL(db), DIMENSION(2) :: ecoll ! storage for collective-flow energy
@@ -445,9 +437,9 @@ CONTAINS
printnow=mprint>0.AND.MOD(iter,mprint)==0
! Step 2: twobody analysis
IF(nof/=2) THEN
- istwobody=.FALSE.
+ istwobody=.FALSE.
ELSE
- CALL twobody_case(dt)
+ CALL twobody_analysis(.FALSE.) ! get distance in separated case
ENDIF
! Step 3: moments calculated
CALL moments
@@ -512,8 +504,6 @@ CONTAINS
ENDIF
! Step 7: print other output
IF(printnow.AND.wflag) THEN
- IF(nof==2.AND.istwobody.AND..NOT.initialcall) &
- CALL twobody_print
WRITE(*,'(/A)') ' Neutron Single Particle States:',header
DO il=1,nstmax
IF(il==npmin(2)) THEN
@@ -526,10 +516,12 @@ CONTAINS
CALL moment_print
ENDIF
DEALLOCATE(ps1)
- ! Step 8: check whether final distance is reached in twobody case
- IF(istwobody.AND.roft>rsep.AND.rdot>=0.D0) THEN
- CALL twobody_print
+ ! Step 8: check whether final distance is reached and it is
+ ! increasing in the twobody case
+ IF(istwobody.AND.roft>rsep.AND.roft>roft_old) THEN
+ CALL twobody_analysis(.TRUE.) ! do complete version
CALL write_wavefunctions
+ CALL write_densities
IF(wflag) WRITE(*,*) ' Final separation distance reached'
STOP ' Final distance reached'
ENDIF
diff --git a/Code/fragments.f90 b/Code/fragments.f90
index f2f5c907d5443144645571fbd426d89a9cf1aad2..2f338bbd61ab141a7309b60b0c68015bb55a5ec7 100644
--- a/Code/fragments.f90
+++ b/Code/fragments.f90
@@ -127,7 +127,6 @@ CONTAINS
!!This calculates the \c fboost values from \c ecm and \c b.
!---------------------------------------------------------------------------
SUBROUTINE getin_fragments
- USE Twobody, ONLY: centerx,centerz
INTEGER :: i
REAL(db) :: fdx,fdy,fdz,fwxyz
CHARACTER(8) :: forcename
@@ -208,11 +207,6 @@ CONTAINS
charge_number=nprot
mass_number=nneut+nprot
IF(.NOT.fix_boost.AND.nof==2.AND.tdynamic) CALL twobody_init
- IF(nof==2) THEN
- ! Give fragment positions to 2-body analysis as initial guess
- centerx=fcent(1,1:2)
- centerz=fcent(3,1:2)
- END IF
END SUBROUTINE getin_fragments
!---------------------------------------------------------------------------
! DESCRIPTION: read_fragments
diff --git a/Code/inout.f90 b/Code/inout.f90
index 75747ebd5cc23ddb581e5e33983d0dda1a465164..e4c199ba14fa9686719a376728a43d8c147ac46d 100644
--- a/Code/inout.f90
+++ b/Code/inout.f90
@@ -104,7 +104,7 @@ CONTAINS
IF(mpi_myproc==0) THEN
OPEN(UNIT=scratch2,FILE=wffile,STATUS='REPLACE',FORM='UNFORMATTED')
WRITE(scratch2) iter,time,f%name,nstmax,nneut,nprot,number,npsi, &
- charge_number,mass_number,cm
+ charge_number,mass_number,cmtot
WRITE(scratch2) nx,ny,nz,dx,dy,dz,wxyz
WRITE(scratch2) x,y,z
WRITE(scratch2) wocc,sp_energy,sp_parity,sp_norm,sp_kinetic, &
diff --git a/Code/meanfield.f90 b/Code/meanfield.f90
index af7791cd0c247d42c0d57aa940f01b295047e20d..da0635e26014a6bf53b1cc6c350260ab398e5cc6 100644
--- a/Code/meanfield.f90
+++ b/Code/meanfield.f90
@@ -150,7 +150,7 @@ CONTAINS
ENDDO
! Step 3: Coulomb potential
IF(tcoul) THEN
- CALL poisson
+ CALL poisson(rho(:,:,:,2))
upot(:,:,:,2)=upot(:,:,:,2)+wcoul
IF(f%ex/=0) &
upot(:,:,:,2)=upot(:,:,:,2)-slate*rho(:,:,:,2)**(1.0D0/3.0D0)
diff --git a/Code/twobody.f90 b/Code/twobody.f90
index 648030d625bb0021067e140ed3013585fe02d3f1..a73df5370f9bfe25badbfbc85dd7fb45dc42f833 100644
--- a/Code/twobody.f90
+++ b/Code/twobody.f90
@@ -3,171 +3,577 @@
!------------------------------------------------------------------------------
! DESCRIPTION:
!> @brief
-!!This module contains the code to analyze the final (and also, though
-!!less useful) initial stage of a heavy-ion reaction. It is applicable
-!!only to the dynamic calculations and only if there are essentially two
-!!separated fragments. It calculates the fragment masses and charges,
-!!their distance, the relative motion kinetic energy, angular momentum,
-!!and the scattering angle.
-!!
-!>
-!>@details
-!!The analysis assumes that the two fragments are separated by a region
-!!of noticeably lower density, tries to find the line connecting their
-!!centers of mass and then divides up space by a plane perpendicular to
-!!this line. Of necessity the results are not of high precision, but
-!!tend to be useful nevertheless.
-!!
-!!<b> The user should be critical and always make sure that the real
-!! situation is a two-fragment one before accepting the results of
-!! these routines. It is assumed that the reaction plane is the
-!! (x,z)-plane. The code could be generalized to a three-dimensional
-!! situation, if necessary, but usually the assumption of a fixed
-!! scattering plane should not be a problem. If the initial nuclei have
-!! nonzero internal angular momentum, e.~g., the reaction plane can
-!! rotate and then a more general analysis cannot be avoided.</b>
+!!This module contains the routines to distinguish between a one and two-body
+!!system and to perform a two-body analysis in the case of a two-body collision
+!!including scattering angles and momenta.
!------------------------------------------------------------------------------
MODULE Twobody
USE Params
USE Grids
- USE Densities, ONLY: rho
- USE Moment, ONLY: cmtot
+ USE Densities
+ USE Coulomb
USE Forces, ONLY: nucleon_mass
IMPLICIT NONE
- SAVE
- REAL(db) :: roft !<separation distance between fragments in fm.
- REAL(db) :: rdot !<relative-motion velocity in units of \f$ c \f$.
- REAL(db),PRIVATE :: xmin !<x-coordinate of the point in
- !!the (x,z)-plane where minimum density is found between the fragments.
- REAL(db),PRIVATE :: zmin !<x-coordinate of the point in
- !!the (x,z)-plane where minimum density is found between the fragments.
- REAL(db),PRIVATE :: slope !<Slope of the line
- !!connecting the fragment centers-of-mass.
- REAL(db),PRIVATE :: slold !<the value from the previous time step or iteration
- !!of \c slope where the 2-body analysis was last performed.
- REAL(db),PRIVATE :: bb !<intercept of the line. The line is thus
- !!given as <tt>z=bb+slope*x</tt>.
- REAL(db) :: centerx(2) !<x-coordinates of the two fragment centers of mass.
- REAL(db) :: centerz(2) !<z-coordinates of the two fragment centers of mass.
- REAL(db),PRIVATE :: mass(2) !<masses of the two fragments.
- REAL(db),PRIVATE :: charge(2) !<charges of the two fragments
- REAL(db),PRIVATE :: tke2body(2) !<total kinetic energy in MeV.
- LOGICAL :: istwobody !<logical, indicates whether this is a
- !!two-body case (\c TRUE) or not.
- REAL(db),PARAMETER :: vacuum=0.03D0 !<parameter indicating the limiting density below
- !!which vacuum is assumed. This value is not critical, since it is
- !!only used in looking for "empty" regions between the nuclei.
- REAL(db) :: xmu !<the reduced mass in MeV.
- REAL(db) :: xlf !<angular momentum of relative motion in \f$ \hbar \f$. It
- !!is calculated assuming two point bodies via \f$ \mu R^2\omega \f$.
- REAL(db) :: ecmf !<final relative motion energy after extrapolated to
- !!infinite separation.
- REAL(db) :: tdotc !<time derivative of scattering angle. Units \f$ c/{\rm fm} \f$.
- REAL(db) :: tketot !<relative-motion kinetic energy in MeV.
- REAL(db) :: teti !< present scattering angle.
- REAL(db) :: tetc !<TODO
- REAL(db) :: tetf !<TODO
- REAL(db) :: tets !<TODO
- REAL(db) :: xcoul !<Coulomb energy, calculated from point charges.
- REAL(db) :: xcent !<centrifugal energy, based on angular momentum and distance.
- PRIVATE :: getslope, divpoint
+ ! Constants
+ REAL(db),PARAMETER :: threshold=1.D-5 !< threshold density to recognze separate fragmenst
+ ! Data describing initial configuration
+ INTEGER :: mass_case !< records the initial mass relation. 0 indicates a symmetric
+ !! collision, 1 that fragment #1 is the lighter one, 2 that fragment #2 is heavier.
+ !! This is needed to order the fragments the same way in the final state.
+ ! Data related to final state
+ REAL(db) :: roft !< fragment separation at the current time, zero means connected system
+ REAL(db) :: roft_old !< fragment separation at the last time step. Needed to check
+ !! whether the motion is outgoing or incoming
+ REAL(db) :: velocity(3,0:2) !< c.m. velocity vectors for the total system and the fragments
+ ! Data for fragments that are needed globally
+ ! index 0 for total, 1:2 for fragments
+ !>@name mass,charge, c.m., momentum, and angular momentum of the total system and the fragments
+ !>@{
+ REAL(db) :: mass(0:2),charge(0:2),cmfrag(3,0:2),momentum(3,0:2),angmom(3,0:2)
+ !>@}
+ ! Data associated with the Coulomb calculation
+ REAL(db),ALLOCATABLE,PRIVATE :: rhotot(:,:,:) !< total density
+ REAL(db),ALLOCATABLE,PRIVATE :: currtot(:,:,:,:) !< total current density
+ INTEGER,ALLOCATABLE,PRIVATE :: frag(:,:,:) !< for each mesh pint, tells whether it is in
+ !! fragment 1 or 2, or below the cutoff density, (value 0)
+ LOGICAL :: istwobody !< principal return value. Tells whether the system is separated.
CONTAINS
-!---------------------------------------------------------------------------
-! DESCRIPTION: twobody_case
+!---------------------------------------------------------------------------
+! DESCRIPTION: twobody_analysis
+!> @brief
+!! For a two-body collision, this subroutine checks whether the system
+!! is separated and in that case determines either only the distance of
+!! the fragments or, for the final state, produces a detailed analysis
+!>
+!> @details
+!! The subroutine starts off by allocating arrays \c rhotot and, if the full
+!! analysis is desired, \c currtot, which are then assigned the total density and
+!! current density, respectively. These are set to zero wherever the density
+!! is below the \c threshold. Then the following steps are taken:
+!! - For the full analysis the total current density and the isospin-dependent
+!! density and current density are also cut off. Then the subroutine
+!! \c analyze is called for the whole system by setting \c frag such that
+!! all points have fragment index 0. The coordinate values are corrected
+!! for a shift in the center-of-mass. The total current density is then
+!! adjusted to remove any overall motion of the system in the grid.
+!! - Following this, subroutine \c cut_system is called to calculation
+!! fragment properties in the case of separation. If \c full_analysis is true,
+!! this should be the case, as it is done only for the final state.
+!! - For \c full_analysis=false, the analysis is finished, since the fragment
+!! distance has been calculated. Therefore there is a \c RETURN after the arrays
+!! have been deallocated.
+!! - After this the subroutine \c analyze is called separately for the two
+!! fragments to determine their properties
+!! - Based on this information, subroutine \c scattering is called to calculate
+!! relative motion and the overall scattering results.
+!>
+!> @param[in] full_analysis
+!> LOGICAL, if TRUE the full analysis is done, otherwise only the
+!> fragment separation is calculated.
+!---------------------------------------------------------------------------
+ SUBROUTINE twobody_analysis(full_analysis)
+ LOGICAL,INTENT(IN) :: full_analysis
+ INTEGER :: i,j,k
+ !***************************************************************
+ ALLOCATE(rhotot(nx,ny,nz),frag(nx,ny,nz))
+ rhotot=rho(:,:,:,1)+rho(:,:,:,2)
+ ! cut off rhotot to threshold
+ WHERE(rhotot<threshold) rhotot=0.D0
+ ! calculate properties of total system
+ IF(full_analysis) THEN
+ ALLOCATE(currtot(nx,ny,nz,3))
+ currtot(:,:,:,:)=current(:,:,:,:,1)+current(:,:,:,:,2)
+ FORALL(i=1:nx,j=1:ny,k=1:nz,rhotot(i,j,k)<threshold)
+ rho(i,j,k,:)=0.D0
+ currtot(i,j,k,:)=0.D0
+ current(i,j,k,:,:)=0.D0
+ END FORALL
+ ! analyze total system
+ frag=0
+ CALL analyze(0)
+ ! correct coordinates to have zero at c.m.
+ x=x-cmfrag(1,0)
+ y=y-cmfrag(2,0)
+ z=z-cmfrag(3,0)
+ cmfrag(:,0)=0.D0
+ ! correct total momentum to zero
+ WRITE(*,'(A,3F10.6)') 'Uncorrected velocity:',momentum(:,0)/mass(0)
+ DO i=1,3
+ momentum(i,0)=momentum(i,0)/mass(0) ! average velocity
+ currtot(:,:,:,i)=currtot(:,:,:,i)-rhotot*momentum(i,0)
+ END DO
+ END IF
+ ! compute dividing plane
+ CALL cut_system(full_analysis)
+ ! return for reduced analysis
+ IF(.NOT.full_analysis) THEN
+ DEALLOCATE(rhotot,frag)
+ RETURN
+ END IF
+ ! calculate properties separately for the two fragments
+ CALL analyze(1)
+ CALL analyze(2)
+ ! finally determine relative motion and total scattering effect
+ CALL scattering
+ END SUBROUTINE twobody_analysis
+!---------------------------------------------------------------------------
+! DESCRIPTION: analyze
+!> @brief
+!! This subroutine calculates properties of the two separate fragment
+!! or of the whole system.
+!>
+!> @details
+!! It evaluates integrated quantities for a subset of the mesh controlled
+!! by its argument \c ifrag in conjunction with the array \c
+!! frag. More specifically, \c ifrag=0 is used to calculate on the
+!! whole mesh while \c ifrag=1 or 2 select the points belonging to the
+!! corresponding fragment. The array \c frag contains the relevant index
+!! for each mesh point.
+!! - Most of the calculated quantities have an index \c ifrag to store
+!! these results separately. Thus \c cm(1:3,0:2) is used to store the
+!! total center of mass in \c cm(:,0) and the centers of mass of the
+!! two fragments in \c cm(:,1) or \c cm(:,2), respectively.
+!! - The first loop and the following statements calculate the mass number,
+!! charge number, center of mass, and total momentum of the system or
+!! fragment. From the latter two velocities are derived: \c velcorr,
+!! which is in the units used for the \c current density in Sky3D,
+!! namely fm\f$^{-4}\f$, while \c velocity is in units of \f$c\f$.
+!! - The next loop calculates the collective orbital angular momentum \c
+!! angmom with respect to the center of mass, subtracting its motion,
+!! and the classical tensor \f$\Theta\f$ of inertia, \c inertia. This allows
+!! calculating also the instantaneous angular velocity vector \c omega
+!! by solving \f$\vec L=\Theta\vec\omega\f$ using the subroutine \c gauss.
+!>
+!> @param[in] ifrag
+!> INTEGER, index of fragment 1 or 2 to be investigated, or 0 for whole system
+!---------------------------------------------------------------------------
+ SUBROUTINE analyze(ifrag)
+ INTEGER,INTENT(IN) :: ifrag
+ REAL(db) :: velcorr(3,0:2),omega(3),inertia(3,3),curr(3),xx,yy,zz
+ INTEGER :: i,j,k
+ mass(ifrag)=0.D0
+ charge(ifrag)=0.D0
+ cmfrag(:,ifrag)=0.D0
+ momentum(:,ifrag)=0.D0
+ DO k=1,nz
+ DO j=1,ny
+ DO i=1,nx
+ IF(frag(i,j,k)/=ifrag) CYCLE
+ mass(ifrag)=mass(ifrag)+rhotot(i,j,k)
+ charge(ifrag)=charge(ifrag)+rho(i,j,k,2)
+ cmfrag(1,ifrag)=cmfrag(1,ifrag)+rhotot(i,j,k)*x(i)
+ cmfrag(2,ifrag)=cmfrag(2,ifrag)+rhotot(i,j,k)*y(j)
+ cmfrag(3,ifrag)=cmfrag(3,ifrag)+rhotot(i,j,k)*z(k)
+ momentum(:,ifrag)=momentum(:,ifrag)+currtot(i,j,k,:)
+ END DO
+ END DO
+ END DO
+ cmfrag(:,ifrag)=cmfrag(:,ifrag)/mass(ifrag)
+ ! fragment velocity in units of current density
+ velcorr(:,ifrag)=momentum(:,ifrag)/mass(ifrag)
+ ! fragment velocity in units of c
+ velocity(:,ifrag)=momentum(:,ifrag)*wxyz*hbc/ &
+ (nucleon_mass*mass(ifrag))
+ ! add volume element to quantities depending on it
+ mass(ifrag)=mass(ifrag)*wxyz
+ charge(ifrag)=charge(ifrag)*wxyz
+ momentum(:,ifrag)=momentum(:,ifrag)*wxyz
+ ! Now calculate orbital angular momentum for fragment, relative
+ ! to its c.m.
+ angmom(:,ifrag)=0.D0
+ inertia=0.D0
+ DO k=1,nz
+ zz=z(k)-cmfrag(3,ifrag)
+ DO j=1,ny
+ yy=y(j)-cmfrag(2,ifrag)
+ DO i=1,nx
+ xx=x(i)-cmfrag(1,ifrag)
+ IF(frag(i,j,k)/=ifrag) CYCLE
+ curr=currtot(i,j,k,:)-rhotot(i,j,k)*velcorr(:,ifrag)
+ angmom(1,ifrag)=angmom(1,ifrag)+yy*curr(3)-zz*curr(2)
+ angmom(2,ifrag)=angmom(2,ifrag)+zz*curr(1)-xx*curr(3)
+ angmom(3,ifrag)=angmom(3,ifrag)+xx*curr(2)-yy*curr(1)
+ inertia(1,1)=inertia(1,1)+rhotot(i,j,k)*(yy**2+zz**2)
+ inertia(2,2)=inertia(2,2)+rhotot(i,j,k)*(xx**2+zz**2)
+ inertia(3,3)=inertia(3,3)+rhotot(i,j,k)*(xx**2+yy**2)
+ inertia(1,2)=inertia(1,2)-rhotot(i,j,k)*xx*yy
+ inertia(1,3)=inertia(1,3)-rhotot(i,j,k)*xx*zz
+ inertia(2,3)=inertia(2,3)-rhotot(i,j,k)*yy*zz
+ END DO
+ END DO
+ END DO
+ angmom(:,ifrag)=angmom(:,ifrag)*wxyz
+ inertia(2,1)=inertia(1,2)
+ inertia(3,1)=inertia(1,3)
+ inertia(3,2)=inertia(2,3)
+ inertia=inertia*wxyz
+ CALL gauss(inertia,angmom,omega)
+ WRITE(*,'(A,I2)') '******* Fragment #',ifrag
+ WRITE(*,'(" c.m.=",3F10.4," mass=",F10.4," charge=",F10.4)') &
+ cmfrag(:,ifrag),mass(ifrag),charge(ifrag)
+ Write(*,'(" Velocity/c =",3f10.4)') velocity(:,Ifrag)
+ WRITE(*,100) 'Inertia tensor',inertia
+ WRITE(*,100) 'Angular momentum',angmom(:,ifrag)
+ WRITE(*,100) 'Omega from L',omega
+100 FORMAT(A/(3D15.6))
+ END SUBROUTINE analyze
+!---------------------------------------------------------------------------
+! DESCRIPTION: gauss
+!> @brief
+!! This subroutine does a simple direct solution of the 3 by 3 linear
+!! system of equations.
+!>
+!> @details
+!! This is an explicit implementation using the determinant algorithm.
+!>
+!> @param[in] a(3,3)
+!> REAL(db), Matrix of the linear system.
+!> @param[in] b(3)
+!> REAL(db), right-hand side of the equations.
+!> @param[out] c(3)
+!> REAL(db), solution vector.
+!---------------------------------------------------------------------------
+ SUBROUTINE gauss(a,b,c)
+ REAL(db) :: a(3,3),b(3),c(3),det
+ INTENT(IN) :: a,b
+ INTENT(OUT) :: c
+ det=a(1,3)**2*a(2,2)-2.D0*a(1,2)*a(1,3)*a(2,3)+a(1,1)*a(2,3)**2+ &
+ a(1,2)**2*a(3,3)-a(1,1)*a(2,2)*a(3,3)
+ IF(ABS(det)<1.D-8) THEN
+ c=0.D0
+ ELSE
+ c(1)=(a(2,3)**2*b(1)-a(2,2)*a(3,3)*b(1)-a(1,3)*a(2,3)*b(2)+ &
+ a(1,2)*a(3,3)*b(2)+a(1,3)*a(2,2)*b(3) &
+ -a(1,2)*a(2,3)*b(3))/det
+ c(2)=(-(a(1,3)*a(2,3)*b(1))+a(1,2)*a(3,3)*b(1)+a(1,3)**2*b(2) &
+ -a(1,1)*a(3,3)*b(2)-a(1,2)*a(1,3)*b(3) &
+ +a(1,1)*a(2,3)*b(3))/det
+ c(3)=(a(1,3)*a(2,2)*b(1)-a(1,2)*a(2,3)*b(1)-a(1,2)*a(1,3)*b(2) &
+ +a(1,1)*a(2,3)*b(2)+a(1,2)**2*b(3) &
+ -a(1,1)*a(2,2)*b(3))/det
+ ENDIF
+ END SUBROUTINE gauss
+ !***************************************************************
+!---------------------------------------------------------------------------
+! DESCRIPTION: scattering
+!> @brief
+!! This subroutine calculates the relative motion and the various energies
+!! in the final state and, by referring back to the initial state before the
+!! collision, also the net scattering results.
+!>
+!> @details
+!! The subroutine calculates the scattering angle as the sum of three contributions:
+!! the ingoing Rutherford angle, the rotation angle during the TDHF calculation,
+!! and the outgoing Rutherford angle.
+!!
+!! The Rutherford trajectories are hyperbolas described in polar
+!! coordinates as \f[r(\phi)=\frac{k}{\epsilon\cos\phi-1}.\f]
+!! with \f$r\f$ and \f$\phi\f$ describing the relative vector of the two-body problem.
+!! It is confined to the angular region \f$-\phi_\infty<\phi<\phi_\infty\f$
+!! with the asymptotic angle \f$\phi_\infty=\cos^{-1}(\epsilon^{-1})\f$. The
+!! angle \f$\phi=0\f$ corresponds to the symmetry axis of the hyperbola.
+!!
+!! A problem is that this axis is not directly connected to the
+!! coordinates used for the TDHF calculation, which is assumed to have
+!! \f$x\f$ and \f$z\f$ as the reaction plane. Since, however, the parameters of
+!! the hyperbola \f$k\f$ and \f$\epsilon\f$ can be calculated from the impact
+!! parameter and energy of the relative motion, it suffices to know \f$r\f$
+!! to calculate the angle corresponding to it in the hyperbola system of
+!! coordinates. This is done slightly differently for the incoming and
+!! outgoing trajectories.
+!!
+!! The following steps are done:
+!! - Examine the initial state before the collision: Since the properties
+!! of the fragments may not be available anymore after a restart, we have to
+!! partially restore them from the original input files. <i>This requires
+!! that these files, or symbolic links to them, be present in current directory.</i>
+!! From \c for005 it reads the center-of-mass energy \f$E_{\rm cm}\f$ and
+!! impact parameter \f$b\f$ as well as the initial positions of the nuclei
+!! for the start of the TDHF calculation, \f$\vec r_{0i},\;i=1,2\f$ called
+!! \c fcent(:,i) in the code. Then the masses \f$A_i\f$ and charges \f$Z_i\f$
+!! are input from the fragment data files.
+!! - As the next step, the variable \c mass_case is determined to make sure the
+!! mass relation is the same in the ingoing and outgoing case. For symmetric
+!! collisions of course the scattering is not uniquely determined.
+!! From these data the following quantities are successively calculated:
+!! -# The initial relative vector for the TDHF calculation
+!! \f[\vec r(t=0)=\vec r_2(t=0)-\vec r_1(t=0),\f]
+!! -# the initial reduced mass
+!! \f[\mu=m_0\frac{A_1A_2}{A_1+A_2},\f]
+!! -# the dimensionless initial orbital angular momentum of relative motion,
+!! \f[L_{\rm init}=\mu v_{\rm init}b=\mu\sqrt{\frac{2E_{\rm cm}}{\mu}}\,
+!! \frac{b}{\hbar}.\f]
+!! From these quantities the parameters of the hyperbola can be
+!! calculated:
+!! -# the parameter \f$ k \f$
+!! \f[ k=\frac{L_{\rm init}^2\hbar^2}{e^2Z_1Z_2\mu},\f]
+!! -# the major axis
+!! \f[a=\frac{e^2Z_1Z_2}{2E_{\rm cm}},\f]
+!! -# from which \f$ \epsilon \f$ results:
+!! \f[\epsilon=\frac{\sqrt{a^2+b^2}}{a}.\f]
+!! Now the angles can be calculated from the equation of the
+!! hyperbola. The asymptotic angle
+!! \f[\phi_\infty=\cos^{-1}(1/\epsilon)\f]
+!! and the angle corresponding to the initial position,
+!! \f[\phi(t=0)=\cos^{-1}\left(\frac{k/|\vec r(t=0)|+1}{\epsilon}\right).\f]
+!! In the code both are converted to degrees. The incoming Rutherford
+!! scattering angle is just the difference,
+!! \f[\phi_{\rm incoming}=\phi_\infty-\phi(t=0).\f]
+!! - The rotation of the system during the TDHF calculation is obtained from the
+!! initial and final positions of the fragments via the scalar product of the
+!! relative vectors.
+!! - The code then calculates the relative motion properties in the final state:
+!! reduced mass \c mu, relative velocity \c vrel, kinetic energy \c erel,
+!! and angular momentum of relative motion \c relangmom.
+!! In addition the point-charge Coulomb interaction energy \c ecoul_point,
+!! the velocity and kinetic energy in the radial direction \c rdot and \c edot
+!! are produced for information.
+!! - For the extrapolation of the relative motion energy to infinite separation
+!! a more refinde calculation of the Coulomb energy is needed. This is done by
+!! using subroutine \c poisson three times: once for the total system and then
+!! for each fragment individually by removing the charge density from all space points
+!! associated with the other fragment. This yields \c ecoul(0:2) and the Coulomb
+!! interaction energy \c ecoul_int can then be obtained as
+!! <tt>ecoul(0)-ecoul(1)-ecoul(2)</tt>. The asymptotic energy is then the sum of \c erel and
+!! \c ecoul_int.
+!! - Finally the deflection angle for the outgoing Rutherford trajectory
+!! is obtained. The main difference to
+!! the incoming trajectory is that now the quantities determining the
+!! hyperbola parameters are taken from the two-body analysis, namely the
+!! masses and charges of the fragments, the \f$y\f$-component of the angular
+!! momentum of relative motion \c relangmom(1:2), and the final energy
+!! of relative motion \c final\_ecm. Otherwise the
+!! calculation proceeds identically, though the variable \c b has to
+!! be renamed \c b, since the impact parameter may be different for
+!! the final state.
+!! - The subroutine then prints all the details.
+!>
+!---------------------------------------------------------------------------
+ SUBROUTINE scattering
+ ! Varaiables needed for the calculation of the initial
+ ! Rutherford trajectory
+ CHARACTER(64) :: filename(2) ! names of fragment wf files
+ REAL(db) :: fcent(3,2) ! c.m. vectors for the fragments
+ REAL(db) :: fboost(3,2) ! boost vectors for the fragments
+ REAL(db) :: b ! impact parameter
+ REAL(db) :: ecm ! relative kinetic energy
+ REAL(db) :: mu ! reduced mass
+ REAL(db) :: k,a,epsilon,p,pmax ! hyperbola parameters
+ REAL(db) :: time ! needed for the namelist
+ REAL(db) :: initial_mass(2) ! masses of the original fragments
+ REAL(db) :: initial_charge(2) ! charges of the initial fragments
+ REAL(db) :: initial_ecm ! c.m. energy in the asymptotic initial state
+ REAL(db) :: initial_L ! initial orbital angular momentum of relative motion
+ REAL(db) :: initial_angle ! scattering angle for the Rutherford trajectory
+ !! from infinity to the starting position
+ REAL(db) :: initial_roft ! fragment distance at start of calculation
+ CHARACTER(8) :: forcename
+ REAL(db) :: init_relvec(3) ! relative vector at the beginning of the calculation
+ LOGICAL :: fix_boost
+ INTEGER :: i,dummy(7),iter
+ ! rotation of the interacting system
+ REAL(db) :: tdhf_angle ! rotation angle of the system during the TDHF calculation
+ ! Variables for the outgoing Rutherford trajectory
+ REAL(db) :: rdot ! relative velocity
+ REAL(db) :: edot ! relative kinetic energy at the present separation
+ REAL(db) :: final_ecm ! asymptotic relative motion kinetic energy
+ REAL(db) :: aa,bb
+ REAL(db) :: final_angle ! scattering angle for the outgoing Rutherford trajectory
+ REAL(db) :: final_relvec(3) ! relative coordinate at the end of the calculation
+ REAL(db) :: ecoul_point,vrel,erel,rhof(nx,ny,nz),ecoul(0:2)
+ INTEGER :: ifrag
+ REAL(db) :: e_centrifugal ! centrifugal energy
+ REAL(db) :: ecoul_int ! Coulomb interaction energy
+ REAL(db) :: relangmom(3) ! angular momentum of relative motion
+ ! First read the original input file defining the initialization of
+ ! the collision
+ NAMELIST /fragments/ filename,fcent,fboost,ecm,b,fix_boost
+ OPEN(10,FILE='for005')
+ READ(10,fragments)
+ CLOSE(10)
+ initial_ecm=ecm
+ DO i=1,2
+ OPEN(10,FILE=filename(i),STATUS='old',FORM=&
+ 'unformatted')
+ READ(10) iter,time,forcename,dummy,initial_charge(i),initial_mass(i)
+ CLOSE(10)
+ END DO
+ ! distinguish cases of light vs. heavy fragment
+ IF(initial_mass(1)==initial_mass(2)) THEN
+ mass_case=0 ! equal masses
+ ELSEIF(initial_mass(1)<initial_mass(2)) THEN
+ mass_case=1 ! lighter fragment is first
+ ELSE
+ mass_case=2 ! heavier fragment is first
+ END IF
+ ! Now compute scattering angle due to incoming Rutherford trajectory
+ init_relvec=fcent(:,2)-fcent(:,1) ! relative vector
+ mu=nucleon_mass*initial_mass(1)*initial_mass(2) &
+ /(initial_mass(1)+initial_mass(2)) ! reduced mass
+ initial_L=mu*SQRT(2.D0*initial_ecm/mu)*b/hbc
+ k=(initial_L*hbc)**2/(e2*initial_charge(1)*initial_charge(2)*mu)
+ a=e2*initial_charge(1)*initial_charge(2)/(2.D0*initial_ecm)
+ epsilon=SQRT(a**2+b**2)/a
+ pmax=ACOS(1.D0/epsilon)*180.D0/pi
+ initial_roft=SQRT(SUM(init_relvec**2))
+ p=ACOS((k/initial_roft+1.D0)/epsilon)*180.D0/pi
+ initial_angle=pmax-p
+ WRITE(*,*) '*** Data for incoming Rutherford hyperbola ***'
+ WRITE(*,'(3(A,f8.3))') 'Hyperbola k=',k,' fm. epsilon=',epsilon, &
+ ' Asym. angle=',pmax
+ WRITE(*,'(3(A,F8.3))') 'Angle at ',initial_roft,' fm: ',p,' Deflection: ', &
+ initial_angle
+ ! get initial and final relative vectors in TDHF, calculate angle
+ init_relvec=init_relvec/SQRT(SUM(init_relvec**2))
+ final_relvec=cmfrag(:,2)-cmfrag(:,1)
+ roft=SQRT(SUM(final_relvec**2))
+ tdhf_angle=ACOS(SUM(init_relvec*final_relvec)/roft)*180.D0/pi
+ WRITE(*,'(A,f8.3)') 'TDHF rotation angle: ',tdhf_angle
+ !
+ ! Calculate relative motion quantities-
+ ! reduced mass
+ mu=nucleon_mass*mass(1)*mass(2)/(mass(1)+mass(2)) ! reduced mass
+ ! kinetic energy of relative motion
+ vrel=SQRT(SUM((velocity(:,1)-velocity(:,2))**2))
+ erel=0.5D0*mu*vrel**2
+ ! Coulomb energy in point charge approximation
+ ecoul_point=e2*charge(1)*charge(2)/roft
+ ! angular momentum of relative motion
+ relangmom=0.D0
+ DO i=1,2
+ relangmom(1)=relangmom(1)+nucleon_mass*mass(i)* &
+ (cmfrag(2,i)*velocity(3,i)-cmfrag(3,i)*velocity(2,i))/hbc
+ relangmom(2)=relangmom(2)+nucleon_mass*mass(i)* &
+ (cmfrag(3,i)*velocity(1,i)-cmfrag(1,i)*velocity(3,i))/hbc
+ relangmom(3)=relangmom(3)+nucleon_mass*mass(i)* &
+ (cmfrag(1,i)*velocity(2,i)-cmfrag(2,i)*velocity(1,i))/hbc
+ END DO
+ ! rotational energy assuming fragments as point masses
+ e_centrifugal=hbc**2*SUM(relangmom**2)/(2.D0*mu*roft**2)
+ ! velocity and kinetic energy of radial motion
+ rdot=SUM((velocity(:,2)-velocity(:,1))*(cmfrag(:,2)-cmfrag(:,1)))/roft
+ edot=0.5D0*mu*rdot**2
+ WRITE(*,*) '*** Data for final configuration ***'
+ WRITE(*,'(2(A,F10.5))') 'Fragment distance ',roft,' relative velocity',rdot
+ WRITE(*,'(A,3F10.5)') 'Relative angular momentum ',relangmom
+ WRITE(*,'(3(A,F10.5))') 'Radial kinetic energy ',edot, &
+ ' centrifugal energy',e_centrifugal,' point Coulomb energy ',ecoul_point
+ WRITE(*,'(2(A,F10.5))') 'Relative kinetic energy',erel,' edot+e_centrifugal', &
+ edot+e_centrifugal
+ ! Coulomb energies with more refined calculation
+ CALL coulinit
+ rhof=rho(:,:,:,2)
+ CALL poisson(rhof)
+ ecoul(0)=0.5D0*wxyz*SUM(rhof*wcoul)
+ DO ifrag=1,2
+ WHERE(frag==ifrag)
+ rhof=rho(:,:,:,2)
+ ELSEWHERE
+ rhof=0.D0
+ END WHERE
+ CALL poisson(rhof)
+ ecoul(ifrag)=0.5D0*wxyz*SUM(rhof*wcoul)
+ END DO
+ ecoul_int=ecoul(0)-SUM(ecoul(1:2))
+ WRITE(*,'(3(A,F10.4))') 'Total Coulomb energy: ',ecoul(0), &
+ ' Fragment 1: ',ecoul(1),' Fragment 2:',ecoul(2), &
+ ' Coulomb interaction energy:',ecoul_int
+ !
+ final_ecm=erel+ecoul_int
+ ! now calculate final Rutherford scattering angle
+ k=(relangmom(2)*hbc)**2/(e2*charge(1)*charge(2)*mu)
+ aa=e2*charge(1)*charge(2)/(2.D0*final_ecm)
+ bb=relangmom(2)*hbc/SQRT(2.D0*mu*final_ecm)
+ epsilon=SQRT(aa**2+bb**2)/a
+ pmax=ACOS(1.D0/epsilon)*180.D0/pi
+ roft=SQRT(SUM(final_relvec**2))
+ p=ACOS((k/roft+1.D0)/epsilon)*180.D0/pi
+ final_angle=pmax-p
+ WRITE(*,*) '*** Data for outgoing Rutherford trajectory ***'
+ WRITE(*,'(3(A,f8.3))') 'Hyperbola k=',k,' fm. epsilon=',epsilon, &
+ ' Asym. angle=',pmax
+ WRITE(*,'(3(A,F8.3))') 'Angle at ',roft,' fm: ',p,' Deflection: ', &
+ final_angle
+ ! produce summary
+ WRITE(*,*) '***************** Summary *************************'
+ WRITE(*,'(2(A20,2F10.3))') 'Initial masses:',initial_mass, &
+ ' Initial charges:',initial_charge
+ WRITE(*,'(2(A20,2F10.3))') 'Final masses:',mass(1:2), &
+ ' Final charges:',charge(1:2)
+ WRITE(*,'(2(A20,F10.3))') 'Initial c.m. energy: ',initial_ecm, &
+ ' Final: ',final_ecm
+ WRITE(*,'(3(A15,F10.3))') 'Radial: ',edot,' Centrifugal:', &
+ e_centrifugal,' Coulomb:',ecoul_int
+ WRITE(*,'(2(A20,F10.3))') 'Initial L: ',initial_L,' Final orbital L: ',&
+ relangmom(2)
+ WRITE(*,'(A20,2F10.3,A20,F10.3)') 'Fragment L: ',angmom(2,1:2), &
+ 'Final L_tot: ',relangmom(2)+angmom(2,1)+angmom(2,2)
+ WRITE(*,'(A20,F10.3)') 'Scattering angle: ', &
+ 180.D0-initial_angle-tdhf_angle-final_angle
+ WRITE(*,'(3(A15,F10.3))') 'Initial: ',initial_angle, &
+ ' TDHF part: ',tdhf_angle,' Final: ',final_angle
+ END SUBROUTINE scattering
+!---------------------------------------------------------------------------
+! DESCRIPTION: cut_system
!> @brief
-!!This routine tries to find a separation of the system into two
-!!fragments and determine their properties as well as those of the
-!!relative motion. It keeps track of previous analysis results, since
-!!the 2-body properties are expected to change slowly and this makes the
-!!analysis easier. Also, the motion of the fragments is determined by
-!!time-differencing.
+!! This subroutine finds out whether the system can be decomposed into two
+!! fragments records the geometry of this decomposition.
!>
!> @details
-!!It uses the following steps:
-!! - <b> Step 1 </b>: the results of the last analysis are partially
-!! saved. This includes the positions of the fragments, their
-!! distance, the slope of the connecting line, and its angle.
-!! - <b> Step 2 </b>: the fragment division is sought in an iterative
-!! process with (at present) a fixed iteration limit of 10. The reason
-!! for the iterations is that the connecting line, which determines the
-!! dividing plane, which is orthogonal to it at the dividing point,
-!! should link the centers-of-mass, but these can be calculated only
-!! assuming a dividing plane. There is thus a self-consistency problem
-!! which as usual is solved iteratively. During the iterations the new
-!! value of the center of mass is kept in \c centerx and \c centerz,
-!! while the previous one is in \c centx and \c centz. Each iteration
-!! has several steps:
-!! -# The connecting line (\c slope and intercept \c bb) is
-!! determined in one of two ways: if this is the first iteration,
-!! subroutine getslope is called to calculate the slope from the
-!! quadrupole tensor; the intercept is then calculated assuming that
-!! the line passes through the center-of-mass (which it should do in
-!! principle, but it might be off in practice). For the later
-!! iterations the line is calculated directly from the two centers of
-!! mass.
-!! -# Next function \c divpoint is called to find out whether
-!! this is indeed a two-body situation. It also returns the midpoint
-!! between the fragments in \c xmin and \c zmin. This result is
-!! not used immediately, since it might be that shifting the
-!! connecting line could change this situation.
-!! -# The slope of the line is now rotated by \f$ 90^\circ \f$ to get the
-!! line defining the dividing plane in the \f$ (x,z) \f$-plane. This has
-!! \c slopev and \c bb as slope and intercept.
-!! -# Now in a simple loop integrals are done separately over the
-!! two regions, summing up \c charge, \c mass, and \c center
-!! of mass for the two fragments. The assignment to the fragments is
-!! recognized by checking whether the point is above or below the
-!! dividing line (variable \c diff). Note that the \f$ y \f$-component
-!! of the centers of mass is not calculated.
-!! -# Finally the iteration process is stopped if no two-body
-!! situation is found or the center-of-mass vectors have converged.
-!! .
-!! - <b> Step 3 </b>: The two-body analysis is now assumed to be complete
-!! and the physical quantities are evaluated. These are defined above
-!! in the list of module variables. The following calculation concerns
-!! the final scattering angle \c tets extrapolated to infinity. It
-!! can be understood using the Rutherford trajectories.
+!! There is an iteration using index \c itcm to make the line connecting
+!! the centers of the fragments consistent. It starts with the quadrupole
+!! tensor principal axis using function \c getslope. After calculating the
+!! fragment centers-of-mass, this is corrected iteratively. Within the loop
+!! the following steps are executed:
+!! -# If this is not the first iteration, calculate the slope of the
+!! center-connecting line from the centers of mass.
+!! -# Call \c divpoint to find whether there is a sufficiently low density
+!! between two fragments along that line. If there is, keep track of the
+!! minimum-density location.
+!! -# Taking the line from that point, calculate the properties of the
+!! line perpendicular to the inter-center connection: the line dividing space
+!! between the fragments.
+!! -# Then assign fragment index \c frag(ix,iy,iz) for each grid point by
+!! checking which side of the dividing line they are on. At the same time,
+!! accumulate data for the fragment masses and centers of mass.
+!! -# If the centers of mass have not changed significantly, stop the
+!! iterations. Otherwise repeat.
+!! -# Record the center distance and return if this is not a full analysis.
+!! -# Finally, make sure that the assignment of fragment index to heavier
+!! or lighter fragment remains the same. This uses array \c swap to interchange
+!! 1 and 2.
!>
-!> @param[in] xdt
-!> REAL(db), takes the current time step.
-!---------------------------------------------------------------------------
- SUBROUTINE twobody_case(xdt)
- REAL(db),INTENT(IN) :: xdt
- REAL(db),SAVE :: xold(2),zold(2),rold,tetold
- REAL(db) :: centx(2),centz(2), &
- angle,slopev,xx,zz,diff,rhotot,ratio,vxx(2),vzz(2), &
- tdotc,epsf,ttt,temp,vol
- INTEGER :: ix,iy,iz,itcm,ifrag
- ! *** Step 1
- xold=centerx
- zold=centerz
- rold=SQRT((xold(2)-xold(1))**2 +(zold(2)-zold(1))**2)
- slold=(zold(2)-zold(1))/(xold(2)-xold(1))
- tetold=ATAN(slold)
- ! *** Step 2
+!> @param[in] full_analysis
+!! LOGICAL, if false, do only what is necessary for calculating the
+!! distance
+!---------------------------------------------------------------------------
+ SUBROUTINE cut_system(full_analysis)
+ LOGICAL,INTENT(IN) :: full_analysis
+ REAL(db) :: cent(3,2),center(3,2),angle,slopev,xx,zz,diff,vol, &
+ bb,xmin,zmin,slope
+ INTEGER :: ix,iy,iz,itcm,ifrag
+ INTEGER,PARAMETER :: swap(0:2)=(/ 0,2,1 /)
Iteration: DO itcm=1,10
- ! substep 1
+ ! calculate slope from centers of mass
IF(itcm>1) THEN
- centx=centerx
- centz=centerz
- slope=(centz(2)-centz(1))/(centx(2)-centx(1))
- bb=centz(1)-slope*centx(1)
+ cent=center
+ slope=(cent(3,2)-cent(3,1))/(cent(1,2)-cent(1,1))
+ bb=cent(3,1)-slope*cent(1,1)
ELSE ! first iteration
- centx=xold
- centz=zold
- CALL getslope
- bb=cmtot(3)-slope*cmtot(1)
+ cent=0.D0
+ slope=getslope()
+ bb=0.D0
ENDIF
- ! substep 2
- istwobody=divpoint()
- ! substep 3
+ ! check whether separation
+ istwobody=divpoint(xmin,zmin,bb,slope)
+ ! determine dividing line: slope and intercept
angle=ATAN(slope)
slopev=dtan(angle+pi/2.0D0)
bb=zmin-slopev*xmin
- ! substep 4
- centerx=0.D0; centerz=0.D0
- mass=0.0D0
- charge=0.0D0
+ ! assign fragment cells
+ center(:,1:2)=0.D0
+ mass(1:2)=0.0D0
+ charge(1:2)=0.0D0
DO iz=1,nz
zz=z(iz)
DO ix=1,nx
@@ -175,104 +581,99 @@ CONTAINS
diff=zz-slopev*xx-bb
DO iy=1,ny
vol=wxyz
- rhotot=rho(ix,iy,iz,1)+rho(ix,iy,iz,2)
- ifrag=1 ! 1 for left and 2 for right fragment
- IF(diff<0.0D0) THEN
- ifrag=1
+ frag(ix,iy,iz)=0
+ IF(rhotot(ix,iy,iz)<threshold) CYCLE
+ IF(diff<0.0D0) THEN
+ frag(ix,iy,iz)=1
+ ifrag=1
ELSE
- ifrag=2
+ frag(ix,iy,iz)=2
+ ifrag=2
ENDIF
- mass(ifrag)=mass(ifrag)+vol*rhotot
- charge(ifrag)=charge(ifrag)+vol*rho(ix,iy,iz,2)
- centerx(ifrag)=centerx(ifrag)+vol*rhotot*xx
- centerz(ifrag)=centerz(ifrag)+vol*rhotot*zz
+ mass(ifrag)=mass(ifrag)+vol*rhotot(ix,iy,iz)
+ center(1,ifrag)=center(1,ifrag)+vol*rhotot(ix,iy,iz)*x(ix)
+ center(2,ifrag)=center(2,ifrag)+vol*rhotot(ix,iy,iz)*y(iy)
+ center(3,ifrag)=center(3,ifrag)+vol*rhotot(ix,iy,iz)*z(iz)
ENDDO
ENDDO
ENDDO
- centerx=centerx/mass
- centerz=centerz/mass
- ! substep 5
+ DO ifrag=1,2
+ center(:,ifrag)=center(:,ifrag)/mass(ifrag)
+ END DO
+ ! stop if this is not a separated case
IF(.NOT.istwobody) EXIT
- IF(MAX(MAXVAL(ABS(centx-centerx)),MAXVAL(ABS(centz-centerz))) &
- <1.0d-05) EXIT
+ ! end iterations if the c.m. remains constant
+ IF(MAX(MAXVAL(ABS(cent(1,:)-center(1,:))), &
+ MAXVAL(ABS(cent(3,:)-center(3,:))))<1.0d-05) EXIT
ENDDO Iteration
- !
- ! *** Step 3
- !
- xmu=nucleon_mass*mass(1)*mass(2)/(mass(1)+mass(2))
- ratio=xmu/hbc**2
- !
- vxx=(centx-xold)/xdt
- vzz=(centz-zold)/xdt
- tke2body=0.5D0*mass*nucleon_mass*(vxx**2+vzz**2)
- tketot=0.5D0*xmu *((vxx(1)-vxx(2))**2+(vzz(1)-vzz(2))**2)
- roft=SQRT((centx(2)-centx(1))**2+(centz(2)-centz(1))**2)
- rdot=(roft-rold)/xdt
- teti=angle
- tdotc=(teti-tetold)/xdt
- !
- xlf=tdotc*xmu/hbc*roft**2
- xcoul=charge(1)*charge(2)*e2/roft
- xcent=xlf**2/(2.0D0*ratio*roft**2)
- ecmf=0.5D0*xmu*rdot**2+xcoul+xcent
- epsf=SQRT(1.0D0+2.0D0*ecmf*xlf**2/ &
- (ratio*(charge(1)*charge(2)*e2)**2))
- ttt=xlf**2/(ratio*charge(1)*charge(2)*e2*roft)
- IF(ABS(slope)>ABS(slold)) THEN
- tetc=pi-teti
- tetf=2.0D0*ACOS(1.0D0/epsf)-teti
- ELSE
- tetc=teti
- teti=pi-tetc
- tetf=ACOS(1.0D0/epsf)-ACOS(MAX(1.0D0/epsf *(1.0D0+ &
- ttt),1.D0))
- ENDIF
- tets=tetc-tetf
- IF(ABS(slope) <ABS(slold)) THEN
- temp=mass(1); mass(1)=mass(2); mass(2)=temp
- temp=charge(1); charge(1)=charge(2); charge(2)=temp
- temp=tke2body(1); tke2body(1)=tke2body(2); tke2body(2)=temp
- ENDIF
- END SUBROUTINE twobody_case
+ ! record separation distance
+ IF(istwobody) THEN
+ roft_old=roft
+ roft=SQRT(SUM(center(:,1)-cent(:,2))**2)
+ WRITE(*,'(A,F10.2,A,F8.4)') 'Time:',time,' Center distance: ',roft
+ ELSE
+ roft=0.D0
+ END IF
+ IF(.NOT.full_analysis) RETURN
+ ! correct ordering of fragments to agree with initialization
+ SELECT CASE(mass_case)
+ CASE(1)
+ IF(mass(1)<mass(2)) RETURN
+ CASE(2)
+ IF(mass(1)>mass(2)) RETURN
+ END SELECT
+ ! incorrect case: need to exchange indices
+ DO iz=1,nz
+ DO iy=1,ny
+ DO ix=1,nx
+ frag(ix,iy,iz)=swap(frag(ix,iy,iz))
+ END DO
+ END DO
+ END DO
+ END SUBROUTINE cut_system
!---------------------------------------------------------------------------
! DESCRIPTION: getslope
!> @brief
-!!This subroutine calculates the slope of the line determined by the
-!!eigenvector of largest quadrupole moment in the (x,z)-plane. The
-!!first loop sums up the quadrupole tensor, which is dimensioned <tt>
-!!q2(3,3)</tt> but of which the index 2 is not actually necessary, since
-!!the y-direction is not involved. The definition is kept
-!!three-dimensional to reduce confusion and make later generalization
-!!easier.
+!!In this function the slope of the fragment connection line in the
+!! \f$x-z \f$ plane is determined.
!>
!> @details
-!!The two-dimensional eigenvalue problem for \c q2 has the secular
-!!equation (remember \f$ q_{31}=q_{13} \f$:
-!!\f[ (q_{11}-\lambda)(q_{33}-\lambda)-q_{13}^2=0, \f]
-!!with \f$ \lambda \f$ the eigenvalue. For the larger eigenvalue we get
-!!\f[ \lambda=\frac1{2}\left(q_{11}+q_{33}+\sqrt{(q_{11}-q_{33})^2
-!!-4q_{13}^2}\right), \f] and solving the equation
-!!\f$ q_{13}x+(q_{33}-\lambda)z=0 \f$ for z yields
-!!\f{eqnarray}{
-!! z&=&\frac{q_{13}}{\lambda-q_{33}}\,x\\
-!! &=& \frac{q_{13}}{\frac1{2}\left(q_{11}-q_{33}+\sqrt{(q_{11}-q_{33})^2
-!! -4q_{13}^2}\right)}.
-!!\f}
-!!The code calls the denominator \c denom and makes sure no division
-!!by zero happens (this could happen for a spherical distribution). The
-!!resulting slope is returned in the module variable \c slope.
+!! The slope is determined by the
+!! eigenvector of largest quadrupole moment in the \f$(x,z)\f$-plane. The
+!! first loop sums up the quadrupole tensor, which is dimensioned <tt>
+!! q2(3,3)</tt> but of which the index 2 is not actually necessary, since
+!! the \f$y\f$-direction is not involved. The definition is kept
+!! three-dimensional to reduce confusion and make later generalization
+!! easier.
+!! The two-dimensional eigenvalue problem for \c q2 has the secular
+!! equation:
+!! \f[(q_{11}-\lambda)(q_{33}-\lambda)-q_{13}^2=0,\f]
+!! with \f$\lambda\f$ the eigenvalue. For the larger eigenvalue we get
+!! \f[\lambda=\frac1{2}\left(q_{11}+q_{33}+\sqrt{(q_{11}-q_{33})^2
+!! -4q_{13}^2}\right),\f] and solving the equation
+!! \f$q_{13}x+(q_{33}-\lambda)z=0\f$ for \f$z\f$ yields
+!! \f{eqnarray*}{
+!! z&=&\frac{q_{13}}{\lambda-q_{33}}\,x\\
+!! &=& \frac{q_{13}}{\frac1{2}\left(q_{11}-q_{33}+\sqrt{(q_{11}-q_{33})^2
+!! -4q_{13}^2}\right)}.
+!! \f}
+!! The code calls the denominator \c denom and makes sure no division
+!! by zero happens (this could happen for a spherical distribution). The
+!! resulting slope is returned in the module variable \c slope.
+!>
+!> @retval newslope REAL(db), calculated slope.
!---------------------------------------------------------------------------
- SUBROUTINE getslope
- REAL(db) :: q2(3,3),xx,yy,zz,vol,denom
+ FUNCTION getslope() RESULT(newslope)
+ REAL(db) :: q2(3,3),xx,yy,zz,vol,denom,newslope
INTEGER :: ix,iy,iz
q2=0.D0
DO iz=1,nz
- zz=z(iz)-cmtot(3)
+ zz=z(iz)
DO iy=1,ny
- yy=y(iy)-cmtot(2)
+ yy=y(iy)
DO ix=1,nx
- xx=x(ix)-cmtot(1)
- vol=wxyz*(rho(ix,iy,iz,1)+rho(ix,iy,iz,2))
+ xx=x(ix)
+ vol=wxyz*rhotot(ix,iy,iz)
q2(1,1)=q2(1,1)+(xx*xx+xx*xx-yy*yy-zz*zz)*vol
q2(2,2)=q2(2,2)+(yy*yy+yy*yy-xx*xx-zz*zz)*vol
q2(3,3)=q2(3,3)+(zz*zz+zz*zz-xx*xx-yy*yy)*vol
@@ -288,64 +689,79 @@ CONTAINS
denom=0.5D0*(q2(1,1)-q2(3,3)+SQRT((q2(1,1) &
-q2(3,3))**2+4.D0*q2(3,1)**2))
IF(ABS(denom)<1.D-4) THEN
- slope=100.D0
+ newslope=100.D0
ELSE
- slope=q2(1,3)/denom
+ newslope=q2(1,3)/denom
ENDIF
- END SUBROUTINE getslope
-!---------------------------------------------------------------------------
+ END FUNCTION getslope
+!---------------------------------------------------------------------------
! DESCRIPTION: divpoint
!> @brief
-!!The function divpoint is a helper routine. It examines the line
-!!determined by \c slope and intercept \c bb and finds the point
-!!<tt>(xmin,zmin)</tt> which is in the center of the void between the two
-!!fragments, returning .true. if this is possible.
+!! The function divpoint examines the line determined by the axis of
+!! largest quadrupole moment to look for a suitable separation
+!! point between two fragments. If one is found with sufficiently low
+!! density, it returns \c TRUE and the properties of that point in its
+!! arguments. Otherwise it returns \c FALSE.
!>
!> @details
!!To this end it looks at the behavior of the densities along this line.
!!Since the line has no relation to the numerical grid, this is not
!!trivial.
-!!
-!! - <b>First loop:</b> Essentially it looks through the (x,z)-plane to
-!! find points closer than half a grid spacing to the desired line with
-!! equation <tt> z=slope*x+bb</tt> (logical variable \c online). If the
-!! slope is larger than one, i.~e., if the nuclei are separating
-!! predominantly in the x-direction, we need to take the equation
-!! <tt>x=(z-bb)/slope</tt> instead to get better resolution. To make the
-!! result monotonic along the line, the do loop in z runs backward
-!! for negative slopes. The points found are collected in index vectors
-!! \c ixl, \c izl with accompanying densities <tt> rhol </tt> stored
-!! in arrays of length \c il.
-!!
-!! - <b>Second loop:</b> now the number of fragments \c nf is counted by
-!! examining this one-dimensional density curve, looking for
-!! disconnected density humps above \c vacuum density. It is also
-!! recorded where the "vacuum" region starts and ends in variables
-!! \c n1 and \c n2. The logic is as follows:
-!! -# The logical variables \c in_vacuum} keeps track of whether
-!! the search is in a vacuum region at the moment or not. It starts
-!! as \c TRUE.
-!! -# Go to the next point. If its density is above \c vacuum,
-!! and we are in the vacuum, a new fragment is starting and we
-!! increase the number of fragments \c nf by1. If it becomes
-!! bigger than 2, exit, because there are three or more fragments.
-!! If it is now 2, record the starting index for the second fragment
-!! in \c n2.
-!! -# If the density is below \c vacuum and we are not in vacuum,
-!! a fragment is being ended. If it is the first fragment, we record
-!! this index in \c n1.
-!! - <b>Final processing</b> At this point we expect a two-fragment
-!! situation if \c nf=2 and in this case the void region between the
-!! fragments extends from \c n1 to \c n2, which are indices into
-!! arrays \c ix1 and \c iz1 giving the position in the
-!! (x,z)-plane. The code calculates the midpoint between the two
-!! positions and returns \c TRUE in this case.
+!! - <b> First loop: </b> Essentially it looks through the
+!! \f$(x,z)\f$-plane to find points closer than half a grid spacing
+!! to the desired line with equation <tt>z=slope*x+bb</tt>
+!! (logical variable \c online). If the slope is larger than one,
+!! i. e., if the nuclei are separating predominantly in the
+!! \f$x\f$-direction, we need to take the equation
+!! <tt>x=(z-bb)/slope</tt> instead to get better resolution. To make the
+!! result monotonic along the line, the do loop in \f$z\f$ runs backward
+!! for negative slopes. The points found are collected in index vectors
+!! <tt>ixl</tt>, <tt>izl</tt> with accompanying densities <tt>rhol
+!! </tt> stored
+!! in arrays of length <tt>il</tt>.
+!! - <b> Second loop: </b> now the number of fragments <tt>nf</tt>
+!! is counted by examining this one-dimensional density curve, looking for
+!! disconnected density humps above <tt>vacuum</tt> density. Where
+!! the ``vacuum'' region starts and ends is recorded in variables
+!! <tt>n1</tt> and <tt>n2</tt>. The logic is as follows:
+!! -# The logical variable <tt>in\_vacuum</tt> keeps track of whether
+!! the search is in a vacuum region at the moment or not. It starts
+!! as <tt>TRUE</tt>.
+!! -# Go to the next point. If its density is above <tt>vacuum</tt>,
+!! and we are in the vacuum, a new fragment is starting and we
+!! increase the number of fragments <tt>nf</tt> by 1. If it becomes
+!! bigger than 2, exit, because there are three or more fragments.
+!! If it is now 2, record the starting index for the second fragment
+!! in <tt>n2</tt>.
+!! -# If the density is below <tt>vacuum</tt> and we are not in vacuum,
+!! a fragment is being ended. If it is the first fragment, we record
+!! this index in <tt>n1</tt>.
+!! - <b> Final processing: </b> At this point we expect a two-fragment
+!! situation if <tt>nf=2</tt> and in this case the void region between the
+!! fragments extends from <tt>n1</tt> to <tt>n2</tt>, which are indices into
+!! arrays <tt>ix1</tt> and <tt>iz1</tt> giving the position in the
+!! \f$(x,z)\f$-plane. The code calculates the midpoint between the two
+!! positions and returns <tt>TRUE</tt> in this case.
+!>
+!> @return
+!> LOGICAL, indicates whether there is a fragment division or not.
+!> @param[out] xmin
+!> REAL(db), \f$x\f$-coordinate of the dividing point.
+!> @param[out] zmin
+!> REAL(db), \f$z\f$-coordinate of the dividing point.
+!> @param[in] bb
+!> REAL(db), intercept of the line joining the fragments..
+!> @param[in] slope
+!> REAL(db), slope of the line joining the fragments.
!---------------------------------------------------------------------------
- LOGICAL FUNCTION divpoint()
+ LOGICAL FUNCTION divpoint(xmin,zmin,bb,slope)
LOGICAL :: online,two,in_vacuum
INTEGER :: iyy,iz1,iz2,idz,il,ix,iz,ixl(nx+ny+nz), &
izl(nx+ny+nz),i,n1,n2,nf
REAL(db) :: deltax,deltaz,xx,zz,rhol(nx+ny+nz)
+ REAL(db),INTENT(OUT) :: xmin,zmin
+ REAL(db),INTENT(IN) :: bb,slope
+ REAL(db),PARAMETER :: vacuum=0.03D0
!
! now the calculation of the dividing plane starts by examining the line
! along the axis of largest quadrupole moment
@@ -355,9 +771,9 @@ CONTAINS
deltaz=0.5*dz
! Loop 1
IF(slope>=0.D0) THEN
- iz1=1; iz2=nz; idz=1
+ iz1=1; iz2=nz; idz=1
ELSE
- iz1=nz; iz2=1; idz=-1
+ iz1=nz; iz2=1; idz=-1
END IF
il=0
DO ix=1,nx
@@ -366,18 +782,18 @@ CONTAINS
zz=z(iz)
online=.FALSE.
IF(ABS(slope)<=1.D0) THEN
- online=ABS(zz-slope*xx-bb)<=deltaz
+ online=ABS(zz-slope*xx-bb)<=deltaz
ELSE
- online=ABS(xx-(zz-bb)/slope)<=deltax
+ online=ABS(xx-(zz-bb)/slope)<=deltax
END IF
IF(online) THEN
- il=il+1
- IF(il>nx+ny+nz) THEN
- IF(wflag) WRITE(*,*) ' Increase dimensioning in function divpoint'
- STOP
- END IF
- ixl(il)=ix; izl(il)=iz
- rhol(il)=rho(ix,iyy,iz,1)+rho(ix,iyy,iz,2)
+ il=il+1
+ IF(il>nx+ny+nz) THEN
+ WRITE(*,*) ' Increase dimensioning in function divpoint'
+ STOP
+ END IF
+ ixl(il)=ix; izl(il)=iz
+ rhol(il)=rhotot(ix,iyy,iz)
END IF
END DO
END DO
@@ -385,62 +801,26 @@ CONTAINS
nf=0; in_vacuum=.TRUE.
DO i=1,il
IF(rhol(i)>vacuum) THEN
- IF(in_vacuum) THEN
- in_vacuum=.FALSE.
- nf=nf+1
- IF(nf==2) n2=i
- IF(nf>2) EXIT
- END IF
+ IF(in_vacuum) THEN
+ in_vacuum=.FALSE.
+ nf=nf+1
+ IF(nf==2) n2=i
+ IF(nf>2) EXIT
+ END IF
ELSE
- IF(.NOT.in_vacuum) THEN
- IF(nf==1) n1=MAX(1,i-1)
- in_vacuum=.TRUE.
- END IF
+ IF(.NOT.in_vacuum) THEN
+ IF(nf==1) n1=MAX(1,i-1)
+ in_vacuum=.TRUE.
+ END IF
END IF
END DO
two=nf==2
! final processing
IF(two) THEN
- xmin=0.5D0*(x(ixl(n1))+x(ixl(n2)))
- zmin=0.5D0*(z(izl(n1))+z(izl(n2)))
+ xmin=0.5D0*(x(ixl(n1))+x(ixl(n2)))
+ zmin=0.5D0*(z(izl(n1))+z(izl(n2)))
END IF
divpoint=two
END FUNCTION divpoint
-!---------------------------------------------------------------------------
-! DESCRIPTION: twobody_print
-!> @brief
-!!This subroutine writes the relevant values determined by the twobody
-!!analysis to the main output.
-!---------------------------------------------------------------------------
- SUBROUTINE twobody_print
- REAL(db) :: centx(2),centz(2)
- INTEGER :: i
- CHARACTER(8),PARAMETER :: name(2)=(/ ' left: ',' right: '/)
- IF(wflag) &
- WRITE(*,'(/A,/(4(A12,F12.4)))') &
- ' Relative motion / Coulomb kinematics:',' red. mass:', &
- xmu/nucleon_mass,' l/hbar:',xlf,' ecmf(MeV):',ecmf, &
- ' roft(fm):',roft,' rdot/c:',rdot,' td/c(°):', &
- tdotc*180.0D0/pi,' trke(MeV):',tketot, &
- ' teti(°):',teti*180.0D0/pi,' tetf(°):', &
- tetf*180.0D0/pi,' tetc(°):',tetc*180.0D0/pi, &
- ' tets(°):',tets*180.0D0/pi,' Vcoul:',xcoul, &
- ' Vcent:',xcent
- IF(ABS(slope)>ABS(slold)) THEN
- centx=centerx; centz=centerz
- ELSE
- centx(1)=centerx(2); centz(1)=centerz(2)
- centx(2)=centerx(1); centz(2)=centerz(1)
- ENDIF
- !
- IF(wflag) THEN
- WRITE(*,'(/a/a)') ' Collision kinematics:', &
- ' Side Mass Charge <x> &
- &<y> tke'
- DO i=1,2
- WRITE(*,'(A,2F12.4,1P,3E12.4)') name(i),mass(i),charge(i), &
- centx(i),centz(i),tke2body(i)
- ENDDO
- ENDIF
- END SUBROUTINE twobody_print
+ !***************************************************
END MODULE Twobody
diff --git a/Doc-doxygen/Doxyfile b/Doc-doxygen/Doxyfile
index 83c3abfaf97e4a6bef847f213d5da7d249432cd2..8fdebd564d9ece49ac290b607905524f900c8b51 100644
--- a/Doc-doxygen/Doxyfile
+++ b/Doc-doxygen/Doxyfile
@@ -1690,7 +1690,7 @@ MAKEINDEX_CMD_NAME = makeindex
# The default value is: NO.
# This tag requires that the tag GENERATE_LATEX is set to YES.
-COMPACT_LATEX = NO
+COMPACT_LATEX = YES
# The PAPER_TYPE tag can be used to set the paper type that is used by the
# printer.
diff --git a/Doc/.DS_Store b/Doc/.DS_Store
deleted file mode 100644
index 60ed63c68832a29a49f41a8c12d97aaab092a56c..0000000000000000000000000000000000000000
Binary files a/Doc/.DS_Store and /dev/null differ
diff --git a/Doc/._.DS_Store b/Doc/._.DS_Store
deleted file mode 100644
index 2e58faeae9aadded645a4ab7ac6a4a5f3c0d547f..0000000000000000000000000000000000000000
Binary files a/Doc/._.DS_Store and /dev/null differ
diff --git a/Doc/._Documentation.pdf b/Doc/._Documentation.pdf
deleted file mode 100644
index 4539595eb41e525ffdcba1bd1a3d55f705e9794a..0000000000000000000000000000000000000000
Binary files a/Doc/._Documentation.pdf and /dev/null differ
diff --git a/Doc/Documentation.pdf b/Doc/Documentation.pdf
deleted file mode 100644
index c0796ee081b1cae6fd8ad4b9479fd6c3992e96f3..0000000000000000000000000000000000000000
--- a/Doc/Documentation.pdf
+++ /dev/null
@@ -1,25042 +0,0 @@
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-<< /S /GoTo /D (subsubsection.5.2.3) >>
-endobj
-164 0 obj
-(Module variables)
-endobj
-165 0 obj
-<< /S /GoTo /D (subsubsection.5.2.4) >>
-endobj
-168 0 obj
-(Subroutine poisson)
-endobj
-169 0 obj
-<< /S /GoTo /D (subsubsection.5.2.5) >>
-endobj
-172 0 obj
-(Subroutine coulinit)
-endobj
-173 0 obj
-<< /S /GoTo /D (subsubsection.5.2.6) >>
-endobj
-176 0 obj
-(Subroutine initiq)
-endobj
-177 0 obj
-<< /S /GoTo /D (subsection.5.3) >>
-endobj
-180 0 obj
-(Module Densities)
-endobj
-181 0 obj
-<< /S /GoTo /D (subsubsection.5.3.1) >>
-endobj
-184 0 obj
-(Module variables)
-endobj
-185 0 obj
-<< /S /GoTo /D (subsubsection.5.3.2) >>
-endobj
-188 0 obj
-(Subroutine alloc\137densities)
-endobj
-189 0 obj
-<< /S /GoTo /D (subsubsection.5.3.3) >>
-endobj
-192 0 obj
-(Subroutine add\137density)
-endobj
-193 0 obj
-<< /S /GoTo /D (subsection.5.4) >>
-endobj
-196 0 obj
-(Module Dynamic)
-endobj
-197 0 obj
-<< /S /GoTo /D (subsubsection.5.4.1) >>
-endobj
-200 0 obj
-(Module variables)
-endobj
-201 0 obj
-<< /S /GoTo /D (subsubsection.5.4.2) >>
-endobj
-204 0 obj
-(Subroutine getin\137dynamic)
-endobj
-205 0 obj
-<< /S /GoTo /D (subsubsection.5.4.3) >>
-endobj
-208 0 obj
-(Subroutine dynamichf)
-endobj
-209 0 obj
-<< /S /GoTo /D (subsubsection.5.4.4) >>
-endobj
-212 0 obj
-(Subroutine tstep)
-endobj
-213 0 obj
-<< /S /GoTo /D (subsubsection.5.4.5) >>
-endobj
-216 0 obj
-(Subroutine tinfo)
-endobj
-217 0 obj
-<< /S /GoTo /D (subsubsection.5.4.6) >>
-endobj
-220 0 obj
-(Subroutine resetcm)
-endobj
-221 0 obj
-<< /S /GoTo /D (subsection.5.5) >>
-endobj
-224 0 obj
-(Module Energies)
-endobj
-225 0 obj
-<< /S /GoTo /D (subsubsection.5.5.1) >>
-endobj
-228 0 obj
-(Module variables)
-endobj
-229 0 obj
-<< /S /GoTo /D (subsubsection.5.5.2) >>
-endobj
-232 0 obj
-(Subroutine integ\137energy)
-endobj
-233 0 obj
-<< /S /GoTo /D (subsubsection.5.5.3) >>
-endobj
-236 0 obj
-(Subroutine sum\137energy)
-endobj
-237 0 obj
-<< /S /GoTo /D (subsection.5.6) >>
-endobj
-240 0 obj
-(Module External)
-endobj
-241 0 obj
-<< /S /GoTo /D (subsubsection.5.6.1) >>
-endobj
-244 0 obj
-(Module variables)
-endobj
-245 0 obj
-<< /S /GoTo /D (subsubsection.5.6.2) >>
-endobj
-248 0 obj
-(Subroutine getin\137external)
-endobj
-249 0 obj
-<< /S /GoTo /D (subsubsection.5.6.3) >>
-endobj
-252 0 obj
-(Subroutine extfld)
-endobj
-253 0 obj
-<< /S /GoTo /D (subsubsection.5.6.4) >>
-endobj
-256 0 obj
-(Subroutine extboost)
-endobj
-257 0 obj
-<< /S /GoTo /D (subsubsection.5.6.5) >>
-endobj
-260 0 obj
-(Subroutine print\137extfield)
-endobj
-261 0 obj
-<< /S /GoTo /D (subsection.5.7) >>
-endobj
-264 0 obj
-(Module Forces)
-endobj
-265 0 obj
-<< /S /GoTo /D (subsubsection.5.7.1) >>
-endobj
-268 0 obj
-(Module variables and types)
-endobj
-269 0 obj
-<< /S /GoTo /D (subsubsection.5.7.2) >>
-endobj
-272 0 obj
-(Subroutine read\137force)
-endobj
-273 0 obj
-<< /S /GoTo /D (subsection.5.8) >>
-endobj
-276 0 obj
-(Module Fourier)
-endobj
-277 0 obj
-<< /S /GoTo /D (subsubsection.5.8.1) >>
-endobj
-280 0 obj
-(Module variables)
-endobj
-281 0 obj
-<< /S /GoTo /D (subsubsection.5.8.2) >>
-endobj
-284 0 obj
-(Subroutine initfft)
-endobj
-285 0 obj
-<< /S /GoTo /D (subsection.5.9) >>
-endobj
-288 0 obj
-(Module Fragments)
-endobj
-289 0 obj
-<< /S /GoTo /D (subsubsection.5.9.1) >>
-endobj
-292 0 obj
-(Module variables)
-endobj
-293 0 obj
-<< /S /GoTo /D (subsubsection.5.9.2) >>
-endobj
-296 0 obj
-(Subroutine getin\137fragments)
-endobj
-297 0 obj
-<< /S /GoTo /D (subsubsection.5.9.3) >>
-endobj
-300 0 obj
-(Subroutine read\137fragments)
-endobj
-301 0 obj
-<< /S /GoTo /D (subsubsection.5.9.4) >>
-endobj
-304 0 obj
-(Subroutine read\137one\137fragment)
-endobj
-305 0 obj
-<< /S /GoTo /D (subsubsection.5.9.5) >>
-endobj
-308 0 obj
-(Subroutine locate)
-endobj
-309 0 obj
-<< /S /GoTo /D (subsubsection.5.9.6) >>
-endobj
-312 0 obj
-(Subroutine phases)
-endobj
-313 0 obj
-<< /S /GoTo /D (subsubsection.5.9.7) >>
-endobj
-316 0 obj
-(Subroutine twobody\137init)
-endobj
-317 0 obj
-<< /S /GoTo /D (subsubsection.5.9.8) >>
-endobj
-320 0 obj
-(Subroutine boost\137fragment)
-endobj
-321 0 obj
-<< /S /GoTo /D (subsection.5.10) >>
-endobj
-324 0 obj
-(Module Grids)
-endobj
-325 0 obj
-<< /S /GoTo /D (subsubsection.5.10.1) >>
-endobj
-328 0 obj
-(Module variables)
-endobj
-329 0 obj
-<< /S /GoTo /D (subsubsection.5.10.2) >>
-endobj
-332 0 obj
-(Subroutine init\137grid)
-endobj
-333 0 obj
-<< /S /GoTo /D (subsubsection.5.10.3) >>
-endobj
-336 0 obj
-(Subroutine init\137coord)
-endobj
-337 0 obj
-<< /S /GoTo /D (subsubsection.5.10.4) >>
-endobj
-340 0 obj
-(Subroutine sder)
-endobj
-341 0 obj
-<< /S /GoTo /D (subsubsection.5.10.5) >>
-endobj
-344 0 obj
-(Subroutine sder2)
-endobj
-345 0 obj
-<< /S /GoTo /D (subsubsection.5.10.6) >>
-endobj
-348 0 obj
-(Subroutine setup\137damping)
-endobj
-349 0 obj
-<< /S /GoTo /D (subsubsection.5.10.7) >>
-endobj
-352 0 obj
-(Subroutine setdmc)
-endobj
-353 0 obj
-<< /S /GoTo /D (subsubsection.5.10.8) >>
-endobj
-356 0 obj
-(Subroutine gauss)
-endobj
-357 0 obj
-<< /S /GoTo /D (subsection.5.11) >>
-endobj
-360 0 obj
-(Module Inout)
-endobj
-361 0 obj
-<< /S /GoTo /D (subsubsection.5.11.1) >>
-endobj
-364 0 obj
-(Subroutine write\137wavefunctions)
-endobj
-365 0 obj
-<< /S /GoTo /D (subsubsection.5.11.2) >>
-endobj
-368 0 obj
-(Subroutine write\137densities)
-endobj
-369 0 obj
-<< /S /GoTo /D (subsubsection.5.11.3) >>
-endobj
-372 0 obj
-(Subroutine write\137one\137density)
-endobj
-373 0 obj
-<< /S /GoTo /D (subsubsection.5.11.4) >>
-endobj
-376 0 obj
-(Subroutine write\137vec\137density@write\137vec\137density \(Private\))
-endobj
-377 0 obj
-<< /S /GoTo /D (subsubsection.5.11.5) >>
-endobj
-380 0 obj
-(Subroutine plot\137density)
-endobj
-381 0 obj
-<< /S /GoTo /D (subsubsection.5.11.6) >>
-endobj
-384 0 obj
-(Subroutine sp\137properties)
-endobj
-385 0 obj
-<< /S /GoTo /D (subsubsection.5.11.7) >>
-endobj
-388 0 obj
-(Subroutine start\137protocol)
-endobj
-389 0 obj
-<< /S /GoTo /D (subsection.5.12) >>
-endobj
-392 0 obj
-(Module Levels)
-endobj
-393 0 obj
-<< /S /GoTo /D (subsubsection.5.12.1) >>
-endobj
-396 0 obj
-(Module variables)
-endobj
-397 0 obj
-<< /S /GoTo /D (subsubsection.5.12.2) >>
-endobj
-400 0 obj
-(Subroutine alloc\137levels)
-endobj
-401 0 obj
-<< /S /GoTo /D (subsubsection.5.12.3) >>
-endobj
-404 0 obj
-(Subroutines cdervx, cdervy, cdervz)
-endobj
-405 0 obj
-<< /S /GoTo /D (subsubsection.5.12.4) >>
-endobj
-408 0 obj
-(Subroutine laplace)
-endobj
-409 0 obj
-<< /S /GoTo /D (subsubsection.5.12.5) >>
-endobj
-412 0 obj
-(Subroutine schmid)
-endobj
-413 0 obj
-<< /S /GoTo /D (subsection.5.13) >>
-endobj
-416 0 obj
-(Module Meanfield)
-endobj
-417 0 obj
-<< /S /GoTo /D (subsubsection.5.13.1) >>
-endobj
-420 0 obj
-(Purpose)
-endobj
-421 0 obj
-<< /S /GoTo /D (subsubsection.5.13.2) >>
-endobj
-424 0 obj
-(Module variables)
-endobj
-425 0 obj
-<< /S /GoTo /D (subsubsection.5.13.3) >>
-endobj
-428 0 obj
-(Subroutine alloc\137fields)
-endobj
-429 0 obj
-<< /S /GoTo /D (subsubsection.5.13.4) >>
-endobj
-432 0 obj
-(Subroutine skyrme)
-endobj
-433 0 obj
-<< /S /GoTo /D (subsubsection.5.13.5) >>
-endobj
-436 0 obj
-(Subroutine hpsi)
-endobj
-437 0 obj
-<< /S /GoTo /D (subsection.5.14) >>
-endobj
-440 0 obj
-(Module Moment)
-endobj
-441 0 obj
-<< /S /GoTo /D (subsubsection.5.14.1) >>
-endobj
-444 0 obj
-(Purpose)
-endobj
-445 0 obj
-<< /S /GoTo /D (subsubsection.5.14.2) >>
-endobj
-448 0 obj
-(Module variables)
-endobj
-449 0 obj
-<< /S /GoTo /D (subsubsection.5.14.3) >>
-endobj
-452 0 obj
-(Subroutine moments)
-endobj
-453 0 obj
-<< /S /GoTo /D (subsubsection.5.14.4) >>
-endobj
-456 0 obj
-(Subroutine moment\137shortprint)
-endobj
-457 0 obj
-<< /S /GoTo /D (subsubsection.5.14.5) >>
-endobj
-460 0 obj
-(Subroutine moment\137print)
-endobj
-461 0 obj
-<< /S /GoTo /D (subsubsection.5.14.6) >>
-endobj
-464 0 obj
-(Subroutine q2diag)
-endobj
-465 0 obj
-<< /S /GoTo /D (subsection.5.15) >>
-endobj
-468 0 obj
-(Module Pairs)
-endobj
-469 0 obj
-<< /S /GoTo /D (subsubsection.5.15.1) >>
-endobj
-472 0 obj
-(Module variables)
-endobj
-473 0 obj
-<< /S /GoTo /D (subsubsection.5.15.2) >>
-endobj
-476 0 obj
-(Subroutine pair)
-endobj
-477 0 obj
-<< /S /GoTo /D (subsubsection.5.15.3) >>
-endobj
-480 0 obj
-(Subroutine pairgap)
-endobj
-481 0 obj
-<< /S /GoTo /D (subsubsection.5.15.4) >>
-endobj
-484 0 obj
-(Subroutine pairdn)
-endobj
-485 0 obj
-<< /S /GoTo /D (subsubsection.5.15.5) >>
-endobj
-488 0 obj
-(Subroutine rbrent)
-endobj
-489 0 obj
-<< /S /GoTo /D (subsubsection.5.15.6) >>
-endobj
-492 0 obj
-(Subroutine bcs\137occupation)
-endobj
-493 0 obj
-<< /S /GoTo /D (subsection.5.16) >>
-endobj
-496 0 obj
-(Module Parallel)
-endobj
-497 0 obj
-<< /S /GoTo /D (subsubsection.5.16.1) >>
-endobj
-500 0 obj
-(Parallelization concept)
-endobj
-501 0 obj
-<< /S /GoTo /D (subsubsection.5.16.2) >>
-endobj
-504 0 obj
-(Module variables)
-endobj
-505 0 obj
-<< /S /GoTo /D (subsubsection.5.16.3) >>
-endobj
-508 0 obj
-(Subroutine alloc\137nodes@alloc\137nodes\(Public\))
-endobj
-509 0 obj
-<< /S /GoTo /D (subsubsection.5.16.4) >>
-endobj
-512 0 obj
-(Subroutine init\137all\137mpi)
-endobj
-513 0 obj
-<< /S /GoTo /D (subsubsection.5.16.5) >>
-endobj
-516 0 obj
-(Subroutine allocate\137nodes)
-endobj
-517 0 obj
-<< /S /GoTo /D (subsubsection.5.16.6) >>
-endobj
-520 0 obj
-(Subroutine collect\137densities)
-endobj
-521 0 obj
-<< /S /GoTo /D (subsubsection.5.16.7) >>
-endobj
-524 0 obj
-(Subroutine collect\137sp\137properties)
-endobj
-525 0 obj
-<< /S /GoTo /D (subsubsection.5.16.8) >>
-endobj
-528 0 obj
-(Subroutine finish\137mpi)
-endobj
-529 0 obj
-<< /S /GoTo /D (subsubsection.5.16.9) >>
-endobj
-532 0 obj
-(Using sequential.f90)
-endobj
-533 0 obj
-<< /S /GoTo /D (subsection.5.17) >>
-endobj
-536 0 obj
-(Module Params)
-endobj
-537 0 obj
-<< /S /GoTo /D (subsubsection.5.17.1) >>
-endobj
-540 0 obj
-(Module variables)
-endobj
-541 0 obj
-<< /S /GoTo /D (subsubsection.5.17.2) >>
-endobj
-544 0 obj
-(General parameters)
-endobj
-545 0 obj
-<< /S /GoTo /D (subsubsection.5.17.3) >>
-endobj
-548 0 obj
-(File names and units)
-endobj
-549 0 obj
-<< /S /GoTo /D (subsubsection.5.17.4) >>
-endobj
-552 0 obj
-(Switches)
-endobj
-553 0 obj
-<< /S /GoTo /D (subsubsection.5.17.5) >>
-endobj
-556 0 obj
-(Output control)
-endobj
-557 0 obj
-<< /S /GoTo /D (subsubsection.5.17.6) >>
-endobj
-560 0 obj
-(Globally used variables)
-endobj
-561 0 obj
-<< /S /GoTo /D (subsubsection.5.17.7) >>
-endobj
-564 0 obj
-(Field output control)
-endobj
-565 0 obj
-<< /S /GoTo /D (subsubsection.5.17.8) >>
-endobj
-568 0 obj
-(Fragment number parameters)
-endobj
-569 0 obj
-<< /S /GoTo /D (subsection.5.18) >>
-endobj
-572 0 obj
-(Module Static)
-endobj
-573 0 obj
-<< /S /GoTo /D (subsubsection.5.18.1) >>
-endobj
-576 0 obj
-(Module variables)
-endobj
-577 0 obj
-<< /S /GoTo /D (subsubsection.5.18.2) >>
-endobj
-580 0 obj
-(Subroutine getin\137static)
-endobj
-581 0 obj
-<< /S /GoTo /D (subsubsection.5.18.3) >>
-endobj
-584 0 obj
-(Subroutine init\137static)
-endobj
-585 0 obj
-<< /S /GoTo /D (subsubsection.5.18.4) >>
-endobj
-588 0 obj
-(Subroutine statichf)
-endobj
-589 0 obj
-<< /S /GoTo /D (subsubsection.5.18.5) >>
-endobj
-592 0 obj
-(Subroutine grstep)
-endobj
-593 0 obj
-<< /S /GoTo /D (subsubsection.5.18.6) >>
-endobj
-596 0 obj
-(Subroutine diagstep)
-endobj
-597 0 obj
-<< /S /GoTo /D (subsubsection.5.18.7) >>
-endobj
-600 0 obj
-(Subroutine sinfo)
-endobj
-601 0 obj
-<< /S /GoTo /D (subsection.5.19) >>
-endobj
-604 0 obj
-(Module Trivial)
-endobj
-605 0 obj
-<< /S /GoTo /D (subsubsection.5.19.1) >>
-endobj
-608 0 obj
-(Subroutines cmulx, cmuly, and cmulz)
-endobj
-609 0 obj
-<< /S /GoTo /D (subsubsection.5.19.2) >>
-endobj
-612 0 obj
-(Function rpsnorm)
-endobj
-613 0 obj
-<< /S /GoTo /D (subsubsection.5.19.3) >>
-endobj
-616 0 obj
-(Function overlap)
-endobj
-617 0 obj
-<< /S /GoTo /D (subsubsection.5.19.4) >>
-endobj
-620 0 obj
-(Subroutines rmulx, rmuly, and rmulz)
-endobj
-621 0 obj
-<< /S /GoTo /D (subsection.5.20) >>
-endobj
-624 0 obj
-(Module Twobody)
-endobj
-625 0 obj
-<< /S /GoTo /D (subsubsection.5.20.1) >>
-endobj
-628 0 obj
-(Module variables)
-endobj
-629 0 obj
-<< /S /GoTo /D (subsubsection.5.20.2) >>
-endobj
-632 0 obj
-(Subroutine twobody\137case)
-endobj
-633 0 obj
-<< /S /GoTo /D (subsubsection.5.20.3) >>
-endobj
-636 0 obj
-(Subroutine getslope)
-endobj
-637 0 obj
-<< /S /GoTo /D (subsubsection.5.20.4) >>
-endobj
-640 0 obj
-(Function divpoint)
-endobj
-641 0 obj
-<< /S /GoTo /D (subsection.5.21) >>
-endobj
-644 0 obj
-(Module User)
-endobj
-645 0 obj
-<< /S /GoTo /D (section.6) >>
-endobj
-648 0 obj
-(Input description)
-endobj
-649 0 obj
-<< /S /GoTo /D (subsection.6.1) >>
-endobj
-652 0 obj
-(Namelist files)
-endobj
-653 0 obj
-<< /S /GoTo /D (subsection.6.2) >>
-endobj
-656 0 obj
-(Namelist force)
-endobj
-657 0 obj
-<< /S /GoTo /D (subsection.6.3) >>
-endobj
-660 0 obj
-(Namelist main)
-endobj
-661 0 obj
-<< /S /GoTo /D (subsection.6.4) >>
-endobj
-664 0 obj
-(Namelist grid)
-endobj
-665 0 obj
-<< /S /GoTo /D (subsection.6.5) >>
-endobj
-668 0 obj
-(Namelist static)
-endobj
-669 0 obj
-<< /S /GoTo /D (subsection.6.6) >>
-endobj
-672 0 obj
-(Namelist dynamic)
-endobj
-673 0 obj
-<< /S /GoTo /D (subsection.6.7) >>
-endobj
-676 0 obj
-(Namelist extern)
-endobj
-677 0 obj
-<< /S /GoTo /D (subsection.6.8) >>
-endobj
-680 0 obj
-(Namelist fragments)
-endobj
-681 0 obj
-<< /S /GoTo /D (subsection.6.9) >>
-endobj
-684 0 obj
-(Namelist user)
-endobj
-685 0 obj
-<< /S /GoTo /D (section.7) >>
-endobj
-688 0 obj
-(Output description)
-endobj
-689 0 obj
-<< /S /GoTo /D (subsection.7.1) >>
-endobj
-692 0 obj
-(Output and analysis)
-endobj
-693 0 obj
-<< /S /GoTo /D (subsection.7.2) >>
-endobj
-696 0 obj
-(File conver.res)
-endobj
-697 0 obj
-<< /S /GoTo /D (subsection.7.3) >>
-endobj
-700 0 obj
-(File monopoles.res)
-endobj
-701 0 obj
-<< /S /GoTo /D (subsection.7.4) >>
-endobj
-704 0 obj
-(File dipoles.res)
-endobj
-705 0 obj
-<< /S /GoTo /D (subsection.7.5) >>
-endobj
-708 0 obj
-(File quadrupoles.res)
-endobj
-709 0 obj
-<< /S /GoTo /D (subsection.7.6) >>
-endobj
-712 0 obj
-(File energies.res)
-endobj
-713 0 obj
-<< /S /GoTo /D (subsection.7.7) >>
-endobj
-716 0 obj
-(Standard output)
-endobj
-717 0 obj
-<< /S /GoTo /D (subsubsection.7.7.1) >>
-endobj
-720 0 obj
-(Static calculation)
-endobj
-721 0 obj
-<< /S /GoTo /D (subsubsection.7.7.2) >>
-endobj
-724 0 obj
-(Dynamic calculation)
-endobj
-725 0 obj
-<< /S /GoTo /D (section.8) >>
-endobj
-728 0 obj
-(Utilities)
-endobj
-729 0 obj
-<< /S /GoTo /D (subsection.8.1) >>
-endobj
-732 0 obj
-(Fileinfo)
-endobj
-733 0 obj
-<< /S /GoTo /D (subsection.8.2) >>
-endobj
-736 0 obj
-(Inertia)
-endobj
-737 0 obj
-<< /S /GoTo /D (subsection.8.3) >>
-endobj
-740 0 obj
-(Cuts)
-endobj
-741 0 obj
-<< /S /GoTo /D (subsection.8.4) >>
-endobj
-744 0 obj
-(Overlap)
-endobj
-745 0 obj
-<< /S /GoTo /D (subsection.8.5) >>
-endobj
-748 0 obj
-(Tdhf2Silo)
-endobj
-749 0 obj
-<< /S /GoTo /D (section.9) >>
-endobj
-752 0 obj
-(Running the code)
-endobj
-753 0 obj
-<< /S /GoTo /D (subsection.9.1) >>
-endobj
-756 0 obj
-(Compilation and linking)
-endobj
-757 0 obj
-<< /S /GoTo /D (subsection.9.2) >>
-endobj
-760 0 obj
-(External libraries needed)
-endobj
-761 0 obj
-<< /S /GoTo /D (subsection.9.3) >>
-endobj
-764 0 obj
-(Running with OPENMP)
-endobj
-765 0 obj
-<< /S /GoTo /D (subsection.9.4) >>
-endobj
-768 0 obj
-(Running under MPI)
-endobj
-769 0 obj
-<< /S /GoTo /D (subsection.9.5) >>
-endobj
-772 0 obj
-(Required input)
-endobj
-773 0 obj
-<< /S /GoTo /D (subsubsection.9.5.1) >>
-endobj
-776 0 obj
-(Static calculation)
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-\journal{Computer Physics Communications}
-\biboptions{sort&compress}
-
-
-\begin{document}
-\begin{frontmatter}
-
-\title{The TDHF Code Sky3D\\{\em Online Documentation}}
-
-\author{J.~A.~Maruhn\corref{author}} \ead{maruhn@th.physik.uni-frankfurt.de}
-\address{Institut f\"ur Theoretische Physik, Goethe-Universit\"at,
- Max-von-Laue-Str. 1, \\60438 Frankfurt am Main, Germany}
-
-\author{P.-G.~Reinhard}\ead{Paul-Gerhard.Reinhard@physik.uni-erlangen.de}
-\address{Institut f\"ur Theoretische Physik II, Universit\"at
- Erlangen-N\"urnberg, \\Staudtstrasse 7, 91058 Erlangen, Germany}
-
-\author{P.~D.~Stevenson}\ead{p.stevenson@surrey.ac.uk}
-\address{Department of Physics, University of Surrey, Guildford,
- Surrey, GU2 7XH, United Kingdom}
-
-\author{A. S. Umar}\ead{sait.a.umar@Vanderbilt.Edu}
-\address{Department of Physics and Astronomy, Vanderbilt University, \\Nashville, Tennessee 37235, USA}
-
-\cortext[author]{Corresponding author. \textit{Phone: +49-69-79847873}}
-
-\begin{abstract}
-The nuclear mean-field model based on Skyrme forces or related density
-functionals has found widespread application to the description of
-nuclear ground states, collective vibrational excitations, and
-heavy-ion collisions. The code Sky3D solves the static or dynamic
-equations on a three-dimensional Cartesian mesh with isolated or
-periodic boundary conditions and no further symmetry
-assumptions. Pairing can be included in the BCS approximation for the
-static case. The
-code is implemented with a view to allow easy modifications for including
-additional physics or special analysis of the results.
-\end{abstract}
-\begin{keyword}
-Hartree-Fock; BCS; Density-functional theory; Skyrme energy functional; Giant Resonances;
-Heavy-Ion collisions.
-\end{keyword}
-\end{frontmatter}
-
-{\bf PROGRAM SUMMARY}
-
-\begin{small}
-\noindent
-{\em Title:} The TDHF Code Sky3D \\
-{\em Authors:} J.~A.~Maruhn, P.-G.~Reinhard, P.~D.~Stevenson, and A.~S.~Umar.\\
-{\em Program Title:} Sky3D\\
-{\em Journal Reference:} \\
- %Leave blank, supplied by Elsevier.
-{\em Catalogue identifier:} \\
- %Leave blank, supplied by Elsevier.
-{\em Licensing provisions:} none\\
- %enter "none" if CPC non-profit use license is sufficient.
-{\em Programming language:} Fortran 90. The {\tt OpenMP} version
-requires a relatively recent compiler; it was found to work using {\tt
- gfortran 4.6.2} or later and the Intel compiler version 12 or later.\\
-{\em Computer:} All computers with a Fortran compiler supporting at
-least Fortran 90. \\
- %Computer(s) for which program has been designed.
-{\em Operating system:} All operating systems with such a
-compiler. Some of the Makefiles and scripts depend on a Unix-like
-system and need modification under Windows.\\
- %Operating system(s) for which program has been designed.
-{\em RAM:} 1~GB\\
- %RAM in bytes required to execute program with typical data.
-{\em Number of processors used:} no built-in limit, runs under both
-OpenMP and MPI\\
- %If more than one processor.
-{\em Keywords:} Nuclear theory, Mean-field models, Nuclear reactions\\
- % Please give some freely chosen keywords that we can use in a
- % cumulative keyword index.
-{\em Classification:} 17.16 Theoretical Methods - General, 17.22
-Hartree-Fock Calculations, 17.23 Fission and Fusion Processes\\
- %Classify using CPC Program Library Subject Index, see (
- % http://cpc.cs.qub.ac.uk/subjectIndex/SUBJECT_index.html)
- %e.g. 4.4 Feynman diagrams, 5 Computer Algebra.
-{\em External routines/libraries:} LAPACK, FFTW3\\
-\\
-{\em Nature of problem:} The time-dependent Hartree-Fock equations can
-be used to simulate nuclear vibrations and collisions between nuclei
-for low energies. This code implements the equations based on a Skyrme
-energy functional and also allows the determination of the
-ground-state structure of nuclei through the static version of the
-equations. For the case of vibrations the principal aim is to
-calculate the excitation spectra by Fourier-analyzing the time
-dependence of suitable observables. In collisions, the formation of a
-neck between nuclei, the dissipation of energy from collective motion,
-processes like charge transfer and the approach to fusion are of
-principal interest.\\
- %Describe the nature of the problem here.
-\\
-{\em Solution method:} The nucleonic wave function spinors are
-represented on a three-dimensional Cartesian mesh with no further
-symmetry restrictions. The boundary conditions are always periodic for
-the wave functions, while the Coulomb potential can also be calculated
-for an isolated charge distribution. All spatial derivatives are
-evaluated using the finite Fourier transform method. The code solves
-the static Hartree-Fock equations with a damped gradient iteration
-method and the time-dependent Hartree-Fock equations with an expansion
-of the time-development operator. Any number of initial nuclei can be
-placed into the mesh in with arbitrary positions and
-initial velocities.\\
-\\
-{\em Restrictions:} The reliability of the mean-field approximation is
-limited by the absence of hard nucleon-nucleon collisions. This limits
-the scope of applications to collision energies about a few MeV per
-nucleon above the Coulomb barrier and to relatively short interaction
-times. Similarly, some of the missing time-odd terms in the implementation
-of the Skyrme interaction may restrict the applications to even-even
-nuclei.\\
-\\
- %Describe any restrictions on the complexity of the problem here.
- \\
-{\em Unusual features:}\\
-The possibility of periodic boundary conditions and the highly
-flexible initialization make the code also suitable for astrophysical
-nuclear-matter applications.\\
- \\
-{\em Running time:} The running time depends strongly on the size of
-the grid, the number of nucleons, and the duration of the
-collision. For a single-processor PC-type computer it can vary between
-a few minutes and weeks.\\
- %Give an indication of the typical running time here.
- \\
-\end{small}
-
-
-\tableofcontents
-
-\section{Introduction}
-The vast majority of microscopic models of many-body systems rely
-on a description in terms of the single-particle (s.p.)
-wave functions. Among them, self-consistent mean-field models (SCMF)
-automatically generate the optimal one-body potentials
-corresponding to the s.p.\ wave functions. A rigorous SCMF is the
-Hartree-Fock theory (HF) where the s.p.\ wave functions are determined
-variationally for a given two-body interaction~\cite{Fet71,Mar10aB}. A
-more practical approach is provided by the Density Functional Theory (DFT),
-which incorporates the involved many-body effects into effective
-interactions, or effective energy-density functionals.
-This is a very efficient and successful scheme, widely used in
-electronic systems~\cite{Dre90}. Straightforward HF is unsuitable
-for nuclei because the free-space two-nucleon force contains a strong
-short-range
-repulsion requiring renormalization in the nuclear medium.
-For this reason, nuclear SCMFs necessarily employ effective
-interactions or functionals although they often carry the label
-HF as, e.g, in the Skyrme Hartree-Fock (SHF) method. There are
-relativistic as well as non-relativistic approaches. For an extensive
-review see~\cite{Ben03aR}.
-
-The description of dynamical processes is even more demanding than the
-modeling of structure. SCMFs are also the first method of choice
-in this domain. The natural extension of HF is time-dependent HF
-(TDHF) which was proposed as early as 1930 in~\cite{Dir30}. Earlier
-applications were restricted to the linearized regime covering small
-amplitude motion, see, e.~g., \cite{Bro71aB}. Large scale TDHF
-calculations became possible in the last few decades with the
-increasing computing capacities. Again, as in the static case, true
-TDHF calculations make sense only for electronic systems and even
-there they are very rare. The overwhelming majority of dynamical SCMF
-calculations employ, in fact, time-dependent DFT (TDDFT). In
-electronic systems, this amounts to the time-dependent
-local density approximation (TDLDA) \cite{Dre90}, which is widely used
-in atoms, molecules, and solids; for examples in nanoparticles see,
-e.~g., \cite{Rei03a}. Dynamical SCMFs in nuclei also stay at the level
-of TDLDA even if they are often named TDHF which happens particularly
-for dynamical calculations using the Skyrme energy functional. Nuclear
-TDHF started about forty years ago \cite{Bon76a} and has developed
-since then into a powerful and versatile tool for simulating a great
-variety of dynamical scenarios. Earlier applications were based
-mainly on non-relativistic TDHF using the effective Skyrme energy
-functional~\cite{Neg82aR,Dav85a}. Due to higher numerical demands,
-relativistic calculations appeared somewhat later~\cite{Bai87a}, but
-have developed meanwhile equally well to a widely used
-tool~\cite{Ber95a,Vre05aR}.
-
-In this paper, we present a code for TDHF calculations on the basis of
-the non-relativistic Skyrme energy functional. The code uses a fully
-three dimensional (3D) representation of wave functions and fields on
-a Cartesian grid in coordinate space. There are no symmetry
-restrictions and the full Skyrme energy functional is used including
-the spin-orbit and most important time-odd terms. Such fully-fledged
-3D calculations became possible only over the last decade with the
-steadily increasing computing capabilities. In fact, early TDHF
-studies all used restricted representations, axial symmetry and/or
-reflection symmetries. This limited the possible applications. TDHF
-experienced a revival during the last ten years when unrestricted 3D
-calculations became possible. There are several groups performing
-large scale TDHF studies for various scenarios of nuclear dynamics,
-see, e.g., \cite{,Has08a,Uma09a,Loe12a}. Aside from these studies of
-nuclear collisions a principal application has been to collective
-vibrations, e.g., \cite{Sim03a,Mar05a,Umar05a,Nak05a}. In the linear regime
-TDHF leads to fully self-consistent RPA, for which though, unlike
-TDHF, often additional approximations like the neglect of the
-Coulomb potential or the spin-orbit terms in the residual interaction
-are made. TDHF can also be used to investigate non-linearities of
-nuclear vibrations.
-
-For a recent review of vibrational and collisional applications see
-\cite{Simenel}. Such calculations have clearly outgrown the
-developmental stage. It is an appropriate time to give a broader
-public access to a 3D TDHF code. This is the goal of this paper.
-Skyrme HF covers such a broad range of physical phenomena and is
-relatively involved that efficient computational treatment of 3D
-simulations requires elaborate numerical methods. We shall make an
-effort to explain the many necessary ingredients in a comprehensive,
-and yet compact, manner.
-
-Most recent Skyrme density functionals contain terms such as
-fractional powers of the density that cannot be related to a two- or
-three-body interaction. In that sense, the present code solves the
-TDDFT rather than TDHF equations. Nevertheless, we prefer to keep the
-name TDHF since it is associated historically with this large field of
-nuclear reaction theory.
-
-Although the code may be run as it is and many innovative applications
-are possible, we also intend it to be used for
-exploring new ideas that need a basic HF and TDHF algorithm, which is
-implemented in a transparent modern style of programming and
-extensively documented (see the accompanying online documentation),
-allowing for relatively easy modification. A guide to some possible
-extensions is given in Sect.~\ref{sec:modify}.
-
-Some recent developments in TDHF which require more extensive
-modification but might be implemented on the basis of this code
-include:
-
-\begin{itemize}
-
-\item The extraction of nucleus-nucleus potentials
-with and without dynamical effects. Approaches include
-density-con\-strained TDHF (DC-TDHF) method~\cite{umar2006d}, the
-density-dependent TDHF (DD-TDHF) approach~\cite{washiyama2008}, and
-the frozen HF method~\cite{simenel2013b}.
-
-\item The extraction of (multi-)nucleon transfer probabilities using
-particle number projection techniques~\cite{simenel2010}.
-
-\item The use of novel spatial distributions to excite low-lying
- dipole states \cite{Goddard2013}.
-
-\item The incorporation of fluctuations of one-body observables using
-the Balian-V\'en\'eroni variational
-principle~\cite{balian1984,Broomfield2008,simenel2011}.
-
-\item The inclusion of dynamical pairing correlations at the TDHF+BCS
-or TDHFB levels~\cite{avez2008,ebata}
-
-\item The inclusion of collision terms with the Extended TDHF or the
-time-dependent density matrix approaches~\cite{lacroix1999,tohyama2002a}.
-
-\item The generalization of the static HF code to odd
-particles by the inclusion of the time-odd terms in the Skyrme
-interaction~\cite{Eng75a,dobaczewski1995,umar2005b}.
-\end{itemize}
-
-\section{General purpose and structure}
-
-\subsection{Intended applications}
-The code Sky3D solves the static Hartree-Fock as well as the
-time-dependent Hartree-Fock (TDHF) equations for interactions of
-Skyrme-force type in a general three-dimensional geometry. No
-symmetries of any kind are assumed, so that the code can be used for a
-wide variety of applications in nuclear structure, collective
-excitations, and nuclear reactions; of course within the limitations
-of mean-field theory.
-
-\subsection{Specific model implemented}
-The code in the presented version contains a useful selection of terms
-in the Skyrme force but by no means all terms that have been included
-in some recent works. It should still be useful, because (1) for many
-interesting applications the interest is semi-quantitative so that a
-Skyrme force fitted with the latest models is not necessary --- usually
-a selection of forces is desired to look at the variability of
-results, but not a high-accuracy fit of data; (2) the code is written
-in such a way that additional terms can be added easily. The coding
-corresponds one-to-one to the analytic formulas in most places except
-where efficiency demands reordering the calculations.
-
-\subsubsection{The single-particle basis}
-
-In a mean field theory one seeks to describe the many-body system
-exclusively in terms of a set of single-particle wave functions
-$\psi_\alpha$ with fractional occupation amplitudes $v_\alpha$, i.e.
-\begin{subequations}
- \begin{equation}
- \left\{\psi_\alpha,v_\alpha,\alpha=1,...,\Omega\right\}
- \end{equation}
- where $\Omega$ denotes the size of the active s.p.\ space. The
- occupation amplitude can take values continuously in the interval
- $[0,1]$. The complementary non-occupation amplitude is
- $u_\alpha=\sqrt{1-v_\alpha^2}$. A formal definition of the BCS
- mean-field state reads
- \begin{equation}
- |\Phi\rangle
- =
- \prod_{\alpha>0}\big(
- u_\alpha^{\mbox{}}+v_\alpha^{\mbox{}}\hat{a}^+_\alpha\hat{a}^{+}_{\bar\alpha}
- \big)|0\rangle
- \label{eq:BCState}
- \end{equation}
-\end{subequations}
-where $|0\rangle$ is the vacuum state, $\hat{a}^+_\alpha$ the
-generator of a Fermion in state $\psi_\alpha$, and $\bar\alpha$ the
-time reverse partner to state $\alpha$. We will use variation of the
-BCS amplitudes $v_\alpha^{\mbox{}}$ only in the static part of
-even-even nuclei where the time reverse partner is unambiguously
-defined. In fact, the pairing calculation assumes that the paired
-states have exactly the same spatial density distribution.
-
-The time-dependent calculation technically can be run with pairing
-included. This is done by keeping the pairing occupation probabilities
-fixed during the time evolution; the pairing between states can simply
-be taken over from the static calculation, although time-reversal
-invariance is lost for boosted nuclei. For a the initial state in a
-collision, boosting a single-particle state and its time-reversed
-partner does not destroy their role for pairing, while the boosted
-states are no-longer time-reversal conjugate. This is a consequence of
-Galilei invariance. For a study of small vibrations the initial
-pairing of states is also correct although again they are no longer
-time-reversal conjugate once a boost or an external potential are
-applied.
-
-The conservation of totalenergy {\em excluding the pairing energy} is
-not impaired for the case of constant occupation, while the pairing
-energy is not even computed.
-
-{\em It should be noted, however, that once the wave functions change
- dynamically, this approach is not correct as the occupation
- probabilities will also change with time. In this case the
- TDHF-Bogolyubov equations should be used. In addition, the pairing
- energies are not computed at all. Using this code as is with pairing
- included in the time-dependent case might be useful for schematic or
- exploratory studies but extreme caution is advised when interpreting
- such results.}
-
-\subsubsection{Local densities and currents \label{sec:dens}}
-
-The Skyrme-energy-density functional is defined in terms of only a
-few local densities and currents. These are the {\em time-even} fields
-\begin{align}
- \rho_q(\vec r)&=\displaystyle
- \sum_{\alpha\in q}\sum_{s}
- v_{\alpha}^2|\psi_{\alpha}(\vec{r},s)|^2
- &&
- \text{density}
- \notag \\
- \vec{J}_q(\vec r) &=\displaystyle
- -\I\sum_{\alpha\in q}\sum_{ss'} v_{\alpha}^2
- \psi_{\alpha}^*(\vec{r},s)
- \nabla\! \times\! \vec{\sigma}_{ss'}
- \psi^{\mbox{}}_{\alpha}(\vec{r},s')
- &&
- \text{spin-orbit dens.}
- \notag\\
- \tau_q(\vec r)&=\displaystyle
- \sum_{\alpha\in q}\sum_{s}
- v_{\alpha}^2|\nabla\!\psi_{\alpha}(\vec{r},s)|^2
- &&
- \mbox{kinetic density,}
- \label{eq:rtjeven}
-\end{align}
-the {\em time-odd} fields
-\begin{align}
- \vec{s}_q(\vec r)&=\displaystyle
- \sum_{\alpha\in q}\sum_{ss'} v_{\alpha}^2
- \psi^*_{\alpha}(\vec{r},s)\vec{\sigma}_{ss'}
- \psi^{\mbox{}}_{\alpha}(\vec{r},s')
- &&
- \mbox{spin density}
- \notag\\
- \vec{\jmath}_q(\vec r)&=\displaystyle
- \Im{m}\left\{\sum_{\alpha\in q}\sum_{s} v_{\alpha}^2
- \psi^*_{\alpha}(\vec{r},s)
- \nabla\!\psi^{\text{}}_{\alpha}(\vec{r},s)\right\}
- &&
- \text{current density,}
- \label{eq:rtjodd}
-\end{align}
-and a field with undefined time parity:
- \begin{align}
- \xi_q(\vec r)
- &=\displaystyle
- {\sum_{\alpha\in q}\sum_{s} u_{\alpha}v_{\alpha}
- \psi_{\overline{\alpha}}(\vec{r},s)
- \psi_{\alpha}}(\vec{r},s)
- % \sum_{\alpha\in q} u_{\alpha}v_{\alpha}|\psi_{\alpha}|^2
- &&
- \text{pairing density}
- \label{eq:rtjpair}
-\end{align}
-where $q$ labels the nucleon species with $q=p$ for protons and $q=n$
-for neutrons. A local density/current without $q$ index stands for the
-total quantity, e.g. $\rho=\rho_p+\rho_n$ is the total density, and
-similarly for the other densities/currents. The
-variable $s$ indicates the two spinor components of the wave
-functions.
-
-\subsubsection{The energy-density functional}\label{sec:efunc}
-
-The mean-field equations solved in the code are based on the widely
-used Skyrme energy functional. For recent reviews see
-\cite{Ben03aR,Erl11a}. The functional at the level at which it is
-used here can be written as
-\begin{subequations}
- \begin{equation}
- \label{eq:efundet}
- E_\mathrm{tot}
- =
- T+(E_0+E_1+E_2+E_3+E_\mathrm{ls})+E_{\rm Coulomb}+E_{\rm pair}+E_{\rm corr},
- \end{equation}
- where the parentheses were used to group the terms arising from the
- Skyrme force. The various terms read in detail (all densities and
- currents defined in Section~\ref{sec:dens} are understood to depend
- on $\vec r$)
- \begin{itemize}
- \item $T$: the total kinetic energy calculated as
- \begin{equation}
- \label{eq:ehft}
- T=\sum_q \frac{\hbar^2}{2m_q}\int\D^3 r\, \tau_q
- \end{equation}
- with $\tau_q$ the isospin-specific kinetic density of Eq.~(\ref{eq:rtjeven}).
- \item $E_0$: The $b_0$ and $b'_0$-dependent part is
- \begin{equation}
- \label{eq:ehf0}
- E_0=\int \D^3\!r\,\left(\frac{b_0}{2}\rho^2-\frac{b_0'}{2}\sum_q\rho_q^2\right).
- \end{equation}
- \item $E_{1}$: kinetic terms containing the coefficients $b_1$ and $b_1'$:
- \begin{equation}
- \label{eq:ehf1}
- E_1=\int \D^3\!r\,\left(b_1[\rho\tau-{\vec\jmath\,}{}^2]
- -b_1'\sum_q[\rho_q\tau_q-{\vec\jmath_q}{}^2]\right).
- \end{equation}
- \item $E_{2}$: terms containing the coefficients $b_2$ and $b_2'$.
- They involve the Laplacians of the densities.
- \begin{equation}
- \label{eq:ehf2}
- E_2=\int \D^3\!r\,\left(-\frac{b_2}{2}\rho\Delta\rho
- +\frac{b_2'}{2}\sum_q\rho_q\Delta\rho_q\right).
- \end{equation}
- \item $E_{3}$: The many-body contribution is given by
- \begin{equation}
- \label{eq:ehf3}
- E_{3}=\int \D^3\!r\,\left(\frac{b_3}{3}\rho^{\alpha+2}
- -\frac{b_3'}{3}\rho^\alpha\sum_q\rho_q^2\right).
- \end{equation}
- \item $E_\mathrm{ls}$: the spin-orbit energy
- \begin{equation}\begin{split}
- E_\mathrm{ls}&=\int \D^3\!r\,\biggl(
- -b_4[\rho\nabla\cdot\vec J+
- \vec s\cdot(\nabla\times\vec\jmath)]\\
- &-b_4'\sum_q[\rho_q\nabla\cdot\vec J_q+
- \vec s_q\cdot(\nabla\times\vec\jmath_q)]\biggr)
- \label{eq:ehfls}
- \end{split}
- \end{equation}
- \item $E_\mathrm{C}$: the Coulomb energy. It consists of the
- standard expression for a charge distribution in its own field
- (Hartree term) plus the exchange term in the Slater approximation
- ~\cite{Sla51}. The formula is
- \begin{equation}
- \label{eq:ehfc}
- E_\mathrm{C}
- =
- \frac{e^2}{2}\int \D^3\!r \D^3\!r'
- \frac{\rho_p(\vec{r})\rho_p(\vec{r'})}{|\vec{r}-\vec{r'}|}
- -
- \int \D^3\!r\,\frac{3e^2}{4}\left(\frac3{\pi}\right)^{\tfrac1{3}}\rho_p^{4/3}
- \end{equation}
- where $e$ is the elementary charge with $e^2=1.43989$ MeV$\cdot$fm.
- \item $E_\mathrm{pair}$: the pairing energy. It consists of a
- contact pairing interaction involving the pairing densities $\xi_q$
- augmented by an optional density dependence. The formula is
- \begin{equation}
- E_{\rm pair}
- =
- \frac{1}{4} \sum_{q\in\{p,n\}}V_\mathrm{pair,q}
- \int \D^3r |\xi_q|^2
- \left[1 -\frac{\rho}{\rho_{0,\mathrm{pair}}}\right]\;.
- \label{eq:epair}
- \end{equation}
- It contains a continuous switch, the parameter
- $\rho_{0,\mathrm{pair}}$. A pure $\delta$-interaction (DI), also
- called volume pairing, is recovered for
- $\rho_{0,\mathrm{pair}}\longrightarrow\infty$. The general case is
- the density dependent $\delta$-interaction (DDDI). A typical
- value near matter equilibrium density
- $\rho_{0,\mathrm{pair}}=0.16$ fm$^{-3}$ concentrates pairing to
- the surface. The most flexible choice is to consider
- $\rho_{0,\mathrm{pair}}$ as an additional free parameter. Actual
- adjustments with this option deliver a form of the pairing
- functional which stays in between the extremes of volume and
- surface pairing~\cite{Klu09a}. The implementation in the code is
- discussed in Section \ref{sec:pairs}.
- \end{itemize}
-\end{subequations}
-
-The term $E_{\rm corr}$ stands for all additional corrections from
-correlations beyond the mean field that might be added. Most
-calculations include at least the center-of-mass correction
-$E_\mathrm{cm}$. For deformed nuclei this should be augmented by a
-rotational correction and for soft nuclei by correlations from all
-low-energy quadrupole motions \cite{Klu08a}. So far, these
-correlations are usually added a posteriori after static
-calculations. This procedure is associated with setting the switch
-{\tt zpe=1} which is the standard option adopted here. A fully
-variational treatment and a dynamical propagation of the
-c.m.\ correction is extremely involved and usually not considered. The
-other strategy is to modify the nucleon mass by
-$m\longrightarrow{m}-m/A$ and to include this simplified correction in
-the variational treatment thus avoiding the a posteriori
-correction. This way is chosen in a couple of traditional
-parametrizations, e.g., in SkM$^*$~\cite{Bar82a}. We keep this option
-in case of static calculations for consistency and associate it with
-{\tt zpe=0}.
-
-Using a center-of-mass correction is desirable for static
-calculations, it is disputable for vibrational excitations, and runs
-fully into inconsistencies in collisions and fragmentation as it
-employs only the total mass number $A$ and cannot account for the
-masses of the fragments. Therefore all dynamical runs are done
-with
-\begin{equation}
- E_\mathrm{cm}=0
- \quad.
-\end{equation}
-An inconsistency may occur if {\tt zpe=0} was used in the static
-calculation providing the input for the dynamical run because the
-setting $m\longrightarrow{m}-m/A$ is not used in dynamics. In order
-to safely suppress the c.m.\ correction in statics and dynamics, we
-introduce an additional switch {\tt turnoff\_zpe} in the input, which
-turns off the zero-point energy correction irrespective of what value
-is given to {\tt zpe} in the force definition. This allows the use of
-forces with {\tt zpe=0} also for collisions. It should be pointed out
-also that there are newer Skyrme parametrizations like Sly4d
-\cite{Chabanat1995,Kim1997} and UNEDF \cite{Kortelainen2012} that are
-fitted without any center-of-mass correction and are thus specially
-intended for collision calculations.
-
-For some further considerations on the c.m.\ correction in TDHF see
-\cite{umar2009c}.
-
-The functional in the above form contains the minimal number of terms
-which are needed to guarantee Galilean invariance~\cite{Eng75a,Erl11a}
-and so to allow performance of TDHF calculations which respect all basic
-conservation laws. We ignore the tensor spin-orbit terms and spin-spin
-couplings~\cite{Ben03aR,Per04a,Erl11a}. These may be important for
-magnetic excitations~\cite{Erl11a} and odd nuclei~\cite{Pot10a} which
-are, however, not the prime focus of TDHF studies.
-
-\subsubsection{Force coefficients}
-The above formulation in terms of the Skyrme energy functional
-introduces the force parameters $b_0$, $b'_0$, ... $b'_4$ naturally
-as the factors in front of each contribution in the terms
-(\ref{eq:ehf0}-\ref{eq:ehfls}). Traditionally, the functional is deduced from a
-Skyrme force which is a density-dependent, zero-range interaction
-\cite{Vau72a}. The $t$ and $x$ coefficients in this Skyrme-force
-definition are related to the $b$ coefficients in the functional
-definition as
-\begin{equation}
- \label{eq:force-coeff}
- \begin{split}
- b_0&=t_0\,\left(1+\tfrac1{2}x_0\right) \\
- b_0'&=t_0\,\left(\tfrac1{2}+x_0\right)\\
- b_1&=\tfrac1{4}\left[t_1\,\left(1+\tfrac1{2}x_1\right)+
- t_2\,\left(1+\tfrac1{2}x_2\right)\right] \\
- b_1'&=\tfrac1{4}\left[t_1\,\left(\tfrac1{2}+x_1\right)-
- t_2\,\left(\tfrac1{2}+x_2\right)\right] \\
- b_2&=\tfrac1{8}\left[3t_1\,\left(1+\tfrac1{2}x_1\right)-
- t_2\,\left(1+\tfrac1{2}x_2\right)\right] \\
- b_2'&=\tfrac1{8}\left[3t_1\,\left(\tfrac1{2}+x_1\right)+
- t_2\,\left(\tfrac1{2}+x_2\right)\right]\\
- b_3&=\tfrac1{4}t_3 \left(1+\tfrac1{2}x_3\right) \\
- b_3'&=\tfrac1{4}t_3 \left(\tfrac1{2}+x_3\right) \\
- b_4&=\tfrac1{2}t_4
- \end{split}
-\end{equation}
-The coefficient $b_4'$ is usually fixed as $b'_4=\frac{1}{2}t_4$ for most
-traditional Skyrme forces. More recent variants of Skyrme forces (SkI3
-etc.) handle it as a separate free parameter~\cite{Rei95a}. In addition
-to the $b$ and $b'$ parameters, there is the power coefficient in the
-(originally) three-body term, which is usually called $\alpha$, but in
-the code is referred to as {\tt power}. The input of the force to the
-code is done in terms of the $t_i$, $x_i$ parameters, see Section
-\ref{sec:Forces}.
-
-\subsubsection{The single-particle Hamiltonian}
-\label{sec:otherfields}
-
-The mean-field Hamiltonian $\hat{h}$ is derived from the energy
-functional of Section~\ref{sec:efunc} by variation
-$\partial_{\psi_\alpha^*}E=\hat{h}\psi_\alpha$. It reads in detail
-\begin{subequations}
- \label{eq:mfpots}
- \begin{equation}\begin{split}
- \hat h_q& =
- U_q(\vec r)-\nabla\cdot\left[B_q(\vec r)\nabla\right]
- +\I\vec W_q\cdot(\vec\sigma\times\nabla)
- +\vec S_q\cdot\vec\sigma\\
- &-\frac{\I}{2} \left[(\nabla\cdot\vec A_q)+2\vec
- A_q\cdot\nabla\right]\,,
- \label{eq:spham}
- \end{split}
-\end{equation}
- with $q\in\{p,n\}$ as usual distinguishing the different
- Hamiltonians for protons and neutrons. For the protons the Coulomb
- potential is also added. The first term is the local part of the
- mean field, which acts on the wave functions like a local
- potential. It is defined as
- \begin{equation}\begin{split}
- U_q
- &=
- b_0\rho-b_0'\rho_q+b_1\tau-b_1'\tau_q-b_2\Delta\rho
- +b_2'\Delta\rho_q
- \\
- &+b_3\tfrac{\alpha+2}{3}\,\rho^{\alpha+1}-b_3'\tfrac2{3}\,\rho^\alpha\rho_q
- -b_3'\tfrac\alpha{3}\,\rho^{\alpha-1}\sum_{q'}\rho_{q'}^2
- \\
- &
- -b_4\nabla\cdot\vec J-b_4'\nabla\cdot \vec{J}_q
- \;.
- \label{eq:upot}
- \end{split}
-\end{equation}
- Next comes the ``effective mass'', which replaces the standard
- $\tfrac{\hbar^2}{2m}$ factor by the isospin and space-dependent
- expression
- \begin{equation}
- \label{eq:bmass}
- B_q=\frac{\hbar^2}{2m_q}+b_1\rho-b_1'\rho_q.
- \end{equation}
- Note that the Skyrme force definitions contain the first term
- (nucleon mass) as a parameter which varies slightly from
- parametrization to parametrization and may be different for protons
- and neutrons. The spin-orbit potential is
- \begin{equation}
- \vec W_q
- =
- b_4\nabla\rho+b_4'\nabla\rho_q
- \quad.
- \label{eq:wlspot}
- \end{equation}
- The above three terms involve the time-even densities. Dynamical
- effects come into play with the next terms which include the
- time-odd contributions from current and spin-density:
- \begin{align}
- \label{eq:aq}
- \vec A_q&=-2b_1\vec\jmath+2b_1'\vec\jmath_q-b_4\nabla\times\vec s
- -b_4'\nabla\times\vec s_q.
- \;, \\
- \label{eq:spot}
- \vec S_q&=-b_4\nabla\times\vec\jmath-b_4'\nabla\times\vec\jmath_q
- \;.
-\end{align}
-\end{subequations}
-
-
-\subsection{Coupling to external fields}
-\label{sec:external}
-
-For the dynamic case, the system can also be coupled to an external
-excitation field, to study collective response such as in giant
-resonances. The present code only implements a very simple case, since
-it is expected that most serious applications will need modifications,
-which are quite easy to incorporate.
-
-The external field is introduced as a time-dependent, local operator
-\begin{subequations}
- \begin{equation}
- \hat{h}_q\longrightarrow\hat{h}_q+U_{q,\mathrm{ext}}(\vec{r},t)
- \quad,\quad
- U_{q,\mathrm{ext}}
- =
- \eta\,f(t)\,F_q(\vec r)\;,
- \end{equation}
- where $f(t)$ carries the temporal profile of the excitation
- mechanism, $F_q(\vec r)$ is some local operator, and $\eta$ tunes
- the overall strength. The spatial distribution $F_q(\vec{r})$, is
- allowed to be different for the two isospins $q$. Typical examples
- are isoscalar and isovector multipole operators as, e.g., the
- isoscalar quadrupole $F_q(\vec{r})=2z^2-x^2-y^2$.
-
- The prefactor $\eta$ is a strength parameter which allows
- scanning different excitation strengths easily while keeping the
- temporal and spatial profiles the same. It should be noted that the
- absolute magnitude of the perturbing potential by itself usually has
- little direct meaning. What counts is the excitation energy caused
- by the perturbation and subsequently the magnitude of vibrations in
- the observables (such as the time-dependent quadrupole moment). An
- exception is, e.~g., the simulation of the close approach of another
- nucleus that stays external to the computational grid, where the
- potential is uniquely defined.
-
- One important point remains to be noted concerning the spatial
- profile $F_q(\vec{r})$. This can be illustrated by the quadrupole
- operator. Let us assume an instant where $A>0$ and $f(t)>0$. For
- then the operator $\propto 2z^2-x^2-y^2$ leads to a perturbing potential
- $U_\mathrm{ext}$ which is binding in $z$-direction but
- asymptotically unbound in the $x$- and $y$-directions. This can cause
- unphysical effects in case of large strengths and/or numerical boxes.
- For this reason it is useful to have a cut-off by a
- Woods-Saxon like function according to~\cite{Rut95a}
- \begin{equation}
- \label{eq:extdamp} §§§
- F_q(\vec r)
- \rightarrow
- \frac{F_q(\vec r)}{1+e^{(r-r_0)/\Delta r}}\;,
- \end{equation}
- where $r_0$ and $\Delta r$ are parameters describing a transition
- region sufficiently outside the nucleus, but also sufficiently small
- to maintain binding.
-
- Another problem associated with the external field is that in
- general it will not be periodic but instead have discontinuities on
- the boundary when crossing into the neighboring cell. If damping is
- sufficiently strong, the field may be practically zero on the
- boundary and thus become periodic. Another solution for this problem
- is to make the field explicitly periodic by replacing the
- coordinates with periodic substitutes. The exact formulation depends
- on the specific field used; for the above-mentioned quadrupole
- operator, which depends only on the squares of the coordinates,
- e.~g., substituting
- \begin{equation}\label{eq:extperiodic}
- x^2\rightarrow\sin^2\left(\pi x/x_L\right),
- \end{equation}
- with $x_L={\tt nx}\,\Delta x$ the period interval, will provide the
- proper behavior as the sine squared has a period of $\pi$ and there
- is no unphysical decrease of this function near the boundary. Of
- course the analogous transformation has to be applied to $y$ and
- $z$.
-
- The time dependence $f(t)$ is modeled as a short pulse centered
- around some excitation frequency $\omega$. The code allows a choice
- between two pulse envelopes:
- \begin{enumerate}
- \item A Gaussian of the form
- \begin{equation}
- \label{eq:extgauss}
- f(t)=\exp\left(-(t-\tau_0)^2/\Delta\tau^2\right)\cos(\omega(\tau-\tau_0))\;,
- \end{equation}
- with peak time $\tau_0$ and width $\Delta\tau$.
- \item A cosine squared function defined via
- \begin{equation}
- \label{eq:extcos}
- f(t)=\cos\left(\frac\pi{2}\left(\frac{t-\tau_0}{\Delta\tau}\right)^2\right)
- \theta\left(\Delta\tau-|t-\tau_0|\right)
- \cos(\omega(\tau-\tau_0))\;,
- \end{equation}
- which is confined to the intervals $t\in
- (\tau_0-\Delta\tau,\tau_0+\Delta\tau)$. This envelope is also
- characterized by a peak time of $\tau_0$ and width $\Delta\tau$.
- \end{enumerate}
-\end{subequations}
-Broad envelopes provide a high frequency resolution and so concentrate
-the excitation around the driving frequency $\omega$. Short pulses
-lose frequency selectivity and excite a broad band of frequencies.
-
-The extreme case is an infinitely short pulse,
-$\Delta\tau\longrightarrow{0}$. It amounts eventually to an
-instantaneous boost of the initial wave functions which can be
-expressed as a phase factor according to
-\begin{equation}
- \label{eq:extboost}
- \psi_k(\vec r,s,t\!=\!0)
- =
- \psi_{k,0}(\vec r,s)\,\exp\left(-\I \eta F_q(\vec r)\right)\;,
-\end{equation}
-where $\psi_{k,0}$ stands for the stationary wave function before
-boost. This instantaneous boost, being infinitely short, is
-insensitive to the problem of asymptotically unstable potentials and
-allows the use of a driving field $F_q$ without damping
- (\ref{eq:extdamp})
-which, in turn, simplifies spectral analysis (see Section
-\ref{sec:collex}).
-
-The effect of the boost (\ref{eq:extboost}) can be understood by
-virtue of the Madelung decomposition of a complex wave function
-$\phi(\vec r)=\chi(\vec r)\exp(\I S(\vec r))$ with $\chi$ and $S$
-being real. A straightforward calculation leads to the probability
-flow density as $\vec{\jmath}=\hbar\chi^2\nabla S/m$. Dividing by the
-density $\chi^2$ then produces the ``probability-flow velocity''
-$\vec{v}=\hbar\nabla S/m$. An illustrative example is the plane wave
-where $\chi=$constant and $S=\vec{k}\cdot\vec{r}$ which yields the
-correct result $\vec{v}=\hbar\vec{k}/m$. In the present case,
-assuming that the static wave functions themselves can be written as
-real functions, we get an initial velocity $\vec{v}=-\nabla F_q$,
-i.~e., just in the direction of the classical force resulting from the
-``velocity potential'' $F_q(\mathbf{r})$.
-
-
-
-
-\subsection{Static Hartree-Fock}
-\label{sec:static}
-
-\subsubsection{The coupled mean-field and BCS equations}
-
-The stationary equations are obtained variationally. Variation with
-respect to the single-particle wave functions $\psi_{\alpha}$ yields
-the mean field equations~\cite{Gre05aB,Mar10aB}
-\begin{subequations}
- \label{eq:mfeq}
- \begin{equation}
- \hat{h}\psi_{\alpha}=\varepsilon_\alpha\psi_{\alpha}\;,
- \label{eq:statmfeq}
- \end{equation}
- where $\hat{h}$ is the mean-field Hamiltonian (\ref{eq:spham}) and
- $\varepsilon_\alpha$ is the single-particle energy of state
- $\alpha$. Simultaneous variation of $\psi_{\alpha}$ together with
- the occupation amplitude $v_\alpha$ yields the
- Hartree-Fock-Bogolyubov equations~\cite{Gre05aB,Rin80aB,Ben03aR}
- which complicate Eq.~(\ref{eq:statmfeq}) by non-diagonal terms on
- the right-hand-side~\cite{Rei97a}. We employ here the BCS
- approximation which exploits time-reversal symmetry where each
- single-particle state has a time reversed partner
- $\psi_{\alpha}\leftrightarrow\psi_{\overline{\alpha}}$ as was
- already implied in the pairing densities $\xi_q$ in Eq.~(\ref{eq:rtjpair}).
- Each pair of time-conjugate states is associated with an occupation
- amplitude $v_\alpha$. These are determined by the BCS equation
- $\left(\varepsilon_\alpha-\epsilon_{\mathrm{F},q_\alpha}\right)
- \left(u_\alpha^2-v_\alpha^2\right) = \Delta_\alpha
- u_\alpha^{\mbox{}}v_\alpha^{\mbox{}} $ which can be resolved to a
- closed expression for the occupation amplitudes as
- \begin{eqnarray}
- v_\alpha^2
- & =&
- \frac{1}{2} \left( 1 -
- \frac{\varepsilon_\alpha-\epsilon_{\mathrm{F},q_\alpha}}
- {\sqrt{(\varepsilon_\alpha-\epsilon_{\mathrm{F},q_\alpha})^2
- +\Delta_\alpha^2}}
- \right)
- \quad,
- \label{eq:va}\\
- \Delta_\alpha
- &
- =
- &
- \frac{1}{2}V_\mathrm{pair,q_\alpha}
- \int \D^3r\,\psi^+_\alpha\psi_\alpha^{\mbox{}}
- \xi^{\mbox{}}_{q_\alpha}
- \left[1 -\frac{\rho}{\rho_{0,\mathrm{pair}}}\right]
- \quad,
- \\
- \epsilon_{\mathrm{F},q}
- &:&
- \sum_{\alpha\in q}v_\alpha^2=N_q
- \quad.
- \label{eq:parnum}
- \end{eqnarray}
- $q_\alpha$ denotes the nucleon type to which state $\alpha$
- belongs, ${\alpha\in q}$ means all states of type $q$, and $N_q$ is
- the number of nucleons of type $q$ (identified as $N_p=Z$ and
- $N_n=N$). The Fermi energies $\epsilon_{\mathrm{F},q_\alpha}$ serve
- to regulate the average particle number to the required values
- $N_q$.
-\end{subequations}
-Here, the space of pairing-active states is just the space of states
-actually included in the calculation. The results of BCS pairing
-depend slightly on the size of the active space
-\cite{Gre05aB,Rin80aB}. We recommend using about
-$$N_q+\frac{5}{3}N_q^{2/3}$$
-single-nucleon states, which comes closest to the dynamical pairing
-space of Ref.~\cite{Ben00a}.
-
-
-\subsubsection{Iterative solution}
-\label{sec:statiter}
-
-The coupled mean-field and BCS equations (\ref{eq:mfeq}) are solved
-iteratively. The wave functions are iterated with a gradient step which
-is accelerated by kinetic-energy damping~\cite{Rei82a,Blu92a}
-\begin{equation}
- \psi_\alpha^{(n+1)}
- =
- \mathcal{O}\left\{
- \psi_\alpha^{(n)}
- -
- \frac{\delta}{\hat{T} + E_0}
- \left( \hat{h}^{(n)} -
- \langle\psi_\alpha^{(n)}|\hat{h}^{(n)}|\psi_\alpha^{(n)}\rangle
- \right)\psi_\alpha^{(n)}\right\}
- \label{eq:dampstep}
-\end{equation}
-where $\hat{T}=\hat{p}^2/(2m)$ is the operator of kinetic energy,
-$\mathcal{O}$ means orthonormalization of the whole set of new wave
-functions, and the upper index indicates the iteration number. Note
-that this sort of kinetic-energy damping is particularly suited for
-the fast Fourier techniques that we use in the present code. The
-damped gradient step has two numerical parameters, the step size
-$\delta$ and the damping regulator $E_0$. The latter should be chosen
-typically of the order of the depth of the local potential $U_q$. In
-practice, we find $E_0=100$ MeV a safe choice. The step size is of
-order of $\delta=0.1...0.8$. Larger values yield faster iteration, but
-can run more easily into pathological conditions. The optimum
-choice depends somewhat on the Skyrme parametrization. Those with
-effective mass $m^*/m\approx 1$ allow larger $\delta$ values. Low
-$m^*/m$ may require a reduction in the step size.
-
-
-After each such wave function step, the BCS equations
-(\ref{eq:va}--\ref{eq:parnum}) are solved with
-$\varepsilon_\alpha=\langle\psi_\alpha|\hat{h}|\psi_\alpha\rangle$,
-the densities are updated (Eqs.~\ref{eq:rtjeven}-\ref{eq:rtjpair}),
-and new mean fields computed (\ref{eq:spham}-\ref{eq:spot}). This then provides the starting point for the next
-iteration. The process is continued until sufficient convergence is
-achieved. We consider as the convergence criterion the average energy
-variance, or fluctuation, of the single particle
-states
-\begin{subequations}
- \label{eq:totvar}
- \begin{eqnarray}
- \overline{\Delta\varepsilon}
- &=&
- \sqrt{\frac{\sum_\alpha\Delta\varepsilon_\alpha^2}{\sum_\alpha 1}}
- \quad,
- \\
- \Delta\varepsilon_\alpha^2
- &=&
- \langle\psi_\alpha|\hat{h}^2|\psi_\alpha\rangle
- -
- \varepsilon_\alpha^2
- \quad,
- \label{eq:spvar}
- \\
- \varepsilon_\alpha
- &=&
- \langle\psi_\alpha|\hat{h}|\psi_\alpha\rangle
- \quad.
- \label{eq:spenerg}
- \end{eqnarray}
-\end{subequations}
-The single particle energy $\varepsilon_\alpha$ is defined here as an
-expectation value. It finally becomes an eigenvalue in
-Eq.~(\ref{eq:statmfeq}) if the iteration has converged to
-$\Delta\varepsilon_\alpha\approx 0$. Vanishing total variance
-$\overline{\Delta\varepsilon}$ signals that we have reached minimum
-energy, i.e. a solution of the mean-field plus BCS equations. One has
-to be aware, however, that this may be only a local minimum (isomeric
-state). It requires experience to judge whether one has found the
-absolute energy minimum. In case of doubt, one should redo a couple of
-static iterations from very different initial configurations.
-
-
-This raises the question of how to initialize the iteration. We take
-as a starting point the wave functions of the deformed harmonic
-oscillator (see point \ref{it:iniho} in Section~\ref{sec:init}). These
-are characterized by $\vec{n}=(n_x,n_y,n_z)$, the number of nodes in
-each direction. We stack the wave functions in order of increasing
-oscillator energy
-$\epsilon_\alpha^{(0)}=\hbar\omega_xn_x+\hbar\omega_yn_y+\hbar\omega_zn_z$
-and stop if the desired number of states is reached. The deformation of
-the initializing oscillator influences the initial state in two ways:
-first, through the deformation of the basis wave functions as such,
-and second, through the energy ordering of the $\epsilon_\alpha^{(0)}$
-and corresponding sequence of levels built. Variation of initial
-conditions means basically a variation of the oscillator deformation.
-For example, the iteration will most probably end up in a prolate
-minimum if the initial state was sufficiently prolate, and in an
-oblate minimum after an oblate initial state. It depends on the
-nucleus which one is the absolute minimum.
-
-\subsection{TDHF}
-
-\subsubsection{The time-dependent mean-field equations}
-
-The TDHF equations are determined from the variation of the action
-\begin{equation*}
- S=\int \D t\left[E[\{\psi_\alpha\}]-
- \sum_\alpha\langle\psi_\alpha|\I \partial_t|\psi_\alpha\rangle
- \right]\;,
-\end{equation*}
-with respect to the wave functions $\psi_\alpha^+$ where the energy is
-given as in Eq.~(\ref{eq:efundet}-\ref{eq:ehfc})~\cite{Mar10aB}.
-This yields the TDHF
-equation
-\begin{equation}
- \I \partial_t\psi_{\alpha}=\hat{h}\psi_{\alpha}\;,
- \label{eq:dynmfeq}
-\end{equation}
-where $\hat{h}$ is, again, the mean-field Hamiltonian
-(\ref{eq:spham}). For simplicity, we are not considering variation of
-the occupation amplitude in the time-dependent case. The occupation
-amplitudes obtained from static iteration are kept frozen during time
-evolution. For studies of mean-field flow at moderate
-excitations (heavy-ion collisions, giant resonances) this
-approximation is legitimate. However, a study of truly low energy
-dynamics in the range of a few MeV (soft vibrations, fission) requires
-a full time-dependent Hartree-Fock-Bogolyubov treatment and should not
-be undertaken with the present code.
-
-
-\subsubsection{Time development algorithm}
-\label{sec:timevol}
-
-
-The TDHF equation (\ref{eq:dynmfeq}) can be formally resolved into an
-integral equation as
-\begin{subequations}
- \begin{eqnarray}
- |\psi_\alpha(t\!+\!\Delta t)\rangle
- &=&
- \hat U(t,t\!+\!\Delta t)|\psi_\alpha(t)\rangle
- \\
- \hat U(t,t\!+\!\Delta t)
- &=&
- \hat{\mathcal{T}}\exp\left(-\frac{\I}{\hbar}\int_t^{t+\Delta t}
- \hat h(t')\,\D t'\right)
- \;,
- \label{eq:timexp}
- \end{eqnarray}
-\end{subequations}
-where $\hat U$ is the time-evolution operator and $\hat{\mathcal{T}}$
-the time-ordering operation. This time evolution is unitary, thus
-conserving orthonormalization of the single-particle wave functions,
-and it conserves the total energy (\ref{eq:efundet}) provided that there are
-no time-dependent external fields. To convert this involved
-operator into an efficiently computable but also sufficiently accurate
-form a predictor-corrector strategy is used:
-\begin{enumerate}
-\item In a first step (predictor), we determine the single-particle
- Hamiltonian at midtime $\hat{h}(t\!+\!\Delta t/2)$.
- To that end, a trial step by $\Delta t$
- \begin{equation}
- \tilde{\psi}_\alpha
- =
- \exp\left(-\tfrac{\I}{\hbar}
- \hat h(t)\,\Delta t\right)\psi_\alpha(t)
- \label{eq:midstep}
- \end{equation}
- is performed using the mean field $\hat{h}(t)$ at initial time $t$.
- The density $\tilde\rho$ and similarly also other densities and
- currents at $t+\Delta t$ are accumulated from the wave functions
- $\tilde{\psi}_\alpha$, which can be discarded immediately, so that
- it is not necessary to store a complete second set of wave
- functions. They are used to compute an estimate for the densities at
- half step $\rho_{\rm pre}=(\rho(t)+\tilde\rho)/2$ and analogously
- for the other densities and currents. These are then used to
- calculate the estimated Hamiltonian $\hat h_{\rm pre}$ at $t+\Delta
- t/2$ according to Eq.~(\ref{eq:spham}-\ref{eq:spot}).
-
-\item This is used to perform the corrector step to advance the wave
- functions to the end of the time step (again with frozen
- Hamiltonian, but now $\tilde{h}_\mathrm{pre}$)
- \begin{equation}
- \psi_\alpha(t\!+\!\Delta t)
- =
- \exp\left(-\tfrac{\I}{\hbar}{\hat h}_\mathrm{pre}\,\Delta t\right)
- \psi_\alpha(t)
- \;.
- \label{eq:finstep}
- \end{equation}
-\item In both cases the operator exponential is evaluated by a Taylor
- series expansion up to order $m$:
- \begin{equation}
- \exp\left(-\tfrac{\I}{\hbar}\hat{h}\,\Delta t\right)\psi
- \approx
- \sum_{n=0}^m
- \frac{(-\I\Delta t)^n}{\hbar^n\,n!}\hat h^n\psi\;,
- \label{eq:timepower}
- \end{equation}
- where $\hat{h}$ is the actual mean field in step (\ref{eq:midstep}),
- or (\ref{eq:finstep}) respectively. $\hat h^n\psi$ is computed
- in straightforward manner by successive application of the mean
- field Hamiltonian, i.e. $\hat
- h^n\psi=\underbrace{\hat{h}(...(\hat{h}}_{n \mbox{\tiny
- times}}\psi)...)$.
-\end{enumerate}
-The Taylor expansion spoils strict unitarity of the exponential
-$\exp\left(-\tfrac{\I}{\hbar}\hat{h}\,\Delta t\right)$ and energy
-conservation. We turn this flaw into an advantage and use norm
-conservation as well as energy conservation (if it applies) as
-counter-check of the quality of the step along the propagation. The
-reliability depends, of course, on a proper choice of the numerical
-parameters in this step which are the step size $\Delta t$ and the
-order of the Taylor expansion $m$. The step size is limited by the
-maximum possible kinetic energy and by the typical time scales of changes in
-the mean field $\hat{h}$. The maximum kinetic energy, in turn, depends
-on the grid spacing as $\propto \Delta x^{-2}$. A choice of $\Delta
-t=0.1-0.2$fm/c is applicable in connection with $\Delta x=0.7-1$/fm.
-For the order of Taylor expansion, one needs at least $m=4$. Although
-there are formal reasons for $m=4$ \cite{bertsch}, in practice
-$m>4$ may also be used, but choosing $m>6$ is not so efficient for
-the values of $\Delta t$ considered here.
-
-\subsubsection{Collective excitations}
-\label{sec:collex}
-
-Giant resonances are prominent excitation modes of nuclei. Best known
-is probably the isovector giant dipole resonance, but there are many more
-modes depending on isospin
-and angular momentum. The typical resonance energies lie in a region
-from 10 to 30 MeV where the present TDHF code with frozen occupation
-numbers is applicable because the relevant energy range lies far above
-the pairing gap (1--2 MeV). The generation of these modes is
-particularly simple within the present TDHF treatment. One first
-produces a stationary state as outlined in Section~\ref{sec:static}
-and then triggers the excitation by a time-dependent external
-field as described in Section~\ref{sec:external}. A broad pulse allows
-triggering particular excitation energies. An infinitely short pulse
-amounts to an instantaneous boost.
-
-
-The boost is a generic excitation of a system which gives the same
-weight to all frequencies. It is thus ideally suited for analyzing in
-an unbiased manner the excitation spectra of a system. This, in turn,
-allows a thorough spectral analysis. To obtain the spectral
-distribution of isovector dipole strength, one applies a boost with
-small strength $\eta$ and $F_q=\hat{D}\propto r^1Y_{10}\tau_z$ the
-isovector dipole operator. The Slater determinant $|\Phi(t)\rangle$ is
-propagated in TDHF for a sufficiently long time while recording the
-dipole moment $D(t)=\langle\Phi(t)|\hat{D}|\Phi(t)\rangle$. The dipole
-strength is finally extracted from the Fourier transform
-$\tilde{D}(\omega)$ as
-$S_D(\omega)=\Im\left\{\tilde{D}(\omega)\right\}/\eta$. Note that
-this is valid only in the linear case, i.~e., if the amplitude of the
-vibration if proportional to the boost velocity \cite{Cal97a,Rei07a}.
-This should be checked in the calculations.
-
-The straightforward Fourier transform leads to artifacts if the dipole
-signal has not fully died out at the end of the simulation time. In
-the general case, some filtering is necessary to suppress artifacts
-from cutting the signal at a certain final time~\cite{Pre92aB}. In
-practice, it is most convenient to use filtering in the time domain by
-damping the signal $D(t)$ towards the final time. A robust choice is
-\begin{equation}
- D(t)
- \quad\longrightarrow\quad
- D_\mathrm{fil}(t)
- =
- D(t)\cos\left(\frac{\pi}{2}\frac{t}{t_\mathrm{final}}\right)^{n_\mathrm{fil}}
-\end{equation}
-where $t_\mathrm{final}$ is the final time of the simulation. This
-guarantees that the effective signal $D_\mathrm{fil}$ vanishes at the
-end of the interval. The $\cos^n$ profile switches off gently and
-leaves as much as possible from the relevant signal at early times.
-The parameter $n_\mathrm{fil}$ determines the strengths of filtering.
-Value of order of 4--6 are recommended to suppress the
-artifacts safely. For a detailed description of this spectral analysis see
-\cite{Cal97a}. For typical applications in nuclear physics see
-\cite{Rei07a}. It is to be noted that the code does not include this
-final step of spectral analysis. The time dependent signals are
-printed on the protocol files {\tt monopoles.res}, {\tt dipoles.res},
-and {\tt quadrupoles.res} (see Section~\ref{sec:filenames}). It is
-left to the user to perform the final steps towards a spectral
-distribution. A word is in order about $t_\mathrm{final}$. It
-determines the resolution of the spectral analysis. The corresponding
-energy bins are given by
-$\delta{E}_\mathrm{exc}=\hbar\pi/t_\mathrm{final}$. Windowing effectively
-reduces the time span in which relevant information is contained
-and roughly doubles the relevant $\delta{E}_\mathrm{exc}$. For
-example, to obtain a spectral resolution of 1 MeV, one needs to
-simulate up to about 1200 fm/c.
-
-
-Although excitation spectra are one of the most basic properties of the system, there
-are many other dynamical features of interest. The multipole signals
-in the time domain (printed in the protocol files) are as such
-interesting quantities. One can have, e.g., a look at cross-talk
-between the multipole channels. It is particularly interesting to
-study excitation dynamics for varying excitation strength $\eta$, from
-the regime of linear response (small $\eta$) deep into the non-linear
-regime. It is inefficient to perform a full three-dimensional TDHF
-calculation to obtain linear-regime excitation spectra for spherical nuclei. This is
-better done in a dedicated RPA calculation on a spherical basis (see,
-e.g.,~\cite{Rei92b}) for which there exist an overwhelming multitude
-of codes. The realm of TDHF calculations of nuclear excitations are
-spectra in deformed systems, stability analysis of exotic
-configurations, and in particular non-linear dynamics.
-
-There are many more details worth looking at. One may check the
-densities and currents to visualize the flow pattern associated with a
-mode. A most elaborate analysis deals with a phase-space picture of
-nuclear dynamics by virtue of the Wigner transformation~\cite{Loe11a}.
-The code allows saving all ingredients needed for such elaborate
-analysis in dedicated output files, see Section~\ref{sec:silo}. It is
-left to the user to work out the further steps of the analysis.
-
-A serious problem that can occur in collective excitation studies is
-the evaporation of particles at higher excitation energies. Some of
-the single-particle wave functions become unbound and move towards the
-boundaries for the computational box with a non-negligible part of
-their probability distributions. Since the code assumes periodic
-boundary conditions for the wave functions (for details see
-Sect~\ref{sec:bc}), effectively the particles reenter the box from the
-opposite side and can interact with the nucleus again, leading to a
-spurious revival of the excitation \cite{Nak05a,Rei06a}. This effect
-can be reduced in several ways; for a brief discussion see
-Sect.~\ref{sec:bc}.
-
-\subsubsection{Nuclear reactions}
-\label{sec:react}
-Collisions of nuclei are a prime application of nuclear TDHF. They
-were, in fact, the major motivation for its realization
-\cite{Bon78a,Dav85a}. The present code is designed to initialize such
-collision scenarios in a most flexible manner. We start by explaining
-the simplest case of a collision of two nuclei. First, we prepare the
-ground states of the two nuclei as explained in
-Section~\ref{sec:statiter}. The static solutions are centered around
-the origin $\vec{r}=0$ of their initial grid. The static wave
-functions $\psi_{\alpha,I}^{\mbox{(stat)}}$ where $I=1,2$ labels the
-two nuclei are shifted to new centers $\vec{R}_I$ where the distance
-$|\vec{R}_2-\vec{R}_1|$ should be sufficiently large that the nuclei
-have negligible overlap and negligible Coulomb distortion from the
-other nucleus (the latter condition usually only loosely fulfilled).
-The shifted wave functions
-$\psi_{\alpha,I}^{\mbox{(stat)}}(\vec{r}-\vec{R}_I,s)$ are obtained by
-interpolation on the grid. The interpolation is done by transforming
-to momentum space, applying translation factors, and going back. It is
-obvious that the collisions need a larger numerical box than the
-static Hartree-Fock calculations. Thus, we may compute the static wave
-functions on a smaller box since we are shifting and interpolating the
-result anyway for dynamical initialization.
-
-It should be noted that the nuclei may be placed at arbitrary
-positions in the new grid. It is highly recommended, however, to
-displace them by an integer number of grid points from the original
-position, since otherwise the interpolation may lead to some degree of
-excitation. In practice, if the center of mass in the static calculation
-was at the origin, the new position should be of the form ($m_x$ {\tt
- dx}, $m_y$ {\tt dy}, $m_z$ {\tt dz}) with integer factors $m_x$
-etc. If for some reason this is not desired, accuracy can be restored
-by placing the nucleus {\em alone} in the larger grid and running this
-as input for a static calculation with a sufficient number of timesteps
-before using the resulting wave functions as input for the dynamic one.
-
-At this point we have the
-nuclei resting at a safe distance. To set them in motion we need to
-give each nucleus a momentum $\vec{P}_1=-\vec{P}_2$ (note that the
-total momentum of the combined system still vanishes). Consequently,
-the initial configuration is given by the Slater state built from the
-shifted and boosted single-particle wave functions (see fragment
-initialization, point \ref{it:frag} in Section~\ref{sec:init})
-\begin{equation}
-\begin{array}{ll}
- \psi_{\alpha,1}(\vec{r},s;t\!=\!0)
- =e^{\I\vec{p}_1\cdot\vec{r}}
- \psi_{\alpha,1}^{\mbox{(stat)}}(\vec{r}-\vec{R}_1,s),&
- \vec{p}_1 =\frac{\vec{P}_1}{A_1}
- \quad,
- \\
- \psi_{\alpha,2}(\vec{r},s;t\!=\!0)
- = e^{\I\vec{p}_2\cdot\vec{r}}
- \psi_{\alpha,2}^{\mbox{(stat)}}(\vec{r}-\vec{R}_2,s),&
- \vec{p}_2=\frac{\vec{P}_2}{A_2}
- \quad.
- \label{eq:inittdhf}
-\end{array}
-\end{equation}
-The distance between the nuclei is large, but inevitably finite. This may
-induce minor violations of orthonormality. Thus the
-full set of wave functions (\ref{eq:inittdhf}) is orthonormalized as a final step of
-initial preparation.
-
-The occupation amplitudes $v_{\alpha,I}$ are taken over from the
-static solution and frozen along the dynamical evolution. In fact,
-most of the collision studies will be principally to explore
-the dynamical features in the regimes of fusion and inelastic
-collisions. It is then recommended to use the conceptually simplest
-and most robust strategy, namely to start from simple Slater states
-(not BCS states) for the two nuclei. This means that in most cases
-the static solution is calculated without pairing, fixing $v_\alpha=1$ and
-including just as many states as there are nucleons.
-
-The above example deals with two initial fragments. The code is more
-flexible than that. It allows an initial state composed from several
-fragments. The strategy remains the same as for the binary system. It
-is simply repeated for each new fragment. For details see
-Section~\ref{sec:fragini}.
-
-
-The time evolution is performed as outlined in Section
-\ref{sec:timevol}. It requires some effort to visualize the complex
-dynamics which emerges in collisions. A rough picture is, again,
-provided by the multipole moments. The quadrupole moment, e.g., can
-serve as a measure of stretching of the total system. Small values
-indicate a compound nucleus while asymptotically growing values signal
-fragmentation. One may want, particularly in case of collisions, more
-detailed pictures of the flow as, e.g., snapshots of the density of
-current distributions and, ultimately, a full phase space picture
-\cite{Loe11a}. Material for that can be output on demand, as detailed
-in Section \ref{sec:silo}. Again, we leave it to the user to extract
-the wanted information and to prepare it for visualization.
-
-In nuclear collisions at higher energies the emission of particles can
-also cause problems; in those cases one better uses absorbing boundary
-conditions. See Sect.~\ref{sec:bc} for details.
-
-\subsection{Observables}
-\label{sec:observables}
-
-It was already mentioned in Sections~\ref{sec:collex} and
-\ref{sec:react} which observables may be used to analyze nuclear
-dynamics. We here briefly summarize the observables computed
-and output in the code and indicate how further observables may be
-extracted. Basic features of the description by the Skyrme
-energy-density functional are, of course, energy and densities.
-
-\subsubsection{Multipole moments}
-\label{sec:mulmom}
-
-The gross features of the density distribution are well characterized
-by the multipole moments. The most important moment is the
-center of mass (c.m.)
-\begin{subequations}
- \label{eq:mulmom}
- \begin{equation}
- \vec{R}^\mathrm{(type)}
- =
- \frac{\int \D^3r\,\vec{r}\,\rho^\mathrm{(type)}(\vec r)}{A}
- \quad,
- \label{eq:cmcart}
- \end{equation}
- where $A=\int \D^3r\,\rho(\vec r)$ is the total mass number and
- ``type'' can refer to proton c.m.\ from $\rho_p$, neutron c.m.\
- from $\rho_n$, isoscalar or total c.m.\ from the total density
- $\rho=\rho_p+\rho_n\equiv\rho_{T=0}$, or isovector moment related to
- the isovector density $\rho_{T=1}=\frac{N}{A}\rho_p-\frac{Z}{A}\rho_n$.
- The
- definition of $\vec{R}^\mathrm{(type)}$ directly employs the
- Cartesian coordinate $r_i$. The same holds for the Cartesian
- quadrupole tensor
- \begin{equation}
- \mathcal{Q}_{kl}^\mathrm{(type)}
- =
- \int \D^3r \left(3(r_k-R_k)(r_l-R_l)-\delta_{kl}\sum_i(r_i-R_i)^2\right)
- \rho^\mathrm{(type)}(\vec r)
- \quad,
- \label{eq:Qcart}
- \end{equation}
- again for the various types as discussed above. The matrix
- $\mathcal{Q}_{kl}$ is not invariant under rotations of the
- coordinate frame. There is a preferred coordinate system: the system
- of principle axes. It is obtained by diagonalizing
- $\mathcal{Q}_{kl}$. The quadrupole matrix in the principle-axis
- frame thus has only three non-vanishing entries $Q_{xx}$, $Q_{yy}$,
- and $Q_{zz}$ together with the trace condition
- $Q_{xx}+Q_{yy}+Q_{zz}=0$.
-
-For the quadrupole case the spherical moments defined as
- \begin{equation}
- Q_{2m}^\mathrm{(type)}
- =
- \int \D^3r\,r^2Y_{2m}\,\rho^\mathrm{(type)}(\vec{r}-\vec{R})\;,
- \label{eq:spherQ}
- \end{equation}
- are also useful, where $r=|\vec{r}|$ and $Y_{2m}$ are the
-spherical harmonics. They are often expressed as dimensionless
- quadrupole moments
- \begin{equation}
- a_m
- =
- \frac{4\pi}{5}
- \frac{Q_{2m}}{AR^2}
- \quad,
- \label{eq:dimlessQ}
- \end{equation}
- with $R=r_0A^{1/3}$ a fixed radius derived from the total mass
- number $A$.
- This, again, could be defined for any ``type'', but is used, in
- practice, mainly for the isoscalar moments.
-
- The dimensionless moments have
- the advantage of being free of an overall scale which was removed by
- the denominator $AR^2$. They allow characterization of the
- shape of the nucleus. However, the general $a_m$ are not invariant
- under rotations of the coordinate frame. We obtain a unique
- characterization by transforming to the principle-axis system. These
- are defined by the conditions $a_{\pm 1}=0$ and $a_2=a_{-2}$. There
- remain only two shape parameters $a_0$ and $a_2$. These are often reexpressed
- as total deformation $\beta$ and triaxiality $\gamma$, often
- called Bohr-Mottelson parameters, through
- \begin{equation}
- \beta
- =
- \sqrt{a_0^2+2a_2^2}
- \quad,\quad
- \gamma
- =
- \mbox{atan}\left(\frac{\sqrt{2}\,a_2}{a_0}\right)
- \quad.
- \label{eq:triax}
- \end{equation}
- Triaxiality $\gamma$ is handled like an angle. It can, in principle,
- take all values between $0^o$ and $360^o$, but physically relevant
- parameters stay in the $0\ldots60^\circ$ range. The other sectors
- correspond to equivalent configurations~\cite{Gre05aB}.
-
- The monopole moment just corresponds to the total particle number,
- so to describe monopole vibrations usually the r.m.s.\ radii or
- their squares are employed. The r.m.s. radii
- are defined as
- \begin{equation}
- r_\mathrm{rms}^\mathrm{(type)}
- =
- \sqrt{\frac{\int \D^3r\,(\vec{r}-\vec{R})^2\,\rho^\mathrm{(type)}(\vec{r})}
- {\int \D^3r\,\rho^\mathrm{(type)}(\vec{r})}}
- \label{eq:rmsrad}
- \end{equation}
- where ``type'' can be proton, neutron, or total. The isovector
- variant does not make sense here.
-
- We supply printouts of all the above variants of multipole moments
- to allow a most flexible analysis. For the reason given above the
- r.m.s.\ radii are output in a file called {\tt monopolesfile}.
-\end{subequations}
-
-\subsubsection{Alternative way to evaluate the total energy}
-\label{sec:alterenerg}
-
-The key observable is the total energy $E_\mathrm{tot}$. It is computed as
-given in Eqs. (\ref{eq:efundet}-\ref{eq:ehfc}). More detailed energy observables are
-provided by the s.p.\ energies (\ref{eq:spenerg}). These can also be
-used to compute the total energy. The traditional HF scheme deals with
-pure two-body interactions and exploits that to simplify~\cite{Gre05aB}
-\begin{equation}
- \label{eq:koopman}
- E_\mathrm{tot,HF}
- =
- \frac{1}{2}\sum_\alpha\left(t_\alpha+\epsilon_\alpha\right)
-\end{equation}
-where $t_\alpha=\langle\psi_\alpha|\hat{T}|\psi_\alpha\rangle$ is the
-s.p.\ kinetic energy. This is possible because
-\begin{equation*}
- \epsilon_\alpha=t_\alpha+u_\alpha
- \quad,\quad
- u_\alpha
- =
- \sum_{\beta}\left[v_{\alpha\beta\alpha\beta}-v_{\alpha\beta\beta\alpha}\right]
- -\tfrac1{2} v_\alpha=\epsilon_\alpha-t_\alpha
- \quad,
-\end{equation*}
-where $u_\alpha$ is the s.p.\ mean-field potential energy and $v$ the
-two-body interaction, and
-\begin{equation*}
- E_\mathrm{tot,HF}
- =
- \sum_\alpha t_\alpha
- +
- \frac{1}{2}\sum_{\alpha\beta}
- \left[v_{\alpha\beta\alpha\beta}-v_{\alpha\beta\beta\alpha}\right]
- \quad.
-\end{equation*}
-The Skyrme force does not simply have this two-body structure. Still
-the total energy is very often computed along the strategy of
-Eq.~(\ref{eq:koopman}). However, the density dependence requires
-augmenting this recipe by a rearrangement energy which accounts for a
-contribution missing in the simple recipe (\ref{eq:koopman}). The
-extension to the Skyrme energy thus reads
-\begin{subequations}
- \label{eq:koopman2}
- \begin{eqnarray}
- E_\mathrm{tot,HF}
- &=&
- \frac{1}{2}\sum_\alpha\left(t_\alpha+\epsilon_\alpha\right)
- + E_{3,\rm corr} + E_\mathrm{C,corr}
- \quad,
- \\
- E_{3,\rm corr}
- & = &
- \int \D^3r \frac{\alpha}{6} \rho^{\alpha}
- \left[b_3\rho^2 - b'_3 (\rho_{p}^2+\rho_{n}^2) \right]
- \quad,
- \label{eq:e3corr}
- \\
- E_\mathrm{C,corr}
- &=& \frac{1}{4} \left( \frac{3}{\pi} \right)^{1/3}
- \int \D^3r\, \rho_{pr}^{4/3}
- \quad.
- \label{eq:ecorc}
- \end{eqnarray}
-\end{subequations}
-In the code the total energy is computed both ways, from the
-straightforward Skyrme energy (\ref{eq:efundet}) as well as from the
-above recipe (\ref{eq:koopman2}), as described in
-Section~\ref{sec:modvar}. Numerically these values are close but not
-identical.
-
-\subsection{Discretization}
-\subsubsection{Data types}
-In the module {\tt Params} a type {\tt db} is defined for 12-digit
-accuracy and on all present machines should amount to double
-precision, i.~e., {\tt REAL(db)} and {\tt COMPLEX(db)} are actually
-{\tt REAL(8)} and {\tt COMPLEX(8)} for most compilers. Keeping the symbolic type
-throughout of course makes the code more flexible for future hardware.
-Using single precision is not recommended.
-
-Since the external libraries are based on C or Fortran-77 coding,
-special care has to be taken in this respect. The FFTW3 library stores
-its plans in 8-byte integers, which in the modules {\tt Coulomb} and
-{\tt Fourier} are defined using the type {\tt C\_LONG} from the
-system-supplied module {\tt ISO\_C\_BINDING}. If this is not
-available, they may be defined as {\tt INTEGER(8)} or if that also
-causes problems, {\tt DOUBLE PRECISION}, which certainly corresponds
-to at least 8 bytes. For the {\tt LAPACK} routines, double-precision
-real and complex variables are necessary, which we also define using
-``{\tt db''}. If {\tt db} should ever be changed in such a way that
-{\tt REAL(db)} no longer corresponds to 8 bytes, different {\tt LAPACK}
-routines must be selected.
-
-Note that data conversion needs some care. If {\tt a} and {\tt b} are
-double precision, {\tt CMPLX(a,b)} is not; according to the standard
-it returns the default accuracy, which on many machines will still be
-single precision. Thus that for example in {\tt EXP(CMPLX(a,b))} the
-exponential would be evaluated in single precision. That is why in the
-code the expression {\tt CMPLX(a,b,db)} is used consistently; this is
-also safe for future changes in accuracy.
-
-The other conversion functions used are safe. {\tt AIMAG} is generic
-and {\tt REAL} reproduces the accuracy of (only) a {\em complex
- argument}.
-
-\subsubsection{Grid definition}
-\label{sec:griddef}
-All wave functions and fields are defined on a three-dimensional
-regular Cartesian grid of {\tt nx} by {\tt ny} by {\tt nz} grid
-points. {\bf {\tt nx}, {\tt ny}, and {\tt nz} must be even numbers}.
-The physical spacing between the points is given as {\tt dx}, {\tt
- dy}, and {\tt dz} (in fm). In principle these could be different for
-the three directions, but since this will lead to a loss of accuracy
-it is highly recommended to give the same value to all of them. A
-typical range is 0.5--1.0~fm.
-
-The grid is automatically arranged in such a way that in each
-direction the same number of grid points are located on both sides of the
-origin. This means that the three-dimensional origin is in the center
-of a cubic cell and has the advantage that exact parity properties for
-the wave functions can be maintained. The coordinate values for e.~g.,
-the $x$-direction are thus:
-\begin{equation}\begin{split}
- -\frac{{\tt nx}-1}{2}&{\tt dx},\quad -\frac{{\tt nx}-1}{2}{\tt
- dx}+{\tt dx},\quad\ldots \quad -\frac{{\tt dx}}{2},\\
- &+\frac{{\tt dx}}{2},\quad \ldots\quad \frac{{\tt nx}-1}{2}{\tt
- dx}.
-\end{split}
-\end{equation}
-The corresponding values are available in the arrays {\tt x(nx)}, {\tt
- y(ny)}, and {\tt z(nz)}.
-
-\subsubsection{Derivatives}
-\label{sec:deriv}
-
-The computation of the kinetic densities and currents and the
-application of the mean-field Hamiltonian require first and second
-derivatives at several places in the code. We define them in Fourier
-space. For simplicity, the strategy is explained here for one
-dimension. The generalization to 3D is obvious.
-
-The {\tt nx} discrete grid points $x_\nu$ in coordinate space are
-related to the same number of grid points $k_n$ in Fourier space
-(physically equivalent to momentum space) as
-\begin{subequations}
-\label{eq:FT}
-\begin{eqnarray}
- x_\nu
- &=&
- \left(-\frac{{\tt nx}-1}{2}+\nu\right){\tt dx}
- \quad,
- \nu=1,...,{\tt nx}
- \quad,
-\\ \label{eq:kvalues}
- k_n &=&
- (n-1) {\tt dk},\quad n=1,\ldots {\tt nx}/2 \quad,
-\\ \nonumber
- k_n&=&(n-{\tt nx}-1)\,{\tt dk},\quad n={\tt nx}/2+1,\ldots,{\tt nx} \quad,
-\\
- {\tt dk}
- &=&
- \frac{2\pi}{{\tt nx}\cdot{\tt dx}}
- \quad.
-\end{eqnarray}
-Note the particular indexing for the $k$-values. In principle, the
-values $k_n=(n-1){\tt dk}$ for all $n$ are equivalent for the Fourier
-transform, but for the second half of this range the negative
-$k$-values should be chosen because of their smaller magnitude. For
-the Fourier expansion, $k=-{\tt dk}$ and $k=({\tt nx}-1){\tt dk}$ are
-equivalent because of periodicity in $k$-space.
-
-A function $f(x_\nu)$ in coordinate space is connected to a
-function $\tilde{f}(k_n)$ in Fourier space by
-\begin{eqnarray}
- \tilde{f}(k_n)
- &=&
- \sum_{\nu=1}^{\tt nx}
- \exp{\left(-\I k_nx_\nu\right)}f(x_\nu)
- \quad,
-\label{eq:FTforward}\\
- f(x_\nu)
- &=&
- \frac{1}{{\tt nx}}\sum_{n=1}^{\tt nx}
- \exp{\left(\I k_nx_\nu\right)}\tilde{f}(k_n)
-\label{eq:FTbackward}
-\end{eqnarray}
-\end{subequations}
-This complex Fourier representation implies that the function $f$ is
-periodic with $f(x+{\tt dx}\cdot{\tt nx})=f(x)$. The appropriate
-integration scheme is the trapezoidal rule which complies with the above
-summations adding up all terms with equal weight.
-The derivatives of the exponential basis functions are
-\begin{equation}\label{eq:expbas}
- \frac{\D^m}{\D x^m}\exp{(\I k_nx)}
- =
- (\I k_n)^m\exp{(\I k_nx)}
- \quad.
-\end{equation}
-Computation of the $m$th derivative thus becomes a trivial
-multiplication by $(\I k_n)^m$ in Fourier space. Time critical
-derivatives are best evaluated in Fourier space using the fast Fourier
-transformation (FFT). To that end a forward transform
-(\ref{eq:FTforward}) is performed, then the values $\tilde{f}(k_n)$
-are multiplied by $(\I k_n)^m$ as given in
-Eq.~(\ref{eq:expbas}) and finally transformed $(\I
-k_n)^m\tilde{f}(k_n)$ back to coordinate space by the transformation
-(\ref{eq:FTbackward}). This strategy is coded in the subroutines
-\Indexp{cdervx}, \Indexp{cdervy}, and \Indexp{cdervz} contained in
-module \Indexm{Levels}. It is used for derivatives of wave functions
-provided the switch {\tt TFFT} is set.
-
-For coding purposes, it is often useful to perform derivatives as a
-matrix operation directly in coordinate space. The derivative matrices
-are built by evaluating the double summation of forward and backward
-transform ahead of time. For the $m$th derivative this reads
-\begin{equation*}\begin{split}
- f^{(m)}(x_\nu)
- &=
- \frac{1}{\tt nx}
- \sum_n\exp{(ik_nx_\nu)}(\I k_n)^m
- \sum_{\nu'}\exp{(-\I k_nx_{\nu'})}f(x_{\nu'})
-\\
- &=
- \sum_{\nu'}\;
- \underbrace{
- \frac{1}{\tt nx}
- \sum_n\exp{(\I k_nx_\nu)}(\I k_n)^m
- \exp{(-\I k_nx_{\nu'})}}_{D^{(m)}_{\nu\nu'}}\;
- f(x_{\nu'})
- \quad.
-\end{split}
-\end{equation*}
-From here, the detailed handling depends on the order of
-derivative. The $k_n$ run over the values
-$k_n=0,
-\pm{\tt dk},\pm 2{\tt dk},...,({\tt nx}/2-1){\tt dk},+{\tt dk}\,{\tt
- nx}/2$.
-Here the index ordering given in Eq.~(\ref{eq:kvalues}) does not matter
-as the index is summed over.
-Note that the first and the last value come alone while all others
-come in pairs of $\pm$ partners. These pairwise terms can be combined
-into a sine function for $n=1$ and a cosine for $n=2$. The derivative
-matrices thus read in detail
-\begin{subequations}
-\begin{align}
-\label{eq:derv1}
- D^{(1)}_{\nu\nu'}
- =&
- -\frac{2{\tt dk}}{\tt nx}
- \sum_{n=1}^{{\tt nx}/2-1}n\sin(k_n{\tt dx}(\nu\!-\!\nu'))
-\notag\\
- &
- -\frac{{\tt dk}}{\tt nx}
- \frac{{\tt nx}}{2}\sin((\nu\!-\!\nu'){\tt dk}\,{\tt nx}/2)
- \quad,
-\\
-\intertext{and}
-\label{eq:derv2}
- D^{(2)}_{\nu\nu'}
- =&
- -\frac{2{\tt dk}^2}{\tt nx}
- \sum_{n=1}^{{\tt nx}/2-1}n^2\cos(k_n{\tt dx}(\nu\!-\!\nu'))
-\notag\\
- &
- -\frac{{\tt dk}^2}{\tt nx}
- \left(\frac{{\tt nx}}{2}\right)^2\cos((\nu\!-\!\nu'){\tt dk}\,{\tt nx}/2)
- \quad.
-\end{align}
-\end{subequations}
-A word is in order about the first derivative. The upper point in the
-$k$-grid, ${\tt dk}\,{\tt nx}/2$, is ambiguous. Exploiting periodicity,
-it could be equally well $-{\tt dk}\,{\tt nx}/2$. In order, to deal
-with a $\pm k$ symmetric derivative we have anti-symmetrized this last
-point. The price for this is a slight violation of hermiticity which,
-however, should be very small as we anyway should not have significant
-wave-function contributions at the upper edge of the $k$-grid.
-
-The derivative matrices $D^{(m)}_{\nu\nu'}$ can be prepared ahead of
-time and are then at disposal for any derivative in the course of the
-program. Actually, the matrices for the first derivative are prepared
-in routine \Indexp{sder} and for the second derivative in routine
-\Indexp{sder2}, both contained in module \Indexm{Grids}. These
-routines are applied to generate the derivative matrices
-\Indexv{derv1x}, \Indexv{derv1y}, \Indexv{derv1z}, \Indexv{derv2x},
-\Indexv{derv2y}, and \Indexv{derv2z}, for first and second derivatives
-in the $x$, $y$ and $z$ directions.
-
-The matrix formulation of the derivatives is used in the code in two
-ways: on the one hand, the derivatives of the real-valued densities,
-currents, and mean-field components are always calculated using these
-derivative matrices, because they are real and the Fourier transform
-method would require converting them to complex values (using special
-Fourier techniques for real arrays is in principle possible but has
-not been worked out yet). In addition the user can switch to using the
-matrix method everywhere, which may give a slight speed advantage for
-small grid dimensions.
-
-\subsubsection{Boundary conditions}\label{sec:bc}
-The code uses a periodic Fourier transform to calculate derivatives.
-This is valid only with {\em periodic boundary conditions}. Thus in
-principle the wave functions and potentials are assumed to be repeated
-periodically in each Cartesian direction. Because of the short range
-of the nuclear force, this is not a serious problem in most cases; at
-higher energies, however, as mentioned in Sect.~\ref{sec:collex}
-and~\ref{sec:react} the emission of low-density material from
-the nuclei can interfere with the dynamics in the neighboring box and
-cause problems in the conservation of energy and angular momentum; for
-a detailed discussion see~\cite{Guo08a}. This is aggravated by
-the fact that even with periodic boundary conditions periodicity is
-truly fulfilled only for the wave functions and mean-field
-components. Since the vector $\vec r$ itself is not periodic but
-jumps at the boundary, operators such as the orbital angular
-momentum are not periodic.
-
-Several ways to solve, or reduce, the problem have been brought up.
-The most obvious and conceptually simplest approach is to increase the
-size of the numerical box. This is, however, not an option in 3D
-calculations as the expense grows cubically with the box length. Very
-recently, a multigrid method has been proposed \cite{deGio12} which
-renders the use of enlarged boxes feasible (although still at the
-boundaries of present days computer capabilities). Perfect removal of
-escaping particles is achieved by radiating or exact boundary
-conditions \cite{Bou97,Man98a,Pardi2013,pardi13} which, again,
-are not yet practicable in 3D calculations. Robust and efficient are
-absorbing boundary layers using an especially tailored imaginary
-potential \cite{Nak05a} or by applying a mask function during time
-evolution \cite{Kra92a}. The latter technique is particularly easy to
-implement and has been widely used in the past. Its robustness and
-efficiency allow developing advanced analyzing techniques on the grid
-as, e.g., the computation of kinetic-energy spectra and angular
-distributions of the emitted particles \cite{Rei06f}. Those who are
-not afraid of a little bit of coding can easily implement the
-mask-function technique for absorbing boundary conditions into the
-present code. A detailed description and discussion of this approach
-and the proper choice of numerical parameters is found in
-\cite{Rei06a}.
-
-On the other hand, for the Coulomb field with its long range periodicity
-would be clearly wrong. Therefore a computation of the Coulomb
-potential for the boundary condition of an isolated charge
-distribution is implemented in addition to the periodic one; see the
-manual for the module {\tt Coulomb}. This is selected by the logical
-input variable {\tt periodic} {\bf and applies only to the Coulomb
- potential}.
-
-\subsubsection{Wave function storage}
-Module {\tt Levels} handles the single-particle wave functions and
-associated quantities.
-
-The principal array for the wave function is called {\tt psi}, which
-is of type {\tt COMPLEX(db)}. Its dimension is {\tt
- (nx,ny,nz,2,nstloc)}, where the first three indices naturally refer
-to the spatial position. The 4th index corresponds to spin: index~1
-refers to spin up and~2 to spin down, quantization being along the
-$z$-direction.
-
-The last index numbers the wave functions. If the code is run on a
-single node, the value is {\tt nstmax}, the total number of
-single-particle wave functions. They are divided up into neutron and
-proton states, with the index range given by {\tt npmin} and {\tt
- npsi}. The sub-ranges are:
-\begin{itemize}
-\item {\tt npmin(1)}$\ldots${\tt npsi(1)} : the neutron states,
-\item {\tt npmin(2)}$\ldots${\tt npsi(2)} : the proton states.
-\end{itemize}
-In the present code {\tt npmin(1)=1} and {\tt npsi(2)=nstmax}.
-
-\begin{figure}[hp]
-\centerline{\includegraphics[width=0.7\linewidth]{Wavefun.pdf}}
-\caption{\label{fig:wavefun}
-Storage arrangement of the single-particle wave functions. On the left
-the case for single-processor or {\tt OpenMP} is shown, which for the
-case of distributed memory under {\tt MPI} is mapped to the individual
-processors as shown on the right. Note that each node will have its
-own values of {\tt nstloc} and {\tt globalindex}.
-}
-\end{figure}
-
-If the code is run in parallel (MPI) on several nodes, only {\tt nstloc}
-single-particle wave functions are stored on a given node, where {\tt
- nstloc} may vary. Pointers are then defined to indicate the
-relationship between the local index and that in the global array of
-wave functions. For details see the section on parallelization; the
-general layout is given in Fig.~\ref{fig:wavefun}.
-
-There are a number of arrays containing the physical properties of the wave
-functions, such as the single-particle energy. The names start with
-{\tt sp\_} and they are defined in module {\tt Levels}. They are not
-split up in the parallel case, but on each node only the pertinent
-index positions are used.
-
-\subsubsection{Densities and currents}
-The various densities necessary for constructing the mean field are
-actually kept in separate arrays and can be output onto data files for
-later analysis (see subroutine {\tt write\_densities}). The
-dimensioning is {\tt (nx,ny,nz,2)} with the last index referring to
-isospin for scalar densities, so that {\tt rho(:,:,:,1)} is the
-neutron density and {\tt rho(:,:,:,2)} the proton density. For vector
-densities there is an additional index with values~1 to~3 for the
-Cartesian direction, thus {\tt sdens(nx,ny,nz,3,2)} containing the
-spin density in each direction for neutrons and protons.
-
-Since it is often not necessary to keep the neutron and proton
-contributions separate, subroutine {\tt write\_densities} has the option of
-adding them up before output.
-
-\subsection{Initialization}
-\label{sec:init}
-A particular strength of the code is its flexible initialization.
-There are essentially three types of initialization, which can be
-selected through the input variable {\tt nof}:
-
-\begin{enumerate}
-\item\label{it:iniho} {\bf Harmonic oscillator:} {\tt nof=0}: this is
- applicable only to static calculations. The initial wave functions
- are generated from harmonic oscillator states with initial radii
- {\tt radinx}, {\tt radiny}, and {\tt radinz} in the three
- directions. It is advisable to choose the three radii different to
- avoid being kept in a symmetric configuration for non-spherical
- nuclei. Note that this is a very simple initialization and has some
- defects; for example, the initial deformation is controlled more by
- the occupation of the oscillator states than by the radius
- parameters. This should eventually be replaced by, e.~g., Nilsson
- wave functions.
-
- For this case the type of nucleus is determined by the input numbers
- {\tt nneut} and {\tt nprot} giving the number of neutrons and
- protons, while {\tt npsi} can be used to add some unoccupied states
- (this sometimes leads to faster convergence).
-
-\item\label{it:frag} {\bf Fragment initialization:} {\tt nof>0}: wave
- functions for a number {\tt nof} of fragments are read in and
- positioned in the grid at certain positions. The wave functions are
- read from files produced by the static code with the file names
- given by the input {\tt filename}, they are positioned at
- center-of-mass positions {\tt fcent} and given an initial velocity
- controlled by {\tt fboost}. The code determines the number of wave
- functions needed from these data files and also checks the agreement of
- Skyrme force and grid used. This initialization is used, e.g., for
- nuclear reactions (see Section~\ref{sec:react}). For details see the
- input description in Section~\ref{sec:fragini}.
-
- The number of fragments read in is arbitrary, but there are two
- special cases:
- \begin{itemize}
- \item for {\tt nof=1} a single fragment is read in. This can be
- useful for initializing with static wave functions to study
- collective vibrations in a nucleus using the TDHF mode.
- \item for {\tt nof=2} a special initialization can be done where the
- initial velocities are not given directly but computed from a
- center-of-mass energy {\tt ecm} and an impact parameter {\tt b}.
- \end{itemize}
-
-\item {\bf User initialization:} a user-supplied routine {\tt
- user\_init} can be employed to set up the wave functions in any
- desired way. The only condition is that the index ranges etc.\ are
- set up correctly and the wave function array {\tt psi} is filled
- with the proper values. It was found useful, e.~g., to use initial
- Gaussians distributed in various geometric patterns for
- $\alpha$-cluster studies.
-\end{enumerate}
-
-\subsection{Restarting a calculation}\label{sec:restart}
-Sometimes it is necessary to continue a calculation that was not run
-to the desired completion because of a machine failure or because the
-number of iterations or time steps was set too low. In such cases the
-last wave function file with name {\tt wffile}, which is generated at
-regular intervals of {\tt mrest} iterations or time steps, can be used
-to initialize a continuation. The program handles this in a simple fashion: if
-the logical variable {\tt trestart} is input as {\tt TRUE}, it sets up
-an initialization with one fragment (read from the initialization file)
-placed at the origin and with zero
-velocity. The only other modifications to the regular setup are then
-to take the initial iteration number and time from that file instead
-of starting at zero, as well as suppressing some unneeded
-initialization steps.
-
-Restarting works for both static and dynamic calculations. To continue
-a calculation that was stopped because the desired number of
-iterations or time steps was reached, a new limit for these should be
-provided in the input.
-
-This flexible restart makes it possible to use a different grid for
-the continuation in the sense that the grid spacings must agree, but
-the new grid can be larger than the old one.
-
-\subsection{Accuracy considerations}\label{sec:accuracy}
-The grid representation and solution methods introduced above depend
-on several numerical parameters. Their proper choice is crucial
-for the accuracy and speed of the calculations. In this Section, we
-want to briefly address the dependence on numerical parameters. An
-extensive discussion of grid representations and static iteration is
-found in~\cite{Blu92a}.
-
-\begin{figure}[h]
-\centerline{\includegraphics[width=\linewidth]{static-grid.pdf}}
-\caption{\label{fig:static-grid} Binding energy (left) and
- r.m.s. radius (right) of $^{16}$O computed for the force SkI3 drawn
- as functions of grid spacing $\Delta x=\Delta y=\Delta z$. A
- logarithmic scale us used for $\Delta x$. The number of grid points
- has been chosen to keep the box size constant at $N_x\Delta x=24$
- fm.}
-\end{figure}
-Figure \ref{fig:static-grid} shows the sensitivity with respect to the grid
-spacings $\Delta x$, $\Delta y$, $\Delta z$. The trend is the same for
-both observables, energy and radius: The results have very high
-quality and change very little up to $\Delta x=$ 0.75 fm. They quickly
-degrade above that spacing. But even at $\Delta x=$ 1 fm, we still
-find an acceptable quality which suffices for most applications,
-particularly for large scale explorations. If high accuracy matters,
-$\Delta x\approx$ 0.75 fm should be chosen; not much is gained
-by going to even finer gridding. This holds for
-ground states and moderate excitations. High excitations and fast
-collisions may require a finer mesh. Note that the maximum
-representable kinetic energy is
-$E_\mathrm{kin,max}=(\hbar^2/2m)(\pi/\Delta x)^2$, which amounts to
-about 200 MeV for $\Delta x=$ 1 fm. The actual energies of interest
-should stay far below this limit. It is an instructive exercise to
-study uniform center-of-mass motion at various velocities to explore the
-limits of a given representation.
-
-The number of grid points $N_x=N_y=N_z$ in the tests of Figure
-\ref{fig:static-grid} were chosen such that the box size was the same
-in all cases. The actual choice of $N_x$ depends sensitively on the
-system, its size and separation energy. As a rule of thumb, the
-density decreases asymptotically as
-$\rho\propto\exp{(-2\sqrt{2m\varepsilon_N}r/\hbar)}$ where
-$\varepsilon_N$ is the single particle energy of the least bound
-state. One should aim for at least $\rho<10^{-8}\,{\rm fm}^{-3}$ at
-the boundaries.
-
-\begin{figure}
-\centerline{\includegraphics[width=0.5\linewidth]{dynamic-grids.pdf}}
-\caption{\label{fig:dynamic-grids} Time evolution of the quadrupole
- momentum for two different grid spacings $\Delta x=\Delta y=\Delta
- z$ and constant box size of 24 fm. The test case is $^{16}$O excited by
- an instantaneous boost computed with the force SkI3.}
-\end{figure}
-Figure \ref{fig:dynamic-grids} explores the effect of grid spacing for
-dynamics. Two different spacings are compared for a quadrupole
-oscillation following an instantaneous quadrupole boost. Practically
-no difference can be seen for the ``safe choice'' $\Delta x=$ 0.75 fm
-and the robust choice $\Delta x=$ 1 fm. Dynamical applications,
-oscillations and collisions, are in general less demanding and can
-be performed very well with $\Delta x=$ 1 fm. This is pleasing as dynamical
-calculations are usually much more costly than purely static ones.
-
-There are two parameters regulating the static iteration according to
-Eq.~(\ref{eq:dampstep}), the damping energy {\tt e0inv} and the step
-size {\tt x0dmp}. {\tt e0inv} should correspond to the depth of
-the binding potential. The overall step size {\tt x0dmp} can be of
-order of one if {\tt e0inv} is well chosen. Nuclear binding is very
-similar all over the chart of nuclei. This allows to develop one safe
-choice for nearly all cases. We recommend {\tt e0inv}$\approx 100$ MeV
-together with {\tt x0dmp}$\approx 1/2$, reducing the latter slightly
-if convergence problems appear. Of course,
- a few percent in iteration speed might be gained by fine-tuning these
-parameters for a given case, but this is not worth the effort unless
-large scale surveys for a given class of nuclei and forces are
-planned.
-
-\begin{figure}[htb]
-\centerline{\includegraphics*[width=0.95\linewidth]{dynamicSV-energ.pdf}}
-\caption{\label{fig:dynamicSV-energ} Time evolution of the binding energy
- of $^{16}$O after an excitation by a soft cos$^2$ pulse of width 20
- fm/c computed for the force SV-bas~\cite{Klu09a} with a grid spacing
- $\Delta x=0.75$ fm and box size of 24 fm. Results are shown for
- different sizes of time step {\tt dt} and different order of Taylor
- expansion {\tt m} of the exponential evolution. The left panel shows
- the full evolution from ground state energy to the excited
- energy. The right panels concentrate on the times after the pulse is
- over in a narrower energy range relevant for this excitation.}
-\end{figure}
-
-The time stepping using the exponential propagator has the two parameters,
-step size {\tt dt} and order {\tt mxp}$=m$ of the Taylor expansion
-(\ref{eq:timepower}) of the exponential. Intuitively, one expects that
-small {\tt dt} and large {\tt mxp} improve the quality of the
-step. An efficient stepping scheme, however, looks for the largest {\tt
- dt} and smallest {\tt mxp} which still provide acceptable and
-stable results. It is hard to give general rules as good working
-values for the parameters depend on all details of the actual calculation:
-gridding, nuclei involved, excitation energy, and kind of
-excitation.
-
-Figure \ref{fig:dynamicSV-energ} demonstrates the dependence of a
-typical dynamical evolution on these time-stepping parameters. We
-consider a time interval up to 1260 fm/c which is a long time for
-heavy-ion collisions and just sufficient for a spectral analysis of
-oscillations~\cite{Rei07a}. The excitation is done by a soft sin$^2$
-pulse of finite extension in time. The energy increases during the
-initial excitation phase, as can be seen from the left panel in the
-figure. After the external pulse is over, energy conservation holds,
-which is nicely seen at plotting resolution in the left
-panel. Normalization should be conserved at all times. Both
-conservation laws serve as tests for the time step. Norm is conserved
-up to at least six digits for all cases and times shown in Figure
-\ref{fig:dynamicSV-energ}. The energy is more critical. The right
-panels show the energy in a small window around the final energy after
-the excitation phase is over. The right lower panel shows a variation
-of the Taylor order $m$ for fixed time step. The most prominent effect
-is the sudden turn to catastrophic failure for $m=6$. In fact,
-propagation by approximate exponential evolution explodes sooner or
-later in all cases. The art is to extend the stable interval by
-a proper choice of the stepping parameters. It is plausible that the
-cases with $m>6$ maintain stability longer because the exponential is
-better approximated. It is surprising that $m=4$ is also stable over
-the whole time interval. There seem to be subtle cancellations of
-error going on. Considering the stable signals, we see very little
-differences between the cases. One may generally be happy with low
-$m$. It is mainly stability demands which could call for larger
-$m$. Note that this is not a generic result. Stability for a given
-test case should be checked once in a while and particularly before
-launching larger surveys.
-
-The right upper panel in Figure \ref{fig:dynamicSV-energ} shows
-results for different {\tt dt} (as we have seen, the $m$ values are
-not important as long as we achieve stable results). Here we see a
-clear dependence on the step size. The energies remain constant in the
-average. But there are energy fluctuations and these depend
-sensitively on {\tt dt}. Smaller {\tt dt} yields smaller fluctuations.
-As far as one can read off from the figure, the amplitude of the
-fluctuations shrink $\propto${\tt dt}$^2$. It depends on the intended
-analysis to which level of precision the time
-evolution should be driven. A value of {\tt dt}$\approx$ 0.4 fm/c will be acceptable in most
-cases because the average trend remains far smaller than the
-fluctuations. Here also it must be emphasized that this is not a
-generic number. Forces with lower effective mass (SV-bas has
-$m^*/m=0.9$) are more demanding and usually require smaller {\tt
- dt}. On the other hand, running propagation without the spin-orbit term
-allows even larger time steps because the spin-orbit potential is the
-most critical piece in the mean-field Hamiltonian. The mix of
-$\mathbf{p}$ and $\mathbf{r}$ imposes high demands on the numerical
-representations. We again strongly recommend running a few
-tests when switching forces or excitation schemes.
-
-\section{Code structure}
-The code is completely modularized to provide as large a degree of
-encapsulation as possible in order to ease modification. Most modules
-read their operating parameters from an associated {\tt NAMELIST} and
-have their local initialization routines. In addition, a modern style
-of programming is used that employs a minimum number of local
-variables and streamlined array calculations that make the code lines
-very close to the physical equations being solved.
-
-Here we give a brief overview over the modules and their purpose. The
-source files containing the modules have the same names, but all in
-lower case, with an extension of {\tt .f90}. The
-higher-level modules are:
-
-\begin{description}
-\item{Main program}: It calls initialization routines, sets up the
- initial wave functions using either harmonic oscillator states or
- reading wave functions of static Hartree-Fock solutions from given
- input files (module {\tt Fragments}). It then calls either {\tt
- statichf} from module {\tt Static} or {\tt dynamichf} from module
- {\tt Dynamic} to run the calculation.
-\item{Static}: This contains the code for the static iterations {\tt
- statichf} and the subroutine {\tt sinfo} to generate output of the
- results.
-\item{Dynamic}: runs the dynamic calculation in {\tt dynamichf} and
- generates output in {\tt tinfo}. Also controls the inclusion of an
- external excitation implemented in module {\tt External}.
-\item{Densities}: calculates the densities and current densities by
- summing over the single-particle states.
-\item{Meanfield}: contains the central physics calculation: the
- computation of the components of the mean field (subroutine {\tt
- skyrme}) and the application of the single-particle Hamiltonian to
- a wave function (subroutine {\tt hpsi}).
-\item{Coulomb}: calculation of the Coulomb potential.
-\item{Energies}: calculation of the total energies and its various
- contributions.
-\item{External}: calculation of the action of an external potential or
- initial collective boost of the wave functions.
-\item{Pairs}: Implementation of the pairing correlations in the BCS
- approximation.
-\item{Moment}: calculation of moments and deformation parameters for
- the bulk density.
-\item{Twobody}: attempts to divide up the system into two separated
- nuclei and to calculate their properties and relative motion.
-\end{description}
-
-The lower-level supporting modules are:
-\begin{description}
-\item{Params}: general parameters used throughout the code.
-\item{Forces}: defines parameters of the Skyrme force and the pairing
- interaction and constructs them according to input.
-\item{Grids}: defines everything associated with the numerical grid
- and sets it up.
-\item{Levels}: definition of the single-particle wave functions and
- elementary operations on them such as derivatives.
-\item{Fragments}: controls the reading of static wave functions from
- precomputed data and setting them up in the grid.
-\item{Inout}: contains the subroutines for I/O of wave functions and
- densities.
-\item{Trivial}: defines some very basic operations on wave functions
- and densities.
-\item{Fourier}: sets up the transform plans for the {\tt FFTW3}
- package to calculate Fourier transforms of wave functions and
- densities.
-\item{Parallel}: This comes in two versions. The source file {\tt
- parallel.f90} contains the routines to handle {\tt MPI} message
- passing, while {\tt sequential.f90} sets up essentially dummy
- replacements for sequential or {\tt OpenMP} mode.
-\item{User}: contains a sample user initialization code which can be
- used as a template for more complicated setups.
-\end{description}
-
-\section{Parallelization}
-For both OpenMP~\cite{Cha08aB} and MPI~\cite{none12} the code can be
-run in parallel mode. Parallelization for the static mode works in
-OpenMP but not in MPI: the reason is in the orthogonalization step
-which is not easily amenable for distributed-memory parallel
-computation. This should be worked on in the future.
-
-MPI and OpenMP can be used jointly if there are computing nodes with
-multiple processors.
-
-In both cases parallelization is done over the wave functions. The
-code applies the time-development operator or the gradient iteration, which use the fixed set
-of mean-field components, to each wave function, and this can naturally be
-parallelized. Computing the mean fields and densities by summing up
-over single-particle wave functions is also easily parallelizable.
-
-The library {\tt FFTW3}~\cite{Fri05a} itself can also run on multiple
-processors in parallel. This can be used in addition to OpenMP or MPI,
-but was found to be helpful only in the sequential version of the
-code.
-
-\subsection{OpenMP}
-The application of the subroutine {\tt tstep} for propagating one wave
-function for one time step, and of {\tt add\_densities} for adding one
-wave function's contribution to the mean fields is done in
-parallel loops. The only complicating factor is that the densities,
-being accumulated in several subsets, must be kept separate using the
-{\tt REDUCTION(+)} clause of OpenMP. The summation cannot be done
-internally in {\tt add\_densities}, because for the half time step the
-wave functions are immediately discarded after adding their
-contribution to the densities to avoid having to store the full set at
-half time. Thus only the combined {\tt tstep}-{\tt add\_densities}
-loop should be parallelized.
-
-The OpenMP program version can be compiled using the appropriate
-compiler option. A separate {\tt Makefile.openmp} is provided which
-just contains the {\tt -fopenmp} option for the GNU compiler. The
-number of parallel threads is not set by the code: the user should set
-the environment variable {\tt OMP\_NUM\_THREADS} to the desired number.
-
-The user should also check compatibility of the locally installed {¸tt
- LAPACK/BLAS} libraries with {\tt OpenMP}.
-
-\subsection{MPI}
-In principle MPI uses the same technique as OpenMP, parallelizing over
-wave functions. In this case, however, each node contains only a
-fraction of the wave functions. This has several consequences:
-\begin{enumerate}
-\item The time-stepping of the wave functions can be done
- independently on each node, but requires that the densities are
- broadcast to all nodes after each half or full time step by summing
- up partial densities from the nodes in subroutine {\tt
- collect\_densities}.
-\item The other calculation that uses wave functions directly is that
- of the single-particle properties. These are calculated on each node
- for the wave functions present on that node and then collected using
- subroutine {\tt collect\_sp\_properties}.
-\item Only one node must be allowed to produce output. This is
- regulated by choosing node \#0 and setting the flag {\tt wflag} to
- {\tt .TRUE.}\ on that node.
-\item The saving of the wave functions is done in the following way:
- Node zero writes a header file containing the job information on
- file {\tt wffile}, then each node writes a separate file {\tt
- wwfile.001}, {\tt wwfile.002}, and so on up to the number of
- nodes. This avoids having to collect the wave functions on one node.
-\item Using these parallel output files as fragment initialization or
- restart files is handled so flexibly that they can be read into a
- different nodal configuration or even a sequential run.
-\end{enumerate}
-
-The MPI version needs the appropriate compiler and linker calls for
-the system used. The sequential or OpenMP versions are obtained simply
-by linking with {\tt sequential.f90} instead of {\tt parallel.f90},
-which replaces the MPI calls with a set of dummy routines and sets up
-the descriptor arrays for the wave function allocation in a trivial
-way.
-
-The {\tt Makefile.mpi} shows the procedure; in practice systems differ
-considerably and the user should look up the compilation commands for
-his particular system.
-
-\newpage
-\section{Program unit descriptions}
-\subsection{Program {\tt Main3D}} \Indexmb{Main3D}
-This is the main program that organizes the reading of the input and
-funnels the calculation into the correct subroutines.
-
-It consists of a number of simple steps. They are:
-
-\begin{itemize}
-\item{\bf Step 1:} initialize {\tt MPI} in case of an {\tt MPI} parallel job.
-Start reading from standard input beginning with
- namelist \Indexn{files} for any changes in the file names.
-
-\item{\bf Step 2:} read the definition of the force to be used and set
- it up (for its parameters see Section~\ref{sec:efunc}).
-
-\item{\bf Step 3:} read namelist \Indexn{main}, which contains the
- overall controlling parameters. The choice of \Indexv{imode} is used to
- set up \Indexv{tstatic} and \Indexv{tdynamic} as logical variables. In
- addition, if this is a restart, the number of fragments is set to
- one. The properties of this one fragment are then filled in in
- subroutine \Indexp{getin\_fragments}.
-
-\item{\bf Step 4:} read the grid definition (see Section
- \ref{sec:griddef}). With the dimensions known, all grid-based
- arrays (but not the wave functions) can be allocated and the
- initialization of the {\tt FFTW} system can be done.
-
-\item{\bf Step 5:} the appropriate namelist is read for the static or
- dynamic case, defining some parameters in those modules.
-
-\item{\bf Step 6:} determine wave function numbers. The way this is
- done depends on the choice of \Indexv{nof}. For positive {\tt nof}, the
- fragment files are consulted to find out the properties of each and
- add up the numbers. For {\tt nof=0} the numbers are taken from
- namelist \Indexn{static}, which was read before. If {\tt nof<0}, they
- are also given in namelist {\tt static}, but the wave functions will
- be replaced by user-calculated ones.
-
-\item{\bf Step 7:} now that the numbers are known, the wave function
- distribution over nodes is computed or set to trivial for a
- sequential calculation, and the the arrays related with wave
- functions are allocated.
-
-\item{\bf Step 8:} the initial values of the wave functions are
- calculated. For \Indexv{nof}{\tt>0} the wave functions are read from the
- fragment files and inserted into the proper positions (see Sections
- \ref{sec:react} and \ref{sec:fragini}). For {\tt nof=0} the routine
- \Indexp{harmosc} in module {\tt Static} is called to calculate
- harmonic-oscillator wave function, and for {\tt nof<0} the routine
- \Indexp{init\_user} does an arbitrary user initialization.
-
-\item{\bf Step 9:} the mean-field arrays are zeroed and the Coulomb
- solver is initialized.
-
-\item{\bf Step 10:} the calculation branches into either static or
- dynamic mode.
-
- In the static calculation, \Indexp{init\_static} just prints some
- information and sets up damping (see Section~\ref{sec:statiter}) and
- \Indexp{statichf} does the real calculation.
-
- In the dynamic case, the subroutine \Indexp{dynamichf} does all the
- work.
-
-\item{\bf Step 11:} finally the {\tt MPI} system is terminated.
-
-\end{itemize}
-
-
-\subsection{Module {\tt Coulomb}}\Indexmb{Coulomb}
-\label{sec:modcoul}
-\subsubsection{Purpose}
-This module offers the subroutine {\tt poisson}, which calculates the
-Coulomb potential from the proton density. The two boundary conditions
-allowed are distinguished by the value of the variable \Indexv{periodic}.
-If it is true, the mesh is assumed to be periodic in all three
-directions and the Poisson equation is solved using the $1/k^2$
-Green's function in momentum space and assuming a homogeneous negative
-background annulling the total charge (jellium approximation).
-Otherwise the boundary condition is for an isolated charge
-distribution. In this case the potential is calculated by the method
-of doubling the grid in all directions with zero density, folding with
-the $1/r$-potential in momentum space and then restricting to the
-physical region~\cite{Eas79}.
-
-
-\subsubsection{Details on the Coulomb solver for an isolated system}
-\newcommand{\bnu}{\mbox{\boldmath$\nu$}}
-\newcommand{\bmu}{\mbox{\boldmath$\mu$}}
-
-The Coulomb solver follows the ideas of \cite{Eas79}. We summarize
-here the way it is realized in the code without detailed proofs.
-
-Two grids are used. We will call them here $\mathcal{G}_1$ and
-$\mathcal{G}_2$. The grid $\mathcal{G}_1$ is our standard working grid
-as given in eq. (\ref{eq:FT}). We repeat here the essentials
-\begin{subequations}
-\begin{align*}
-\mathcal{G}_1:\;&
- \nu_x=1,...,{\tt nx}\;,\;
- \nu_y=1,...,{\tt ny}\;,\;
- \nu_z=1,...,{\tt nz}\;;\;
-\nonumber\\&
- \mbox{coordinate spacing: } {\tt dx}, {\tt dy}, {\tt dz};
-\nonumber\\&
- \mbox{momentum spacing: }
- {\tt dk}_x
- =
- \frac{2\pi}{{\tt nx}\!\cdot\!{\tt dx}},
- {\tt dk}_y
- =
- \frac{2\pi}{{\tt ny}\!\cdot\!{\tt dy}},
- {\tt dk}_z
- =
- \frac{2\pi}{{\tt nz}\!\cdot\!{\tt dz}};
-\end{align*}
-\end{subequations}
-The second grid amounts to a doubled box in each direction, thus
-\begin{subequations}
-\begin{align*}
-\mathcal{G}_2:\;&
- \nu_x=1,...,2{\tt nx}\;,\;
- \nu_y=1,...,2{\tt ny}\;,\;
- \nu_z=1,...,2{\tt nz}\;;\;
-\nonumber\\&
- x_{\nu_x}=(\nu_x\!-\!{\tt nx}){\tt dx}\,,\,
- y_{\nu_y}=(\nu_y\!-\!{\tt ny}){\tt dy}\,,\,
- z_{\nu_z}=(\nu_z\!-\!{\tt nz}){\tt dz}\,;
-\nonumber\\&
- \mbox{momentum spacing: }
- {\tt dk}_x
- =
- \frac{2\pi}{{2\tt nx}\!\cdot\!{\tt dx}},
- {\tt dk}_y
- =
- \frac{2\pi}{{2\tt ny}\!\cdot\!{\tt dy}},
- {\tt dk}_z
- =
- \frac{2\pi}{{2\tt nz}\!\cdot\!{\tt dz}};
-\end{align*}
-\end{subequations}
-The momenta $k_{n_i}$ in $\mathcal{G}_2$ are arranged around $k=0$
-similar as in $\mathcal{G}_1$, but with half spacing ${\tt dk}_i$.
-For simplicity, we will use compact notation
-$\mathbf{r}_{\bnu}=\left(x_{\nu_x},y_{\nu_y},z_{\nu_z}\right)$
-and similarly for $\mathbf{k}_\mathbf{n}$.
-
-First, the Coulomb solver has to be initialized by defining the
-appropriate Greens function which is done on $\mathcal{G}_2$.
-The two necessary steps are:
-\begin{enumerate}
- \item
- Prepare the Greens function in $r$ space as
- \begin{equation*}
- G(\mathbf{r}_{\bnu})=\frac{e^2}{|\mathbf{r}_{\bnu}|}
- \quad.
- \end{equation*}
-Special consideration has to be given to the singularity of the $1/r$
-potential. In the discretized version, the value at $\vec r=0$ should
-be replaced by the average of $1/r$ over a cuboid of size $\Delta
-x\times\Delta y\times\Delta z$. For equal spacing in all three
-directions this would lead to a value of $2.38/\Delta x$. Practical
-experimentation, however, showed that the underlying assumption of a
-point charge can be improved by some smeared-out density for nuclear
-applications; a value of $2.84/\Delta x$ was found to be optimal.
-We use the expression
- \begin{equation}\label{eq:coulsing}
- G(0)
- =
- \frac{2.84\sqrt{3}}{\sqrt{{\tt dx}^2+{\tt dy}^2+{\tt dz}^2}}
- \quad.
- \end{equation}
-
- \item
- Prepare the Greens function in $k$-space by
- 3D fast Fourier transformation (FFT) on the double grid $\mathcal{G}_2$.
- $\tilde{G}(\mathbf{k}_{\bmu})={\tt FFT}\{G(\mathbf{r}_{\bnu})\}$.
- The array $\tilde{G}(\mathbf{k}_{\bmu})$ is stored
- for further continued use.
-\end{enumerate}
-Once properly prepared, the practical steps for computing
-the Coulomb field ${U}_\mathrm{Coul}(\mathbf{r}_{\bnu})$ for a
-density $\rho(\mathbf{r}_{\bnu})$ given on $\mathcal{G}_1$ are
-\begin{enumerate}
- \item
- Extend $\rho_{\bnu}$ from $\mathcal{G}_1$ to $\mathcal{G}_2$
- by zero padding:
- \begin{equation*}
- \rho_2(\mathbf{r}_{\bnu})
- =
- \left\{\begin{array}{lcl}
- \rho(\mathbf{r}_{\bnu})
- &\mbox{ if }& 1\leq \nu_i \leq {\tt n}i \mbox{ for } i\in\{x,y,z\}
- \\
- 0 &\mbox{ else }
- \end{array}\right.
- \end{equation*}
- \item
- Fourier transform the density in $\mathcal{G}_2$:
- \begin{equation*}
- \rho_2(\mathbf{r}_{\bnu})\quad\longrightarrow\quad
- \tilde{\rho}_2(\mathbf{k}_{\bmu})={\tt FFT}\{\rho_2(\mathbf{r}_{\bnu})\}
- \end{equation*}
- \item
- Compute solution in $k$-space by multiplication with the
- formerly prepared Greens function
- \begin{equation*}
- \tilde{U}_2(\mathbf{k}_{\bmu})
- =
- \tilde{G}_2(\mathbf{k}_{\bmu})\tilde{\rho}_2(\mathbf{k}_{\bmu})
- \end{equation*}
- \item
- Compute solution in $r$-space by Fourier back transformation
- in $\mathcal{G}_2$:
- \begin{equation*}
- U_2(\mathbf{r}_{\bnu})={\tt FFT}^{-1}\{\tilde{U}_2(\mathbf{k}_{\bmu})\}
- \end{equation*}
-
- \item
- Map to standard grid $\mathcal{G}_1$:
- \begin{equation*}
- {U}_\mathrm{Coul}(\mathbf{r}_{\bnu})
- =
- U_2(\mathbf{r}_{\bnu})
- \quad\mbox{for}\quad
- \bnu\in\mathcal{G}_1
- \end{equation*}
-\end{enumerate}
-
-\subsubsection{Module variables}
-\begin{description}
-\item{{\tt nx2, ny2, nz2:}} (INTEGER,PRIVATE) dimensions of the grid
- on which the Fourier transform is calculated. For the periodic case
- they are identical to the regular dimensions, for the isolated case
- they are doubled.
-\item{{\tt coulplan1, coulplan2:}} (INTEGER,PRIVATE) plans for {\tt
- FFTW} complex forward and reverse transforms with array dimensions
- depending on the boundary condition.
-\item{\Indexvb{wcoul}:} the Coulomb potential as a three-dimensional
- array. Units: MeV.
-\item{{\tt q:}} (COMPLEX,PRIVATE) array for the
- complex Green's function (isolated) or array of $1/r$ values. Its
- dimension also depends on the boundary condition.
-\end{description}
-
-\subsubsection{Subroutine {\tt poisson}}\Indexpb{poisson}
-This subroutine solves the Poisson equation by the Fourier method. For
-the periodic case, this means to just Fourier transform, multiply by
-$1/k^2$, and the transform back. Note that the coefficient for
-momentum zero must be zero also: this means that the total charge in
-the box vanishes, corresponding to the jellium approximation.
-
-In the non-periodic case we use use the trick of doubling the box size
-in every direction and then folding with $1/r$ my multiplying the
-Fourier transforms. The result in the physical box is then the correct
-solution for the boundary condition of zero potential at infinity.
-The key to success is not to use simply $1/k^2$ for the Fourier
-transform of $1/r$, but to compute it on the actual grid the same way
-as the densities and potentials are transformed~\cite{Eas79}.
-
-\subsubsection{Subroutine {\tt coulinit}}\Indexpb{coulinit}
-This subroutine does the necessary initialization. It calculates the
-dimension for the Fourier transform depending on the boundary
-condition, allocates the necessary arrays and sets up the {\tt FFTW}
-plans.
-
-Then it composes the array {\tt q}. For the periodic case this
-contains the values of the Green's function $1/k^2$, with zero at the
-origin, while for the isolated case it first calculates the inverse
-shortest distance $1/r$ from the origin with indices (1,1,1), replaces
-the value at the origin according to Eq.~(\ref{eq:coulsing}) and then
-Fourier-transforms this to use it for folding the density with the
-coordinate-space Green's function.
-
-\subsubsection{Subroutine {\tt initiq}}\Indexpb{initiq}
-This subroutine calculates the contributions of each Cartesian index
-to $r^2$ and $k^{-2}$. This depends on the boundary condition. For
-the periodic case, the values of the momenta are given by
-$$ k_i=\frac{2\pi}{n\Delta x}\bigl(0,\;1,\;2,\;\ldots \frac{n}2-1,\;
--\frac{n}2,\;-\frac{n}2+1,\;\ldots -1\bigr).
-$$
-For an isolated distribution the shortest distances to the point with
-index 1 are calculated (periodicity used):
-$$ d_i=d\bigl(0,\;1,\;2,\;\ldots \frac{n}2-1,\;
--\frac{n}2,-\frac{n}2+1,\;\;\ldots 1\bigr).
-$$
-The input is the dimension along the coordinate direction given, the
-output is the one-dimensional array {\tt iq} containing these values
-{\bf squared}.
-
-
-\newpage
-\subsection{Module {\tt Densities}} \Indexmb{Densities}
-This module has two purposes: it defines and allocates the densities
-and currents making up the mean field according to Sect.~\ref{sec:dens},
-and also contains the subroutine \Indexp{add\_density} which accumulates
-the basic densities over the single-particle wave functions.
-Subroutine \Indexp{skyrme} in module \Indexm{Meanfield} then uses these
-densities to build up the components of the single-particle Hamiltonian.
-
-\subsubsection{Module variables}
-\begin{itemize}
-\item{\bf Scalar densities:} These are dimensioned {\tt (nx,ny,nz,2)},
- where the last index is~1 for neutrons and~2 for protons.
- \begin{description}
- \item{\Indexvb{rho}:} the density, separately for each isospin (in
- fm$^{-3}$). The definition is:
- $$ \rho_q(\vec r)=\sum_{k\in q}w_k^2\sum_s|\phi_k(\vec r,s)|^2,\qquad
- q=n,p $$
- \item{\Indexvb{tau}:} the kinetic energy density, also separately
- for each isospin. It is defined as the sum of the spin
- contributions and all particles of the given isospin
- $$ \tau_q(\vec r)=\sum_{k\in q}w_k^2\sum_s|\nabla\phi_k(\vec r,s)|^2,
- \qquad q=n,p$$ Note that it does not include the factor
- $\hbar^2/2m$. Units: fm$^{-5}$.
- \end{description}
-\item{\bf Vector densities} These are dimensioned {\tt
- (nx,ny,nz,3,2)}, where the last index is~1 for neutrons and~2 for
- protons, and the next-to-last stands for the Cartesian direction.
- \begin{description}
- \item{\Indexvb{sdens}:} the spin density. It is defined as
- $$\vec\sigma_q(\vec r)=
- \sum_{\alpha\in q} w_\alpha^2 \sum_{ss'}\psi_\alpha^*(\vec
- r,s)\,\sigma_{ss'} \,\psi_\alpha(\vec r,s').$$ Note that it does
- not include the factor $\hbar/2$. Units: fm$^{-3}$.
- \item{\Indexvb{current}:} this is the total probability current
- density, defined in the familiar way as
- $$ \vec\jmath_q(\vec r)
- = \frac{1}{2\I}\sum_{\alpha\in q}w_\alpha^2
- \sum_s\left(\psi_\alpha^*(\vec r,s)\nabla\psi_\alpha(\vec
- r,s)-\psi_\alpha(\vec r,s) \nabla\psi_\alpha^*(\vec r,s)\right).
- $$
- Note that the factor $\frac{\hbar}{m}$ is not included. Its units
- are therefore fm$^{-4}$.
- \item{\Indexvb{sodens}:} the spin-orbit density, defined as
- $$ \vec J_q(\vec r)=\frac{1}\I\sum_{\alpha\in q}w_\alpha^2
- \sum_{ss'}\left(\psi_\alpha^*(\vec r,s)\nabla\times\sigma_{ss'}
- \psi_\alpha(\vec r,s')\right).
- $$
- Its units are also fm$^{-4}$.
- \end{description}
-\end{itemize}
-
-\subsubsection{Subroutine {\tt alloc\_densities}}
-\Indexpb{alloc\_densities}
-This is simply a short routine to allocate all the arrays defined in
-this module.
-
-\subsubsection{Subroutine {\tt add\_density}}
-\Indexpb{add\_density}
-This subroutine is given a single-particle wave function {\tt psin}
-with its isospin index {\tt iq} and occupation $w_\alpha^2=${\tt weight} and adds
-its contribution to the density arrays.
-
-The reason for not including the loop over states in the subroutine is
-that in the dynamic code, the contribution of a new single-particle
-wave function (calculated by \Indexp{tstep}) to the densities is added
-without saving that wave function, eliminating the requirement for a
-second huge wave-function array.
-
-It may seem strange that {\tt add\_density} has the densities
-themselves as parameters, which are readily available in the module.
-The reason for this is {\tt OPENMP} parallelization. The loop over
-wave functions is done in parallel under {\tt OPENMP}. Since any of
-the parallel tasks must add up the contributions of its assigned wave
-functions, each task must have a copy of the densities to work on;
-otherwise they would try to update the same density at the same time.
-The separate copies are then combined using the {\tt OPENMP} {\tt
- REDUCE(+)} directive.
-
-The local copies of the densities passed as arrays are denoted with
-the prefixed letter ``l'' for {\em local}; they are {\tt lrho}, {\tt
- ltau}, {\tt lcurrent}, {\tt lsdens}, and {\tt lsodens}.
-
-If the weight is zero, there is nothing to do and the subroutine
-returns immediately. Otherwise, the contributions not involving
-derivatives are first computed and added to the affected densities,
-i.~e., number and spin density.
-
-After this the derivative terms are evaluated by computing each
-Cartesian direction separately. In all three cases the derivative is
-evaluated first and put into {\tt ps1}, after which the contributions
-are added straightforwardly. They involve the wave function itself,
-the derivative, and for the spin-orbit density also a Pauli matrix, so
-that different spin projections have to be combined properly.
-
-The complex products always in the end evaluate to something real and
-the expressions are simplified to take this into account. For example,
-the following transformation is done:
-\begin{eqnarray*}
- \frac{1}{2\I}(\psi^*\nabla\psi-\psi\nabla\psi^*)&=&
- \frac{1}{2\I}\left(\psi^*\nabla\psi-(\psi^*\nabla\psi)^*\right)\\
- &=&\frac{1}{2\I}\left(2\I\Im(\psi^*\nabla\psi)\right)\\
- &\rightarrow&{\tt AIMAG(CONJG(psin)*psi1)}
-\end{eqnarray*}
-and similarly for the other expressions.
-
-The efficiency of this relies on the FORTRAN compiler recognizing that
-only the imaginary part of the complex product is needed and not
-computing the real part at all. This seems to be the case with all
-present compilers.
-
-\newpage
-\subsection{Module {\tt Dynamic}}\Indexmb{Dynamic}
-This module contains the routines needed for time propagation of the
-system (see Section~\ref{sec:timevol}). All the logic needed for this
-case is concentrated here; all other modules except for \Indexm{External}
-are equally used in the static calculation.
-
-\subsubsection{Module variables}
-
-\begin{itemize}
-\item \Indexvb{nt}: the number of the final time step to be
- calculated. In case of a restart this is smaller than the total
- number of time steps.
-\item \Indexvb{dt}: the physical time increment in units of fm/c.
-\item \Indexvb{mxpact}: the number of terms to be taken in the
- expansion of the potential according to Eq.~(\ref{eq:timepower}).
-\item \Indexvb{rsep}: the final separation distance. The calculation
- is stopped if there has been a reseparation into two fragments and
- their distance exceeds {\tt rsep}.
-\item \Indexvb{texternal}: this logical variable indicates that an
- external field is present. See module {\tt External}.
-\item \Indexvb{text\_timedep}: this logical variable indicates that the
- external field is time-dependent and does not describe an
- instantaneous boost.
-\item \Indexvb{esf}: this is the energy shift for the call to {\tt
- hpsi}. Since it is not used in the dynamics part of the code, it
- is here set to the constant value of zero.
-\end{itemize}
-
-\subsubsection{Subroutine {\tt getin}\_dynamic}
-\Indexpb{getin\_dynamic}
-This is a relatively simple routine that reads the input for namelist
-\Indexn{dynamic} and prints it on standard output. If \Indexv{texternal} is
-true, it also calls \Indexp{getin\_external} to read the external field
-parameters.
-
-\subsubsection{Subroutine {\tt dynamichf}}\Indexpb{dynamichf}
-This subroutine performs the main time-integration algorithm starting
-at time step 0 and then iterating the desired number of steps. Its
-building blocks are:
-\begin{itemize}
-\item {\bf Step 1: preparation phase}: this phase consists of several
- substeps.
- \begin{enumerate}
- \item If this is not a restart, the time and iteration number are
- zeroed (for a restart only the physical time is taken from the
- {\tt wffile}). The wave functions are saved in the
- \Indexv{wffile}, to save setup time in case the calculation has to
- be restarted from this initial point.
- \item The instantaneous external boost (\ref{eq:extboost}) is applied using
- \Indexp{extboost}. If this subroutine has applied a boost, it sets
- \Indexv{text\_timedep} to {\tt .FALSE.} so no further calls to
- external-field routines are made (except for {\tt print\_extfield}).
- \item The protocol files {\tt *.res} are initialized with
- their header lines.
- \item The densities and current are calculated in a loop over wave
- function by calling \Indexp{add\_densities}.
- They are first set to zero and then accumulated in a
- loop over the set on the local node, followed by collecting them
- over all nodes.
- \item The mean field and (if {\tt texternal} is true) the
- external field are calculated for time zero using routines
- \Indexp{skyrme} and \Indexp{extfld}.
- \item Then \Indexp{tinfo} is called to calculate and print the
- single-particle quantities and the total energies at time 0 or
- iteration~0.
- \item Finally preparations are made for the time-stepping loop: the
- starting index is set to iter+1: this is either after the end of a
- previous job in the case of a restart, or just one for a new
- calculation. The physical time is either~0 or the time taken from
- a restart file.
- \end{enumerate}
-
-\item{\bf Step 2: predictor time step}: the loop over the iteration
- index {\tt iter} is started. Then the densities and mean-field
- components are estimated, i.~ e., in effect the Hamiltonian $\hat
- h(t+\tfrac1{2}\Delta t)$ required by the numerical method (see
- Eq.~(\ref{eq:midstep}) and Section~\ref{sec:timevol}). This is done
- by evolving the wave functions for a full {\tt dt} using the old
- Hamiltonian and averaging the densities between old and new ones to
- obtain the mid-time Hamiltonian for propagation. In detail the
- procedure is as follows:
- \begin{enumerate}
- \item The densities are not set to zero in order to allow adding the
- contributions of the wave functions at the end of the time step.
- \item In the MPI version, the densities are divided by the number of
- nodes. In this way, adding up contributions from all nodes, the
- densities from the beginning of the time step will be included
- correctly.
- \item Subroutine \Indexp{tstep} is used to propagate the wave functions
- to the end of the time step. Note that truncation in the
- exponential happens at \Indexv{mxpact/2}, since the accuracy need not
- be as high as in the full step. For each wave function its
- contribution is added to the densities. The wave functions
- themselves do not need to be saved as they are not used for
- anything else.
- \item After the loop, contributions from all the nodes are added up
- for the MPI case using subroutine \Indexp{collect\_densities}.
- \item The densities are multiplied by one half to form the average
- of the values at $t$ and $t+\Delta t$.
- \item These average densities are then used to calculate the mean
- field at half time using subroutine \Indexp{skyrme}; also the
- external field is obtained for the half time using \Indexp{extfld}.
- \end{enumerate}
-
-\item {\bf Step 3: full time step}: Now that the single-particle
- Hamiltonian has been estimated for the middle of the time step, the
- propagation can be carried out to the end of the time step. This is
- quite analogous to the half step with only three crucial
- differences:
- \begin{enumerate}
- \item The densities are reset to zero before the wave function loop,
- so the densities summed up are the purely the densities at the end
- of the time step,
- \item the series expansion in \Indexp{tstep} now uses the full
- \Indexv{mxpact} terms, and
- \item the new wave functions are copied back into \Indexv{psi} to be
- available for the next time step.
- \end{enumerate}
-
-\item {\bf Step 4: Center-of-mass correction}
-If a center-of-mass correction is desired by the user by setting
-\Indexv{mrescm}{\tt /=0},
-subroutine \Indexp{resetcm} is called every {\tt mrescm}'th time step to
-reset the center-of-mass velocity to zero.
-
-\item {\bf Step 5: generating some output} {\bf At this point the time is
- advanced by {\tt dt} because the physical time is now the end of
- the time step, and this must be printed out correctly by the
- following output routines.} \Indexp{tinfo} is called to calculate
- single-particle properties, total energies, and so on.
-
-\item{\bf Step 6: finishing up the time step}: {\tt tinfo} is called
- to output the calculated data, then \Indexp{skyrme} and \Indexp{extfld}
- calculate the mean field and the external field, respectively, for the
- end of the time step, after which the wave functions are written
- onto \Indexv{wffile} depending on \Indexv{mrest}.
-\end{itemize}
-This ends the time loop and subroutine {\tt dynamichf} itself.
-
-\subsubsection{Subroutine {\tt tstep}} \Indexpb{tstep}
-In this subroutine one wave function given as the argument {\tt psout}
-is stepped forward in time by the interval {\tt dt}. The method used
-is the expansion of the exponential time-development operator
-according to Eq.~(\ref{eq:timepower}), cut off at the power of {\tt
- mxp}, which in practice is usually around~6. Suppressing the argument of $\hat
-h$ for brevity, we can write
-\begin{equation}
- \label{eq:timepower2}
- \hat U(t,t+\Delta t)\,\phi\approx
- \sum_{n=0}^m \phi^{(n)}
-\end{equation}
-with
-\begin{equation}
- \phi^{(0)}=\phi,\qquad \phi^{(k+1)}=\frac{-\I\,\Delta t}{\hbar c
- k}\,\hat h\,\phi^{(k)},
- \quad k=0,\ldots,m-1.
-\end{equation}
-
-Thus the application of the polynomial to a single-particle wave
-function can be evaluated simply in a loop applying $\hat h\phi_k$
-repeatedly and accumulating the results in wave function {\tt psout}.
-
-The argument {\tt iq} is only necessary because {\tt hpsi} needs
-information about the isospin of the wave function.
-
-\subsubsection{Subroutine {\tt tinfo}} \Indexpb{tinfo}
-\label{sec:info}
-
-This subroutine is used to output various pieces of information
-relevant especially to the dynamic mode of the code. It is called at
-the end of the full time step and consists of the following steps:
-\begin{itemize}
-\item {\bf Step 1: initialization}: the flag \Indexv{printnow} is
- calculated to keep track of whether this is the proper time step for
- a full printout (determined by \Indexv{mprint}).
-\item {\bf Step 2: twobody analysis} the twobody analysis is
- performed, but only if the calculation started as a twobody
- scenario.
-
-\item {\bf Step 3: moments}: the moments of the distribution are
- calculated using subroutine {\tt moments}. This includes total mass,
- momenta, and angular momenta. They are printed out if indicated by
- \Indexv{printnow}, both on the large output and in the specialized
- files \Indexv{dipolesfile}, \Indexv{momentafile}, and
- \Indexv{spinfile}. If there is an external field, the routine
- \Indexp{print\_extfield} is called to print the current expectation
- value of the external field.
-
- {\bf Note that the moments need to be calculated every time step,
- because some of the logic may depend on them, especially the
- twobody-analysis, which needs the correct c.m., for example and is
- calculated at every time step and on every node.}
-
-\item {\bf Step 4: single-particle quantities}: the single-particle
- energies are calculated straightforwardly as expectation values
- of the Hamiltonian.
- The routine \Indexp{sp\_properties} is then called to obtain the other
- single-particle properties like angular momenta. They are
- communicated between the processors. The angular momenta are written
- to \Indexv{spinfile}.
-\item{\bf Step 5: total energies}: The integrated energy \Indexv{ehfint}
- and its contributions are calculated in \Indexp{integ\_energy}. The
- subroutine {\tt sum\_energy} is called to calculate the three-body
- energy and the single-particle based total energy {\tt ehf}. The
- collective kinetic energy \Indexvb{ecoll} is computed directly
- here, because it is needed only in the dynamic calculations and only
- for output. It is defined as
- \begin{equation}
- \label{eq:ecoll}
- E_{\rm coll}=\frac{\hbar^2}{2m}\int \D^3r \frac{{\vec \jmath\,}^2}{\rho}
- \end{equation}
- The energies are protocolled in \Indexv{energiesfile} and on standard
- output.
-
-\item {\bf Step 6: density output}: at intervals of \Indexv{mprint} or in
- the first time step the density printer plot is generated using
- \Indexp{plot\_densities} and the binary densities are written onto {\tt *.tdd}
- files using \Indexp{write\_densities}.
-
-\item {\bf Step 7: other output}: in the proper \Indexv{mprint} interval
- the two-body analysis results, the single-particle state
- information, and the moments are printed on standard output, using
- also the routines \Indexp{twobody\_print} and
- \Indexp{moment\_print}.
-
- It is important to note that when {\tt
- tinfo} is called before the time step iteration starts, the
- two-body analysis cannot work because the fragment centers of mass
- from the previous time step are either not known yet (restart) or
- identical to the present ones (initialization). The subroutine {\tt
- twobody\_case} is still called to find the fragment properties,
- but the derived kinetic energy etc.\ will be incorrect. Therefore
- the call to {\tt twobody\_print} is suppressed in this case. The
- logical variable {\tt initialcall} is used to recognize this case.
-
-
-\item {\bf Step 8: check for final separation}: for the twobody case
- it is checked whether the separation found between the two fragments
- is larger than the input quantity \Indexv{rsep} with positive time
- derivative \Indexv{rdot} of the separation distance, in which case the
- program is terminated with an appropriate message.
-\end{itemize}
-
-\subsubsection{Subroutine {\tt resetcm}}\Indexpb{resetcm}
-This subroutine resets the center-of-mass velocity to zero. The
-velocity, or rather the corresponding wave vector, is calculated from
-the current density using
-\begin{equation*}
-\rho(\vec r) \vec k=\frac{1}{2\I}\sum_{\alpha\in q}w_\alpha^2
- \sum_s\left(\psi_\alpha^*(\vec r,s)\nabla\psi_\alpha(\vec
- r,s)-\psi_\alpha(\vec r,s) \nabla\psi_\alpha^*(\vec r,s)\right).
-\end{equation*}
-The wave functions are then multiplied by a common plane-wave phase
-factor $\exp(-\I\vec k\cdot \vec r)$ to give a counter boost.
-
-\newpage
-\subsection{Module {\tt Energies}}\Indexmb{Energies}
-This module computes the total energy and the various contributions to
-it in two ways. The first method evaluates the density functional,
-Eq.~(\ref{eq:efundet}), by direct integration to compute the {\em integrated
- energy} {\tt ehfint}. The second method uses the sum of
-single-particle energies plus the rearrangement energies, $E_{3,\rm
- corr}$ for the density dependent part and $E_{C,\rm corr}$ for
-Coulomb exchange as explained in Section~\ref{sec:alterenerg}.
-
-The two ways of calculating the energy are assigned to the subroutines
-{\tt integ\_energy}, which also calculates the rearrangement energies,
-and {\tt sum\_energy}. Note that since {\tt integ\_energy} is always
-called briefly before {\tt sum\_energy}, the rearrangement energies
-are correctly available.
-
-In addition subroutine {\tt sum\_energies} also calculates the summed
-spin, orbital, and total angular momenta.
-
-\subsubsection{Module variables}
-\label{sec:modvar}
-All the variables in this module are given in MeV unless otherwise
-noted.
-\begin{itemize}
-\item \Indexvb{ehft} denotes the total kinetic energy of
-Eq.~(\ref{eq:ehft}).
-\item \Indexvb{ehf0} corresponds to the $b_0$ and $b_0'$-dependent
- terms of Eq.~(\ref{eq:ehf0}).
-\item \Indexvb{ehf1}: the contribution depending on $b_1$ and $b_1'$
- of Eq.~(\ref{eq:ehf1}).
-\item \Indexvb{ehf2} accounts for the $b_2$, $b_2'$-dependent
- contributions of Eq.~(\ref{eq:ehf2}).
-\item \Indexvb{ehf3} is the $b_3$, $b_3'$-dependent part of
- Eq.~(\ref{eq:ehf3}) which models density dependence.
-\item \Indexvb{ehfls} is the spin-orbit energy (\ref{eq:ehfls}).
-\item \Indexvb{ehfc} is the Coulomb energy of Eq.~(\ref{eq:ehfc}).
-\item \Indexvb{ecorc} is the rearrangement correction
- (\ref{eq:ecorc}) to Coulomb exchange in the Slater approximation.
-\item {\tt e3corr} is the rearrangement energy (\ref{eq:e3corr}) to
- the density-dependent part of the Skyrme energy.
-\item \Indexvb{orbital}: the three components of the total orbital
- angular momentum in units of $\hbar$.
-\item \Indexvb{spin}: the three components of the total spin in
- units of $\hbar$.
-\item \Indexvb{total\_angmom}: the three components of the total
- angular momentum in units of $\hbar$.
-\end{itemize}
-
-\subsubsection{Subroutine {\tt integ\_energy}}\Indexpb{integ\_energy}
-The purpose of this subroutine is to calculate the integrated energy
-of Eq.~(\ref{eq:efundet}). This is implemented pretty straightforwardly
-using the integrals defined in Section~\ref{sec:efunc}. The only
-programming technique worth noting is that intermediate variables such
-as {\tt rhot} for the total density are used to avoid repeating the
-lengthy index lists. Compilers will eliminate these by optimization.
-
-In principle the integration loops in the subroutine could be
-combined, but some space is saved by using the array {\tt worka} for
-different purposes in different loops.
-
-The calculation proceeds in the following steps:
-
-\begin{itemize}
-\item {\bf Step 1}: the Laplacian of the densities is calculated in
- {\tt worka}, then the integrals of
- Eqs.~(\ref{eq:ehf0},\ref{eq:ehf2}, and \ref{eq:ehf3}) are performed.
- After the loop the result for {\tt ehf3} is also used to calculate
- {\tt e3corr} of Eq.~(\ref{eq:e3corr}).
-\item {\bf Step 2}: the integral of Eq.~(\ref{eq:ehf1}) is evaluated
- using {\tt worka} for the $\vec\jmath_q{}^2$ term.
-\item {\bf Step 3}: the spin-orbit contribution of
-Eq.~(\ref{eq:ehfls}) is calculated using {\tt worka} as storage for
- $\nabla\cdot\vec J_q$.
-\item {\bf Step 4}: the Coulomb energy is evaluated from
- Eq.~(\ref{eq:ehfc}) with the Slater correction taken into account if the
- force's \Indexv{ex} is nonzero. At the same time the Coulomb correction
- for the summed energy is calculated according to
- Eq.~(\ref{eq:ecorc}) and stored in {\tt ecorc}. It will be used in
- the subroutine {\tt sum\_energy}.
-\item {\bf Step 5}: the kinetic energy is integrated for
-Eq.~(\ref{eq:ehft}). Note that only at this point the correct prefactor
- $\hbar^2/2m$ is added; the use of {\tt tau} in other expressions
- assumes its absence.
-\item {\bf Step 6}: Finally all terms are added to produce the total
- energy, Eq.~(\ref{eq:efundet}), from which the pairing energies are
- subtracted.
-\end{itemize}
-
-\subsubsection{Subroutine {\tt sum\_energy}}
-\Indexpb{sum\_energy}
-This subroutine mainly computes the Koopman sum of
-Eq.~(\ref{eq:koopman}), but also sums up a number of other single-particle
-properties. For systematics, the latter should be done in a different
-place, but at present is left here.
-
-The summation of the total energy uses Eq.~(\ref{eq:spenerg}) to
-compute
-$$ \sum_k (\epsilon_k-\tfrac1{2}v_k)=\tfrac1{2}\sum_k(2t_k+v_k)=
-\tfrac1{2}\sum_k(t_k+\epsilon_k). $$ The last sum is calculated,
-the rearrangement corrections are added and the pairing energies
-subtracted.
-
-The subroutine then sums up the single-particle energy fluctuation
-\Indexv{sp\_efluct1} and \Indexv{sp\_efluct2}, dividing them by the nucleon
-number. Finally the orbital and spin angular
-momentum components are summed to form the total ones.
-
-\newpage
-\subsection{Module {\tt External}}\Indexmb{External}
-This module allows the coupling of the nucleonic wave functions to an
-external field. As described in Sect.~\ref{sec:external}, this can be
-done either by adding a time-dependent external (i.~e., not
-self-consistent) potential to the single-particle Hamiltonian, or by
-giving an initial ``boost'' to each wave function.. Since this is very
-easy to modify and will probably have to be adjusted for most
-applications, the present version just contains a sample for a
-quadrupole coupling. The logic for different time-dependence
-assumptions and the isospin are however, fully functional and should
-be useful in many cases.
-
-\subsubsection{Module variables}
-Most of these just describe the behavior of the external field. They
-are all declared as private, so that the user modifying this code can
-be sure that the internals are not used anywhere else. The variables
-are:
-
-\begin{itemize}
-\item{\Indexvb{amplq0}:} a strength parameter for the perturbation.
- As explained in Sect-~\ref{sec:external}, its numerical magnitude is
- usually not important by itself, but varying it allows studying the effects
- of different strengths of the excitation.
-\item{\Indexvb{textfield\_periodic}:} if this is set to {\tt true}, the
- external field is made periodic according to
- Eq.~(\ref{eq:extperiodic}), otherwise a damping factor is used as
- defined in Eq.~(\ref{eq:extdamp}).
-\item{\Indexvb{radext},\Indexvb{widext}:} parameters $r_0$ and
- $\Delta r$ for the cutoff of the field according to
- Eq.~(\ref{eq:extdamp}). Dimension: fm.
-\item{\Indexvb{isoext}:} isospin behavior of the external field: {\tt
- isoext=0} denotes the same action on protons and neutrons, {\tt
- isoext=1} that with opposing signs.
-\item{\Indexvb{ipulse}:} type of pulse. {\tt ipulse=0} denotes the
- initial boost configuration, {\tt ipulse=1} a Gaussian pulse
- according to Eq.~(\ref{eq:extgauss}), and {\tt ipulse=2} a cosine
- squared behavior as defined in Eq.~(\ref{eq:extcos}).
-\item{\Indexvb{omega}, \Indexvb{tau0}, and \Indexvb{taut}:} these
- correspond to the parameters $\omega$, $\tau_0$, and $\Delta\tau$ in
- Eqs.~(\ref{eq:extgauss}) and (\ref{eq:extcos}).
-\item{\Indexvb{extfield}:} this is the time-independent field
- generated according to the parameters {\tt amplq0}, {\tt
- textfield\_periodic}, and depending on the latter, possibly {\tt radext}
- and {\tt widext}. It is used either to calculate the initial boost
- or is added to the mean field multiplied with the time-dependent
- factor.
-\end{itemize}
-
-\subsubsection{Subroutine {\tt getin\_external}}
-\Indexpb{getin\_external}
-This is quite straightforward. It reads in all the parameters of the
-external field and immediately does some consistency checks. The
-relative strength for the neutron and proton fields is set equal for
-{\tt isoext=0} but reduced by the corresponding number of particles in
-the {\tt isoext=1}. This avoids a shift of the center of mass in this
-case.
-
-Then the array {\tt extfield} is allocated and the time-independent
-spatial potential calculated for both isospin cases and in either the
-periodic or damped versions, depending on the value of {\tt
- textfield\_periodic}.This is just an illustrative sample field of
-type $Q_{zz}$; in a real calculation there is no need to provide both
-versions.
-
-\subsubsection{Subroutine {\tt extfld}}\Indexpb{extfld}
-This is again a very straightforward routine. It calculates the
-time-dependent prefactor {\tt time\_factor} depending on the
-parameters, and adds the time-independent field {\tt extfield}
-multiplied by this factor to {\tt upot}. The physical time is here
-given as an argument, because it needs to be evaluated at both full
-and half time steps.
-
-\subsubsection{Subroutine {\tt extboost}}\Indexpb{extboost}
-This performs the initial boost according to Eq.~(\ref{eq:extboost})
-on all single-particle wave functions. Note that it is always called
-from {\tt dynamichf} and checks itself whether {\tt ipulse} is zero.
-This enables {\tt ipulse} to also be made a private variable. Its
-argument is used to communicate whether it has actually done anything:
-if it has applied a boost, it sets its argument to {\tt .FALSE.} so that in {\tt
- dynamichf} the variable {\tt text\_timedep} makes it possible to
-distinguish this case..
-
-\subsubsection{Subroutine {\tt print\_extfield}}
-\Indexpb{print\_extfield}
-This subroutine calculates the expectation of the coupling energy to
-the external field,
-$$\sum_q\int\,\D^3r\,\rho_q(\vec r)F_q(\vec r)$$
-and prints one line containing the present time and this value onto
-the file \Indexv{extfieldfile}.
-
-\newpage
-\subsection{Module {\tt Forces}}\Indexmb{Forces}
-\label{sec:Forces}
-Module {\tt Forces} describes the interactions used in the code. The
-idea is to produce a library of Skyrme forces that can be called up
-simply by name, but for exploratory purposes a force can also be input
-using individual parameter values. A force is usually fitted together
-with a prescription for the pairing and center-of-mass correction, so
-that these properties are here defined as part of the force.
-
-\subsubsection{Module variables and types}
-\label{sec:types}
-To deal with forces and pairing as a unit, derived types are used:
-\begin{itemize}
-\item{\tt Pairing}: This contains the following parameters for
- pairing:
- \begin{description}
- \item[{\tt v0prot}]: the strength of pairing for protons in MeV.
- \item[{\tt v0neut}]: the strength of pairing for neutrons in MeV.
- \item[{\tt rho0pr}]: the density parameter for the density-dependent
- delta pairing (see description for module \Indexm{Pairs}).
- \end{description}
-\item{\tt Force}: the description of a Skyrme force. It contains the
- following values:
- \begin{description}
- \item[name]: the name of the force used to identify it.
- \item[{\tt ex}]: some forces are fitted excluding the Coulomb
- exchange term. For {\tt ex=1} it is included (this is the normal
- case), for {\tt ex=0} not.
- \item[\Indexvb{zpe}]: index for the treatment of the center-of-mass
- correction (see end of Section~\ref{sec:efunc}).
- \item[\Indexvb{h2m}] value of $\frac{\hbar^2}{2m}$, separately
- for neutrons and protons.
- \item[\Indexvb{t0}, \Indexvb{t1}, \Indexvb{t2}, \Indexvb{t3},
- \Indexvb{t4}]: Skyrme parameters $t_0$, $t_1$, $t_2$, $t_3$, and
- $t_4$. Defined in the usual way.
- \item[\Indexvb{x0}, \Indexvb{x1}, \Indexvb{x2}, \Indexvb{x3}]
- Skyrme exchange parameters $x_0$, $x_1$, $x_2$, and $x_3$.
- \item[\Indexvb{b4p}] spin-orbit modification parameter $b_4'$
- introduced in the ``SkI'' series of forces (see Sect.~{xx}).
- \item[\Indexvb{power}]: exponent in the nonlinear (originally
- three-body) term.
- \item[\Indexvb{vdi}]: parameter set for the volume-delta
- pairing case.
- \item[\Indexvb{dddi}]: parameter set for the
- density-dependent delta pairing case.
- \end{description}
-\end{itemize}
-The variables defined in the module are
-\begin{itemize}
-\item \Indexvb{ipair}: selects one of several pairing modes. For
- historical reasons the values are 0: no pairing, 5: VDI pairing, and
- 6: DDDI pairing. In the input the symbolic names are used so these
- numerical values are hidden to the user. For details see the input
- description and module \Indexm{Pairs}.
-\item \Indexvb{f}: this contains parameters for
- the Skyrme force actually used in the present calculation, packed
- into the derived-type {\tt Force}.
-\item \Indexvb{p} : the pairing parameters used in the
- present calculation. This is separate from the force itself: the
- force definition usually contains suggestions for the associated
- pairing, but this often overridden, e.g, by turning off
- pairing.
-\item \Indexvb{h2ma}: the average of the two {\tt h2m} values for
- protons and neutrons.
-\item \Indexvb{nucleon\_mass}: the mass of the nucleon (average of
- neutron and Proton) in MeV calculated from {\tt h2ma} and \Indexv{hbc}.
-\item The {\tt b} coefficients: these are the coefficients actually
- used for the mean-field and single-particle Hamiltonian calculations
- in \Indexp{skyrme} and \Indexp{integ\_energy}. Note that only {\tt
- b4p} is also included in the Skyrme-force definition; the others
- are derived from the {\tt t} coefficients.
-\end{itemize}
-
-
-The predefined Skyrme forces are contained in {\tt forces.data}, which
-contains an array {\tt pforce} of {\tt TYPE(Force)} data. This is a
-bit unreliable since the Fortran standard restricts the length of
-statements; it should be replaced by reading from a data file in case
-this limit causes problems. The present version is, however, still preferred as
-a data file would have to be replicated in every application directory
-(or an absolute path would have to be defined in the {\tt OPEN}
-statement).
-
-\subsubsection{Subroutine {\tt read\_force}}\Indexpb{read\_force}
-The purpose of the subroutine is to read the force and pairing
-definitions. The {\tt NAMELIST} \Indexn{force} contains all the
-defining values for a Skyrme force (but now as individual variables,
-not in a derived type) plus a selection of pairing type and strengths
-(see input description). In addition there is a logical variable
-\Indexvb{turnoff\_zpe} which allows turning off the center-of-mass
-correction.
-
-Some quantities are first set negative to see whether required input
-is missing. Then a predefined Skyrme force with the given name is
-sought; if it is found, it is simply copied into {\tt f}. If no force
-of this name is found, a new one is composed from the numbers given in
-the input.
-
-If the input varable \Indexv{turnoff\_zpe} is true, the indicator {\tt
- f\%zpe} is set to~1, which implies not doing anything about the
-center-of-mass correction. Actually, in the current version this
-affects only one statement in the static module.
-
-Now the ``b'' coefficients as given in Eq.~(\ref{eq:force-coeff}) are
-calculated straightforwardly. There is one additional coefficient
-{\tt slate} used for the Slater approximation to the Coulomb exchange
-term. This is not a free parameter, but precomputed for convenience in
-{\tt forces.f90}. The $b$ and $b'$ coefficients are used in
-subroutine \Indexp{skyrme} (module \Indexm{Meanfield}) and \Indexp{energy}
-(module \Indexm{Energies}).
-
-The variable {\tt pairing} from the namelist then determines the
-pairing. If it is set to {\tt 'NONE'}, no pairing is
-included. Otherwise the strength parameters are taken from the input
-or from the predefined force. If this process does not find a
-reasonable pairing combination, stop with an error message.
-
-
-Finally the routine calculates the values of \Indexv{nucleon\_mass} and
-\Indexv{h2ma}. It then prints out a description of the force and pairing
-parameters.
-
-\newpage
-\subsection{Module {\tt Fourier}}\Indexmb{Fourier}
-This module initializes the {\tt FFTW3} package for doing Fourier
-transforms~\cite{Fri05a}. It needs the file {\tt fftw3.f} from the
-{\tt FFTW3} package to define the symbolic parameters for the
-initialization calls.
-
-Note that {\tt FFTW} is used only for the complex transforms of the
-wave functions and the Coulomb solver; the Fourier derivatives of the
-real fields are handled by explicit matrix multiplication (see module
-\Indexm{Trivial}, routines \Indexp{rmulx}, \Indexp{rmuly}, and \Indexp{rmulz}).
-The definition of the plans for the Coulomb solver is contained in
-subroutine \Indexp{coulinit} in module \Indexm{Coulomb}.
-
-\subsubsection{Module variables}
-The module variables describe the various {\tt FFTW} plans used in the
-code.
-\begin{description}
-\item{\Indexvb{pforward}, \Indexvb{pbackward}:} plans for full
- three-dimensional forward and backward transforms for both spin
- components.
-\item{\Indexvb{xforward}, \Indexvb{xbackward}:} one-dimensional
- forward and backward transform in the $x$-direction, but for all
- values of $y$, $z$, and spin $s$.
-\item{\Indexvb{yforward}, \Indexvb{ybackward}:} one-dimensional
- forward and backward transform in the $y$-direction, but for all
- values of $x$, $z$, and spin $s$.
-\item{\Indexvb{zforward}, \Indexvb{zbackward}:} one-dimensional forward
- and backward transform in the $z$-direction, but for all values of
- $x$, $y$, and spin $s$.
-\end{description}
-
-\subsubsection{Subroutine {\tt initfft}}\Indexpb{initfft}
-In this subroutine the {\tt FFTW} system is initialized and a 2-wave
-function array {\tt p} is allocated temporarily for performing the
-tests to make the plans efficient (we do not use dynamic allocation,
-since having this array on the stack may produce different results
-than on the heap --- a point which has not been tested, though).
-
-This is followed by the calls to set up the plans. These have
-different variations, since the way in which the index or indices over
-which the transform is done are intertwined with the unaffected
-indices is quite different for the different directions. Understanding
-this Section requires a thorough familiarity with the FFTW
-documentation and Fortran indexing.
-
-{\bf An important point to consider when modifying the code:} the
-setting-up of a plan must agree with its later use with respect to
-whether the input and output arrays are the same (in-place transform)
-or different. This is why {\tt p} sometimes has a last index of 2 in
-these calls: in the code all transforms are in-place except for {\tt
- xforward}, {\tt yforward}, and {\tt zforward}. It was found that
-very strange things happen if this rule is not obeyed.
-
-\newpage
-\subsection{Module {\tt Fragments}}\Indexmb{Fragments}
-\label{sec:fragini}
-This module is concerned with setting up the initial condition from
-data files containing fragment wave functions, usually obtained in a
-previous static calculation. A typical application is the
-initialization of a heavy-ion reaction, see Section~\ref{sec:react}.
-Beyond that, any number of fragments (limited by the parameter
-variable \Indexv{mnof}) can be put into the grid at prescribed positions
-with given initial velocities. A condition is, however, that the grid
-they are defined in is smaller than the new grid. If they are put
-close to the boundary, they may have density appearing on the other
-side because of periodicity; this may be acceptable if the boundary
-condition is periodic. A calculation may be restarted on a larger
-grid, but this may be accompanied by some loss of accuracy, since the
-wave function outside the original regions is set to a constant small value.
-
-For the case of parallel {\tt MPI} calculations, there is logic to read wave functions
-distributed over a series of files as described in connection with
-subroutine \Indexp{write\_wavefunctions}. Since only the dynamic case
-can run under {\tt MPI} at present, the only application of this is to
-restart a dynamic calculation. The number of processors may be
-different in the restart.
-
-\subsubsection{Module variables}
-\begin{description}
-\item[\Indexvb{fix\_boost} (input)]: if this is set to true, the boost
- (initial kinetic energy) values are used unchanged from the input;
- otherwise they are calculated from the initial kinetic energy of
- relative motion {\tt ecm} and the impact parameter {\tt b}. This
- implies a two-body initial configuration. The initial motion is
- assumed to be in the $(x,z)$-plane in this case. Note that for two
- initial fragments fix\_boost can also be set to true for special
- initial conditions.
-\item[\Indexvb{filename} (input)]: for each fragment this indicates the
- name of the file with the associated wave functions. One file can be
- used several times in the case of identical fragments, abut the code
- does not treat that as a special case.
-\item[\Indexvb{fcent} (input)] for fragment no. {\tt i} the initial
- three-dimensional position is determined by {\tt fcent(:,i)}.
-\item[\Indexvb{fboost} (input)] gives the initial motion {\tt
- fboost(:,i)} of fragment {\tt i} in 3 dimensions. This is not a
- velocity, but the total kinetic energy of the fragment {\em in that
- Cartesian direction} in MeV. The sign indicates the direction,
- positive or negative along the corresponding axis. The sum of
- absolute values thus is the total kinetic energy of the fragment. These
- values are used only if {\tt fix\_boost} is true.
-\end{description}
-The following variables are used only if fix\_boost is false, i.~e.,
-for two-body initialization from relative motion.
-\begin{description}
-\item[\Indexvb{ecm}, \Indexvb{b} (input)] The kinetic energy of
- relative motion in MeV and the impact parameter in fm.
-\item[{\tt vx}, {\tt vz}] The components of the relative velocity in
- the $(x,z)$-plane
-\item[{\tt xli}] The angular momentum of relative motion in units of
- $\hbar$.
-\end{description}
-The following quantities have names of other variables prefixed by an
-``f'' and {\em are indexed by fragment number}; for the most part they duplicate
-variables describing the whole system on a fragment-by-fragment basis.
-\begin{description}
-\item[{\tt fcmtot}, {\tt fmass}, {\tt fcharge}] Centers of mass, masses,
- and charges of the fragments.
-\item[{\tt fnneut}, {\tt fnprot}, {\tt fnstmax}, {\tt fnpmin}, {\tt
- fnpsi}] These have the same meaning as the variable without ``f'',
- but apply to the set of wave functions for one specific fragment.
-\item[{\tt fnumber}] gives the number of wave functions in the
- fragment for each isospin. For initialization of a dynamic
- calculation only wave functions with a nonzero occupation are taken
- into account.
-\item[{\tt fnewnpmin}, {\tt fnewnpsi}] are the starting and ending
- indices for each fragment's wave functions in the total set of
- wave functions combining all the fragments.
-\item[{\tt fnx}, {\tt fny}, {\tt fnz}] indicate the grid size on which
- the fragment wave functions are defined.
-\item[{\tt fnode} and {\tt flocalindex}] are used for MPI. When the
- wave functions were written by the parallel code, each node produced
- a file named ``{\em nnn}{\tt .wffile}'', which contains the wave
- functions present on that node. For the wave function with index {\tt
- nst}, {\tt fnode(nst)} indicates the number of the file and {\tt
- flocalindex(nst)} the position in the file. These are analogous to
- the variables \Indexv{node} and \Indexv{localindex} for the complete
- calculation and are explained more thoroughly in the description of
- module \Indexm{Parallel}, see section~\ref{sec:parallel}.
-\end{description}
-
-\subsubsection{Subroutine {\tt getin\_fragments}}
-\Indexpb{getin\_fragments}
-This subroutine reads the input variables, makes some consistency
-checks, and then prepares for the initialization.
-
-A restart is set up by placing one fragment taken from {\tt wffile} at
-the origin with zero velocity (see Sect.~\ref{sec:restart}).
-
-The main loop is over fragments. The fragment files are opened a first
-time to obtain information about the fragments, which is stored in
-fragment-specific arrays. The checks include agreement of the forces
-and grid spacings as well as that the fragment grid is not larger than
-the new grid.
-
-The code at this point makes a distinction between static and dynamic
-modes: for a static calculation it is assumed that the data may be
-needed for a restart. In this case all wave functions are read in even
-if not occupied even fractionally. For the dynamic case only those
-with non-zero occupation are input as determined from \Indexv{fwocc}.
-This yields a reduced value for \Indexv{fnumber}.
-
-{\bf Note that at present it is assumed that the static wave functions
- are ordered in ascending energy, so that the occupied ones will
- start at index one and all empty states will be at the uppermost
- index positions.}
-
-Following this, the index positions in the new wave function array are
-calculated in \Indexv{fnewnpmin} and \Indexv{fnewnpsi} by adding the
-numbers of wave functions of each fragment-specific successively. Note
-that at this stage the proton indices are still counted from one.
-
-After this input loop, the indices \Indexv{npmin} and \Indexv{npsi} as well
-as the total number \Indexv{nstmax} for the combined system are
-calculated and finally the proton indices where the fragment wave
-functions should be inserted are shifted to behind the neutrons,
-i.~e., starting at \Indexv{npmin(2)}.
-
-At the end the two-body initialization \Indexp{twobody\_init} is called
-for the dynamic case with two fragments and \Indexv{fix\_boost}{\tt =.FALSE.}.
-This calculates the \Indexv{fboost} values from \Indexv{ecm} and {\tt b}.
-
-\subsubsection{Subroutine {\tt read\_fragments}}
-\Indexpb{read\_fragments}
-This subroutine prints summary information about which fragment wave
-functions occupy which range of indices. Then it does a loop over
-fragments to read in their wave functions using
-{\tt read\_one\_fragment}, followed by applying the boost to them using
-{\tt boost\_fragment}.
-
-\subsubsection{Subroutine {\tt read\_one\_fragment}}
-\Indexpb{read\_one\_fragment} This subroutine reopens a fragment file
-for the fragment indexed by {\tt iff} and reads the wave functions,
-inserting them at the correct index into the new wave function array
-while also moving them to the desired center-of-mass position.
-
-The principal loop is over isospin. The index limits in the fragment
-file for that isospin are put into {\tt il} and {\tt iu}, and the new
-indices into {\tt newil} and {\tt newiu}. In the next step the grid
-coordinates and the single-particle quantities are read. The latter
-are copied into the new arrays and the isospin is also recorded. Then
-this information is printed.
-
-Once this loop is concluded, the spatial shift is prepared. The shift
-is calculated from the difference between the desired position {\tt
- fcent} with respect to the origin of the new coordinates given by
-\Indexv{x} etc., and the fragment center-of-mass \Indexv{fcmtot} with
-respect to the origin in fragment coordinates {\tt fx} etc.
-Subroutine {\tt phases} is used to calculate essentially the shift
-phase factor $\exp(-\I\vec k\cdot\Delta\vec r)$, which is a product of
-phases in each coordinate direction {\tt akx}, {\tt aky}, and {\tt
- akz}. These have an index corresponding to $\vec k$ in the Fourier
-transform.
-
-Now the index arrays \Indexv{fnode} and \Indexv{flocalindex} are read,
-which indicate where the wave functions are to be found in case of an
-MPI job. The logical variable {\tt multifile} records whether this is
-the case by testing whether any of the wave functions was stored on
-other than node~0.
-
-The following loop runs over the new index positions.
-If the wave function is not on the current node, nothing is done
-except for ignoring the input record.
-
-For the case of multiple files for one fragment (which is recognized
-by not all the indices {\tt fnode} being zero, a short subroutine {\tt
- locate} is used to position input at the correct location.
-
-Otherwise the wave function is read from the fragment file using the
-fragment grid dimensions into a variable {\tt ps1} defined with the
-full new dimension, then it is Fourier transformed, multiplied by the
-phase factor, and transformed back. Finally it is inserted into the
-total wave function array {\tt psi}, where any zeroes are replaced by
-a small number, presently set to $10^{-20}$.
-
-\subsubsection{Subroutine {\tt locate}}\Indexpb{locate}
-This has the task to position the file for reading wave functions at
-the correct place. It has as arguments the number of the file
-(corresponding to the node number in the previous calculation) and the
-index of the wave function in this file. The variable {\tt
- presentfile} keeps track of which file is currently open.
-
-If we are starting to read a new file (which is always true initially,
-as no wave functions are written into the header file {\tt wffile}
-itself), it closes the present file, composes the file name for the
-new one and opens it. Then the proper number of records are ignored
-to position at the correct one. If the file has not changed, it simply
-returns and reading continues sequentially. The logic of course
-assumes that files are stored sequentially in each partial file, but
-the division into partial files need not be the same as in the
-previous run.
-
-\subsubsection{Subroutine {\tt phases}}\Indexpb{phases}
-This subroutine calculates the phase factors for a one-dimensional
-translation $\Delta x$ as
-$$ \exp\left(-\frac{2\pi\I k_j \Delta x}{L}\right).$$
-The shift was calculated in \Indexp{read\_one\_fragment}, the
-argument {\tt c} includes a denominator $L={\tt nx*dx}$, the total
-length of the grid. The momentum $k_j$ is determined in the usual way
-for the finite Fourier transform.
-
-\subsubsection{Subroutine {\tt twobody\_init}}
-\Indexpb{twobody\_init}
-The purpose of this subroutine is to calculate the boost values from
-\Indexv{ecm} and \Indexv{b} in the two-body case. The calculation can be
-followed with an elementary understanding of the two-body system
-kinematics, so we just give a brief overview.
-
-The reduced mass \Indexv{xmu}, the relative velocity \Indexv{vrel} and the
-angular momentum \Indexv{xli} are calculated first. Then the components
-of the vector linking the two centers of mass {\tt dix} and {\tt diz}
-are divided by the length of this vector {\tt roft} to get the
-direction cosines.
-
-The Coulomb energy is calculated assuming two point charges at
-distance {\tt roft} and is subtracted from {\tt ecm} to yield the
-kinetic energy remaining at this distance, from which the relative
-velocity {\tt vrel\_d} is
-calculated. Since the total center
-of mass is assumed to be at rest, the velocities of the fragments {\tt v1}
-and {\tt v2} can then be simply obtained. The instantaneous impact parameter
-{\tt b\_d } results from the angular momentum and the relative
-velocity, and the angle by which the two fragments need to miss each other in
-order to realize this impact parameter is computed in {\tt sint} and {\tt
- cost}.
-
-This now makes it possible to calculate the boost velocity components,
-which are converted into kinetic energies to conform to the definition
-of \Indexv{fboost} as given in the input. Note that these energies are
-signed to indicate the direction of motion.
-
-Both velocities and energies are printed.
-
-\subsubsection{Subroutine {\tt boost\_fragment}}
-\Indexpb{boost\_fragment}
-This subroutine multiplies the configuration-space wave functions of
-fragment no.~{\tt iff} by plane-wave factors to give them
-translational motion. This is done by calculating the wave number
-vector {\tt akf} from the kinetic energy. There is one subtle point:
-since the boost is applied to {\em single-particle} wave functions,
-the momentum $k$ should be the correct one for {\em one nucleon}. Thus
-the total kinetic energy of the fragment is
-$$T=A\frac{\hbar^2}{2m}k^2,$$
-where $A$ is the mass number of the fragment and $m$ the nucleon mass.
-Solving for $k$ and taking the sign into account yields the expression
-in the code.
-
-The rest is then straightforward.
-
-\newpage
-\subsection{Module {\tt Grids}}\Indexmb{Grids}
-This module deals with the definition of the spatial grid and associated
-operations.
-\subsubsection{Module variables}
-\begin{description}
-\item{\Indexvb{nx}, \Indexvb{ny}, \Indexvb{nz}:} Number of points
- in each Cartesian direction. All must be even numbers to preserve
- reflection symmetry.
-
-\item{\Indexvb{dx}, \Indexvb{dy}, \Indexvb{dz}:} The spacing
- between grid points (in fm) in the three Cartesian directions.
- Usually these should be identical.
-
-\item{\Indexvb{periodic}:} logical variable indicating whether the
- situation is triply periodic in three-dimensional space.
-
-\item{\Indexvb{wxyz}}: the volume element {\tt wxyz=dx*dy*dz}.
-
-\item{\Indexvb{x}, \Indexvb{y}, \Indexvb{z}:} arrays containing the
- actual coordinate values in fm. They are dimensioned as {\tt x(nx)},
- {\tt y(ny)}, {\tt z(nz)} and are allocated dynamically, thus
- allowing dynamic dimensioning through input values of {\tt nx}, {\tt ny},
- and {\tt nz}.
-
-\item{\Indexvb{der1x}, \Indexvb{der1y}, \Indexvb{der1z}}: matrices
- describing the first spatial derivatives in the $x$-, $y$-, and
- $z$-direction, respectively. They are dynamically allocated with
- dimensions {\tt (nx,nx)} etc.\ and are calculated in subroutine {\tt
- sder}.
-
-\item{\Indexvb{der2x}, \Indexvb{der2y}, \Indexvb{der2z}}: matrices
- describing the second spatial derivatives in the $x$-, $y$-, and
- $z$-direction, respectively. They are dynamically allocated with
- dimensions {\tt (nx,nx)} etc.\ and are calculated in subroutine {\tt
- sder2}.
-
-\item{\Indexvb{cdmpx}, \Indexvb{cdmpy}, \Indexvb{cdmpz}}: matrices
- describing the damping operation in the $x$-, $y$-, and
- $z$-direction, respectively. They are dynamically allocated with
- dimensions {\tt (nx,nx)} etc.\ and are calculated using subroutine
- {\tt setup\_damping}, which in turn calls {\tt setdmc}.
-
-\end{description}
-
-\subsubsection{Subroutine {\tt init\_grid}}\Indexpb{init\_grid}
-This subroutine is called during the initialization for both static
-and dynamic calculations. It reads the grid dimension and spacing
-information using namelist {\tt Grid}, allocates the necessary arrays,
-and calculates the coordinate values and the derivative and damping
-matrices.
-
-This is done by calling the subroutine {\tt init\_coord} once for each
-direction. It does everything needed except the calculation of
-the volume element.
-
-\subsubsection{Subroutine {\tt init\_coord}}
-\Indexpb{init\_coord}
-In this subroutine the defining information for a grid direction
-(generically called {\tt v}, which can be replaced by {\tt x}, {\tt
- y}, or {\tt z}) in the form of the number of points {\tt nv} and the
-spacing {\tt dv} is used to generate the associated data. The arrays
-of coordinate values, derivative and damping matrices are allocated.
-Since all the quantities that are later used in the code are passed as
-arguments, this subroutine can handle all three directions in a
-unified way. To print the information intelligibly, it is also passed
-the name of the coordinate as {\tt name}.
-
-It is assumed that the coordinate zero is in the center of the grid,
-i.~e., since the dimension is even the number of points to each side
-of zero is equal andthe origin is in the center of a cell. The special
-position of the origin is used in static calculations, e.~g., for the
-parity determination. In other situations, the position of the center
-of mass is more important, this is defined in module {\tt Moment}.
-
-If a difference location of the origin in the grid is desired, it can
-be done by changing the statement generating the values of {\tt v}.
-
-Finally the derivative matrices and damping matrix are computed using
-{\tt sder}, {\tt sder2}, and {\tt setdmc}.
-
-\subsubsection{Subroutine {\tt sder}}\Indexpb{sder}
-This subroutine calculates the matrix for the first derivative using
-equation~\ref{eq:derv1}. It receives the dimension {\tt nmax} and the
-grid spacing {\tt d} as arguments and returns the matrix in {\tt der}.
-
-\subsubsection{Subroutine {\tt sder2}}\Indexpb{sder2}
-This subroutine calculates the matrix for the second derivative using
-equation~\ref{eq:derv2}. It receives the dimension {\tt nmax} and the
-grid spacing {\tt d} as arguments and returns the matrix in {\tt der}.
-
-
-\subsubsection{Subroutine {\tt setup\_damping}}
-\Indexpb{setup\_damping}
-This sets up the damping matrices by calls to {\tt setdmc} for each
-coordinate direction. The reason for not including this in {\tt
- init\_grid} is that it used only in the static calculation and
-requires the damping parameter {\tt e0dmp}, which is in the static
-module. It has to be passed as a parameter because
-circular dependence of modules would result otherwise.
-
-\subsubsection{Subroutine {\tt setdmc}}\Indexpb{setdmc}
-This subroutine calculates the matrices corresponding to the
-one-dimensional operators
-$$\frac{1}{1+\hat t/E_0} {\rm~with~} \hat
-t=-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}.$$ Here $E_0$ is
-the damping parameter called {\tt e0dmp} in the code. Since the
-kinetic energy operator $\hat t$ contains the parameter $\hbar^2/2m$,
-which is force-dependent, this subroutine depends on module {\tt
- Forces}.
-
-The calculation proceeds simply by constructing the unit matrix,
-adding the operator to it to form the denominator, and then
-calculating the inverse matrix using subroutine {\tt gauss}.
-
-\subsubsection{Subroutine {\tt gauss}}\Indexpb{gauss}
-This is a Fortran 95 implementation of the standard Gauss algorithm
-with pivoting. It is simplified for the special case of computing
-$B=B^{-1}A$ with both matrices dimensioned {\tt (n,n)}.
-
-\newpage
-\subsection{Module {\tt Inout}}\Indexmb{Inout}
-This module contains the procedures for binary input and output
-of the larger fields. There are two variants, both of which are written
-at regular intervals: the wave function file \Indexv{wffile} and the
-files containing the densities and currents. Since the former is
-extremely space-consuming, each output normally overwrites the
-previous one. These files are intended to be used for a restart, as
-initialization input (static solution for one fragment) for another
-run, or for a final analysis of the wave functions.
-
-The densities, on the other hand, are written on a series of file {\tt
- nnnnnn.tdd}, where {\tt nnnnnn} is the number of the time step or
-iteration. This is useful for later graphical or other types of
-analysis.
-
-{\bf Note: } where the variable name {\tt wffile} is used inside file
-names in the following, it should not be taken literally but is
-replaced by the character string it contains.
-
-In addition the routine for printer plots, {\tt plot\_density}, is
-included in this module, as well as the subroutines {\tt
- sp\_properties} and {\tt start\_protocol}, which do not completely
-match the purpose of this module but are placed here for convenience.
-
-\subsubsection{Subroutine {\tt write\_wavefunctions}}
-\Indexpb{write\_wavefunctions} This subroutine has two modes of
-operation depending on whether the code runs on distributed-memory
-systems in {\tt MPI} mode or on a shared-memory or single-processor
-machine. In both cases it first determines the number of filled
-single-particle states {\tt number(iq)}, which need not be the same as
-either the number of particles or the number of states, since pairing
-may lead to partial occupation and in addition there can be empty
-states.
-
-\begin{description}
-\item{{\bf Sequential operation:}}
- this case is recognized recognized by \Indexv{mpi\_nprocs}{\tt ==1}. Open
-\Indexv{wffile}, then write four records containing general information.
- \begin{itemize}
- \item[Record 1:] \Indexv{iter}, \Indexv{nstmax}, \Indexv{nneut},
- \Indexv{nprot}, \Indexv{number}, \Indexv{npsi},
- \Indexv{charge\_number},
- \Indexv{mass\_number}, \Indexv{cm}.
- \item[Record 2:] \Indexv{nx}, \Indexv{ny}, \Indexv{nz}, \Indexv{dx}, \Indexv{dy},
- \Indexv{dz}, \Indexv{wxyz}.
- \item[Record 3:] \Indexv{x}, \Indexv{y}, \Indexv{z}.
- \item[Record 4:] \Indexv{wocc}, \Indexv{sp\_energy}, \Indexv{sp\_parity},
- \Indexv{sp\_norm}, \Indexv{sp\_kinetic}, \Indexv{sp\_efluct1}.
- \end{itemize}
- These are followed by one record containing information for the {\tt
- MPI} case, which is included here only for compatibility:
- \Indexv{node}, \Indexv{localindex}. This is then followed by a series of
- \Indexv{nstloc} records (in the sequential case, {\tt nstloc} equals
- {\tt nstmax}), containing the array of {\tt nx*ny*nz*2} wave
- function values for each single-particle state (including spin).
-
-\item{{\bf MPI operation:}} in this case processor \#0 writes the same
- general data as in the sequential case onto file {\tt wffile}, which
- is then closed. The purpose of record 5 in this case is to record
- for each wave function (in global index space) which node it is in
- and what the index on that node is. Since each node produces a
- separate output file with only its wave functions, this allows reading
- any wave function correctly from the set of files.
-
- Each processor thus only writes the wave function data for its
- locally stored set of {\tt nstloc} wave functions onto files with
- the names composed (in variable {\tt rsfp}) of the number of the
- processor and {\tt wffile} in the form {\tt nnn.wffile}. For
- example, if {\tt wffile} has the value 'Ca40', these files will be
- {\tt 000.Ca40}, {\tt 001.Ca40}, {\tt 002.Ca40}, etc.\ up to the
- number of processors.
-\end{description}
-
-\subsubsection{Subroutine {\tt write\_densities}}
-\Indexpb{write\_densities}
-This subroutine produces a file {\tt iter.tdd} with density and
-current data for the present time step or iteration with number {\tt
- iter}. In the file name {\tt iter} is given in 6 decimal digits. The
-record structure is as follows:
-
-\begin{itemize}
-\item{Record 1:} this contains the variables \Indexv{iter},
- \Indexv{time}, \Indexv{nx}, \Indexv{ny}, and \Indexv{nz} to define
- the dimensions of the fields.
-\item{Record 2:} contains the variables \Indexv{dx}, \Indexv{dy},
- \Indexv{dz}, \Indexv{wxyz}, \Indexv{x}, \Indexv{y}, and \Indexv{z}
- to allow proper labelling of axes in plots, etc.
-\item{Further records:} for each field to be written, a record is
- produced with the following information:
- \begin{enumerate}
- \item Name of the field with up to 10 characters
- \item Logical value {\tt scalar} to indicate whether it is a scalar
- ({\tt .FALSE.}) or a vector field ({\tt .TRUE.}).
- \item Logical value {\tt write\_isospin} to indicate whether the
- field is summed over protons and neutrons (({\tt .FALSE.} or not
- ({\tt .TRUE.}). In the latter case the field has a last index
- running from~1 to~2 for neutrons and protons, respectively. {\bf
- This selection applies to all fields equally (except the Coulomb
- potential)}.
- \end{enumerate}
- After this identification record, the corresponding field itself is
- written.
- The dimension varies in the following way:
- \texttt{
- \begin{center}\begin{tabular}{c|c|c}
- scalar&write\_isospin&dimension\\
- \hline
- .FALSE.&.FALSE&(nx,ny,nz)\\
- .FALSE.&.TRUE.&(nx,ny,nz,2)\\
- .TRUE.&.FALSE.&(nx,ny,nz,3)\\
- .TRUE.&.TRUE.&(nx,ny,nz,3,2)\\
- \hline
- \end{tabular}\end{center}
- }
-\end{itemize}
-
-The {\bf selection of fields to be output} is handled through variable
-\Indexv{writeselect} consisting of \Indexv{nselect} characters. Each field
-is selected by a one-character code, where both lower and upper case
-are acceptable. At present the choices are:
-\begin{itemize}
-\item{R}: density {\tt rho} (scalar). Name {\tt Rho}.
-\item{T}: kinetic energy density {\tt tau} (scalar). Name {\tt Tau}
-\item{U}: local mean field {\tt upot}. Name {\tt Upot}.
-\item{W}: Coulomb potential {\tt wcoul} (scalar). This has to be
- handled specially, since it has no isospin index. Name {\tt Wcoul}.
-\item{C}: current density {\tt current} (vector). Name {\tt Current}.
-\item{S}: spin density {\tt sdens} (vector). Name {\tt Spindens}.
-\item{O}: spin-orbit density {\tt sodens}. Name {\tt s-o-Dens}.
-\end{itemize}
-
-This system is set up to be easily modified for writing additional fields.
-
-\subsubsection{Subroutine {\tt write\_one\_density}}
-\Indexpb{write\_one\_density}
-This subroutines does the actual output for {\tt write\_densities} in
-the case of a scalar field. Its functioning should be clear from the
-description above.
-
-\subsubsection{Subroutine \Indexpb{write\_vec\_density} (Private)}
-This also does the actual output for subroutines {\tt
- write\_densities} for the case of a vector field.
-Its functioning should be clear from the
-description above.
-
-\subsubsection{Subroutine {\tt plot\_density}}\Indexpb{plot\_density}
-Produces a simple printer plot of the density distribution in the
-reaction plane. This is not supposed to replace better plotting codes,
-but simply allows a quick glance at what is happening in the code,
-even while it is running.
-
-It is based on a very old routine found at ORNL and was translated
-into modern Fortran. It uses helper function \Indexpb{bplina} for
-interpolation.
-
-\subsubsection{Subroutine {\tt sp\_properties}}
-\Indexpb{sp\_properties}
-In this routine the kinetic energy, orbital and spin angular momenta
-expectation values, \Indexv{sp\_kinetic}, \Indexv{sp\_orbital} and
-\Indexv{sp\_spin} of the single-particle states are calculated. The
-latter are both three-dimensional vectors.
-
-Note that the single-particle energy \Indexv{sp\_energy} itself is not
-calculated here but in the main static and dynamic routines, since it
-is obtained by applying the single-particle Hamiltonian, which is
-done more conveniently there.
-
-The procedure is quite simple: in a loop over wave functions the
-active one is copied into {\tt pst} for convenience. Then its three
-directional derivatives {\tt psx}, {\tt psy}, and {\tt psz} and
-Laplacian {\tt psw} are calculated. In the big loop over the grid they
-are combined to the desired matrix elements; the only technical point
-to remark is that since the result must be real, efficiency can be
-achieved by formulating the complex products in an explicit way. Then
-{\tt kin} contains the kinetic energy (without the $\hbar^2/2m$), {\tt
- cc} the orbital and {\tt ss} then spin matrix elements.
-
-Finally only the volume element, the factor of one half for the
-spin ad the prefactor of the kinetic energy are added.
-
-\subsubsection{Subroutine {\tt start\_protocol}}
-\Indexpb{start\_protocol} This is given a file name and a character
-string for a header line to start the file contents. It is used for
-the {\tt *.res} files. If the file already exists, nothing is done,
-since this probably a restart job and output should just be added at
-the end of the file.
-
-\newpage
-\subsection{Module {\tt Levels}}\Indexmb{Levels}
-This module is concerned with the wave function data: definition of
-the pertinent arrays, allocating their storage and simple operations on them.
-
-\subsubsection{Module variables}
-First there are some general variables describing the arrays:
-\begin{description}
-\item{\Indexvb{nstmax}:} is the total number of wave functions
- present in the calculation. For the MPI version only {\tt nstloc}
- wave functions are present on each node. Note that for all other
- wave-function related arrays, such as single-particle energies, the
- full set is stored on each node.
-\item{\Indexvb{nstloc}:} the number of wave functions stored on the
- present node. In principle this should be defined in module
-\Indexm{Parallel}, but this would lead to a circular module dependence.
-\item{\Indexvb{nneut}, \Indexvb{nprot};} the physical numbers of
- neutrons and protons. These may be smaller than the number of
- single-particle states.
-\item{\Indexvb{npmin}, \Indexvb{npsi}:} the neutron states are
- numbered {\tt npmin(1)} through {\tt npsi(1)} and the proton states
- run from {\tt npmin(2)} through {\tt npsi(2)}. Protons follow
- neutrons, so {\tt npmin(1)=1} and {\tt npmin(2)=npsi(1)+1}. {\em
- Note that for each particle type the number of states can be
- larger than the particle number, as states may be fractionally
- occupied or even empty.} If initialization is not from fragments,
- {\tt npsi(2)} {\em as an input value} refers to the total number of
- proton states, it is later updated (in {\tt init.f90}) to its normal
- meaning as the final index for proton states, which coincides with
- the total number of states, {\tt npsi(2)=nstmax}.
-\item{\Indexvb{charge\_number}, \Indexvb{mass\_number}:} the physical
- charge and mass numbers.
-\end{description}
-We have to distinguish three different numbers: the number of physical
-particles {\tt nneut} or {\tt nprot}, the number of single particle
-states read as {\tt npsi(1:2)}, and the number of states actually
-having nonzero occupation, which can differ from the particle number
-if pairing is used. Since the occupation given by \Indexv{wocc} changes
-with iteration, this latter number may be iteration dependent. It becomes
-important for the output of the wave functions, as unfilled states
-need not be read from fragment data files in the dynamic case.
-Therefore the numbers of states with nonzero occupation are computed
-only in subroutine \Indexp{write\_wavefunctions}, where they are
-called \Indexv{number}{\tt (1:2)}..
-
-Next are the arrays for the wave functions themselves and the
-single-particle Hamiltonian matrix. Each wave
-function is complex dimensioned {\tt (nx,ny,nz,2)} with the last index
-denoting spin (1=up, 2=down for the $z$-direction).
-\begin{description}
-\item{\Indexvb{psi}:} this is the main array for the wave functions.
- It has an additional last index counting the states. In the
- sequential case it runs from $1\ldots{\tt nstmax}$, in the MPI
- version each node has only {\tt nstloc} wave functions.
-\item{\Indexvb{hmatr}:} this is dimensioned {\tt (nstmax,nstmax)} and
- is used for the single-particle Hamiltonian in the diagonalization
- step.
-\end{description}
-Finally arrays dimensioned {\tt nstmax} to describe properties of each
-wave function.
-\begin{description}
-\item{\Indexvb{sp\_energy}:} single-particle energy in MeV.
-\item{\Indexvb{sp\_efluct1}:} single-particle energy fluctuation
- [MeV] calculated as
- $$\sqrt{\langle\psi|\hat h^2|\psi\rangle
- -\langle\psi|\hat h|\psi\rangle^2}.$$ Used only as informational
- printout in the static part.
- \item{\Indexvb{sp\_efluct2}:} single-particle energy fluctuation
- [MeV] calculated as
- $$\sqrt{\langle\hat h\psi|\hat h\psi\rangle
- -\langle\psi|\hat h|\psi\rangle^2}.$$ Used only as informational
- printout in the static part.
- \item{\Indexvb{sp\_kinetic}:} single-particle kinetic energy in
- units of MeV.
- \item{\Indexvb{sp\_norm}:} norm of single-particle wave function;
- should be unity normally.
- \item{\Indexvb{sp\_parity}:} single-particle parity w.r.t.
- three-dimensional reflection at the origin; calculated as
- $$\sum_s\int\D^3r\, \psi^*(\vec r,s)\psi_s(-\vec r,s).$$
- \item{\Indexvb{wocc}:} occupation probability of single-particle
- state, may be fractional because of pairing. In the equations this
- is usually denoted as $w_k^2$, the square added because of the pairing
- notation.
- \item{\Indexvb{sp\_orbital}:} dimensioned {\tt (3,nstmax)}:
- expectation values of three components of single-particle orbital
- angular momentum, in units of $\hbar$.
- \item{\Indexvb{sp\_spin}:} dimensioned {\tt (3,nstmax)}:
- expectation values of three components of single-particle spin, in
- units of $\hbar$.
- \item{\Indexvb{isospin}:} keeps track of isospin of particle,
- 1=neutron, 2=proton.
- \end{description}
-
-\subsubsection{Subroutine {\tt alloc\_levels}}\Indexpb{alloc\_levels}
-This subroutine allocates all the arrays associated with
-single-particle wave functions. Note that while most have dimension
-{\tt nstmax}, {\tt psi} itself is dimensioned for the number {\tt
- nstloc} of wave functions on one specific processor. It also records the
-isospin value.
-
-\subsubsection{Subroutines {\tt cdervx}, {\tt cdervy}, {\tt cdervz}}
-\Indexpb{cdervx}\Indexpb{cdervy}\Indexpb{cdervz}
-These three routines calculate
-derivatives of wave functions using the FFT method explained in
-section~\ref{sec:deriv}, in the $x$-, $y$-, and
-$z$-direction, respectively. The first argument is the wave function
-$\psi$ to be differentiated, the second returns the first derivative
-$\frac{\partial\psi}{\partial x}$, and the third one the second
-derivative $\frac{\partial^2\psi}{\partial x^2}$, where $x$ can be
-replaced by $y$ or $z$, of course. The last argument can be omitted
-and no second derivative is calculated in this case. The derivatives
-add the proper dimension of fm$^{-1}$ and fm$^{-2}$,
-respectively. Note the dependence of the $k$-value on index as
-explained in Eq.~\ref{eq:kvalues}.
-
-\subsubsection{Subroutine {\tt laplace}}\Indexpb{laplace}
-depending on the presence of the third argument \Indexv{e0inv}, it can
-calculate two things using FFT methods:
-\begin{itemize}
-\item if {\tt e0inv} is not present, it calculates the Laplacian
-$$\frac{\partial^2\psi}{\partial x^2}+\frac{\partial^2\psi}
-{\partial y^2}+\frac{\partial^2\psi} {\partial z^2}.$$
-\item if {\tt e0inv} is present and positive, it calculates
-$$\frac1{E_{0\rm inv}+\hat t}\psi,$$
-with the kinetic-energy operator
-$$\hat t=-\frac{\hbar^2}{2m}\left(\frac{\partial^2}
- {\partial x^2}+\frac{\partial^2} {\partial
- y^2}+\frac{\partial^2}{\partial z^2}\right),
-$$
-which is of course expressed through $\vec k$ in momentum space.
-
-The methods used in this routine are similar to those for the
-derivatives {\tt cdervx} etc.
-\end{itemize}
-
-\subsubsection{Subroutine {\tt schmid}}\Indexpb{schmid}
-This performs a Gram-Schmidt orthogonalization of the single particle
-levels in the straightforward way: loop over states and subtract the
-components of all lower-indexed states from a wave function. It can be
-parallelized under {\tt OpenMP}, albeit not very efficiently, but
-becomes too slow under {\tt MPI}, so that {\tt MPI} is not enabled for
-the static calculations.
-
-\newpage
-\subsection{Module {\tt Meanfield}}\Indexmb{Meanfield}
-\subsubsection{Purpose}
-This module calculates all the ingredients needed for the
-energy functional and for applying the single-particle Hamiltonian to
-a wave function.
-
-The work is done by two subroutines: {\tt skyrme} for the calculation
-of all the fields, which can be scalar or vector and
-isospin-dependent. {\tt hpsi} then is the routine applying the
-single-particle Hamiltonian to one single-particle wave function.
-
-Note the division of labor between {\tt skyrme} and
-\Indexp{add\_density} of module \Indexm{Densities}: everything that
-constructs fields --- densities and current densities --- from the
-single-particle wave functions is done in {\tt add\_density}, which is
-called in a loop over the states, while {\tt skyrme} does the further
-manipulations to complete the fields entering the single-particle
-Hamiltonian by combining the densities and their derivatives. It does
-not need access to the wave functions.
-
-\subsubsection{Module variables}
-All of the variables in this module are fields, either of scalar or
-vector character. Their dimension is {\tt (nx,ny,nz,2)} for the scalar
-and {\tt (nx,ny,nz,3,2)} for the vector fields with the last index
-indicating isospin in all cases. They are derived from the densities
-calculated in module {\tt Densities}.
-
-\begin{description}
-\item{\Indexvb{upot}:} this is the local part of the mean field $U_q$
- as defined in Eq.~(\ref{eq:upot}). It is a scalar field with isospin
- index.
-\item{\Indexvb{bmass}:} this is the effective mass $B_q$ as defined
- in Eq.~(\ref{eq:bmass}). It is a scalar, isospin-dependent field.
-\item{\Indexvb{aq}:} This is the vector filed $\vec A_q$ as defined
- in Eq.~(\ref{eq:aq}). It is a vector, isospin-dependent field.
-\item{\Indexvb{divaq}:} this is simply the divergence of {\tt aq},
- i.~e., $\nabla\cdot\vec A_q$. Its is a scalar, isospin-dependent
- field.
-\item{\Indexvb{spot}:} the field $\vec S_q$ as defined in
- Eq.~(\ref{eq:spot}). It is a vector, isospin-dependent field.
-\item{\Indexvb{wlspot}:} the field $\vec W_q$ as defined in
- Eq.~(\ref{eq:wlspot}). It is a vector, isospin-dependent field.
-\end{description}
-
-\subsubsection{Subroutine {\tt alloc\_fields}}\Indexpb{alloc\_fields}
-This subroutine has the simple task of allocating all the fields that
-are local to the module {\tt Meanfield}.
-
-\subsubsection{Subroutine {\tt skyrme}}\Indexpb{skyrme}
-In this subroutine the various fields are calculated from the
-densities that were previously generated in module \Indexm{Densities}.
-The method of calculation is pretty much a direct application of the
-equations given in Section~\ref{sec:otherfields}, but with a few
-modifications for efficiency. The expressions divide up the
-contributions into an isospin-summed part with $b$-coefficients
-followed by the isospin-dependent one with $b'$-coefficients. As it
-would be a waste of space to store the summed densities and currents,
-the expressions are divided up more conveniently. If we denote the
-isospin index $q$ by {\tt iq} and the index for the opposite isospin
-$q'$ by {\tt ic} (as {\tt iq} can take the values 1~or 2, it can
-conveniently be calculated as {\tt 3-iq}), this can be written for
-example as
-
-\begin{quote}
-$b_1\rho-b_1'\rho_q\longrightarrow
-b_1(\rho_q+\rho_{q'})-b_1'\rho_q$\\
-$\longrightarrow $\texttt{(b1-b1p)*rho(:,:,:,iq)+b1*rho(:,:,:,ic)}
-\end{quote}
- This decomposition is used in all applicable cases.
-
-For intermediate results the fields {\tt workden} (scalar) and {\tt
- workvec} (vector) are used.
-
-Now the subroutine proceeds in the following steps:
-\begin{itemize}
-\item{\bf Step 1:} all the parts in \Indexv{upot} (see
- Eq.~(\ref{eq:upot}) involving an $\alpha$-dependent power of the
- density are collected. Note that in order to avoid having to
- calculate different powers, $\rho^\alpha$ is factored out. The
- division by the total density uses the small number {\tt epsilon} to
- avoid division by zero.
-\item{\bf Step 2:} the divergence of $\vec J$ (\Indexv{sodens}) is
- calculated for both isospins in {\tt workden} and the contributions
- are added to {\tt upot}.
-\item{\bf Step 3:} the Coulomb potential is calculated using
- subroutine \Indexp{poisson} (see module \Indexm{Coulomb}). It and the
- Slater exchange correction (only if the \Indexv{ex} parameter in the
- force is nonzero) are added to {\tt upot} for protons, {\tt
- iq=2}.
-\item{\bf Step 4:} the Laplacian is applied to the densities and
- the result stored in {\tt workden}. Then the remaining terms of
- Eq.~(\ref{eq:upot}) are constructed. Note that the {\tt iq}-loop is
- combined with the following steps.
-\item{\bf Step 5:} the effective mass is calculated according to
- Eq.~(\ref{eq:bmass}).
-\item{\bf Step 6:} the gradient of the density is calculated and the
- spin-orbit vector $\vec W_q$ is constructed in \Indexv{wlspot}
- according to Eq.~(\ref{eq:wlspot}).
-\item{\bf Step 7:} the curl of the spin density vector is calculated
- and stored in {\tt workvec}.
-\item{\bf Step 8:} the vector $\vec A_q$ is calculated according to
- Eq.~(\ref{eq:aq}) from the current density and the curl of the spin
- density.
-\item{\bf Step 9:} the curl of the current density is calculated and
- stored in \Indexv{spot}.
-\item{\bf Step 10:} now the two isospin contributions in \Indexv{spot}
- are combined in the proper way according to Eq.~(\ref{eq:spot}).
- This way of handling it avoids the introduction of an additional
- work vector for $\nabla\times\vec\jmath_q$.
-\item{\bf Step 11:} the divergence of $\vec A_q$ is calculated and
- stored in \Indexv{divaq}.
-\item{\bf Step 12:} finally, the gradient of the effective mass term
- $B_q$ (Eq.~(\ref{eq:bmass}) is calculated and stored in the vector
- variable \Indexv{dbmass}.
-\end{itemize}
-This concludes the calculation of all scalar and vector fields needed
-for the application of the Skyrme force.
-
-\subsubsection{Subroutine {\tt hpsi}}\Indexpb{hpsi}
-This subroutine applies the single-particle Hamiltonian to a
-single-particle wave function {\tt pinn} to produce an output wave
-function {\tt pout}. The argument {\tt iq} indicates the isospin for
-the wave function and {\tt eshift} is an energy shift which is zero in
-the dynamic calculation but crucial to the static algorithm (see
-\Indexp{grstep} in module \Indexm{Static}).
-
-For an understanding of this module the role of the following local
-variables is crucial. They are
-\begin{itemize}
-\item{{\tt is}}: this is used in the loops over spin to indicate the
- spin component: {\tt is=1} for spin up and {\tt is=2} for spin down.
-\item{{\tt ic}}: denotes the index for the opposite spin; it is
- calculated as {\tt ic=3-is}. Note the similar handling of
- the two isospin projections using {\tt iq} and {\tt icomp} in
- subroutine {\tt skyrme}.
-\item{{\tt sigis}}: this variable denotes the sign of the spin
- projection. It is calculated as {\tt sigis=3-2*is} and thus is {\tt
- +1} for spin up ({\tt is=1}) and {\tt -} for spin down ({\tt
- is=2}).
-\end{itemize}
-
-The general structure of the subroutine is as follows: first the part
-of the Hamiltonian not involving derivatives is applied, followed by the
-terms involving derivatives in order $x$, $y$, $z$.
-Since the structure of the Hamiltonian involves only first or second
-derivatives in one spatial direction in each term, the derivatives can
-be calculated for one direction and then the working space can be
-reused for the next one.
-
-The expressions for the different spatial derivatives are quite
-analogous, so that only the $x$-direction will be discussed at length
-below.
-
-When examining the expressions one by one, please refer to
-Eq.~(\ref{eq:spham}), which for convenience is repeated here:
-\begin{equation}
- \hat h=U_q(\vec r)-\nabla\cdot\left[B_q(\vec r)\nabla\right]
- +\I\vec W_q\cdot(\vec\sigma\times\nabla)
- +\vec S_q\cdot\vec\sigma
- -\frac{\I}{2} \left[(\nabla\cdot\vec A_q)+2\vec A_q\cdot\nabla\right].
-\end{equation}
-
-\begin{itemize}
-\item{\bf Step 1:} the non-derivative parts not involving spin. These
- arise from $U_q$ and $-\tfrac{\I}{2}\,\nabla\cdot\vec A_q$, which
- are combined into a complex expression. The energy shift {\tt
- eshift} is also included.
-\item{\bf Step 2:} the spin current coupling is constructed by simply
- using the explicit definition of the Pauli matrices and multiplying
- the resulting matrix onto the spinor wave function.
-\item{\bf Step 3:} the first and second derivative in the
- $x$-direction are evaluated and stored in the arrays {\tt pswk} and
- {\tt pswk2}. The last term in the Hamiltonian gives rise to the two
- contributions
-$$ -\frac{\partial B_q}{\partial x}\frac{\partial}{\partial x}-B_q
-\frac{\partial^2}{\partial x^2}, $$ of which the second is evaluated
-straightforwardly, while the first one is combined with the spin-orbit
-contribution. The part of $\I\vec W_q\cdot(\vec\sigma\times\nabla)$
-that contains an $x$-derivative is
-$$
-(\I W_y\sigma_z-\I W_z\sigma_y)\frac{\partial}{\partial x}=
-\begin{pmatrix} \I W_y&-W_z\\ W_z&-\I W_y
-\end{pmatrix}\frac{\partial}{\partial x}
-$$
-This is programmed employing the variable {\tt sigis} to account for
-the different signs in the rows of the matrix.
-\item{\bf Step 4:} for the derivatives in the $y$-direction the
- procedure is similar; the spin-orbit part is now
-$$
-(\I W_z\sigma_x-\I W_x\sigma_z)\frac{\partial}{\partial y}=
-\begin{pmatrix} -\I W_x&\I W_z\\ \I W_z&\I W_x
-\end{pmatrix}\frac{\partial}{\partial y}
-$$
-\item{\bf Step 5:} for the derivatives in the $z$-direction the
- procedure is again similar; the spin-orbit part is now
-$$
-(\I W_x\sigma_y-\I W_y\sigma_x)\frac{\partial}{\partial z}=
-\begin{pmatrix} 0&W_x-\I W_y\\ -W_x-\I W_y & 0
-\end{pmatrix}\frac{\partial}{\partial z}
-$$
-\end{itemize}
-
-
-\newpage
-\subsection{Module {\tt Moment}}\Indexmb{Moment}
-\subsubsection{Purpose}
-In this module various moments of the density distribution are
-calculated. Most of them come in two versions: an isospin-dependent
-one characterized by a final isospin index of dimension~2, and a
-summed one distinguished by the ending ``{\tt tot}'' in its name.
-Thus, e.~g., three variants of center of mass (\ref{eq:cmcart}) are
-stored~: that of the neutrons {\tt cm(1:3,1)}, of the protons {\tt
- cm(1:3,2)}, and of the total mass distribution {\tt cmtot(1:3)}.
-
-Since the geometrical arrangement in space can be arbitrary with
-respect to the Cartesian coordinate system --- this is certainly true
-for non-central collision situations --- some quantities associated
-with the axes become meaningless in the general situation. For this
-reason, the code also calculates the complete Cartesian quadrupole
-tensor (\ref{eq:Qcart}) and diagonalizes it to obtain the principal
-axes of the nucleus. In this frame then we compute the spherical
-quadrupole moments $Q_{2m}$, see Eq.~(\ref{eq:spherQ}), with their
-dimensionless counterparts $a_o$, $a_2$ defined in
-Eq.~(\ref{eq:dimlessQ}) and the deformation parameters $\beta$ and
-$\gamma$ from Eq.~(\ref{eq:triax}).
-
-\subsubsection{Module variables}
-\begin{description}
-\item{\Indexvb{pnr}, \Indexvb{pnrtot}:} the numbers of neutrons {\tt
- pnr(1)}=$N$, protons {\tt pnr(2)}=$Z$, and the total particle
- number {\tt pnrtot}=$A$. These are obtained by a simple integration
- of the densities {\tt rho}. Dimensionless.
-\item{\Indexvb{cm}, \Indexvb{cmtot}:} the center of mass vectors of
- the neutron, proton, and total mass distribution, $\vec R_n$, $\vec
- R_p$, and $\vec R$. Dimension: fm.
-\item{\Indexvb{pcm}:} the integrated momentum vectors, not containing
- the nucleon mass. They thus correspond to an integral over the
- current density only and have a dimension of velocity: $c$.
-\item{\Indexvb{rms}(private), \Indexvb{rmstot}:} these are the root
- mean-square radii (\ref{eq:rmsrad}) of neutron, proton, and total
- mass distribution. Dimension: fm.
-\item{\Indexvb{x2m} (private), \Indexvb{x2mtot} (private):} the
- vectors of second moments of the radii in the three coordinate
- directions, i.~e.,
- $$\langle r_i^2\rangle=\frac1{A} \int \D^3r\, r_i^2\rho(\vec r).$$
- They are useful to get an idea of the shape of the nucleus in
- static calculations. Dimension: fm$^2$
- \item{\Indexvb{q20} (private), \Indexvb{q20tot} (private):} the
- $m=0$ components of the spherical quadrupole tensor $Q_{20}$ in
- the principal-axes frame.
- \item{\Indexvb{q22} (private), \Indexvb{q22tot} (private):} the
- $m=2$ components of the spherical quadrupole tensor $Q_{22}$ in
- the principal-axes frame.
- \item{\Indexvb{beta20} (private), \Indexvb{beta20tot} (private):}
- the quadrupole deformation parameters $a_0$.
- \item{\Indexvb{beta22} (private), \Indexvb{beta22tot} (private):}
- the quadrupole deformation parameters $a_2$.
- \item{\Indexvb{beta}, \Indexvb{gamma}:} the Bohr-Mottelson
- deformation parameters $\beta$ and $\gamma$. These are calculated
- only for the total mass distribution. {\tt beta} is dimensionless
- whereas {\tt gamma} is expressed in degrees.
- \end{description}
-
-\subsubsection{Subroutine {\tt moments}}\Indexpb{moments}
-This is the principal subroutine for calculating the geometric
-quantities. It consists of two loops, both over isospin and space, and
-a final analysis Section. In detail:
-
-\begin{enumerate}
-\item The first loop calculates the particle numbers, centers of mass,
- and, for the dynamic case only, the momenta divided by nucleon mass.
-\item The second loop is separate because the center-of mass must be
- known to use it as the origin for the vectors. The r.m.s.\ radii
-\Indexv{rms} and the quantities \Indexv{x2m} are calculated as well as the
- components {\tt qmat} of the Cartesian quadrupole tensor
- $Q_{kl}=\int \D^3r \left(3x_kx_l-r^2\delta_{kl}\right) \rho(\vec
- r)$, see Eq.~(\ref{eq:Qcart}).
-\item After this, the subroutine {\tt q2diag} is used to determine the
- spherical components $Q_{20}$ and $Q_{22}$ according to
-Eq.~(\ref{eq:spherQ}) in the principal-axes frame (see Section
- \ref{sec:mulmom}), also generating some printout in the process.
- These are then multiplied with a scale factor to yield the
- dimensionless deformation parameters $a_0$ and $a_2$ according to
- Eq.~(\ref{eq:dimlessQ}) and finally by conversion to polar
- coordinates the Bohr-Mottelson parameters $\beta$ and $\gamma$ with
- Eq.~(\ref{eq:triax}).
-\item The Cartesian and polar deformation parameters are then printed.
-\end{enumerate}
-
-\subsubsection{Subroutine {\tt moment\_shortprint}}
-\Indexpb{moment\_shortprint}
-This subroutine simply prints some information into the specialized
-output files. {\tt monopolesfile} receives the r.m.s.\ radii and also
-the difference of neutron minus proton radius, while {\tt
- quadrupolesfile} receives the spherical quadrupole components {\tt
- q20} and {\tt q20tot} as well as the moments {\tt x2m}. The physical time
-starts each line to enable easy time-curve plotting.
-
-\subsubsection{Subroutine {\tt moment\_print}}
-\Indexpb{moment\_print}
-This subroutine prints a somewhat more detailed information. Particle
-number , r.m.s.\ radius, $Q_{20}$, \Indexv{x2m}, and center-of-mass are
-printed for the total distribution and also separately for neutrons
-and protons. This output goes to the regular output unit.
-
-\subsubsection{Subroutine {\tt q2diag}}\Indexpb{q2diag}
-This subroutine diagonalizes the Cartesian quadrupole tensor. To this
-purpose it calls the {\tt LAPACK} routine {\tt DSYEV}.
-
-The eigenvalues are obtained as {\tt q\_eig(i)} in ascending order of
-magnitude, and the corresponding eigenvectors as {\tt q\_vec(:,i)}.
-Both are printed. Then the corresponding spherical moments are
-calculated assuming the $z$-axis is selected as that of largest
-quadrupole moment:
-$$Q_{20}=\sqrt{\frac{5}{16\pi}}\,Q_{zz},\qquad
-Q_{22}=\sqrt{\frac{5}{96\pi}}\,\left(Q_{yy}-Q_{xx}\right).$$ They are
-returned in {\tt q20x} and {\tt q22x}.
-
-\newpage
-\subsection{Module {\tt Pairs}}\Indexmb{Pairs}
-\label{sec:pairs}
-The principal part of this module is the subroutine {\tt pair}, which
-computes the pairing solution based on the BCS model. It is the only
-public part of this module. The other subroutines are helper routines
-that directly use the single-particle properties defined in module
-\Indexm{Levels}. The module variable {\tt iq} controls whether the
-solution is sought for neutrons ({\tt iq=1} or protons {\tt iq=2}
-and accordingly the single-particle levels from {\tt npmin(iq)} to
-{\tt npsi(iq)} are affected.
-
-The principal procedure followed is to first calculate the pairing gap
-for each single-particle state. This determines the occupation numbers
-\Indexv{wocc}, which of course are used throughout the program. Then the
-Fermi energy for the given isospin is determined such that the correct
-particle number results.
-
-{\bf Note that there is a factor of one half in many formulas compared
- to what is usually found in textbooks. This is because here the sum
- over states $\sum_k\ldots$ runs over all states, while in textbooks
- the sum is over pairs, giving half of that result.}
-
-\subsubsection{Module variables}
-\begin{description}
-\item{\Indexvb{eferm}(2):} Fermi energy in MeV for the two isospins.
-\item{\Indexvb{epair}(2):} Pairing energy in MeV for the two isospins. It
- is given by
-$$E_{\rm pair}=\frac1{2}\sum_k \Delta_k u_k v_k.$$
-This is a public variable and the sum of the two values is subtracted from the
-total energies in module \Indexm{Energies}.
-\item{\Indexvb{avdelt}(2):} Average gap in MeV for the two isospins. It is
- given by $$ \frac{\sum_k\Delta_ku_kv_k}{\sum_k u_kv_k}$$ where the
- sum is over states with the given isospin.
-\item{\Indexvb{avg}(2):} The average pairing force for each isospin, given by
-$$\frac{E_{\rm pair}}{\sum_k u_k v_k/2}.$$
-\item{\Indexvb{deltaf}(nstmax):} single-particle gap in MeV for each
- single-particle state.
-\end{description}
-
-\subsubsection{Subroutine {\tt pair}}\Indexpb{pair}
-This is the only routine visible from outside the module. It solves
-the pairing problem and prints out summary information. The principal
-results used in the rest of the code are the BCS occupation numbers
-$v_k^2\rightarrow \Indexv{wocc}$ and the pairing energies \Indexv{epair}.
-
-The subroutine is structured straightforwardly: it first calculates
-the pairing gaps \Indexv{deltaf} by calling {\tt pairgap}. Then for the
-two isospin values {\tt pairdn} is called with the correct particle
-number as argument. This does the real work of solving the equations.
-Finally summary information is printed.
-
-\subsubsection{Subroutine {\tt pairgap}}\Indexpb{pairgap}
-This subroutine calculates the pairing gaps $\Delta_k$ stored in the
-array \Indexv{deltaf} for all single-particle states.
-
-First a simplified version is returned if {\tt ipair=1} or for the
-initial {\tt itrsin} (at present set to 10) iterations of a static
-calculation. All gaps are set equal to $11.2\,{\rm MeV}/\sqrt{A}$ in
-this case.
-
-In the general case, there is a loop over the two isospin values {\tt
- iq}. The pairing density is obtained by evaluation of
-\begin{equation}
- {\tt work}(\vec r)=\sum_k u_kv_k\left|\phi_k(\vec r)\right|^2
-\end{equation}
-where the simple conversion
-\begin{equation}
- u_kv_k=v_k\sqrt{1-v_k^2}=\sqrt{v_k^2(1-v_k^2)}=\sqrt{{\tt wocc}-{\tt wocc}^2}
-\end{equation}
-is used.
-
-The isospin-dependent pairing strength {\tt v0act} is obtained from
-the force definition. The pairing field $V_P(\vec r)$ is then given by
-two different expressions: for \Indexv{VDI} pairing ({\tt ipair=5}),
-the pairing density is simply multiplied by {\tt v0act}, while for
-\Indexv{DDDI} pairing ({\tt ipair=6}) it is
-\begin{equation}
- V_P(\vec r)={\tt v0act}\cdot{\tt work}\cdot (1-\rho(\vec r))/{\tt rho0pr}
-\end{equation}
-involving the {\em total} density $\rho$ and the parameter
-\Indexv{rho0pr} from the pairing force definition.
-
-In the final step the gaps are computed as the expectation values of
-the pairing field,
-\begin{equation}
- \Delta_k=\int\D^3r V_P(\vec r)\left|\phi_k(\vec r)\right|^2.
-\end{equation}
-
-\subsubsection{Subroutine {\tt pairdn}}\Indexpb{pairdn}
-The subroutine {\tt pairdn} determines the pairing solution by using
-{\tt rbrent} to find the correct Fermi energy for the given particle
-number. After that a few averaged or integral quantities are
-calculated.
-
-As the starting value for the Fermi energy the one for gap zero is
-used, i. e., the average of the first unfilled and last filled
-single-particle energies. Then {\tt rbrent} is called to calculate the
-correct solution, after which there is a loop for a straightforward
-evaluation of the module variables \Indexv{epair}, \Indexv{avdelt},
-and \Indexv{avg}.
-
-\subsubsection{Subroutine {\tt rbrent}}\Indexpb{rbrent}
-This subroutine is an adapted version of the {\em Van
- Wijngaarden-Dekker-Brent} method for finding the root of a function
-(see~\cite{Pre92aB}). Given the desired particle number as an
-argument, it searches for the value of the Fermi energy that makes
-this particle number agree with that returned by {\tt
- bcs\_occupation}.
-It is clear that this subroutine is in a very antiquated style of
-Fortran; it will be replaced at some time in the future.
-
-\subsubsection{Subroutine {\tt bcs\_occupation}}
-\Indexpb{bcs\_occupation}
-For a given Fermi energy $\epsilon_F$ passed as argument {\tt efermi},
-this subroutine evaluates the particle number that would result with
-such a Fermi energy and returns it as its second argument, {\tt
- bcs\_partnum}. The isospin is controlled by module variable {\tt
- iq}. First the occupation probabilities are calculated using the
-standard BCS expression
-\begin{equation}
- v_k^2=\frac1{2}\left(1-\frac{\epsilon_k-\epsilon_F}
- {\sqrt{(\epsilon_k-\epsilon_F)^2+\Delta_k^2}}\right).
-\end{equation}
-They are stored in \Indexv{wocc}. A small correction is added, so
-that they are not exactly identical to~1 or~0. The particle number
-is finally obtained as $N=\sum_k v_k^2$.
-
-\newpage
-\subsection{Module {\tt Parallel}}\Indexmb{Parallel}
-\label{sec:parallel}
-This module organizes the execution on distributed-memory machines
-using {\tt MPI}. Its interface is made such that on sequential
-machines {\tt parallel.f90} can simply be replaced by a special
-version {\tt sequential.f90} which contains a definition of this
-module generating trivial data.
-
-\subsubsection{Parallelization concept} {\tt MPI} parallelization is
-based on distributing the wave functions onto the different nodes.
-Thus all operations acting directly on the wave functions can be done
-in parallel, not only the time evolution by the application of the
-single-particle Hamiltonian, but also the summing up of the densities
-and currents over those wave function stored on the node. This means
-that only the final summation of the densities and the calculations
-done with them have to be communicated across the nodes.
-
-It is important that the single-particle properties also defined in
-\Indexm{Levels}, e.g. \Indexv{sp\_energy} are not split up up for the nodes
-but the full set is present on each node. The values are communicated
-by summing from all nodes with zeroes in those index positions not
-present on a particular one. This method of handling them avoids having to
-communicate many small arrays.
-
-Since an efficient way of dealing with Gram-Schmidt orthogonalization
-in this case was yet not found, at present the code can be run in {\tt
- MPI} parallel mode only for the dynamic case.
-
-\subsubsection{Module variables}
-\begin{description}
-\item{\Indexvb{tmpi}:} a logical variable set to true if {\tt MPI}
- parallelization is activated. It is used to turn the calling of all
- the {\tt MPI} routines in the code on or off.
-\item{\Indexvb{node}, \Indexvb{localindex}:} these are vectors of
- integers with dimension {\tt nstmax}. For the single-particle state
- with index {\tt i} the wave function is stored on computing node
- {\tt node(i)} and its index on that node is {\tt localindex(i)}.
-\item{\Indexvb{globalindex}:} tells the index of the single-particle
- state in the whole array of {\tt nstmax} states (it could be
- dimensioned {\tt nstloc} but is dimensioned as {\tt nstmax}
- to make its allocation simpler). So for wave function index {\tt i}
- {\em on the local node}, {\tt i=1..nstloc}, the
- single-particle energy must be obtained using {\tt
- sp\_energy(globalindex(i))}.
-\item{\Indexvb{mpi\_myproc}, {\tt mpi\_nprocs}, {\tt mpi\_ierror}:}
- these are variables associated with MPI and the MPI documentation should be consulted.
-\end{description}
-
-\subsubsection{Subroutine \Indexpb{alloc\_nodes}(Public)}
-This subroutine merely allocates the internal arrays of module {\tt
- Parallel}.
-\subsubsection{Subroutine {\tt init\_all\_mpi}}
-\Indexpb{init\_all\_mpi} This subroutine initializes {\tt MPI} and
-finds out the number of processors {\tt mpi\_nprocs} as well as the
-index of the current one {\tt mpi\_myproc}. The flag \Indexv{wflag} is
-set to true only for the processor numbered 0.
-
-\subsubsection{Subroutine {\tt allocate\_nodes}}
-\Indexpb{allocate\_nodes}
-The first loop in this subroutine distributes the wave functions over
-the nodes. This is done by looping over the wave functions and
-assigning one to each processor in turn. When the number of processors
-has been reached, it restarts from processor~0. This way of allocation
-is to some extent arbitrary and can be changed.
-
-The second loop then calculates which wave functions are present on
-the local node and records their index \Indexv{gobalindex} in the
-complete sequence. The third loop sets up the reverse pointers
-\Indexv{localindex}, which has to be done in a loop over all processors to
-set up the information for the proper global indices.
-
-\subsubsection{Subroutine {\tt collect\_densities}}
-\Indexpb{collect\_densities}
-This subroutine uses the {\tt MPI} routine {\tt mpi\_allreduce} to sum
-up the partial densities from the different nodes, using temporary
-arrays {\tt tmp\_rho} and {\tt tmp\_current} (depending on whether it
-is a scalar or vector field) in the process.
-
-\subsubsection{Subroutine {\tt collect\_sp\_properties}}
-\Indexpb{collect\_sp\_properties}
-This subroutine collects the single-particle properties calculated
-from the wave functions and thus available only for the local wave
-functions on each node. It uses a simple trick: the arrays like {\tt
- sp\_energy} are defined for the full set of indices but set to zero
-before the calculation of these properties. On each node then the
-local values are calculated but inserted at the proper index for the
-full set of wave functions. In this subroutine the results from all
-the nodes are added up using {\tt mpi\_reduce}, so that effectively
-for each index one node contributes the correct value and the others
-zeroes. This process sounds inefficient but considering the small size
-of the arrays that does not matter.
-
-\subsubsection{Subroutine {\tt finish\_mpi}}\Indexpb{finish\_mpi}
-This is just a wrapper for the {\tt MPI} finalization call.
-
-\subsubsection{Using {\tt sequential.f90}}
-In the sequential case module variables are added to simulate the {\tt
- MPI} environment. Thus variables with names starting {\tt mpi\_}
-have to be supplied. The only ones whose value is important are the
-processor number {\tt mpi\_myproc} which is always zero in this case,
-and the total number of processors {\tt mpi\_nprocs}, which is~1.
-
-All subroutines with names starting with {\tt mpi\_} stop the code,
-because the calls to {\tt MPI} routines should only happen in parallel
-execution (they are conditioned by {\tt tmpi}. Thus it is an error if
-the code runs into one of these.
-
-The indexing arrays become trivial: \Indexv{node}{\tt(i)=0},
-\Indexv{localindex}{\tt (i)=i}, and \Indexv{globalindex}{\tt (i)=i} for
-all {\tt i}.
-
-\newpage
-\subsection{Module {\tt Params}}
-\Indexmb{Params}
-This module contains some general parameters critical to controlling
-the code and some mathematical and physical constants used everywhere.
-\subsubsection{Module variables}
-\subsubsection{General parameters}
-\begin{description}
-\item{\Indexvb{db}:} this is a constant determining the precision of
- real numbers in the code in a portable manner. In practice it will usually be {\tt
- REAL(8)}.
-\item{\Indexvb{pi}:} the constant $\pi$.
-\item{\Indexvb{hbc}:} the constant $\hbar c$ in units of MeV*fm.
-\item{\Indexvb{e2}:} the electron charge squared. It is calculated as
- $\alpha\hbar c$ and has units of MeV*fm.
-\end{description}
-\subsubsection{File names and units}\label{sec:filenames}
-These variables allow the user to change the names of some of the
-files used by the code. Except for {\tt wffile}, output is produced
-for an iteration or a time step at selected intervals.
-\begin{description}
-\item{\Indexvb{wffile}:} file to contain the static single-particle
- wave functions plus some additional data. It is also used for
- restarting an interrupted calculation and is rewritten regularly
- (see variables {\tt trestart} and {\tt mrest}). It can be turned off
- completely using the name {\tt 'NONE'}.
-\item{\Indexvb{converfile}:} contains convergence information for the
- static calculation. Default: {\tt conver.res}.
-\item{\Indexvb{monopolesfile}:} contains moment values of monopole
- type. Default: {\tt monopoles.res}.
-\item{\Indexvb{dipolesfile}:} contains moment values of dipole type.
- Default: {\tt dipoles.res}.
-\item{\Indexvb{quadrupolesfile}:} contains moment values of
- quadrupole type. Default: {\tt quadrupoles.res}.
-\item{\Indexvb{momentafile}:} contains components of the total
- momentum. Default: {\tt momenta.res}.
-\item{\Indexvb{energiesfile}:} energy data for time-dependent
-\item{\Indexvb{spinfile}:} time-dependent total, orbital, and spin
- angular-momentum data as three-dimensional vectors. calculations.
- Default: {\tt energies.res}.
-\item{\Indexvb{extfieldfile}:} contains the time-dependence of the
- expectation value of the external field. Present only if an
- external field for boost or time-dependent excitation
- is used. Default: {\tt extfield.res}.
-\item{\Indexvb{scratch}, \Indexvb{scratch2}:} the unit numbers used
- for temporary storage. Default: 11 and 12.
-\end{description}
-\subsubsection{Switches}
-These are logical variables to turn various features on or off.
-\begin{description}
-\item{\Indexvb{tcoul}:} indicates whether the Coulomb field should be
- included or not.
-\item{\Indexvb{tstatic}, \Indexvb{tdynamic}:} these are set {\tt
- true} for a static or dynamic job, respectively. They are not input
- directly but from the input variable \Indexv{imode}, which is 1 for the
- static and 2 for the dynamic case.
-\item{\Indexvb{trestart}:} if {\tt true}, restarts the calculation
- from the {\tt wffile}.
-\item{\Indexvb{tfft}:} if {\tt true}, the derivatives of the wave
- functions, but not of the densities, are done directly through FFT.
- Otherwise matrix multiplication is used, but with the matrix also
- obtained from FFT. Default is {\tt true}.
-\end{description}
-\subsubsection{Output control}
-\begin{description}
-\item{\Indexvb{mprint}:} control for printer output. If {\tt mprint}
- is greater than zero, more detailed output is produced every {\tt
- mprint} iterations or time steps on standard output.
-\item{\Indexvb{mplot}:} if {\tt mplot} is greater than zero, a
- printer plot is produced and the densities are dumped onto {\tt
- *.tdd} files every {\tt mplot} time steps or iterations.
-\item{{\tt mrest}:} if greater than zero, a \Indexv{wffile} is produced
- every {\tt mrest} iterations or time steps.
-\end{description}
-\subsubsection{Globally used variables}
-\begin{description}
-\item{\Indexvb{iter}:} number of the current time step or iteration.
- Used in both static and dynamic modes.
-\item{\Indexvb{time}:} the simulation time of the current step in
- fm/c; only meaningful in a dynamic calculation.
-\item{\Indexvb{wflag}:} indicates whether printing is allowed. This
- is necessary for the parallel job to have only one processor print
- and concerns both the standard output and the {\tt *.res} files.
-\item{\Indexvb{printnow}:} this variable is set to true if conditions
- for printing are met, such as a certain interval in iteration
- number.
-\end{description}
-\subsubsection{Field output control}
-\begin{description}
-\item{\Indexvb{nselect}:} parameter limiting how many fields can be
- selected for binary output, it is just the length of the character
- string {\tt writeselect}.
-\item{\Indexvb{writeselect}:} it is used to determine which fields
- should be output under the control of {\tt mplot}.
-\item{\Indexvb{write\_isospin}:} if this is {\tt .FALSE.}, the proton
- and neutron contributions of a field are added up before output.
- Otherwise both are written.
-\end{description}
-\subsubsection{Fragment number parameters}
-\begin{description}
-\item{\Indexvb{nof}, \Indexvb{mnof}:} number of fragments for the
- initialization and maximum allowed. These should really be
- determined in module {\tt Fragments}; the reason for putting these
- here and reading {\tt nof} early is that {\tt nof} must be known and
- the arrays for fragments allocated before reading the other fragment
- input. Setting {\tt mnof} to a large number is no problem since the
- arrays are quite small.
-\item{\Indexvb{r0}:} nuclear radius parameter. The nuclear radius
- $R=r_0A^{1/3}$ is used to compute the $\beta$ and $\gamma$
- deformation parameters in subroutine \Indexp{moments}. Units: fm,
- default value 1.2~fm.
-\end{description}
-
-\newpage
-\subsection{Module {\tt Static}}\Indexmb{Static}
-\subsubsection{Module variables}
-\begin{description}
-\item{\Indexvb{tdiag}:} if {\tt true}, there is a diagonalization of
- the Hamiltonian during the later (after the 20th) static iterations.
- The 20 is hard coded in {\tt static.f90}. Default is {\tt false}.
-\item{\Indexvb{tlarge}:} if {\tt true}, during the diagonalization
- the new wave functions are temporarily written on disk to avoid
- doubling the memory requirements. Default is {\tt false}.
-\item{\Indexvb{maxiter}} maximum number of iterations allowed.
-\item{\Indexvb{serr}} convergence criterion. Iterations are stopped
- if the fluctuation in single-particle energies falls below this
- value (see near the end of subroutines {\tt statichf}).
-\item{\Indexvb{e0dmp},\Indexvb{x0dmp}}: these correspond to the
- parameters $E_0$ and $x_0$ appearing in the damped gradient
- iteration of Eq.~(\ref{eq:dampstep}).
-\item{\Indexvb{radinx}, \Indexvb{radiny}, \Indexvb{radinz}}: these
- are the radius parameters used in the harmonic-oscillator
- initialization (see subroutine {\tt harmosc}).
-\item{\Indexvb{delesum}}: used to sum up the changes in
- single-particle energies during one iteration; it is calculated in
- {\tt statichf} but printed in {\tt sinfo} so that it is a module
- variable.
-\end{description}
-
-\subsubsection{Subroutine {\tt getin\_static}}
-\Indexpb{getin\_static}
-This routine reads the input for the static calculation using namelist
-\Indexn{static}. This includes the module variables of this module, but
-also the numbers of particles, which are input quantities only in the
-static mode if initialization is not done via fragment files. In the
-case of user initialization they are also needed to correctly allocate
-the fields; only the wave functions are then calculated in
-\Indexp{user\_init}. The values given in the input are overwritten by
-fragment data otherwise.
-
-Thus only if {\tt nof<=0} the input numbers are used. If \Indexv{npsi} is
-not given in the input, the values of \Indexv{nneut} and \Indexv{nprot} are
-used for the number of wave functions, except for the pairing case,
-when they are computed from a formula.
-
-The variables \Indexv{charge\_number} and \Indexv{mass\_number} are also set
-for this case.
-
-\subsubsection{Subroutine {\tt init\_static}}
-\Indexpb{init\_static}
-This subroutine essentially just prints the static input and then
-initializes the header files with their header lines. This is not
-included in {\tt getin\_static}, because the particle and state
-numbers may have been changed by fragment input.
-
-In addition, the damping matrices are constructed by calling
-\Indexp{setup\_damping}. Finally for some Skyrme forces the effective mass
-is changed to account for the center-of-mass correction. This should
-only be used if necessary and not in dynamic calculations.
-
-\subsubsection{Subroutine {\tt statichf}}\Indexpb{statichf}
-This is the principal routine for the static iterations. It applies
-the gradient step repeatedly until convergence is achieved or the
-maximum number of iterations is reached. The following local variables
-are worth defining:
-\begin{description}
-\item[\Indexvb{sumflu}] keeps track of the fluctuations (\ref{eq:totvar})
- in single-particle energies summed over the states. It is used as
- the convergence criterion.
-\item[\Indexvb{addnew}, {\tt addco}] are simple factors with standard
- values 0.2 and 0.8 used for relaxation (see Step 8). They are
- defined as parameter variables so they can be changed easily if
- desired.
-\item[\Indexvb{denerg}] is used as an argument to {\tt grstep} to contain
- the relative change in energy of the single-particle state.
-\end{description}
-
-\begin{itemize}
-\item {\bf Step 1: Initialization}: if this is a restart, the number
- of the initial iteration is set to the value of {\tt iter+1}
- obtained from {\tt wffile}. In this case the single-particle
- quantities do not have to be set to zero and orthogonalization is
- not necessary. If this is not a restart, the initialization is done
- as zeroth iteration and the first iteration number for the loop is
- set to one. some variables are initialized to zero and the matrix
- for the diagonalization \Indexv{hmatr} is allocated. Then \Indexp{schmid}
- is called for initial orthogonalization.
-\item {\bf Step2: calculating densities and mean fields}: the
- densities are reset to zero and then in a loop over states the
- contributions of the single-particle states are added up. The
- subroutine \Indexp{skyrme} is called to compute the mean-field
- components.
-\item{\bf Step 3: initial gradient step}: in a loop over the
- single-particle wave functions the gradient step (see Section
- \ref{sec:statiter}) is applied --- see subroutine \Indexp{grstep}. The
- sum of relative changes in single-particle energies and fluctuations
- are accumulated in \Indexv{delesum} and {\tt sumflu}. This loop is
- followed by the pairing calculation (which needs the single-particle
- energies calculated in the gradient step) and renewed
- orthogonalization. Then the detailed single-particle properties and
- energies are calculated using \Indexp{sp\_properties} and \Indexp{sinfo}.
-\item{\bf Step 4: start iteration}: this is
- the principal loop for the static calculation. The iteration number
- is printed.
-\item{\bf Step 5: gradient step}: this is identical to the initial
- gradient step in ``Step 3''.
-\item{\bf Step 6: diagonalization}: after 20 iterations and only if
- the switch \Indexv{tdiag} is true, \Indexp{diagstep} is called to
- diagonalize the single-particle Hamiltonian.
-\item{\bf Step 7: pairing and orthogonalization}: these are called for
- the new wave functions.
-\item{\bf Step 8: calculate densities and fields with relaxation}: the
- old density \Indexv{rho} and kinetic energy density \Indexv{tau} are saved
- in \Indexv{upot} and \Indexv{bmass}, which are here used purely as work
- arrays. Then the new densities are accumulated from the wave
- functions and for {\tt rho} and {\tt tau} they are mixed with the
- old densities in a ratio given by \Indexv{addnew} and \Indexv{addco}, if
- this is turned on by \Indexv{taddnew}. \Indexp{skyrme} is called to
- calculate the new fields. Then the detailed single-particle
- properties and energies are calculated and printed using
-\Indexp{sp\_properties} and \Indexp{sinfo}.
-\item{\bf Step 9: finalizing the loop}: convergence is checked by
- comparing \Indexv{sumflu} per particle to \Indexv{serr}, if it is smaller,
- the job terminates after writing the final wave functions.
- Otherwise, the wave functions are written if indicated by
- \Indexv{mrest}
-and the loop continues.
-\end{itemize}
-
-\subsubsection{Subroutine {\tt grstep}}\Indexpb{grstep}
-This subroutine applies the damped gradient step (\ref{eq:dampstep}) to a wave
-function given as argument {\tt psin}. Its index is {\tt nst} and
-isospin {\tt iq} --- these data are needed for the construction of the
-Hamiltonian matrix {\tt hmatr}. The argument {\tt spe}
-$\rightarrow\epsilon$ represents the single-particle energy which is
-used a an energy shift in the calculation.
-
-The work is done in the following steps:
-\begin{itemize}
-\item {\bf Step 1}: apply $\hat h-\epsilon$ to the wave function {\tt
- psin} to obtain {\tt ps1}.
-\item {\bf Step 2}: calculate the diagonal matrix element of $\hat h$
-$$\Indexv{xnormb}=\langle {\tt psin}|{\tt ps1}\rangle=\langle {\tt
- psin}|\hat h|{\tt psin}\rangle$$ and the squared norm of {\tt psin}
-(in \Indexv{xnorm}). The expression {\tt xnormb/xnorm} thus corresponds
-to the expectation value $\|\hat h\|$ in {\tt psin}.
-
-Then the matrix elements of $\hat h$ with all other states of the same
-isospin are calculated and inserted into {\tt hmatr}; in the diagonal
-matrix elements the energy shift $\epsilon$ is added back.
-\item {\bf Step 3}: for the calculation of the fluctuation in the
- single-particle energy, the quantity
-$${\tt exph2}=\langle {\tt psin}|\hat h^2|{\tt psin}\rangle$$
-is used to compute
-$$ \Indexv{sp\_efluct1}=\sqrt{\|\hat h^2\|-\|\hat h\|^2} $$
-and the squared norm
-$${\tt varh2}=\|\hat h\,{\tt psin}\|^2$$
-similarly for the second fluctuation measure
-$$\Indexv{sp\_efluct2}=\sqrt{\|\hat h\,{\tt psin}\|^2/\|{\tt psin}\|^2-\|\hat
- h\|^2}.
-$$
-They are calculated only for time steps with output turned on.
-\item {\bf Step 4}: now the damping is performed. We first compute
-$$|{\tt ps1}\rangle-{\tt xnormb}\,|{\tt psin}\rangle=\left(\hat h
- -\langle {\tt psin}|\hat h|{\tt psin}\rangle\right)\,|{\tt
- psin}\rangle$$ replacing {\tt ps1}, on which then the real damping
-operator acts. If {\tt FFT} is being used for the derivatives, we use
-the routine \Indexp{laplace} from module \Indexm{levels} to compute
-$$\frac{x_{0\rm dmp}}{E_{0\rm inv}+\hat t}\,|{\tt ps1}\rangle.$$
-If derivatives are to done by matrices, the damping matrices
-\Indexv{cdmpx} etc.\ are used (see subroutine \Indexp{setdmc} in
-module \Indexm{Grids}. The factors \Indexv{x0dmp} and \Indexv{e0dmp} have to be
-manipulated a bit in this case.
-% ### This needs an explanation under numerical methods ????
-Finally we subtract this result multiplied by \Indexv{x0act} from the
-original wave function to get the damped one.
-
-\item {\bf Step 5}: the single-particle energy is calculated from its
- new expectation value with the energy shift restore, and the
- comparison with the initial value yields the relative change
-\Indexv{denerg}, which is passed back to the caller.
-\end{itemize}
-
-\subsubsection{Subroutine {\tt diagstep}}\Indexpb{diagstep}
-This subroutine performs a diagonalization of the single-particle
-Hamiltonian using the LAPACK routine {\tt ZHBEVD}. This is of course
-done separately for protons and neutrons indexed by {\tt iq}. The
-matrix \Indexv{hmatr} is produced in {\tt grstep}.
-
-The details here depend on the requirements of this routine, which is
-written in Fortran-77 and so has a complicated calling sequence with
-many arguments used for intermediate storage.
-
-We therefore do not give excessive detail here but summarize the main
-points, which should be easy to analyze. The steps are:
-
-\begin{itemize}
-\item {\bf Step 1}: the matrix is copied into lower-diagonal form into
- array {\tt hmatr\_lin}. The {\tt ZHBEVD} is called, which leaves as
- main results the vector of eigenvalues {\tt eigen} and the unitary
- matrix {\tt unitary} describing the transformation from the
- original states to the diagonalized ones. The latter matrix is also
- stored in lower-diagonal form.
-\item {\bf Step 2: transform states}: the matrix {\tt unitary} is used
- to form the appropriate linear combinations of the original
- single-particle states. If \Indexv{tlarge} is true, each state if
- formed independently and written onto a scratch file, otherwise an
- intermediate array {\tt ps1}, dimensioned to hold either only
- protons or only neutrons (number of states through the argument {\tt
- nlin}), is allocated to receive the new states, which are then
- copied back into \Indexv{psi}.
-\end{itemize}
-
-\subsubsection{Subroutine {\tt sinfo}}\Indexpb{sinfo}
-This subroutine computes the data that are not needed for the
-calculation itself, but only for informative output and writes them
-onto the appropriate output files.
-
-First \Indexp{moments}, \Indexp{integ\_energy}, and
-\Indexp{sum\_energy} are called to compute the relevant physical
-quantities. Output lines are added to \Indexv{converfile},
-\Indexv{dipolesfile}, \Indexv{momentafile}, and
-\Indexv{spinfile}. Information on the current energy \Indexv{ehf}, the
-total kinetic energy \Indexv{tke}, the relative change in energy over
-the last iteration \Indexv{delesum}, the fluctuations in
-single-particle energy, and the energy corrections are printed on
-standard output.
-
-At intervals of \Indexv{mplot} iterations the density printer plot is
-produced and the {\tt *.tdd} file containing the densities written.
-
-Finally, on standard output more detailed output is given: an overview
-of the different contributions to the energy, a list of
-single-particle state properties, and a listing of various moments
-using \Indexp{moment\_print}.
-
-\newpage
-\subsection{Module {\tt Trivial}}\Indexmb{Trivial}
-This module contains some basic calculations with wave functions and
-densities, which are similar enough to be grouped together.
-
-\subsubsection{Subroutines {\tt cmulx}, {\tt cmuly}, and {\tt cmulz}}
-\Indexpb{cmulx}\Indexpb{cmuly}\Indexpb{cmulz}
-The routines do a complex matrix multiplication on a wave function
-along one of the Cartesian directions, which we denote as {\tt
- [x,y,z]} to indicate the possible choices. The first argument {\tt
- [x,y,z]mat} is the matrix dimensioned as a square matrix with the
-number of points in the direction affected. The second argument {\tt
- pinn} is the input wave function, and the third one, {\tt pout}, the
-output wave function. The result of the multiplication is added to the
-output wave function if the final argument, {\tt ifadd}, is nonzero.
-This allows accumulating results but is not used in the present code.
-If {\tt ifadd} is zero, the output wave function is cleared to
-zeroes and thus is simply the result of the matrix multiplication.
-
-The operation carried out here includes a loop over the spin index.
-
-The effective calculation is, e.~g., for the $x$-direction
-$${\tt pout(i,j,k,s)}=\sum_{i'=1}^{nx}{\tt xmat(i,i')pin(i',j,k,s)}$$
-for all values of {\tt i,j,k,s}.
-
-Not that for the $z$-direction an explicit {\tt DO}-loop is programmed
-instead of the {\tt FORALL} used in the other cases. This is because
-the compiler at some point did not optimize well. This should be
-reexamined in the future.
-
-\subsubsection{Function {\tt rpsnorm}}\Indexpb{rpsnorm}
-This function with one argument calculates the norm of a wave
-function, i.~e.,
-$$ |\psi|=\sum_{s=\pm{1\over2}}\int |\psi(\vec r,s)|^2\,\D^3 r\rightarrow
-{\tt wxyz}\sum_{s=1}^2 \sum_{i=1}^{nx}\sum_{j=1}^{ny}\sum_{k=1}^{nz}
-|{\tt psi(i,j,k,s)}|^2,
-$$
-with \Indexv{wxyz} the volume element.
-
-This could be calculated with the built-in {\tt SCALAR\_PRODUCT}
-function, but this was found to be less efficient. Again, future
-compilers may change that.
-
-\subsubsection{Function {\tt overlap}}\Indexpb{overlap}
-This calculates the overlap of two wave functions $psi_L$ and $psi_R$
-given as arguments {\tt pl} and {\tt pr}. This is defined as
-\begin{eqnarray*}
- \langle\psi_L|\psi_R\rangle&=&\sum_{s=\pm{1\over2}}\int \psi_L^*(\vec
- r)\,\psi_R(\vec r)\,\D^3 r \\
- &\rightarrow&
- {\tt wxyz}\sum_{s=1}^2
- \sum_{i=1}^{nx}\sum_{j=1}^{ny}\sum_{k=1}^{nz}
- {\tt CONJG(pl(i,j,k,s))pr(i,j,k,s)}.
-\end{eqnarray*}
-
-This also might be replaced by {\tt SCALAR\_PRODUCT} eventually.
-
-\subsubsection{Subroutines {\tt rmulx}, {\tt rmuly}, and {\tt rmulz}}
-\Indexpb{rmulx}\Indexpb{rmuly}\Indexpb{rmulz}
-The subroutines work quite analogously to {\tt cmulx} etc., with two
-important differences:
-\begin{itemize}
-\item The array {\tt finn} operated on is not a complex wave function,
- but a real field in space as is the result {\tt fout}. Thus the
- arithmetic is real and there is no summation over spin.
-\item The final argument {\tt ifadd} has an additional function
- depending on its sign. If it is zero, {\tt fout} is set to zero
- before the calculation. If it is {\em positive or zero}, the result
- is {\em added} to {\tt pout}, otherwise it is {\em subtracted}.
-\end{itemize}
-This facility is used in the code to calculate the curl of vector
-fields, for example, by applying the derivative matrices to the
-Cartesian components of the field and adding and subtracting
-appropriately.
-
-\newpage
-\subsection{Module {\tt Twobody}}\Indexmb{Twobody}
-\label{sec:twobody}
-
-This module contains the code to analyze the final (and also, though
-less useful) initial stage of a heavy-ion reaction. It is applicable
-only to the dynamic calculations and only if there are essentially two
-separated fragments. It calculates the fragment masses and charges,
-their distance, the relative motion kinetic energy, angular momentum,
-and the scattering angle.
-
-The analysis assumes that the two fragments are separated by a region
-of noticeably lower density, tries to find the line connecting their
-centers of mass and then divides up space by a plane perpendicular to
-this line. Of necessity the results are not of high precision, but
-tend to be useful nevertheless.
-
-{\bf The user should be critical and always make sure that the real
- situation is a two-fragment one before accepting the results of
- these routines.} {\bf It is assumed that the reaction plane is the
- $(x,z)$ plane. The code could be generalized to a three-dimensional
- situation, if necessary, but usually the assumption of a fixed
- scattering plane should not be a problem. If the initial nuclei have
- nonzero internal angular momentum, e.~g., the reaction plane can
- rotate and then a more general analysis cannot be avoided.}
-
-\subsubsection{Module variables}
-\begin{description}
-\item{\Indexvb{roft}:} separation distance between fragments in fm.
-\item{\Indexvb{rdot}:} relative-motion velocity in units of $c$.
-\item{{\tt xmin}, {\tt zmin} (private):} coordinates of the point in
- the $(x,z)$-plane where minimum density is found between the
- fragments.
-\item{{\tt slope}, {\tt slold} (private):} Slope of the line
- connecting the fragment centers-of-mass. {\tt slold} is the value
- from the previous time step or iteration where the 2-body analysis
- was last performed.
-\item{{\tt bb} (private):} intercept of the line. The line is thus
- given as {\tt z=bb+slope*x}.
-\item{{\tt centerx}, {\tt centerz} (both private):} coordinates of the
- two fragment centers of mass.
-\item{\Indexvb{istwobody}:} logical, indicates whether this is a
- two-body case ({\tt TRUE}) or not.
-\item{{\tt vacuum}:} parameter indicating the limiting density below
- which vacuum is assumed. This value is not critical, since it is
- only used in looking for ``empty'' regions between the nuclei.
-\end{description}
-The two-body properties finally calculated are the following:
-\begin{description}
-\item[{\tt xmu}] $\mu$: the reduced mass in MeV.
-\item[{\tt vxx}, {\tt vzz}]: $x$- and $z$-velocities of the fragments,
- calculated by differencing the center-of-mass positions.
-\item[{\tt tke2body}]: total kinetic energy in MeV.
-\item[{\tt tketot}]: relative-motion kinetic energy in MeV.
-\item[{\tt roft}]: $R$: distance between the fragments in fm.
-\item[{\tt rdot}]: time-derivative of the separation distance, units
- of $c$.
-\item[{\tt teti}]: present scattering angle.
-\item[{\tt tdotc}]: $\omega$: time derivative of scattering angle.
- Units $c/{\rm fm}$.
-\item[{\tt xlf}]: angular momentum of relative motion in $\hbar$. It
- is calculated assuming two point bodies via $\mu R^2\omega$.
-\item[{\tt xcoul}]: Coulomb energy, calculated from point charges.
-\item[{\tt xcent}]: centrifugal energy, based on angular momentum and
- distance.
-\item[{\tt ecmf}]: final relative motion energy after extrapolated to
- infinite separation.
-\end{description}
-
-\subsubsection{Subroutine {\tt twobody\_case}}
-\Indexpb{twobody\_case}
-This routine tries to find a separation of the system into two
-fragments and determine their properties as well as those of the
-relative motion. It keeps track of previous analysis results, since
-the 2-body properties are expected to change slowly and this makes the
-analysis easier. Also, the motion of the fragments is determined by
-time-differencing.
-
-The parameter {\tt xdt} is the current time step. It cannot be
-imported directly from module \Indexm{Dynamic} since that would lead to a
-circular module dependence.
-
-It uses the following steps:
-\begin{itemize}
-\item{\bf Step 1}: the results of the last analysis are partially
- saved. This includes the positions of the fragments, their
- distance, the slope of the connecting line, and its angle.
-\item{\bf Step 2:} the fragment division is sought in an iterative
- process with (at present) a fixed iteration limit of~10. The reason
- for the iterations is that the connecting line, which determines the
- dividing plane, which is orthogonal to it at the dividing point,
- should link the centers-of-mass, but these can be calculated only
- assuming a dividing plane. There is thus a self-consistency problem
- which as usual is solved iteratively. During the iterations the new
- value of the center of mass is kept in {\tt centerx} and {\tt
- centerz}, while the previous one is in {\tt centx} and {\tt
- centz}. Each iteration has several steps:
- \begin{enumerate}
- \item The connecting line ({\tt slope} and intercept {\tt bb}) is
- determined in one of two ways: if this is the first iteration,
- subroutine getslope is called to calculate the slope from the
- quadrupole tensor; the intercept is then calculated assuming that
- the line passes through the center-of-mass (which it should do in
- principle, but it might be off in practice). For the later
- iterations the line is calculated directly from the two centers of
- mass.
- \item Next function \Indexp{divpoint} is called to find out whether
- this is indeed a two-body situation. It also returns the midpoint
- between the fragments in {\tt xmin} and {\tt zmin}. This result is
- not used immediately, since it might be that shifting the
- connecting line could change this situation.
- \item The slope of the line is now rotated by $90^\circ$ to get the
- line defining the dividing plane in the $(x,z)$-plane. This has
- {\tt slopev} and {\tt bb} as slope and intercept.
- \item Now in a simple loop integrals are done separately over the
- two regions, summing up {\tt charge}, {\tt mass}, and {\tt center}
- of mass for the two fragments. The assignment to the fragments is
- recognized by checking whether the point is above or below the
- dividing line (variable {\tt diff}). Note that the $y$-component
- of the centers of mass is not calculated.
- \item Finally the iteration process is stopped if no two-body
- situation is found or the center-of-mass vectors have converged.
- \end{enumerate}
-
-\item{\bf Step 3:} The two-body analysis is now assumed to be complete
- and the physical quantities are evaluated. These are defined above
- in the list of module variables. The following calculation concerns
- the final scattering angle {\tt tets} extrapolated to infinity. It
- can be understood using the Rutherford trajectories.
-\end{itemize}
-
-\subsubsection{Subroutine {\tt getslope}}\Indexpb{getslope}
-This subroutine calculates the slope of the line determined by the
-eigenvector of largest quadrupole moment in the $(x,z)$-plane. The
-first loop sums up the quadrupole tensor, which is dimensioned {\tt
- q2(3,3)} but of which the index $2$ is not actually necessary, since
-the $y$-direction is not involved. The definition is kept
-three-dimensional to reduce confusion and make later generalization
-easier.
-
-The two-dimensional eigenvalue problem for {\tt q2} has the secular
-equation (remember $q_{31}=q_{13}$:
-$$(q_{11}-\lambda)(q_{33}-\lambda)-q_{13}^2=0,$$
-with $\lambda$ the eigenvalue. For the larger eigenvalue we get
-$$\lambda=\frac1{2}\left(q_{11}+q_{33}+\sqrt{(q_{11}-q_{33})^2
- -4q_{13}^2}\right),$$ and solving the equation
-$q_{13}x+(q_{33}-\lambda)z=0$ for $z$ yields
-\begin{eqnarray*}
- z&=&\frac{q_{13}}{\lambda-q_{33}}\,x\\
- &=& \frac{q_{13}}{\frac1{2}\left(q_{11}-q_{33}+\sqrt{(q_{11}-q_{33})^2
- -4q_{13}^2}\right)}.
-\end{eqnarray*}
-The code calls the denominator {\tt denom} and makes sure no division
-by zero happens (this could happen for a spherical distribution). The
-resulting slope is returned in the module variable \Indexv{slope}.
-
-\subsubsection{Function {\tt divpoint}}\Indexpb{divpoint}
-The function divpoint is a helper routine. It examines the line
-determined by {\tt slope} and intercept {\tt bb} and finds the point
-{\tt(xmin,zmin)} which is in the center of the void between the two
-fragments, returning .true. if this is possible.
-
-To this end it looks at the behavior of the densities along this line.
-Since the line has no relation to the numerical grid, this is not
-trivial.
-
-\begin{itemize}
-\item{First loop:} Essentially it looks through the $(x,z)$-plane to
- find points closer than half a grid spacing to the desired line with
- equation {\tt z=slope*x+bb} (logical variable {\tt online}). If the
- slope is larger than one, i.~e., if the nuclei are separating
- predominantly in the $x$-direction, we need to take the equation
- {\tt x=(z-bb)/slope} instead to get better resolution. To make the
- result monotonic along the line, the do loop in $z$ runs backward
- for negative slopes. The points found are collected in index vectors
- {\tt ixl}, {\tt izl} with accompanying densities {\tt rhol} stored
- in arrays of length {\tt il}.
-
-\item{Second loop:} now the number of fragments {\tt nf} is counted by
- examining this one-dimensional density curve, looking for
- disconnected density humps above {\tt vacuum} density. It is also
- recorded where the ``vacuum'' region starts and ends in variables
- {\tt n1} and {\tt n2}. The logic is as follows:
- \begin{enumerate}
- \item The logical variables {\tt in\_vacuum} keeps track of whether
- the search is in a vacuum region at the moment or not. It starts
- as {\tt TRUE}.
- \item Go to the next point. If its density is above {\tt vacuum},
- and we are in the vacuum, a new fragment is starting and we
- increase the number of fragments {\tt nf} by~1. If it becomes
- bigger than~2, exit, because there are three or more fragments.
- If it is now~2, record the starting index for the second fragment
- in {\tt n2}.
- \item If the density is below {\tt vacuum} and we are not in vacuum,
- a fragment is being ended. If it is the first fragment, we record
- this index in {\tt n1}.
- \end{enumerate}
-\item{Final processing} At this point we expect a two-fragment
- situation if {\tt nf=2} and in this case the void region between the
- fragments extends from {\tt n1} to {\tt n2}, which are indices into
- arrays {\tt ix1} and {\tt iz1} giving the position in the
- $(x,z)$-plane. The code calculates the midpoint between the two
- positions and returns {\tt TRUE} in this case.
-\end{itemize}
-
-\newpage
-\subsection{Module {\tt User}}\Indexmb{User}
-This is included as a place to insert arbitrary user initialization of
-the wave functions. It really does the same job as subroutine {\tt
- harmosc}, which is a relatively complicated example. For this reason
-a sample user initialization is also provided in the file {\tt
- user\_sample.f90}. It produces Gaussian wave functions for three
-alpha-particle-like nuclei separated by a distance {\tt d} and with
-radii {\tt r}.
-
-The only routine that has to be defined is {\tt user\_init}
-\Indexpb{user\_init}, but
-for more complicated initializations there can be any number of
-additional procedures accompanying it in {\tt user.f90}.
-
-The setup assumes that the relevant particle numbers {\tt nneut} and
-{\tt nprot} and numbers of states {\tt npmin}, {\tt npsi}, and {\tt
- nstmax} are set correctly using the static input. In this case we
-assume {\tt nprot=6}, {\tt nneut=6}, {\tt npmin=1,7}, {\tt npsi=6,12},
-and {\tt nstmax=12}. Note that the occupation numbers {\tt wocc}
-still have to be set explicitly, in this case they are all unity.
-
-The routine then reads the parameters for the setup from namelist {\tt
- user}. This namelist can be used to read anything desired. If there
-is no user initialization, it is simply omitted from the input file.
-
-Now there is a loop over center positions with index {\tt ic}. For
-each of them, the appropriate Gaussian is calculated and put into wave
-function \#{\tt ic} in the spin-up component; the spin-down component
-is set to zero. Then the Gaussian is copied into index position {\tt
- ic+3}, spin-down component, and finally the complete wave functions
-are copied to the proton indices by adding~6.
-
-There is no need to orthonormalize the wave functions, since {\tt
- schmid} is called before the static iterations are started. User
-initialization for the dynamic case does not appear useful; it could
-be easily done without modifying the code by running one static
-iteration and using the wave-function file generated at the beginning
-to initialize the dynamic calculation.
-
-\newpage
-\section{Input description}
-All the input is through {\tt NAMELIST} and many variables have
-default values. {\bf The {\tt NAMELIST}s should be in this file in the
- order in which they are described here, any {\tt NAMELIST} not used
- for a particular job may be omitted or left in the input file, in
- which case it is ignored}. The input is from a file called {\tt
- for005}, so the input data have to be produced with an editor. If the
-large output listing is to go into {\tt output}, the code should be
-run using, e.~g.,
-\begin{verbatim*}
-./sky3d.seq > output
-\end{verbatim*}
-The reason for not using redirected input is that in most MPI
-implementations for an input through ``{\tt < for005}''
-is passed only to
-node~0, while all nodes can read the same file in parallel using an
-explicit {\tt OPEN} statement.
-
-\subsection{Namelist {\tt files}}\Indexnb{files}
-This {\tt NAMELIST} contains names for the files used in the code.
-They are defined in module \Indexm{Params} and are:
-\begin{description}
-\item{\Indexvb{wffile}:} file to contain the static single-particle
- wave functions plus some additional data. This can be used for
- fragment initialization or for restarting a job. Default: {\tt
- 'none'}, i.~e., nothing is written.
-\item{\Indexvb{converfile}:} contains convergence information for the
- static calculation. Default: {\tt conver.res}.
-\item{\Indexvb{monopolesfile}:} contains moment values of monopole
- type. Default: {\tt monopoles.res}.
-\item{\Indexvb{dipolesfile}:} contains moment values of dipole type.
- Default: {\tt dipoles.res}.
-\item{\Indexvb{quadrupolesfile}:} contains moment values of
- quadrupole type. Default: {\tt quadrupoles.res}.
-\item{\Indexvb{momentafile}:} contains components of the total
- momentum. Default: {\tt momenta.res}.
-\item{\Indexvb{energiesfile}:} energy data for time-dependent calculations.
- Default: {\tt energies.res}.
-\item{\Indexvb{spinfile}:} time-dependent total, orbital, and spin
- angular-momentum data as three-dimensional vectors.
-\item{\Indexvb{extfieldfile}:}
-time dependence of expectation value of the external field.
-\end{description}
-
-\subsection{Namelist {\tt force}} \Indexnb{force}
-This defines the Skyrme force to be used. In most cases it should just
-uses the input values:
-\begin{description}
-\item[{\tt name}]: the name of the force, referring to the predefined
- forces in {\tt forces.data}.
-\item[{\tt pairing}]: the type of pairing, at present either {\tt
- NONE} for no pairing, \Indexv{VDI} for the volume-delta
- pairing, or \Indexv{DDDI} for density-dependent delta pairing.
- The pairing parameters are included in the force definition. {\bf Note
- that the pairing type must be written in upper case.}
-\item[{\tt turnoff\_zpe}]: if this is input as {\tt .TRUE.}, the
- zero-point energy correction is turned off independent of the setting
- of {\tt zpe} in the force definition. Its default value is {\tt .FALSE.}.
-\end{description}
-There is also the possibility for inputting a user-defined force; this
-is described in detail with module \Indexm{Forces}, see Section
-\ref{sec:Forces}.
-
-\subsection{Namelist {\tt main}}\Indexnb{main}
-This contains general variables applicable to both static and dynamic
-mode. They are mostly defined in module \Indexm{Params}.
-\begin{description}
-\item[\Indexv{tcoul}:] determines whether the Coulomb field should be
- included. Default is {\tt true}.
-\item[\Indexv{trestart}:] if {\tt true}, restarts the calculation from
- {\tt wffile}. Default is {\tt false}.
-\item[\Indexv{tfft}:] if {\tt true}, the derivatives of the wave
- functions, but not of the densities, are done directly through FFT.
- Otherwise matrix multiplication is used, but with the matrix also
- obtained from FFT. Default is {\tt true}.
-\item[\Indexv{mprint}:] control for printer output. If {\tt mprint} is
- greater than zero, more detailed output is produced every {\tt
- mprint} iterations or time steps on standard output.
-\item[\Indexv{mplot}:] if {\tt mplot} is greater than zero, a printer plot is
- produced and the densities are dumped every {\tt mplot} time steps
- or iterations. Default is~0.
-\item{\Indexv{mrest}:} if greater than zero, a {\tt wffile} is produced
- every {\tt mrest} iteration or time step. Default is~0.
-\item[\Indexv{writeselect}]: selects the output of densities by giving
- a string of characters choosing them (see subroutine
-\Indexp{write\_densities} for details. Default is {\tt 'r'}, i.~e., only
-the density is written.
-\item[\Indexv{write\_isospin}]: determines whether the densities should
- be output isospin-summed ({\tt false}) or separately for neutrons
- and protons ({\tt true}). Default is {\tt false}.
-\item[\Indexv{imode}]: selects a static {\tt imode=1} or dynamic {\tt
- imode=2} calculation.
-\item[\Indexv{nof}]: (number of fragments) selects the initialization.
- {\tt nof=0}: initialization from harmonic oscillator, only for the
- static case; {\tt nof<0}: user-defined initialization by subroutine
- \Indexp{init\_user} in module \Indexm{User}; {\tt nof>0}:
- initialization from fragment data as determined in {\tt NAMELIST}
- {\tt fragments}.
-\item{\Indexv{r0}:} nuclear radius parameter. The nuclear radius
- $R=r_0A^{1/3}$ is used to compute the $\beta$ and $\gamma$
- deformation parameters in subroutine \Indexp{moments}. Units: fm,
- default value 1.2~fm.
-\end{description}
-
-\subsection{Namelist {\tt grid}}\Indexnb{grid}
-This defines the properties of the numerical grid.
-\begin{description}
-\item[\Indexv{nx}, \Indexv{ny}, \Indexv{nz}]: number of grid points in the
- three Cartesian directions. They must be even numbers.
-\item[\Indexv{dx}, \Indexv{dy}, \Indexv{dz}]: spacing between grid points in fm.
- If only {\tt dx} is given in the input, all three grid spacings
- become equal.
- The grid positions are then set up to be symmetric with the
- coordinate zero centrally between point number {\tt nx/2} and {\tt
- nx/2+1}.
-\item[\Indexv{periodic}]: chooses a periodic ({\tt true}) or isolated
- ({\tt false}) system.
-\end{description}
-
-\subsection{Namelist {\tt static}}\Indexnb{static}
-These input variables control the static calculations.
-\begin{description}
-\item[\Indexv{tdiag}:] if {\tt true}, there is a diagonalization of the
- Hamiltonian during the later (after the 20th) static iterations. This
- 20 is hard coded in {\tt static.f90}. Default is {\tt false}.
-\item[\Indexv{tlarge}:] if {\tt true}, during the diagonalization the new
- wave functions are temporarily written on disk to avoid doubling the
- memory requirements. Default is {\tt false}.
-\item[\Indexv{nneut}, \Indexv{nprot}:] The numbers of neutrons and protons
- in the nucleus. These are used for the harmonic-oscillator
-and user initialization.
-\item[\Indexv{npsi}:] the numbers of neutron ({\tt npsi(1)}) and proton
- ({\tt npsi(2)}) wave functions actually used including unfilled
- orbitals. Again, useful only for harmonic-oscillator or user initialization.
-\item[\Indexv{radinx}, \Indexv{radiny}, \Indexv{radinz}:] the radius parameters of the
- harmonic oscillator in the three Cartesian directions, in fm.
-\item[\Indexv{e0dmp}:] the damping parameter. For its use see subroutine
- \Indexp{setdmc}. The default value is 100~MeV.
-\item[\Indexv{x0dmp}:] parameters controlling the relaxation. The
- default value is 0.2. In special cases it may be desirable to
- change this to accelerate convergence.
-\item[\Indexv{serr}:] this parameter is used for a convergence
- check. If the sum of fluctuations in the single-particle energies,
- \Indexv{sumflu} goes below this value, the calculation stops. A
- typical value is {\tt 1.E-5}, but for heavier systems and with
- pairing this may be too demanding.
-\end{description}
-
-\subsection{Namelist {\tt dynamic}}\Indexnb{dynamic}
-These are variables controlling the dynamic (TDHF) calculation.
-
-\begin{description}
-\item[\Indexv{nt}]: number of time steps to be run.
-\item[\Indexv{dt}]: the time step in fm/c. A standard value is of the
- order of 0.2 to 0.3~fm/c, it depends somewhat on the value of {\tt
- mxpact}. If the combination of these two is not good enough, the
- calculation becomes unstable after some time, in the sense that the
- norm of the wave functions and the energy drift off and can diverge
- (see Sect.~\ref{sec:accuracy}).
-\item[\Indexv{mxpact}]: the order of expansion for the exponential
- time-development operator. The predictor (trial) step calculation
- uses {\tt mxpact/2} as the order. For more information see
- Sect.~\ref{sec:accuracy}.
-\item[\Indexv{rsep}]: termination condition. If the final state in a
- two-body reaction is also of two-body character, the calculation is
- terminated as soon as the separation distance exceeds {\tt rsep}.
- Units: fm. No default. The purpose of this variable is to prevent
- the calulation of continuing into meaningless configurations, like
- crossing of the boundary.
-\item[\Indexv{texternal}]: indicates that an external perturbing field is
- used. In this case the namelist {\tt extern} must be
- present. Default: {\tt false}.
-\end{description}
-
-\subsection{Namelist {\tt extern}}\Indexnb{extern}
-The variables read here describe the external field that is applied to
-get the nucleus into a collective vibration. Details can be found in
-the description of module \Indexm{External}. It is read only if the
-parameter \Indexv{texternal} read in namelist \Indexn{dynamic} is
-true.
-\begin{description}
-\item[\Indexv{ipulse}]: the type of pulse applied. For {\tt ipulse=0} the
- wave function is multiplied with a phase factor that produces an
- initial excitation. For {\tt ipulse=1} a Gaussian time dependence is
- used, for {\tt ipulse=2} a $\cos^2$ one. Default: 0. Details are
- given in Eqs.~(\ref{eq:extgauss}) and (\ref{eq:extcos}).
-\item[\Indexv{isoext}]: isospin character of the excitation. If this is
- zero, protons and neutrons are exited in the same way. For a value
- of 1, they behave oppositely but with a coupling that leaves the
- center-of-mass invariant. Default: 0.
-\item[\Indexv{tau0}, \Indexv{taut}]: time at which the excitation field
- reaches its maximum, and width of the pulse. No defaults.
-\item[\Indexv{omega}]: if this is nonzero, the time-dependence of the
- external field gets an additional cosine factor with frequency {\tt
- omega}.
-\item[\Indexv{radext}, \Indexv{widext}]: radius and width of a
- Woods-Saxon-type cutoff factor in radius for the external field.
- Defaults: 100~fm and 1~fm, which practically implies no damping.
-Definition in Eq.~(\ref{eq:extdamp}).
-\item[\Indexv{amplq0}]: amplitude for quadrupole excitation of the
- $Q_{20}$ type. Defined as usual with respect to the $z$-axis.
-\end{description}
-
-\subsection{Namelist {\tt fragments}}\Indexnb{fragments}
-The variables in this namelist control fragment initialization for the
-case of \Indexv{nof}{\tt >0}. Most quantities are dimensioned for the fragments
-and we indicate this by index ``{\tt i}'' in the following.
-\begin{description}
-\item[\Indexv{filename}{\tt (i)}]: the name of the file containing the wave
- functions of fragment {\tt i}.
-\item[\Indexv{fcent}{\tt (1:3,i)}]: initial position of fragment {\tt i} given
- as three Cartesian coordinate values in fm. The position must be
- such that the complete fragment grid fits inside the new
- computational grid.
-\item[\Indexv{fix\_boost}]: used only for the two-fragment case. if this
- logical variable is {\tt TRUE}, the initial velocities are
- calculated from the {\tt fboost} values; otherwise from the relative
- motion quantities {\tt ecm} and {\tt b}.
-\item[\Indexv{fboost}{\tt (1:3,i)}]: the initial boost of the fragment in the three
- Cartesian directions. It is given as the total kinetic energy in
- each direction in MeV, with the sign indicating positive or negative
- direction. Thus {\tt SUM(ABS(fboost(:,i)))} is the total kinetic
- energy of fragment {\tt i}.
-\item[\Indexv{ecm}, \Indexv{b}]: center-of-mass kinetic energy in MeV and
- impact parameter in fm. Used only if {\tt fix\_boost} is {\tt FALSE}.
- These are the values at infinite distance
- and are corrected using Rutherford trajectories (assuming spherical
- nuclei) for initialization at the finite distance given by
- the {\tt fcent} coordinates.
-\end{description}
-
-\subsection{Namelist {\tt user}}\Indexnb{user}
-This namelist is read only if needed for user initialization (see
-module \Indexm{User}). Its contents depend on the specific user
-initialization and the only thing to be said here is that it should
-appear last in the input file. Since the namelist is defined and used
-only in module {\tt User}, its name can also be changed arbitrarily,
-of course.
-
-\section{Output description}
-\subsection{Output and analysis}
-The code produces a number of output files containing various pieces
-of information.
-The bulky observables, such as densities or currents, are selectively
-output at certain time steps into special binary output files
-{\em nnnnnn}{\tt .tdd}, where {\em nnnnnn} indicates the iteration or
-time step number. These files can then be used for further analysis or
-converted to be used as input in visualization codes. Examples of this
-are found among the utility codes provided.
-
-The complete set of wave functions is saved at regular
-intervals of {\tt mrest} iterations or time steps. Because this leads
-to large storage requirements, only the last such file in a run is
-kept. It can be used for restarting the calculation or for inputting
-fragment wave functions for initializing another calculation.
-
-In {\tt MPI} mode the wave functions are distributed over several
-files, each containing only those present on a specific processor. An
-additional header file contains the remaining information and can be
-used to read the wave functions even on a different processor
-configuration.
-
-Aside from these binary files there are a number of text files. The
-{\tt *.res} files contain one line for each time step or iteration
-where output is triggered according to the value of \Indexv{mprint}.
-There is an explanatory header line in these that has a leading '\#',
-so that is treated as a comment by {\tt gnuplot} --- thus {\tt
- gnuplot} can be used immediately to plot the behavior of any one
-column of numbers, i.~e., the dependence of a physical quantity on
-iteration or time. In addition more complicated output is printed on
-standard output, which can be redirected into a file using shell
-redirection.
-
-The names of the output files can be adjusted using input variables as
-listed in Section~\ref{sec:filenames}, so that the files are here
-denoted by the default names given there. The {\tt *.res} files are
-relatively small, so that no mechanism was implemented to suppress
-them.
-
-\subsection{File {\tt conver.res}}
-This is produced in \Indexp{sinfo} only in the static calculation and its
-purpose is to give a quick impression of the convergence behavior. The
-numbers given in each line are the iteration count, the total energy
-in MeV, the relative change in energy from one iteration to the next,
-the average uncertainties in the single-particle energies {\tt efluct1} and
-{\tt efluct2}, the root-mean-square radius in fm and finally the
-deformation parameters $\beta$ and $\gamma$ (see Section
-\ref{sec:mulmom}). The latter give an impression as to where the
-nuclear shape is ending up.
-
-For the judging of convergence, the {\tt efluct} values are more
-important than the change in total energy, since the energy can remain
-constant while the wave functions still change considerably.
-
-\subsection{File {\tt monopoles.res}}
-At present this file is generated in subroutine
-\Indexp{moment\_shortprint}, but only in the dynamic mode. It contains
-the time, the neutron, proton, and total root-mean-square radii, and
-the difference of neutron minus proton root-mean-square radii.
-
-\subsection{File {\tt dipoles.res}}
-This file is produced both in the static and dynamic calculations in
-subroutines \Indexp{sinfo} and \Indexp{tinfo}. It contains the iteration or
-time step number followed by the three components of the center of
-mass vector $\vec{R}$ and those of the difference of proton minus
-neutron center-of-mass vectors $\vec{R}^{(T=1)}$, both in fm, for the
-definition see Eq.~(\ref{eq:cmcart}). The first of these is useful as
-a check to see whether the center of mass drifts off during the
-calculation, while the second vector may be useful to look at proton
-vs. neutron vibrations.
-
-\subsection{File {\tt quadrupoles.res}}
-This is also generated in \Indexp{moment\_shortprint} and thus only in
-dynamic mode. It contains the Cartesian quadrupole moments
-(\ref{eq:Qcart}) for neutrons, protons, and the full mass distribution
-followed by the expectation values of $x^2$, $y^2$, and $z^2$ for
-neutrons and protons, all in fm$^2$.
-
-\subsection{File {\tt energies.res}}
-This is the important monitoring file for the dynamic calculation. It
-is written in subroutine \Indexp{tinfo} and each line contains the
-simulation time, the number of neutrons and protons in the system
-(these should be constant, so this is a stability check), the total
-energy (again, this should be conserved), the total kinetic energy,
-and finally the collective energy {\tt ecoll} separately for neutrons
-and protons --- see Eq.~(\ref{eq:ecoll}. Units for the energies are
-all MeV.
-
-\subsection{Standard output}
-This contains all the additional information that in most cases is not
-needed directly for further processing in, e.~g., graphics programs. If it should
-be found necessary to utilize some data from this file, it is in most
-cases easy to use {\tt grep} or a scripting language like {\tt Perl}
-or {\tt Python} to extract the necessary data. Of course the code can
-also be modified to produce additional output files.
-
-The initial part of the output essentially echoes all the data from
-the {\tt NAMELIST}s in tabular form, to enable checking the
-correctness of input data. In the case of fragment initialization this
-is more involved and is discussed below for the dynamic case, since it
-is not so common for static calculations.
-
-In general the layout of the information is compact with sequences of
-``*'' characters to provide separation between input groups as some
-guidance for the eye.
-
-\subsubsection{Static calculation}
-The code first prints the current iteration number. Iteration ``0'' refers
-to the state before iterations are started, for the later iteration
-numbers, the information refers to the end of the iteration.
-
-The overview of the various energy contributions is printed: the
-first part is similar to what is in file {\tt conver.res}, while a
-second list shows the energies calculated from the density functional
-and split up for the various contributions.
-
-Next there is a simple printer plot of the density distribution in the
-$(x-z)$-plane. This is often quite helpful, since it shows what is
-going on in the calculation without the need to start a graphics
-program, which requires converting the data first.
-
-Next there is a listing of single-particle states. For each state this
-shows its parity, occupation probability {\tt wocc} (which is called
-$v^2$, as it is interesting mostly in the pairing case), the energy
-fluctuations \Indexv{sp\_efluct1} and \Indexv{sp\_efluct2}, the norm, the kinetic and total
-energies of the state, and finally the expectation values of the three
-components of the orbital and spin angular momentum, respectively.
-
-Finally a summary of some integrated quantities is given, separately for
-neutrons, protons, and all nucleons: the particle number, the
-root-mean-square radius, quadrupole moment, and the average of the
-coordinates squared, followed by the center-of-mass components.
-
-Then iterations continue and only one line is printed for each, as
-this may be quite slow and it is important to be able to check
-progress while the code is running. After \Indexv{mprint} iterations
-the detailed information is repeated.
-
-\subsubsection{Dynamic calculation}
-After echoing the parameters for the dynamic calculation, the fragment
-definitions are given and all the resulting information is printed:
-the computed boost values in case of twobody initialization, the
-properties of the single-particle states read in, including which
-index in the fragment file is transferred to which index in the total
-set of wave functions.
-
-In case of an external field, the input data is also echoed in a
-detailed form.
-
-The time stepping starts and detailed output is produced every
-\Indexv{mprint} steps at the end of the time step. Much of it is similar to
-the static case, so only the differences are pointed out.
-
-Because in the dynamic case the situation can have a general
-three-dimensional character, the full information on the quadrupole
-tensor (\ref{eq:Qcart}) is printed, separately for the neutron,
-proton, and total mass distributions. The three eigenvalues and
-associated normalized eigenvectors are given, followed by Cartesian
-and polar deformation parameters $a_{0}$, $a_{2}$, $\beta$, and
-$\gamma$, as defined in Section~\ref{sec:mulmom}.
-
-The separation of the two fragments and its time derivative is printed
-next {\bf and repeated every time step}, as examining these quantities
-is meaningful only with more frequent sampling.
-
-The energy information is shortened by omitting the quantities not of
-interest in the dynamic case. After the printer plot the results of
-the two-body analysis are given (for the meaning of the various
-quantities see Section~\ref{sec:twobody}). The Section on ``collision
-kinematics'' shows the mass, charge, position, and kinetic energy of
-the two fragments. {\bf It should be kept in mind that the two-body
- analysis is only valid if the reaction plane is the $(x-z)$-plane
- and the results printed may not be useful if the physical situation
- is not of two-body nature but the code does not recognize that.}
-
-The single-particle property list and the integrated quantities are as
-in the static case, but the energy fluctuations are omitted.
-
-The next time step is then indicated and the fragment separation data
-are printed for every time step until after \Indexv{mprint} steps the
-full output recurs.
-
-\section{Utilities}
-A set of short programs is designed to help with further processing of
-output from the code. The currently available set is described here.
-
-Most of these routines contain a loop to input a file name from the
-terminal. If it is desired to do this in a loop over a set of files, a
-simple trick can be used: generate a list of file names using, e.~g.,
-
-{\tt ls -1 *.tdd > list}
-
-and then execute the program with ``list'' as input, e.~g.,
-
-{\tt ./fileinfo < list}
-
-For {\tt Tdhf2Silo} a script {\tt convert} is provided that handles
-this (see below). It can easily be adapted to the other utilities and
-must be stored in the same directory as the executable utility program
-itself in order for the {\tt dirname} command to work properly.
-
-\subsection{Fileinfo}
-This is a short program to print information about binary files
-generated by Sky3D. It takes the name of either a {\tt *.tdd} or a
-wave function file as input and prints out essentially all the
-information contained in the header. It can be compiled simply by
-executing
-
-{\tt gfortran -o fileinfo fileinfo.f90}
-
-\subsection{Inertia}
-This program calculates the tensor of inertia relative to the center
-of mass from the density distribution. It is intended as an example of
-an analysis code reading {\tt *.tdd} files and doing some computation,
-which can be used as a model for doing similar things. It illustrates
-looking for the desired field in the file and taking into account
-whether it is stored as a total density or isospin-separated. Being
-given a filename as input, it reads the density and calculates the
-inertia tensor to print all its 9~components. Compile it using
-
-{\tt gfortran -o Inertia Inertia.f90}
-
-\subsection{Cuts}
-This utility reads the density from a file {\em nnnnnn}{\tt .tdd} file and produces
-output files named {\em nnnnnn}{\tt rxy.tdd}, {\em nnnnnn}{\tt
- rxz.tdd}, and {\em nnnnnn}{\tt ryz.tdd}, which contains
-two-dimensional cuts through the system in the $(x,y)$, $(x,z)$, and
-$(y,z)$ plane, respectively. The cuts are evaluated at the origin for
-the third coordinate by averaging the two neighboring planes.
-
-These data files are written in such a format that they can be read
-by {\tt gnuplot} for use in its commands for 2-dimensional plotting.
-
-This program is intended again as a template that can be modified for
-other applications.
-
-\subsection{Overlap}
-This is a code to calculate the overlap of two Slater determinants.
-Given the names of two wave function files (which must contain
-compatible data: dimensions, force, etc.) it reads the wave functions,
-generates the matrices of overlaps between one set and the other
-separately for neutrons and protons, and then calculates the
-determinant of each, which is the overlap between the two Slater
-determinants. It prints some summary information: distance between the
-centers of mass of each set, minimum and maximum diagonal elements,
-maximum absolute value of off-diagonal elements, and finally the
-overlaps for protons and neutrons as well as their product.
-
-This code uses subroutines from {\tt LINPACK} (stored at {\tt
- NETLIB.ORG}), which are included in the file {\tt det.f} with
-appropriate copyright. It can be compiled using
-
-{\tt gfortran -o overlap overlap.f90 det.f}.
-
-\subsection{Tdhf2Silo}
-\label{sec:silo}
-This program is quite complicated. It reads a set of {\tt *.tdd} files
-and converts them into {\tt Silo} files. {\tt Silo} is a library for
-handling scientific datasets developed at Lawrence Livermore National
-Laboratory ({\tt https://wci.llnl.gov/codes/silo/index.html}). This is
-the most appropriate library to use in conjunction with the LLNL
-graphics visualization tool {\tt VisIt} ({\tt
- https://wci.llnl.gov/codes/visit/home.html}), which was found to be
-highly suitable for plotting {\tt Sky3D} results and producing
-movies. It should be noted that a copy of the included file {\tt
- silo.inc} is provided, which defines symbolic names for the various
-parameters used in the library calls. This file is from Silo version
-4.9. {\bf Older versions of this file may cause problems as they use
-fixed-format Fortran style; the present version of {\tt silo.inc}
-should, however, also work with older
-versions of the library.}
-
-The conversion code is quite flexible in that it decides what
-to produce for the different field types: isospin-summed or not,
-vector or scalar. They are given appropriate names for {\tt Silo} with
-suffixes p and n for protons and neutrons, and x, y, z for the vector
-components. In the case of vector fields a variable containing the
-vector definition is also written so that the field can be plotted
-immediately as a vector field in {\tt VisIt}.
-
-If the user wants another dataset handling method, the code should be
-readily adaptable to other libraries. {\tt VisIt} itself has many ways
-of importing data, but of course there are also alternative 3D
-visualization systems.
-
-\newpage
-\section{Running the code}
-\subsection{Compilation and linking}
-To produce executable files the code comes with several {\tt
- Makefile}s. The standard {\tt Makefile} produces a sequential code
-{\tt sky3d.seq}, the file {\tt Makefile.openmp} a parallel code using
-{\tt OpenMP}, while {\tt Makefile.mpi} should produce an {\tt MPI}
-distributed system code.
-
-The {\tt Makefile}s are written for the {\tt gfortran} compiler and
-the commands and options must be adapted if other compilers are used.
-{\bf The user may also have to modify the library names and execution
- of the code under MPI will require consulting the local
- documentation or system administrator.}
-
-No attempt was made to select the compiler and linker options
-optimally for speed, since experience has shown that optimization at
-the cutting edge is highly time-dependent. Thus users should do some
-speed tests before embarking on major calculations.
-
-\subsection{External libraries needed}
-The {\tt LAPACK} library is used in the code to supply the routines
-{\tt ZHBEVD} and {\tt DSYEV}. {\tt LAPACK} should be installed in most
-scientific computing centers; if not, the files can be obtained from
-{\tt www.netlib.org} and just be added as additional source files to
-the code. Note that a complete set of routines called by these two
-subroutines must be downloaded.
-
-The other external routine library that is used is {\tt FFTW3}. Again,
-it will be preinstalled in most systems. If not, there are two
-possibilities:
-\begin{enumerate}
-\item Download the source code from {\tt www.fftw.org} and compile the
- library yourself. In our experience this worked smoothly. The
- generated library can be installed in a system library directory or
- kept in a user account. In the latter case the use of {\tt -lfftw3}
- in the makefiles does not work anymore and the full path name of the
- library file must be given.
-\item Replace it by another FFT routine. This requires quite a bit of
- work: {\tt FFTW} organizes its calls around ``plans'', which
- describe a set of operations to be done on the three-dimensional
- arrays. In \Indexp{init\_fft} quite sophisticated plans are set up to,
- for example, transform in the y-direction for all x- and
- z-values. This means that all calls to subroutines beginning with
- {\tt dfftw} have to be examined and possibly replaced by loops over
- one-dimensional FFT transforms. This should be relatively
- straightforward, but there are two more important points to
- consider: 1)~normalization differs between FFT codes. For {\tt FFTW}
- transformation followed by inverse transformation multiplies the
- original data by {\tt nx*ny*nz} and this factor is taken into
- account in several places. 2)~For the non-periodic case the Fourier
- transform in the Coulomb solver uses doubled dimensions in all three
- directions. Some FFT codes have an initialization that sets up the
- transformation factors depending on the dimension; in such cases
- the initialization may have to be repeated.
-\end{enumerate}
-
-\subsection{Running with {\tt OPENMP}}
-The {\tt OPENMP} version can be compiled using the file {\tt
- Makefile.openmp}, which produces an executable {\tt sky3d.omp}. The
-main difference to the sequential makefile is the addition of an
-openmp compiler option. Since this depends on the compiler used, it
-may have to be modified. For {\tt gfortran} the option is {\tt
- -fopenmp}, while for Intel Fortran it is simply {\tt -openmp}.
-
-For controlling the running of the code the user should set the
-environment variable {\tt OMP\_NUM\_THREADS} to the number of parallel
-threads to be used (usually the number of processors). In addition it
-may be necessary to set {\tt OMP\_STACKSIZE}. The two parallel loops
-in {\tt dynamichf} need to store all the density fields in parallel,
-and the second loop adds {\tt ps4} to that. Taking into account vector
-fields and isospin, a total of~24 three-dimensional {\tt COMPLEX(8)}
-fields need to be stored, amounting to {\tt nx*ny*nz*24*16} bytes,
-which is the stack size needed.
-
-\subsection{Running under {\tt MPI}}
-The situation for {\tt MPI} is a bit more complex than for {\tt
- OPENMP}, so that the file {\tt Makefile.mpi} will almost certainly
-have to be modified. One crucial difference to the other makefiles is
-that the module {\tt Parallel} is now generated from the source file
-{\tt parallel.f90}. In addition the compilation commands have to be
-adapted; something like {\tt mpif90} will be needed but is
-installation dependent. In addition a command like {\tt mpirun} will
-be needed for execution; the user is advised to consult local
-documentation.
-
-\subsection{Required input}
-Here it is just summarized what input is needed for a static or dynamic
-calculation. A full description can be found with the documentation
-for the {\tt NAMELIST}s.
-
-\subsubsection{Static calculation}
-
-The {\tt NAMELIST}s needed are, in that order:
-\begin{quote}
-\Indexn{files}, \Indexn{force}, \Indexn{main}, \Indexn{grid},
-\Indexn{static}.
-In addition, if initialization is from fragments, \Indexn{fragments},
-and for user initialization possibly \Indexn{user} (only if the user
-initialization requires input).
-\end{quote}
-
-\subsubsection{Dynamic calculation}
-The {\tt NAMELIST}s needed are, in that order:
-\begin{quote}
-\Indexn{files}, \Indexn{force}, \Indexn{main}, \Indexn{grid},
-\Indexn{dynamic}, For external field excitation \Indexn{extern}, and
-in all cases \Indexn{fragments}.
-\end{quote}
-
-\subsection{Test cases}
-To allow checking the proper behavior of the code, we provide three
-test cases exercising different functions: a static calculation for
-the ground state of $^{16}$O, a dynamic calculation using an external
-excitation to stimulate a giant resonance in $^{16}$O, and finally a
-sample deep-inelastic collision of two $^{16}$O nuclei. The test cases
-directory contains a more detailed description of these cases and what
-to look for principally in the results. Note that since the
-calculations are quite large-scale, differences in roundoff errors may
-lead to the output not being quite identical to the samples provided.
-
-
-\newpage
-\section{Caveats concerning the code}
-The user should be aware of the limitations of the code in various
-respects but also note some less straightforward procedures for
-improving accuracy in certain cases.
-
-\subsection{Static calculations}
-Since the main use of the code is expected to be in time-dependent
-calculations, the static part is less highly developed. The omission
-of all symmetry restrictions, while very useful for innovative
-applications with time-dependence, can cause some problems in the
-static case.
-\begin{enumerate}
-\item The spin is not aligned along a fixed direction. Since for
- even-even nuclei Kramers degeneracy operates, two degenerate levels
- will mix in an uncontrolled way to produce an arbitrary spin
- alignment. This can be remedied by diagonalizing the spin operators
- in such a two-level subspace.
-\item The center of mass may move away from the origin during the
- iterations. This is typically a very small effect and will be
- corrected when the wave functions are placed at a given position in
- the dynamic initialization. The calculation of observables, however,
- should always use coordinates relative to the real center of mass.
-\item In very heavy nuclei sometimes even a rotation was seen, as the
- reoccupation of high-lying levels can change the geometric
- orientation. If this is a serious problem, constraints should be
- introduced or another code used for the static calculation.
-\item The harmonic oscillator initialization can be quite deficient
- for heavier nuclei. It is planned to develop a Nilsson-model
- alternative; meanwhile in case of problems the use of wave functions
- from axial or symmetry-restricting codes could be implemented by
- generating a wave function file from such results. Since this will
- involve interpolation, a number of static iterations should then be
- performed using the present code to improve stability.
-\end{enumerate}
-
-\subsection{Dynamic calculations}
-Here it is important what the required accuracy will be. Most
-exploratory calculations will not pose high accuracy demands. There
-are several possible ways in which accuracy can be improved if needed:
-\begin{enumerate}
-\item The initial configuration may be improved. If the fragment
- nuclei are not situated at grid points, one should run a number of
- static iterations with each fragment situated alone in the new
- grid.
-\item If the fragments are deformed, the energy estimate from the
- Rutherford trajectory will not be reliable; in this case it is
- recommended to run a number of dynamic iterations, observe the
- change in relative distance, and then correct the boost energies to
- match the correct velocity of relative motion.
-\end{enumerate}
-
-\section{Modifying the code}\label{sec:modify}
-Since the code was developed with a view for easy modification, in this
-Section we give some advice on how to add new things to it and how to
-run simulations.
-
-\subsection{Modifying the Skyrme force}
-The code comes with a quite large database of Skyrme force
-parametrizations together with appropriate pairing parameters built
-in. This is certainly useful to avoid mistakes in the input by having
-to indicate only the name of the force. Still there will be a need to
-add new forces and even forces with a different density functional to
-it, for which we suggest three different approaches.
-
-\subsubsection{Direct parameter input}
-If a specific parametrization is not expected to be a permanent
-addition to the code, for example if one or several parameters are
-varied to study the sensitivity of the results to specific parts of
-the density functional, the best way is to use the facility for giving
-the parameters directly in the input. This is triggered by using some
-force name that is not in the database, in which case the routine
-expects all parameters to be given in {\tt NAMELIST forces}.
-
-\subsubsection{Expanding the database}
-At present the database of forces is contained in the file {\tt
- forces.data} as a long initialization statement. This has the
-advantage of readability and avoids having to make a database file
-accessible in every directory used for code applications.
-
-So a new force which will be used more permanently can be added simply
-by adding the appropriate lines in {\tt forces.data} and increasing
-the number assigned to {\tt nforce} accordingly. There is a slight
-danger that the total length of the list will exceed the 255 lines
-allowed by the Fortran standard; in this case either remove some
-outdated forces, remove the separation lines of all stars, or if there
-is still a problem, convert the initialization to {\tt DATA}
-statements initializing a smaller number of forces in each case.
-
-\subsubsection{Adding new physics to the density functional}
-This can of course require a lot more modifications to the code.
-Generally speaking, such a new term will appear not only in the
-density functional, but also leads to new contributions in the
-single-particle Hamiltonian and may require new types of densities and
-currents. In addition, it will involve new force parameters. So in
-general the following steps will be needed:
-\begin{enumerate}
-\item {\bf Parameter definition:} The new parameter can be added to
- the general {\tt Force} type definition in {\tt forces.f90}. This is
- the most logical and systematic way, but we recommend it only if the
- new physics is to be there permanently, since in this case the whole
- database has to be updated to give a default value --- probably zero
- --- to the new parameter for each existing force. The alternative is
- to leave this parameter as a separate entity, which can still be a
- module variable in {\tt Forces} and be input using the same
- namelist. Derived parameters should also be calculated here.
-
-\item {\bf New densities and currents:} everything that can be defined
- directly as a sum over the occupied single-particle wave functions
- should be defined and calculated in module {\tt Densities}. It
- should be a module variable and allocated during initialization
- similar to the densities already defined. Then the contributions of
- the different derivatives of the wave function can be accumulated
- separately as is done for the existing contributions.
-
-\item {\bf Calculation of mean-field components:} subroutine {\tt
- skyrme} in module {\tt Meanfield} is the place where the fields
- appearing in the single-particle Hamiltonian are calculated. The
- difference to module {\tt Densities} is that wave functions are not
- involved directly, but only combinations and derivatives of the
- densities and currents need to be evaluated. The new fields should
- be defined and allocated and then calculated in subroutine {\tt
- skyrme}. The handling of derivatives again can be imitated based
- on the existing terms.
-
-\item {\bf Single-particle Hamiltonian:} The additional contribution
- to the single-particle Hamiltonian must be calculated in subroutine
- {\tt hpsi} of module {\tt Meanfield}. Code has to be added to calculate
- how the additional terms act on the input wave function {\tt pinn}
- and the result has to be added to the output wave function {\tt
- pout}. Again, for efficiency the spatial derivatives can be
- handled in separate loops.
-
-\item {\bf Contribution to the energy:} subroutine {\tt integ\_energy}
- in module {\tt Energies} must include an additional contribution of
- the new term, which in this case means simply computing the
- expression for the energy functional. Probably it will be useful to
- add this up in some new variable (defined as a module variable), so
- that the contribution of the new term can be printed out together
- with the other contributions in subroutines {\tt sinfo} of module
- {\tt Static} or subroutine {\tt tinfo} of module {\tt Dynamic},
- respectively.
-
-\item {\bf Output of densities:} it may be desirable to write out the
- new densities, currents, or mean-field contributions into the {\tt
- *.tdd} files. To do that, subroutine {\tt write\_densities} in
- module {\tt Inout} must be modified. A new letter to use for {\tt
- writeselect} must be defined and selected in the {\tt SELECT CASE}
- statement, and then depending on whether it is a scalar or a vector
- field, {\tt write\_one\_density} or {\tt write\_vec\_density} is
- called with a descriptive name given to the field.
-
- The complication that occurs here is that these writing routines
- assume isospin-dependent fields and output either isospin-separate
- or isospin-summed fields depending on the value of {\tt
- write\_isospin}. If the field has no isospin dependence, the
- statements used to output {\tt wcoul} should be imitated.
-\end{enumerate}
-
-\subsection{Using constraints in the static calculation}
-There are many situations in which it is useful to solve the static
-Hartree-Fock equations with added constraints. The most well-known
-application is a quadrupole constraint, in which the expectation value
-$\langle\Psi|\hat H-\lambda\hat Q_{20}|\Psi\rangle$ is minimized to
-obtain deformed states of the nucleus. In the simplest case the
-Lagrange multiplier $\lambda$ may be kept fixed, in which case one has
-to accept the resulting quadrupole moment, or, alternatively, some
-iterative change in $\lambda$ is applied to make the solution converge
-to a desired value for $\langle\Psi|\hat Q_{20}|\Psi\rangle$.
-
-The various ways of introducing a constraint are discussed extensively
-in chapter 7.6 of \cite{Rin80aB}, so that here we only briefly discuss
-practical considerations for adding a constraint to Sky3D.
-
-Adding a constraint corresponds to including another potential term in
-the single-particle Hamiltonian, e.~g., for the quadrupole case
-\begin{equation}
- \hat h\longrightarrow\hat h-\lambda (2z^2-x^2-y^2)
-\end{equation}
-This could be implemented essentially analogously to the use of an
-external exciting field in the time-dependent case. Construct a
-subroutine to compute this external potential and add it to the
-mean-field potential {\tt upot}. This should be applied every time the
-single-particle Hamiltonian is applied to a wave function in module
-{\tt Static}, i.~e., after each call to subroutine {\tt skyrme}. Then
-an additional subroutine should be written which is called at the end
-of each iteration to compute the expectation value of the constraining
-operator (if not already done elsewhere in the code) and adjust the
-value of $\lambda$ if necessary. The method for adjusting $\lambda$
-will depend on the specific constraint used \cite{Cus85a,Sta10a}.
-
-If the constraint is generated by some thing more complicated than a
-scalar potential, like the orbital and spin operators in the case of
-cranking, it will be more convenient to implement the constraint
-inside the subroutine {\tt hpsi} at those places where the appropriate
-derivatives are available.
-
-Since the numerical method used does not restrict deformation, care
-must be taken for unstable constraints like the quadrupole, which can
-go to large values at the edges of the computational box and may pull
-the wave functions there. If that becomes a problem, the operator
-should be damped suitably \cite{Rut95a}.
-
-\subsection{Analyzing the results in new ways}
- The code provides quite a large number of physical observables in
- its output files. For new applications it may be necessary to look
- at additional ones. There are essentially two ways to implement
- this:
- \begin{enumerate}
- \item If the new observable depends only on density and mean-field
- components, the easiest way is to use the {\tt *.tdd} files, where
- if necessary more fields can be output by modifying the subroutine
- \Indexp{write\_densities}. As an example for reading the {\tt *.tdd}
- files, we provide a code calculating the tensor of inertia among
- the utilities.
- \item It becomes more complicated if the wave functions have to be
- used. Here one possibility is to add a call to a user-written
- routine at the beginning of the static or dynamic loops, which
- then can use the array \Indexv{psi} in any way desired. This may
- not be a good option if the calculations are lengthy and the
- analysis routine may have to be modified several times. In this
- case it is better to generate a new file name for the {\tt wffile}
- each time \Indexp{write\_wavefunctions} is called, similar to the
- way it is done in \Indexp{write\_densities}, and to do the
- analysis by reading the wave function files.
- \end{enumerate}
-
-\section*{Acknowledgments}
-This work was supported by the Bundesministerium f\"ur Bildung und
-Forschung under contract No.\ 05P12RFFTG, by the UK STFC under grant
-number ST/J000051/1, and by the U.S. Department of Energy under grant
-No. DE-FG02-96ER40975 with Vanderbilt University.
-
-\bibliographystyle{elsarticle-num}
-\bibliography{Documentation}
-
-\printindex{M}{Index of modules} \printindex{P}{Index of procedures}
-\printindex{V}{Index of variables} \printindex{N}{Index of namelists}
-\end{document}
diff --git a/Doc/Makedoc b/Doc/Makedoc
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+ {clw x1 x0 gt {/x1}{/x0}ifelse addto clw 2 div dup /y0 addto /y1 addto}
+ {clw y1 y0 gt {/y1}{/y0}ifelse addto clw 2 div dup /x0 addto /x1 addto}
+ ifelse
+ x0 y0 M
+ x1 y1 L
+ mstroke
+}bdef
+
+/FillPath{ % graylevel FillPath -- fills current path and leaves path alive
+ gsave
+ dup 1 exch sub % glev 1-glev
+ rbak 2 index mul rfor 2 index mul add
+ gbak 3 index mul gfor 3 index mul add
+ bbak 4 index mul bfor 4 index mul add
+ Mysetrgbcolor
+ pop pop
+ fill
+ grestore
+}bdef
+
+/StrokePath{ % linejoin StrokePath -- strokes current path and leaves path alive
+ gsave
+ setlinejoin
+ mstroke
+ grestore
+}bdef
+
+/RectPath{ % x0 y0 x1 y1 RectPath -- sets current path to a rectange
+ /y1 xdef /x1 xdef /y0 xdef /x0 xdef
+ newpath
+ x0 y0 M
+ x1 y0 L
+ x1 y1 L
+ x0 y1 L
+ closepath
+}bdef
+
+/PaintRect{ % x0 y0 x1 y1 graylevel PaintRect --
+ gsave
+ 5 1 roll
+ RectPath
+ FillPath
+ grestore
+}bdef
+
+
+
+
+% **** begin image stuff *****
+% The following are used for images. This has to be kept
+% in synch with corresponding code in ImageDisp.c.In particular
+% the expansion factor, 8, has to be in synch. LH010606: now to
+% fix colorscales in layouts, expansion factor is in variable.
+% packedcolor is (red*256+green)*256+blue
+
+/ExpFact 8 def % default expansion factor
+
+/SetPackedColor{ % packedcolor SetPackedColor --
+ dup 256 mod /tmp0 xdef % extract blue
+ 256 idiv
+ dup 256 mod /tmp1 xdef % extract green
+ 256 idiv % extract red
+ 255 div /packedRed xdef
+ tmp1 255 div /packedGreen xdef
+ tmp0 255 div /packedBlue xdef
+}bdef
+
+/IR_FillRect{
+ gsave
+ newpath
+ rx0 ry0 M
+ rx1 ry0 L
+ rx1 ry1 L
+ rx0 ry1 L
+ closepath
+ packedRed packedGreen packedBlue Mysetrgbcolor
+ fill
+ grestore
+}bdef
+
+/NewRectImageLine{ % x0 y0 x1 y1 packedcolor NewRectImageLine --
+ SetPackedColor
+
+ /ry1 xdef /rx1 xdef /ry0 xdef /rx0 xdef % remember starting rectange for following calls
+
+ IR_FillRect
+}bdef
+
+/rx{ % xdist rx --
+ ExpFact div
+ dup 0 lt
+ { rx0 /rx1 xdef rx1 add /rx0 xdef } % neg delta steps left
+ { rx1 /rx0 xdef rx0 add /rx1 xdef }
+ ifelse
+ IR_FillRect
+}bdef
+
+/ry{ % ydist rr --
+ ExpFact div
+ dup 0 lt
+ { ry1 /ry0 xdef ry0 exch sub /ry1 xdef } % neg delta steps down
+ { ry0 /ry1 xdef ry1 exch sub /ry0 xdef }
+ ifelse
+ IR_FillRect
+}bdef
+
+
+/cx{ % xdist packedcolor cx --
+ SetPackedColor
+ rx
+}bdef
+
+/cy{ % ydist packedcolor cx --
+ SetPackedColor
+ ry
+}bdef
+
+% **** end image stuff *****
+
+
+
+/EraseRect{ % x0 y0 x1 y1 EraseRect -- fills rect with background color
+ gsave
+ RectPath
+ UseBack fill
+ grestore
+}bdef
+
+/FrameRect{ % x0 y0 x1 y1 FrameRect --
+ RectPath
+ mstroke
+}bdef
+
+/FrameMacRect{ % x0 y0 x1 y1 FrameMacRect -- Mac-like version
+ RectPath
+ mstroke
+}bdef
+
+
+/ellipse{
+ /savematrix matrix currentmatrix def
+ translate scale 0 0 1 0 360 arc
+ savematrix setmatrix
+}bdef
+
+
+
+/OvalPath{ % x0 y0 x1 y1 OvalPath -- sets current path to an oval inside rectange
+ /y1 xdef /x1 xdef /y0 xdef /x0 xdef
+ newpath
+ x0 x1 sub abs 2 div y0 y1 sub abs 2 div x0 x1 add 2 div y0 y1 add 2 div ellipse
+ closepath
+}bdef
+
+
+/PaintOval{ % x0 y0 x1 y1 graylevel PaintOval --
+ gsave
+ 5 1 roll
+ OvalPath
+ FillPath
+ grestore
+}bdef
+
+/EraseOval{ % x0 y0 x1 y1 EraseOval -- fills with background color
+ gsave
+ OvalPath
+ UseBack fill
+ grestore
+}bdef
+
+/FrameOval{ % x0 y0 x1 y1 FrameOval --
+ OvalPath
+ mstroke
+}bdef
+
+
+/RRectPath{ % x0 y0 x1 y1 r OvalPath -- sets current path to rounded rectange, radius r
+ /radius xdef
+ /y1 xdef /x1 xdef /y0 xdef /x0 xdef
+ newpath
+ x0 radius add y0 M
+ x1 y0 x1 y1 radius arcto 4{pop}repeat
+ x1 y1 x0 y1 radius arcto 4{pop}repeat
+ x0 y1 x0 y0 radius arcto 4{pop}repeat
+ x0 y0 x1 y0 radius arcto 4{pop}repeat
+ closepath
+}bdef
+
+
+/PaintRRect{ % x0 y0 x1 y1 r graylevel PaintRRect --
+ gsave
+ 6 1 roll
+ RRectPath
+ FillPath
+ grestore
+}bdef
+
+/EraseRRect{ % x0 y0 x1 y1 r EraseRRect -- fills rect with background color
+ gsave
+ RRectPath
+ UseBack fill
+ grestore
+}bdef
+
+/FrameRRect{ % x0 y0 x1 y1 r FrameRect --
+ RRectPath
+ mstroke
+}bdef
+
+
+
+
+/CapLine{ % x0 y0 x1 y1 linecap CapLine -- draws line using given line cap
+ gsave
+ setlinecap
+ /y1 xdef /x1 xdef /y0 xdef /x0 xdef
+ newpath
+ x0 y0 M
+ x1 y1 L
+ mstroke
+ grestore
+}bdef
+
+/Arrow{ % x0 y0 x1 y1 width height graylevel Arrow -- draws arrow at end of line from P0 to P1
+ gsave
+ /gl xdef 1 add /aw xdef 1 add /ah xdef /y0 xdef /x0 xdef /y1 xdef /x1 xdef
+ x0 y0 translate
+ /x1 x1 x0 sub def
+ /y1 y1 y0 sub def
+ y1 x1 atan rotate
+ newpath
+ 0 0 M
+ aw ah 2 div L
+ aw ah 2 div neg L
+ closepath
+ gl FillPath
+% 10 setmiterlimit
+% 1 ps
+% 0 StrokePath
+ grestore
+}bdef
+
+
+% ************ fonts ************
+
+
+/normal 0 def
+/bold 1 def
+/italic 2 def
+/underline 4 def
+/outline 8 def
+/shadow 16 def
+
+/FACE1 bold italic or def
+/FACE2 outline shadow or def
+
+/Shadow {
+ gsave
+ siz 0.08 mul 1 index mul x add
+ exch siz 0.07 mul mul y add M
+ c show
+ grestore
+} def
+
+/Outline {
+ gsave
+ siz .02 mul setlinewidth
+ x y M c true charpath
+ gsave UseBack fill grestore stroke
+ grestore
+} def
+
+/Underline{
+ gsave
+ siz 0.08 mul ps
+ x y clw add M x c stringwidth pop add y clw add L stroke
+ grestore
+}bdef
+
+/DoShow{ % assumes x,y,c are already set
+ fontkind FACE2 and
+ dup normal eq {c x y M show} if
+ dup outline eq {Outline} if
+ dup shadow eq {1 Shadow Outline} if
+ outline shadow or eq {1 Shadow 1.5 Shadow Outline} if
+ fontkind underline and 0 ne { Underline } if
+}bdef
+
+/DoWShow{ % assumes x,y,c & w are already set
+ gsave
+ x y translate
+ /x 0 def /y 0 def
+ w c stringwidth pop div 1 scale
+ DoShow
+ grestore
+}bdef
+
+/S{
+ /y xdef /x xdef /c xdef
+ DoShow
+}bdef
+
+/WS{ % string x y desiredwidth WS -
+ /w xdef /y xdef /x xdef /c xdef
+ DoWShow
+}bdef
+
+/F{
+ /fontkind xdef
+ findfont exch /siz xdef [siz 0 0 siz neg 0 0] makefont setfont
+}bdef
+
+
+
+/Re-encode {
+ dup length array copy
+ /eap 0 def
+ exch {
+ dup type /integertype eq
+ { /eap xdef }
+ {
+ 1 index
+ exch
+ eap exch
+ put
+ /eap eap 1 add def
+ }
+ ifelse
+ } forall
+} bind def
+
+[ 39/quotesingle 96/grave 128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis
+/Udieresis/aacute/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute
+/egrave/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde
+/oacute/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex
+/udieresis/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls
+/registered/copyright/trademark/acute/dieresis/notequal/AE/Oslash
+/infinity/plusminus/lessequal/greaterequal/yen/mu/partialdiff/summation
+/product/pi/integral/ordfeminine/ordmasculine/Omega/ae/oslash
+/questiondown/exclamdown/logicalnot/radical/florin/approxequal/Delta/guillemotleft
+/guillemotright/ellipsis/space/Agrave/Atilde/Otilde/OE/oe
+/endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide/lozenge
+/ydieresis/Ydieresis/fraction/Euro/guilsinglleft/guilsinglright/fi/fl
+/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand
+/Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex/Idieresis/Igrave
+/Oacute/Ocircumflex/apple/Ograve/Uacute/Ucircumflex/Ugrave/dotlessi/circumflex/tilde
+/macron/breve/dotaccent/ring/cedilla/hungarumlaut/ogonek/caron]
+/zzz xdef
+
+/OurEncoding{ zzz StandardEncoding Re-encode }bdef
+
+/EncodeFont {
+ findfont
+ dup length dict begin
+ {
+ 1 index /FID ne
+ { def }
+ { pop pop }
+ ifelse
+ } forall
+ /FontName xdef
+ /Encoding xdef
+ FontName currentdict
+ end
+ definefont pop
+} bind def
+
+%%EndResource
%%BeginFont: /OF0
11 dict begin
/FontName /OF0 def
% TT Name: Geneva
/FontType 3 def
/FontMatrix [0.001 0 0 -0.001 0 0] def
/FontBBox [0 -803 830 182] def
/Encoding 256 array def 0 1 255{Encoding exch/.notdef put}for
Encoding 32 /s32 put
Encoding 33 /s33 put
Encoding 34 /s34 put
Encoding 35 /s35 put
Encoding 36 /s36 put
Encoding 37 /s37 put
Encoding 38 /s38 put
Encoding 39 /s39 put
Encoding 40 /s40 put
Encoding 41 /s41 put
Encoding 42 /s42 put
Encoding 43 /s43 put
Encoding 44 /s44 put
Encoding 45 /s45 put
Encoding 46 /s46 put
Encoding 47 /s47 put
Encoding 48 /s48 put
Encoding 49 /s49 put
Encoding 50 /s50 put
Encoding 51 /s51 put
Encoding 52 /s52 put
Encoding 53 /s53 put
Encoding 54 /s54 put
Encoding 55 /s55 put
Encoding 56 /s56 put
Encoding 57 /s57 put
Encoding 58 /s58 put
/BoundingBoxes 27 dict def
BoundingBoxes begin
/s32 [38 -345 341 -269] def
/s33 [148 -758 392 0] def
/s34 [87 -129 216 0] def
/s35 [99 -758 568 23] def
/s36 [57 -780 610 23] def
/s37 [72 -758 526 0] def
/s38 [45 -561 504 15] def
/s39 [72 -561 830 0] def
/s40 [0 0 0 0] def
/s41 [114 -803 360 182] def
/s42 [72 -758 716 0] def
/s43 [45 -561 522 15] def
/s44 [15 -758 712 0] def
/s45 [87 -803 333 182] def
/s46 [83 -780 561 23] def
/s47 [87 -780 576 0] def
/s48 [74 -780 593 23] def
/s49 [87 -780 595 23] def
/s50 [53 -758 614 0] def
/s51 [45 -788 538 15] def
/s52 [72 -758 163 0] def
/s53 [38 -561 464 15] def
/s54 [34 -674 405 15] def
/s55 [53 -561 504 15] def
/s56 [72 -561 519 0] def
/s57 [42 -788 378 0] def
/s58 [38 -780 303 -515] def
end
/Metrics 27 dict def
Metrics begin
/s32 379 def
/s33 666 def
/s34 303 def
/s35 666 def
/s36 666 def
/s37 576 def
/s38 545 def
/s39 894 def
/s40 333 def
/s41 447 def
/s42 788 def
/s43 576 def
/s44 727 def
/s45 447 def
/s46 666 def
/s47 666 def
/s48 666 def
/s49 666 def
/s50 666 def
/s51 602 def
/s52 235 def
/s53 509 def
/s54 443 def
/s55 553 def
/s56 583 def
/s57 390 def
/s58 341 def
end
/CharProcs 27 dict def
CharProcs begin
/.notdef {} def
/s32 {
38 -269 moveto
38 -345 38 -269 38 -345 curveto
341 -345 38 -345 341 -345 curveto
341 -269 341 -345 341 -269 curveto
38 -269 341 -269 38 -269 curveto
closepath
} bind def
/s33 {
301 0 moveto
301 -664 301 0 301 -664 curveto
148 -603 301 -664 148 -603 curveto
148 -687 148 -603 148 -687 curveto
301 -758 148 -687 301 -758 curveto
392 -758 301 -758 392 -758 curveto
392 0 392 -758 392 0 curveto
301 0 392 0 301 0 curveto
closepath
} bind def
/s34 {
216 0 moveto
87 0 216 0 87 0 curveto
87 -129 87 0 87 -129 curveto
216 -129 87 -129 216 -129 curveto
216 0 216 -129 216 0 curveto
closepath
} bind def
/s35 {
148 -390 moveto
148 -758 148 -390 148 -758 curveto
538 -758 148 -758 538 -758 curveto
538 -674 538 -758 538 -674 curveto
231 -674 538 -674 231 -674 curveto
231 -470 231 -674 231 -470 curveto
254 -470 231 -470 254 -470 curveto
333 -470 394 -460 437 -441 curveto
479 -421 512 -394 535 -359 curveto
557 -324 568 -282 568 -232 curveto
568 -166 548 -107 507 -55 curveto
466 -3 403 23 319 23 curveto
253 23 199 8 159 -21 curveto
119 -51 99 -86 99 -128 curveto
99 -133 99 -128 99 -133 curveto
189 -133 99 -133 189 -133 curveto
189 -128 189 -133 189 -128 curveto
189 -106 201 -88 224 -74 curveto
247 -60 278 -53 318 -53 curveto
371 -53 410 -70 436 -104 curveto
461 -139 474 -177 474 -220 curveto
474 -268 457 -308 423 -341 curveto
389 -374 320 -390 217 -390 curveto
148 -390 217 -390 148 -390 curveto
closepath
} bind def
/s36 {
333 23 moveto
251 23 184 -13 133 -85 curveto
82 -157 57 -255 57 -379 curveto
57 -503 82 -600 133 -672 curveto
184 -744 251 -780 333 -780 curveto
416 -780 483 -744 533 -672 curveto
584 -600 610 -503 610 -379 curveto
610 -255 584 -157 533 -85 curveto
483 -13 416 23 333 23 curveto
closepath
333 -53 moveto
397 -53 444 -86 472 -152 curveto
501 -218 515 -294 515 -379 curveto
515 -464 501 -539 472 -605 curveto
444 -672 397 -705 333 -705 curveto
269 -705 223 -672 194 -605 curveto
166 -539 151 -464 151 -379 curveto
151 -294 166 -218 194 -152 curveto
223 -86 269 -53 333 -53 curveto
closepath
} bind def
/s37 {
72 0 moveto
72 -758 72 0 72 -758 curveto
500 -758 72 -758 500 -758 curveto
500 -674 500 -758 500 -674 curveto
170 -674 500 -674 170 -674 curveto
170 -436 170 -674 170 -436 curveto
458 -436 170 -436 458 -436 curveto
458 -360 458 -436 458 -360 curveto
170 -360 458 -360 170 -360 curveto
170 -83 170 -360 170 -83 curveto
526 -83 170 -83 526 -83 curveto
526 0 526 -83 526 0 curveto
72 0 526 0 72 0 curveto
closepath
} bind def
/s38 {
413 -140 moveto
504 -140 413 -140 504 -140 curveto
504 -135 504 -140 504 -135 curveto
504 -111 495 -86 476 -62 curveto
458 -37 434 -18 404 -5 curveto
374 9 338 15 296 15 curveto
216 15 154 -10 111 -61 curveto
67 -112 45 -183 45 -273 curveto
45 -363 67 -433 111 -484 curveto
154 -535 216 -561 296 -561 curveto
338 -561 374 -554 404 -541 curveto
434 -528 458 -510 476 -487 curveto
495 -464 504 -438 504 -410 curveto
504 -405 504 -410 504 -405 curveto
413 -405 504 -405 413 -405 curveto
413 -410 413 -405 413 -410 curveto
414 -430 405 -447 386 -462 curveto
368 -477 340 -485 301 -485 curveto
256 -485 217 -468 187 -435 curveto
156 -401 140 -347 140 -273 curveto
140 -198 156 -144 187 -111 curveto
217 -77 256 -61 301 -61 curveto
340 -61 368 -68 386 -83 curveto
404 -98 413 -115 413 -135 curveto
413 -140 413 -135 413 -140 curveto
closepath
} bind def
/s39 {
163 0 moveto
72 0 163 0 72 0 curveto
72 -545 72 0 72 -545 curveto
163 -545 72 -545 163 -545 curveto
163 -447 163 -545 163 -447 curveto
181 -480 199 -503 216 -518 curveto
232 -532 251 -543 271 -550 curveto
290 -557 312 -561 334 -561 curveto
356 -561 378 -557 398 -550 curveto
419 -543 437 -533 452 -519 curveto
467 -505 481 -481 496 -447 curveto
515 -478 533 -502 551 -516 curveto
569 -531 588 -542 607 -550 curveto
626 -557 646 -561 667 -561 curveto
698 -561 726 -553 751 -539 curveto
777 -524 796 -504 809 -477 curveto
823 -450 830 -405 830 -341 curveto
830 0 830 -341 830 0 curveto
738 0 830 0 738 0 curveto
738 -303 738 0 738 -303 curveto
738 -367 734 -408 727 -425 curveto
719 -442 709 -455 695 -464 curveto
682 -473 665 -477 646 -477 curveto
619 -477 594 -469 571 -452 curveto
549 -435 530 -413 517 -384 curveto
503 -356 496 -316 496 -265 curveto
496 0 496 -265 496 0 curveto
405 0 496 0 405 0 curveto
405 -303 405 0 405 -303 curveto
405 -367 402 -408 395 -425 curveto
388 -442 377 -455 361 -464 curveto
346 -473 330 -478 312 -478 curveto
287 -478 263 -469 239 -451 curveto
215 -434 196 -411 183 -382 curveto
169 -353 163 -314 163 -265 curveto
163 0 163 -265 163 0 curveto
closepath
} bind def
/s41 {
360 106 moveto
360 182 360 106 360 182 curveto
114 182 360 182 114 182 curveto
114 -803 114 182 114 -803 curveto
360 -803 114 -803 360 -803 curveto
360 -727 360 -803 360 -727 curveto
197 -727 360 -727 197 -727 curveto
197 106 197 -727 197 106 curveto
360 106 197 106 360 106 curveto
closepath
} bind def
/s42 {
155 0 moveto
72 0 155 0 72 0 curveto
72 -758 72 0 72 -758 curveto
168 -758 72 -758 168 -758 curveto
404 -216 168 -758 404 -216 curveto
620 -758 404 -216 620 -758 curveto
716 -758 620 -758 716 -758 curveto
716 0 716 -758 716 0 curveto
628 0 716 0 628 0 curveto
628 -452 628 0 628 -452 curveto
631 -587 628 -452 631 -587 curveto
628 -587 631 -587 628 -587 curveto
580 -451 628 -587 580 -451 curveto
445 -114 580 -451 445 -114 curveto
346 -114 445 -114 346 -114 curveto
200 -453 346 -114 200 -453 curveto
155 -576 200 -453 155 -576 curveto
153 -576 155 -576 153 -576 curveto
155 -453 153 -576 155 -453 curveto
155 0 155 -453 155 0 curveto
closepath
} bind def
/s43 {
428 -140 moveto
519 -140 428 -140 519 -140 curveto
519 -135 519 -140 519 -135 curveto
519 -111 509 -87 490 -62 curveto
471 -36 446 -17 414 -4 curveto
382 9 346 15 305 15 curveto
222 15 158 -12 113 -65 curveto
68 -119 45 -187 45 -269 curveto
45 -353 69 -423 117 -478 curveto
165 -533 226 -561 301 -561 curveto
343 -561 380 -552 414 -534 curveto
447 -516 474 -488 493 -450 curveto
513 -412 522 -359 522 -292 curveto
522 -261 522 -292 522 -261 curveto
135 -261 522 -261 135 -261 curveto
135 -203 151 -155 182 -117 curveto
214 -79 257 -61 313 -61 curveto
353 -61 382 -68 401 -84 curveto
419 -99 428 -116 428 -135 curveto
428 -140 428 -135 428 -140 curveto
closepath
141 -330 moveto
436 -330 141 -330 436 -330 curveto
436 -340 436 -330 436 -340 curveto
436 -366 429 -390 417 -414 curveto
405 -438 388 -456 367 -468 curveto
345 -479 320 -485 292 -485 curveto
258 -485 226 -471 198 -444 curveto
169 -417 150 -379 141 -330 curveto
closepath
} bind def
/s44 {
620 -758 moveto
712 -758 620 -758 712 -758 curveto
403 0 712 -758 403 0 curveto
304 0 403 0 304 0 curveto
15 -758 304 0 15 -758 curveto
123 -758 15 -758 123 -758 curveto
363 -123 123 -758 363 -123 curveto
620 -758 363 -123 620 -758 curveto
closepath
} bind def
/s45 {
87 -727 moveto
87 -803 87 -727 87 -803 curveto
333 -803 87 -803 333 -803 curveto
333 182 333 -803 333 182 curveto
87 182 333 182 87 182 curveto
87 106 87 182 87 106 curveto
250 106 87 106 250 106 curveto
250 -727 250 106 250 -727 curveto
87 -727 250 -727 87 -727 curveto
closepath
} bind def
/s46 {
193 -379 moveto
193 -455 193 -379 193 -455 curveto
250 -455 193 -455 250 -455 curveto
294 -455 329 -462 354 -476 curveto
379 -490 399 -507 414 -528 curveto
428 -549 436 -573 436 -599 curveto
436 -632 424 -657 399 -676 curveto
375 -695 343 -705 304 -705 curveto
281 -705 260 -702 240 -696 curveto
220 -690 205 -680 194 -666 curveto
184 -652 178 -637 178 -622 curveto
178 -617 178 -622 178 -617 curveto
91 -617 178 -617 91 -617 curveto
91 -622 91 -617 91 -622 curveto
91 -662 108 -698 142 -731 curveto
175 -764 231 -780 309 -780 curveto
382 -780 438 -764 475 -733 curveto
512 -701 531 -661 531 -611 curveto
531 -583 525 -558 515 -534 curveto
504 -511 488 -490 467 -471 curveto
446 -453 416 -438 376 -426 curveto
438 -409 484 -383 515 -348 curveto
545 -313 561 -270 561 -218 curveto
561 -157 539 -102 496 -52 curveto
454 -2 389 23 302 23 curveto
226 23 171 7 136 -26 curveto
101 -58 83 -95 83 -135 curveto
83 -140 83 -135 83 -140 curveto
174 -140 83 -140 174 -140 curveto
174 -135 174 -140 174 -135 curveto
174 -108 187 -87 213 -74 curveto
238 -60 271 -53 311 -53 curveto
342 -53 370 -59 394 -71 curveto
417 -82 435 -101 448 -127 curveto
460 -153 466 -181 466 -212 curveto
466 -259 448 -299 410 -331 curveto
373 -363 316 -379 239 -379 curveto
193 -379 239 -379 193 -379 curveto
closepath
} bind def
/s47 {
87 0 moveto
87 -79 87 0 87 -79 curveto
87 -120 100 -164 125 -213 curveto
150 -261 193 -308 253 -354 curveto
304 -394 253 -354 304 -394 curveto
373 -447 416 -486 435 -512 curveto
453 -538 462 -565 462 -594 curveto
462 -623 450 -648 425 -671 curveto
401 -693 366 -705 320 -705 curveto
278 -705 247 -697 227 -680 curveto
207 -664 197 -646 197 -626 curveto
197 -621 197 -626 197 -621 curveto
106 -621 197 -621 106 -621 curveto
106 -626 106 -621 106 -626 curveto
106 -653 115 -678 133 -702 curveto
151 -725 176 -744 206 -759 curveto
237 -773 276 -780 323 -780 curveto
408 -780 469 -762 506 -726 curveto
542 -690 561 -646 561 -595 curveto
561 -568 555 -541 543 -514 curveto
532 -486 510 -458 476 -428 curveto
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diff --git a/Doc/Documentation.bib b/Documentation-Source/Userguide.bib
similarity index 99%
rename from Doc/Documentation.bib
rename to Documentation-Source/Userguide.bib
index 5aa56f793b3e74385dc8c6c527d4588e734a0a17..318a8c111ead66c0ced2605dd6502616a315c03e 100644
--- a/Doc/Documentation.bib
+++ b/Documentation-Source/Userguide.bib
@@ -1,3 +1,12 @@
+@article{Eas79,
+ author = {J. W. Eastwood and D. R. K. Brownrigg},
+ title = {Remarks on the solution of poisson's equation for isolated systems},
+ journal = {J. Comp. Phys.},
+ volume = 32,
+ pages = 24,
+ year = 1979
+}
+
@article{umar2009c,
author = {A. S. Umar and V. E. Oberacker},
title = {Center-of-mass motion and cross-channel coupling in the time-dependent
@@ -1169,10 +1178,3 @@ abstract={ We present a fully microscopic study of the 16 O + 208 Pb
pages={1267},
year={1997},
}
-@article{Eas79,
- author = "J. W. Eastwood and D. R. K. Brownrigg",
- year = "1979",
- journal = "J. Comp. Phys.",
- volume = "32",
- pages = "24"
-}
diff --git a/Documentation-Source/Userguide.pdf b/Documentation-Source/Userguide.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..336491ab71353e6a6f5181298b7d10a907e79b2a
Binary files /dev/null and b/Documentation-Source/Userguide.pdf differ
diff --git a/Documentation-Source/Userguide.tex b/Documentation-Source/Userguide.tex
new file mode 100755
index 0000000000000000000000000000000000000000..0694fad8444b9df0d02af2f45560987bc5ddb661
--- /dev/null
+++ b/Documentation-Source/Userguide.tex
@@ -0,0 +1,3161 @@
+\documentclass[final,5p,times]{elsarticle}
+\usepackage{graphicx}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+%\usepackage{bm}
+\newcommand{\D}{{\mathrm d}}
+\newcommand{\I}{{\mathrm i}}
+\renewcommand*\arraystretch{1.4}
+\newcommand{\bnu}{\mbox{\boldmath$\nu$}}
+\newcommand{\bmu}{\mbox{\boldmath$\mu$}}
+
+\sloppy
+\newcounter{bla}
+\newenvironment{refnummer}{%
+\list{[\arabic{bla}]}%
+{\usecounter{bla}%
+ \setlength{\itemindent}{0pt}%
+ \setlength{\topsep}{0pt}%
+ \setlength{\itemsep}{0pt}%
+ \setlength{\labelsep}{2pt}%
+ \setlength{\listparindent}{0pt}%
+ \settowidth{\labelwidth}{[9]}%
+ \setlength{\leftmargin}{\labelwidth}%
+ \addtolength{\leftmargin}{\labelsep}%
+ \setlength{\rightmargin}{0pt}}}
+ {\endlist}
+
+\journal{Computer Physics Communications}
+\biboptions{sort&compress}
+
+\begin{document}
+\begin{frontmatter}
+
+\title{Sky3D User Guide}
+
+\author{B.~Schuetrumpf\corref{gsi} }\ead{b.schuetrumpf@gsi.de}
+\address{National Superconducting Cyclotron Laboratory\\
+Michigan State University, East Lansing, Michigan 48824, USA}
+
+\author{P.-G.~Reinhard}\ead{Paul-Gerhard.Reinhard@physik.uni-erlangen.de}
+\address{Institut f\"ur Theoretische Physik II, Universit\"at
+ Erlangen-N\"urnberg, \\Staudtstrasse 7, 91058 Erlangen, Germany}
+
+\author{P.~D.~Stevenson}\ead{p.stevenson@surrey.ac.uk}
+\address{Department of Physics, University of Surrey, Guildford,
+ Surrey, GU2 7XH, United Kingdom}
+
+\author{A. S. Umar}\ead{sait.a.umar@Vanderbilt.Edu}
+\address{Department of Physics and Astronomy, Vanderbilt University, \\Nashville, Tennessee 37235, USA}
+
+\author{J.~A.~Maruhn\corref{author}} \ead{maruhn@th.physik.uni-frankfurt.de}
+\address{Institut f\"ur Theoretische Physik, Goethe-Universit\"at,
+ Max-von-Laue-Str. 1, \\60438 Frankfurt am Main, Germany}
+
+\cortext[gsi]{Present address: Gesellschaft f\"ur
+ Schwerionenforschung, 64291 Darmstadt, Germany}
+\cortext[author]{Corresponding author.}
+
+\end{frontmatter}
+\tableofcontents
+
+\section{Introduction}
+This document outlines the physics addressed in the Sky3D code and
+explains how to run actual applications. A detailed documentation of
+the code itself, its modules, variables, etc., is contained in {\tt
+ Sky3D\_doxygen.pdf} and, alternatively, in {\tt Sky3D\_doxygen.html}
+
+The vast majority of microscopic models of many-body systems rely
+on a description in terms of the single-particle (s.p.)
+wave functions. Among them, self-consistent mean-field models (SCMF)
+automatically generate the optimal one-body potentials
+corresponding to the s.p.\ wave functions. A rigorous SCMF is the
+Hartree-Fock theory (HF) where the s.p.\ wave functions are determined
+variationally for a given two-body interaction~\cite{Fet71,Mar10aB}. A
+more practical approach is provided by the Density Functional Theory (DFT),
+which incorporates the involved many-body effects into effective
+interactions, or effective energy-density functionals.
+This is a very efficient and successful scheme, widely used in
+electronic systems~\cite{Dre90}. Straightforward HF is unsuitable
+for nuclei because the free-space two-nucleon force contains a strong
+short-range
+repulsion requiring renormalization in the nuclear medium.
+For this reason, nuclear SCMFs necessarily employ effective
+interactions or functionals although they often carry the label
+HF as, e.g, in the Skyrme Hartree-Fock (SHF) method. There are
+relativistic as well as non-relativistic approaches. For an extensive
+review see~\cite{Ben03aR}.
+
+The description of dynamical processes is even more demanding than the
+modeling of structure. SCMFs are also the first method of choice
+in this domain. The natural extension of HF is time-dependent HF
+(TDHF) which was proposed as early as 1930 in~\cite{Dir30}. Earlier
+applications were restricted to the linearized regime covering small
+amplitude motion, see, e.~g., \cite{Bro71aB}. Large scale TDHF
+calculations became possible in the last few decades with the
+increasing computing capacities. Again, as in the static case, true
+TDHF calculations make sense only for electronic systems and even
+there they are very rare. The overwhelming majority of dynamical SCMF
+calculations employ, in fact, time-dependent DFT (TDDFT). In
+electronic systems, this amounts to the time-dependent
+local density approximation (TDLDA) \cite{Dre90}, which is widely used
+in atoms, molecules, and solids; for examples in nanoparticles see,
+e.~g., \cite{Rei03a}. Dynamical SCMFs in nuclei also stay at the level
+of TDLDA even if they are often named TDHF which happens particularly
+for dynamical calculations using the Skyrme energy functional. Nuclear
+TDHF started about forty years ago \cite{Bon76a} and has developed
+since then into a powerful and versatile tool for simulating a great
+variety of dynamical scenarios. Earlier applications were based
+mainly on non-relativistic TDHF using the effective Skyrme energy
+functional~\cite{Neg82aR,Dav85a}. Due to higher numerical demands,
+relativistic calculations appeared somewhat later~\cite{Bai87a}, but
+have developed meanwhile equally well to a widely used
+tool~\cite{Ber95a,Vre05aR}.
+
+In this paper, we present a code for TDHF calculations on the basis of
+the non-relativistic Skyrme energy functional. The code uses a fully
+three dimensional (3D) representation of wave functions and fields on
+a Cartesian grid in coordinate space. There are no symmetry
+restrictions and the full Skyrme energy functional is used including
+the spin-orbit and most important time-odd terms. Such fully-fledged
+3D calculations became possible only over the last decade with the
+steadily increasing computing capabilities. In fact, early TDHF
+studies all used restricted representations, axial symmetry and/or
+reflection symmetries. This limited the possible applications. TDHF
+experienced a revival during the last ten years when unrestricted 3D
+calculations became possible. There are several groups performing
+large scale TDHF studies for various scenarios of nuclear dynamics,
+see, e.g., \cite{Has08a,Uma09a,Loe12a}. Aside from these studies of
+nuclear collisions a principal application has been to collective
+vibrations, e.g., \cite{Sim03a,Mar05a,Umar05a,Nak05a}. In the linear regime
+TDHF leads to fully self-consistent RPA, for which though, unlike
+TDHF, often additional approximations like the neglect of the
+Coulomb potential or the spin-orbit terms in the residual interaction
+are made. TDHF can also be used to investigate non-linearities of
+nuclear vibrations.
+
+For a recent review of both vibrational and collisional applications
+see \cite{Simenel}. Such calculations have clearly outgrown the
+developmental stage. It is an appropriate time to give a broader
+public access to a 3D TDHF code. This is the goal of this paper.
+Skyrme HF covers such a broad range of physical phenomena and is
+relatively involved that efficient computational treatment of 3D
+simulations requires elaborate numerical methods. We shall make an
+effort to explain the many necessary ingredients in a comprehensive,
+and yet compact, manner.
+
+Most recent Skyrme density functionals contain terms such as
+fractional powers of the density that cannot be related to a two- or
+three-body interaction. In that sense, the present code solves the
+TDDFT rather than TDHF equations. Nevertheless, we prefer to keep the
+name TDHF since it is associated historically with this large field of
+nuclear reaction theory.
+
+Although the code may be run as it is and many innovative applications
+are possible, we also intend it to be used for
+exploring new ideas that need a basic HF and TDHF algorithm, which is
+implemented in a transparent modern style of programming and
+extensively documented (see the accompanying online documentation),
+allowing for relatively easy modification. A guide to some possible
+extensions is given in Sect.~\ref{sec:modify}.
+
+Some recent developments in TDHF which require more extensive
+modification but might be implemented on the basis of this code
+include:
+
+\begin{itemize} \item The extraction of nucleus-nucleus potentials
+with and without dynamical effects. Approaches include
+density-con\-strained TDHF (DC-TDHF) method~\cite{umar2006d}, the
+density-dependent TDHF (DD-TDHF) approach~\cite{washiyama2008}, and
+the frozen HF method~\cite{simenel2013b}.
+
+\item The extraction of (multi-)nucleon transfer probabilities using
+particle number projection techniques~\cite{simenel2010}.
+
+\item The use of novel spatial distributions to excite low-lying
+ dipole states \cite{Goddard2013}.
+
+\item The incorporation of fluctuations of one-body observables using
+the Balian-V\'en\'eroni variational
+principle~\cite{balian1984,Broomfield2008,simenel2011}.
+
+\item The inclusion of dynamical pairing correlations at the TDHF+BCS
+or TDHFB levels~\cite{avez2008,ebata}
+
+\item The inclusion of collision terms with the Extended TDHF or the
+time-dependent density matrix approaches~\cite{lacroix1999,tohyama2002a}.
+
+\item The generalization of the static HF code to odd
+particles by the inclusion of the time-odd terms in the Skyrme
+interaction~\cite{Eng75a,dobaczewski1995,umar2005b}.
+\end{itemize}
+
+\section{General purpose and structure}
+
+\subsection{Intended applications}
+The code Sky3D solves the static Hartree-Fock as well as the
+time-dependent Hartree-Fock (TDHF) equations for interactions of
+Skyrme-force type in a general three-dimensional geometry. No
+symmetries of any kind are assumed, so that the code can be used for a
+wide variety of applications in nuclear structure, collective
+excitations, and nuclear reactions; of course within the limitations
+of mean-field theory.
+
+\subsection{Specific model implemented}
+The code in the presented version contains a useful selection of terms
+in the Skyrme force but by no means all terms that have been included
+in some recent works. It should still be useful, because (1) for many
+interesting applications the interest is semi-quantitative so that a
+Skyrme force fitted with the latest models is not necessary --- usually
+a selection of forces is desired to look at the variability of
+results, but not a high-accuracy fit of data; (2) the code is written
+in such a way that additional terms can be added easily. The coding
+corresponds one-to-one to the analytic formulas in most places except
+where efficiency demands reordering the calculations.
+
+\subsubsection{The single-particle basis}
+
+In a mean field theory one seeks to describe the many-body system
+exclusively in terms of a set of single-particle wave functions
+$\psi_\alpha$ with fractional occupation amplitudes $v_\alpha$, i.e.
+\begin{subequations}
+ \begin{equation}
+ \left\{\psi_\alpha,v_\alpha,\alpha=1,...,\Omega\right\}
+ \end{equation}
+ where $\Omega$ denotes the size of the active s.p.\ space. The
+ occupation amplitude can take values continuously in the interval
+ $[0,1]$. The complementary non-occupation amplitude is
+ $u_\alpha=\sqrt{1-v_\alpha^2}$. A formal definition of the BCS
+ mean-field state reads
+ \begin{equation}
+ |\Phi\rangle
+ =
+ \prod_{\alpha>0}\big(
+ u_\alpha^{\mbox{}}+v_\alpha^{\mbox{}}\hat{a}^+_\alpha\hat{a}^{+}_{\bar\alpha}
+ \big)|0\rangle
+ \label{eq:BCState}
+ \end{equation}
+\end{subequations}
+where $|0\rangle$ is the vacuum state, $\hat{a}^+_\alpha$ the
+generator of a Fermion in state $\psi_\alpha$, and $\bar\alpha$ the
+time reverse partner to state $\alpha$. We will use variation of the
+BCS amplitudes $v_\alpha^{\mbox{}}$ only in the static part of
+even-even nuclei where the time reverse partner is unambiguously
+defined. In fact, the pairing calculation assumes that the paired
+states have exactly the same spatial density distribution.
+
+The time-dependent calculation technically can be run with pairing
+included. This is done by keeping the pairing occupation probabilities
+fixed during the time evolution; the pairing between states can simply
+be taken over from the static calculation, although time-reversal
+invariance is lost for boosted nuclei. For a the initial state in a
+collision, boosting a single-particle state and its time-reversed
+partner does not destroy their role for pairing, while the boosted
+states are no-longer time-reversal conjugate. This is a consequence of
+Galilei invariance. For a study of small vibrations the initial
+pairing of states is also correct although again they are no longer
+time-reversal conjugate once a boost or an external potential are
+applied.
+
+The conservation of total energy {\em excluding the pairing energy} is
+not impaired for the case of constant occupation, while the pairing
+energy is not even computed.
+
+{\em It should be noted, however, that once the wave functions change
+ dynamically, this approach is not correct as the occupation
+ probabilities will also change with time. In this case the
+ TDHF-Bogolyubov equations should be used. In addition, the pairing
+ energies are not computed at all. Using this code as is with pairing
+ included in the time-dependent case might be useful for schematic or
+ exploratory studies but extreme caution is advised when interpreting
+ such results.}
+
+
+\subsubsection{Local densities and currents \label{sec:dens}}
+
+The Skyrme-energy-density functional is defined in terms of only a
+few local densities and currents. These are the {\em time-even} fields
+\begin{align}
+ \rho_q(\vec r)&=\displaystyle
+ \sum_{\alpha\in q}\sum_{s}
+ v_{\alpha}^2|\psi_{\alpha}(\vec{r},s)|^2
+ &&
+ \text{density}
+ \notag \\
+ \vec{J}_q(\vec r) &=\displaystyle
+ -\I\sum_{\alpha\in q}\sum_{ss'} v_{\alpha}^2
+ \psi_{\alpha}^*(\vec{r},s)
+ \nabla\! \times\! \vec{\sigma}_{ss'}
+ \psi^{\mbox{}}_{\alpha}(\vec{r},s')
+ &&
+ \text{spin-orbit dens.}
+ \notag\\
+ \tau_q(\vec r)&=\displaystyle
+ \sum_{\alpha\in q}\sum_{s}
+ v_{\alpha}^2|\nabla\!\psi_{\alpha}(\vec{r},s)|^2
+ &&
+ \mbox{kinetic density,}
+ \label{eq:rtjeven}
+\end{align}
+the {\em time-odd} fields
+\begin{align}
+ \vec{s}_q(\vec r)&=\displaystyle
+ \sum_{\alpha\in q}\sum_{ss'} v_{\alpha}^2
+ \psi^*_{\alpha}(\vec{r},s)\vec{\sigma}_{ss'}
+ \psi^{\mbox{}}_{\alpha}(\vec{r},s')
+ &&
+ \mbox{spin density}
+ \notag\\
+ \vec{\jmath}_q(\vec r)&=\displaystyle
+ \Im{m}\left\{\sum_{\alpha\in q}\sum_{s} v_{\alpha}^2
+ \psi^*_{\alpha}(\vec{r},s)
+ \nabla\!\psi^{\text{}}_{\alpha}(\vec{r},s)\right\}
+ &&
+ \text{current density,}
+ \label{eq:rtjodd}
+\end{align}
+and a field with undefined time parity:
+ \begin{align}
+ \xi_q(\vec r)
+ &=\displaystyle
+ {\sum_{\alpha\in q}\sum_{s} u_{\alpha}v_{\alpha}
+ \psi_{\overline{\alpha}}(\vec{r},s)
+ \psi_{\alpha}}(\vec{r},s)
+ % \sum_{\alpha\in q} u_{\alpha}v_{\alpha}|\psi_{\alpha}|^2
+ &&
+ \text{pairing density}
+ \label{eq:rtjpair}
+\end{align}
+where $q$ labels the nucleon species with $q=p$ for protons and $q=n$
+for neutrons. A local density/current without $q$ index stands for the
+total quantity, e.g. $\rho=\rho_p+\rho_n$ is the total density, and
+similarly for the other densities/currents. The
+variable $s$ indicates the two spinor components of the wave
+functions.
+
+\subsubsection{The energy-density functional}\label{sec:efunc}
+
+The mean-field equations solved in the code are based on the widely
+used Skyrme energy functional. For recent reviews see
+\cite{Ben03aR,Erl11a}. The functional at the level at which it is
+used here can be written as
+\begin{subequations}
+ \begin{equation}
+ \label{eq:efundet}
+ E_\mathrm{tot}
+ =
+ T+(E_0+E_1+E_2+E_3+E_\mathrm{ls})+E_{\rm Coulomb}+E_{\rm pair}+E_{\rm corr},
+ \end{equation}
+ where the parentheses were used to group the terms arising from the
+ Skyrme force. The various terms read in detail (all densities and
+ currents defined in Section~\ref{sec:dens} are understood to depend
+ on $\vec r$)
+ \begin{itemize}
+ \item $T$: the total kinetic energy calculated as
+ \begin{equation}
+ \label{eq:ehft}
+ T=\sum_q \frac{\hbar^2}{2m_q}\int\D^3 r\, \tau_q
+ \end{equation}
+ with $\tau_q$ the isospin-specific kinetic density of
+ Eq.~(\ref{eq:rtjeven}).
+ \item $E_0$: The $b_0$ and $b'_0$-dependent part is
+ \begin{equation}
+ \label{eq:ehf0}
+ E_0=\int \D^3\!r\,\left(\frac{b_0}{2}\rho^2-\frac{b_0'}{2}\sum_q\rho_q^2\right).
+ \end{equation}
+ \item $E_{1}$: kinetic terms containing the coefficients $b_1$ and $b_1'$:
+ \begin{equation}
+ \label{eq:ehf1}
+ E_1=\int \D^3\!r\,\left(b_1[\rho\tau-{\vec\jmath\,}{}^2]
+ -b_1'\sum_q[\rho_q\tau_q-{\vec\jmath_q}{}^2]\right).
+ \end{equation}
+ \item $E_{2}$: terms containing the coefficients $b_2$ and $b_2'$.
+ They involve the Laplacians of the densities.
+ \begin{equation}
+ \label{eq:ehf2}
+ E_2=\int \D^3\!r\,\left(-\frac{b_2}{2}\rho\Delta\rho
+ +\frac{b_2'}{2}\sum_q\rho_q\Delta\rho_q\right).
+ \end{equation}
+ \item $E_{3}$: The many-body contribution is given by
+ \begin{equation}
+ \label{eq:ehf3}
+ E_{3}=\int \D^3\!r\,\left(\frac{b_3}{3}\rho^{\alpha+2}
+ -\frac{b_3'}{3}\rho^\alpha\sum_q\rho_q^2\right).
+ \end{equation}
+ \item $E_\mathrm{ls}$: the spin-orbit energy
+ \begin{equation}\begin{split}
+ E_\mathrm{ls}&=\int \D^3\!r\,\biggl(
+ -b_4[\rho\nabla\cdot\vec J+
+ \vec s\cdot(\nabla\times\vec\jmath)]\\
+ &-b_4'\sum_q[\rho_q\nabla\cdot\vec J_q+
+ \vec s_q\cdot(\nabla\times\vec\jmath_q)]\biggr)
+ \label{eq:ehfls}
+ \end{split}
+ \end{equation}
+ \item $E_\mathrm{C}$: the Coulomb energy. It consists of the
+ standard expression for a charge distribution in its own field
+ (Hartree term) plus the exchange term in the Slater approximation
+ ~\cite{Sla51}. The formula is
+ \begin{equation}
+ \label{eq:ehfc}
+ E_\mathrm{C}
+ =
+ \frac{e^2}{2}\int \D^3\!r \D^3\!r'
+ \frac{\rho_p(\vec{r})\rho_p(\vec{r'})}{|\vec{r}-\vec{r'}|}
+ -
+ \int \D^3\!r\,\frac{3e^2}{4}\left(\frac3{\pi}\right)^{\tfrac1{3}}\rho_p^{4/3}
+ \end{equation}
+ where $e$ is the elementary charge with $e^2=1.43989$ MeV$\cdot$fm.
+ \item $E_\mathrm{pair}$: the pairing energy. It consists of a
+ contact pairing interaction involving the pairing densities $\xi_q$
+ augmented by an optional density dependence. The formula is
+ \begin{equation}
+ E_{\rm pair}
+ =
+ \frac{1}{4} \sum_{q\in\{p,n\}}V_\mathrm{pair,q}
+ \int \D^3r |\xi_q|^2
+ \left[1 -\frac{\rho}{\rho_{0,\mathrm{pair}}}\right]\;.
+ \label{eq:epair}
+ \end{equation}
+ It contains a continuous switch, the parameter
+ $\rho_{0,\mathrm{pair}}$. A pure $\delta$-interaction (DI), also
+ called volume pairing, is recovered for
+ $\rho_{0,\mathrm{pair}}\longrightarrow\infty$. The general case is
+ the density dependent $\delta$-interaction (DDDI). A typical
+ value near matter equilibrium density
+ $\rho_{0,\mathrm{pair}}=0.16$ fm$^{-3}$ concentrates pairing to
+ the surface. The most flexible choice is to consider
+ $\rho_{0,\mathrm{pair}}$ as an additional free parameter. Actual
+ adjustments with this option deliver a form of the pairing
+ functional which stays in between the extremes of volume and
+ surface pairing~\cite{Klu09a}.
+ \end{itemize}
+\end{subequations}
+
+The term $E_{\rm corr}$ stands for all additional corrections from
+correlations beyond the mean field that might be added. Most
+calculations include at least the center-of-mass correction
+$E_\mathrm{cm}$. For deformed nuclei this should be augmented by a
+rotational correction and for soft nuclei by correlations from all
+low-energy quadrupole motions \cite{Klu08a}. So far, these
+correlations are usually added a posteriori after static
+calculations. This procedure is associated with setting the switch
+{\tt zpe=1} which is the standard option adopted here. A fully
+variational treatment and a dynamical propagation of the
+c.m.\ correction is extremely involved and usually not considered. The
+other strategy is to modify the nucleon mass by
+$m\longrightarrow{m}-m/A$ and to include this simplified correction in
+the variational treatment thus avoiding the a posteriori
+correction. This way is chosen in a couple of traditional
+parametrizations, e.g., in SkM$^*$~\cite{Bar82a}. We keep this option
+in case of static calculations for consistency and associate it with
+{\tt zpe=0}.
+
+Using a center-of-mass correction is desirable for static
+calculations, it is disputable for vibrational excitations, and runs
+fully into inconsistencies in collisions and fragmentation as it
+employs only the total mass number $A$ and cannot account for the
+masses of the fragments. Therefore all dynamical runs are done
+with
+\begin{equation}
+ E_\mathrm{cm}=0
+ \quad.
+\end{equation}
+An inconsistency may occur if {\tt zpe=0} was used in the static
+calculation providing the input for the dynamical run because the
+setting $m\longrightarrow{m}-m/A$ is not used in dynamics. In order
+to safely suppress the c.m.\ correction in statics and dynamics, we
+introduce an additional switch {\tt turnoff\_zpe} in the input, which
+turns off the zero-point energy correction irrespective of what value
+is given to {\tt zpe} in the force definition. This allows the use of
+forces with {\tt zpe=0} also for collisions. It should be pointed out
+also that there are newer Skyrme parametrizations like Sly4d
+\cite{Chabanat1995,Kim1997} and UNEDF \cite{Kortelainen2012} that are
+fitted without any center-of-mass correction and are thus specially
+intended for collision calculations.
+
+For some further considerations on the c.m.\ correction in TDHF see
+\cite{umar2009c}.
+
+The functional in the above form contains the minimal number of terms
+which are needed to guarantee Galilean invariance~\cite{Eng75a,Erl11a}
+and so to allow performance of TDHF calculations which respect all basic
+conservation laws. We ignore the tensor spin-orbit terms and spin-spin
+couplings~\cite{Ben03aR,Per04a,Erl11a}. These may be important for
+magnetic excitations~\cite{Erl11a} and odd nuclei~\cite{Pot10a} which
+are, however, not the prime focus of TDHF studies.
+
+\subsubsection{Force coefficients}
+The above formulation in terms of the Skyrme energy functional
+introduces the force parameters $b_0$, $b'_0$, ... $b'_4$ naturally
+as the factors in front of each contribution in the terms
+(\ref{eq:ehf0}-\ref{eq:ehfls}). Traditionally, the functional is deduced from a
+Skyrme force which is a density-dependent, zero-range interaction
+\cite{Vau72a}. The $t$ and $x$ coefficients in this Skyrme-force
+definition are related to the $b$ coefficients in the functional
+definition as
+\begin{equation}
+ \label{eq:force-coeff}
+ \begin{split}
+ b_0&=t_0\,\left(1+\tfrac1{2}x_0\right) \\
+ b_0'&=t_0\,\left(\tfrac1{2}+x_0\right)\\
+ b_1&=\tfrac1{4}\left[t_1\,\left(1+\tfrac1{2}x_1\right)+
+ t_2\,\left(1+\tfrac1{2}x_2\right)\right] \\
+ b_1'&=\tfrac1{4}\left[t_1\,\left(\tfrac1{2}+x_1\right)-
+ t_2\,\left(\tfrac1{2}+x_2\right)\right] \\
+ b_2&=\tfrac1{8}\left[3t_1\,\left(1+\tfrac1{2}x_1\right)-
+ t_2\,\left(1+\tfrac1{2}x_2\right)\right] \\
+ b_2'&=\tfrac1{8}\left[3t_1\,\left(\tfrac1{2}+x_1\right)+
+ t_2\,\left(\tfrac1{2}+x_2\right)\right]\\
+ b_3&=\tfrac1{4}t_3 \left(1+\tfrac1{2}x_3\right) \\
+ b_3'&=\tfrac1{4}t_3 \left(\tfrac1{2}+x_3\right) \\
+ b_4&=\tfrac1{2}t_4
+ \end{split}
+\end{equation}
+The coefficient $b_4'$ is usually fixed as $b'_4=\frac{1}{2}t_4$ for most
+traditional Skyrme forces. More recent variants of Skyrme forces (SkI3
+etc.) handle it as a separate free parameter~\cite{Rei95a}. In addition
+to the $b$ and $b'$ parameters, there is the power coefficient in the
+(originally) three-body term, which is usually called $\alpha$, but in
+the code is referred to as {\tt power}. The input of the force to the
+code is done in terms of the $t_i$, $x_i$ parameters.
+
+\subsubsection{The single-particle Hamiltonian}
+\label{sec:otherfields}
+
+The mean-field Hamiltonian $\hat{h}$ is derived from the energy
+functional of Section~\ref{sec:efunc} by variation
+$\partial_{\psi_\alpha^*}E=\hat{h}\psi_\alpha$. It reads in detail
+\begin{subequations}
+ \begin{equation}\begin{split}
+ \hat h_q& =
+ U_q(\vec r)-\nabla\cdot\left[B_q(\vec r)\nabla\right]
+ +\I\vec W_q\cdot(\vec\sigma\times\nabla)
+ +\vec S_q\cdot\vec\sigma\\
+ &-\frac{\I}{2} \left[(\nabla\cdot\vec A_q)+2\vec
+ A_q\cdot\nabla\right]\,,
+ \label{eq:spham}
+ \end{split}
+\end{equation}
+ with $q\in\{p,n\}$ as usual distinguishing the different
+ Hamiltonians for protons and neutrons. For the protons the Coulomb
+ potential is also added. The first term is the local part of the
+ mean field, which acts on the wave functions like a local
+ potential. It is defined as
+ \begin{equation}\begin{split}
+ U_q
+ &=
+ b_0\rho-b_0'\rho_q+b_1\tau-b_1'\tau_q-b_2\Delta\rho
+ +b_2'\Delta\rho_q
+ \\
+ &+b_3\tfrac{\alpha+2}{3}\,\rho^{\alpha+1}-b_3'\tfrac2{3}\,\rho^\alpha\rho_q
+ -b_3'\tfrac\alpha{3}\,\rho^{\alpha-1}\sum_{q'}\rho_{q'}^2
+ \\
+ &
+ -b_4\nabla\cdot\vec J-b_4'\nabla\cdot \vec{J}_q
+ \;.
+ \label{eq:upot}
+ \end{split}
+\end{equation}
+Next comes the ``effective mass'', which replaces the standard
+ $\tfrac{\hbar^2}{2m}$ factor by the isospin and space-dependent
+ expression
+ \begin{equation}
+ \label{eq:bmass}
+ B_q=\frac{\hbar^2}{2m_q}+b_1\rho-b_1'\rho_q.
+ \end{equation}
+ Note that the Skyrme force definitions contain the first term
+ (nucleon mass) as a parameter which varies slightly from
+ parametrization to parametrization and may be different for protons
+ and neutrons. The spin-orbit potential is
+ \begin{equation}
+ \vec W_q
+ =
+ b_4\nabla\rho+b_4'\nabla\rho_q
+ \quad.
+ \label{eq:wlspot}
+ \end{equation}
+ The above three terms involve the time-even densities. Dynamical
+ effects come into play with the next terms which include the
+ time-odd contributions from current and spin-density:
+ \begin{align}
+ \label{eq:aq}
+ \vec A_q&=-2b_1\vec\jmath+2b_1'\vec\jmath_q-b_4\nabla\times\vec s
+ -b_4'\nabla\times\vec s_q.
+ \;, \\
+ \label{eq:spot}
+ \vec S_q&=-b_4\nabla\times\vec\jmath-b_4'\nabla\times\vec\jmath_q
+ \;.
+\end{align}
+\end{subequations}
+
+
+\subsection{Coupling to external fields}
+\label{sec:external}
+
+For the dynamic case, the system can also be coupled to an external
+excitation field, to study collective response such as in giant
+resonances. The present code only implements a very simple case, since
+it is expected that most serious applications will need modifications,
+which are quite easy to incorporate.
+
+The external field is introduced as a time-dependent, local operator
+\begin{subequations}
+ \begin{equation}
+ \hat{h}_q\longrightarrow\hat{h}_q+U_{q,\mathrm{ext}}(\vec{r},t)
+ \quad,\quad
+ U_{q,\mathrm{ext}}
+ =
+ \eta\,f(t)\,F_q(\vec r)\;,
+ \end{equation}
+ where $f(t)$ carries the temporal profile of the excitation
+ mechanism, $F_q(\vec r)$ is some local operator, and $\eta$ tunes
+ the overall strength. The spatial distribution $F_q(\vec{r})$, is
+ allowed to be different for the two isospins $q$. Typical examples
+ are isoscalar and isovector multipole operators as, e.g., the
+ isoscalar quadrupole $F_q(\vec{r})=2z^2-x^2-y^2$.
+
+ The prefactor $\eta$ is a strength parameter which allows
+ scanning different excitation strengths easily while keeping the
+ temporal and spatial profiles the same. It should be noted that the
+ absolute magnitude of the perturbing potential by itself usually has
+ little direct meaning. What counts is the excitation energy caused
+ by the perturbation and subsequently the magnitude of vibrations in
+ the observables (such as the time-dependent quadrupole moment). An
+ exception is, e.~g., the simulation of the close approach of another
+ nucleus that stays external to the computational grid, where the
+ potential is uniquely defined.
+
+ One important point remains to be noted concerning the spatial
+ profile $F_q(\vec{r})$. This can be illustrated by the quadrupole
+ operator. Let us assume an instant where $A>0$ and $f(t)>0$. For
+ then the operator $\propto 2z^2-x^2-y^2$ leads to a perturbing potential
+ $U_\mathrm{ext}$ which is binding in $z$-direction but
+ asymptotically unbound in the $x$- and $y$-directions. This can cause
+ unphysical effects in case of large strengths and/or numerical boxes.
+ For this reason it is useful to have a cut-off by a
+ Woods-Saxon like function according to~\cite{Rut95a}
+ \begin{equation}
+ \label{eq:extdamp} §§§
+ F_q(\vec r)
+ \rightarrow
+ \frac{F_q(\vec r)}{1+e^{(r-r_0)/\Delta r}}\;,
+ \end{equation}
+ where $r_0$ and $\Delta r$ are parameters describing a transition
+ region sufficiently outside the nucleus, but also sufficiently small
+ to maintain binding.
+
+ Another problem associated with the external field is that in
+ general it will not be periodic but instead have discontinuities on
+ the boundary when crossing into the neighboring cell. If damping is
+ sufficiently strong, the field may be practically zero on the
+ boundary and thus become periodic. Another solution for this problem
+ is to make the field explicitly periodic by replacing the
+ coordinates with periodic substitutes. The exact formulation depends
+ on the specific field used; for the above-mentioned quadrupole
+ operator, which depends only on the squares of the coordinates,
+ e.~g., substituting
+ \begin{equation}\label{eq:extperiodic}
+ x^2\rightarrow\sin^2\left(\pi x/x_L\right),
+ \end{equation}
+ with $x_L={\tt nx}\,\Delta x$ the period interval, will provide the
+ proper behavior as the sine squared has a period of $\pi$ and there
+ is no unphysical decrease of this function near the boundary. Of
+ course the analogous transformation has to be applied to $y$ and
+ $z$.
+
+ The time dependence $f(t)$ is modeled as a short pulse centered
+ around some excitation frequency $\omega$. The code allows a choice
+ between two pulse envelopes:
+ \begin{enumerate}
+ \item A Gaussian of the form
+ \begin{equation}
+ \label{eq:extgauss}
+ f(t)=\exp\left(-(t-\tau_0)^2/\Delta\tau^2\right)\cos(\omega(\tau-\tau_0))\;,
+ \end{equation}
+ with peak time $\tau_0$ and width $\Delta\tau$.
+ \item A cosine squared function defined via
+ \begin{equation}
+ \label{eq:extcos}
+ f(t)=\cos\left(\frac\pi{2}\left(\frac{t-\tau_0}{\Delta\tau}\right)^2\right)
+ \theta\left(\Delta\tau-|t-\tau_0|\right)
+ \cos(\omega(\tau-\tau_0))\;,
+ \end{equation}
+ which is confined to the intervals $t\in
+ (\tau_0-\Delta\tau,\tau_0+\Delta\tau)$. This envelope is also
+ characterized by a peak time of $\tau_0$ and width $\Delta\tau$.
+ \end{enumerate}
+\end{subequations}
+Broad envelopes provide a high frequency resolution and so concentrate
+the excitation around the driving frequency $\omega$. Short pulses
+lose frequency selectivity and excite a broad band of frequencies.
+
+The extreme case is an infinitely short pulse,
+$\Delta\tau\longrightarrow{0}$. It amounts eventually to an
+instantaneous boost of the initial wave functions which can be
+expressed as a phase factor according to
+\begin{equation}
+ \label{eq:extboost}
+ \psi_k(\vec r,s,t\!=\!0)
+ =
+ \psi_{k,0}(\vec r,s)\,\exp\left(-\I \eta F_q(\vec r)\right)\;,
+\end{equation}
+where $\psi_{k,0}$ stands for the stationary wave function before
+boost. This instantaneous boost, being infinitely short, is
+insensitive to the problem of asymptotically unstable potentials and
+allows the use of a driving field $F_q$ without damping
+ (\ref{eq:extdamp})
+which, in turn, simplifies spectral analysis (see Section
+\ref{sec:collex}).
+
+The effect of the boost (\ref{eq:extboost}) can be understood by
+virtue of the Madelung decomposition of a complex wave function
+$\phi(\vec r)=\chi(\vec r)\exp(\I S(\vec r))$ with $\chi$ and $S$
+being real. A straightforward calculation leads to the probability
+flow density as $\vec{\jmath}=\hbar\chi^2\nabla S/m$. Dividing by the
+density $\chi^2$ then produces the ``proba\-bility-flow velocity''
+$\vec{v}=\hbar\nabla S/m$. An illustrative example is the plane wave
+where $\chi=$constant and $S=\vec{k}\cdot\vec{r}$ which yields the
+correct result $\vec{v}=\hbar\vec{k}/m$. In the present case,
+assuming that the static wave functions themselves can be written as
+real functions, we get an initial velocity $\vec{v}=-\nabla F_q$,
+i.~e., just in the direction of the classical force resulting from the
+``velocity potential'' $F_q(\mathbf{r})$.
+
+
+
+
+\subsection{Static Hartree-Fock}
+\label{sec:static}
+
+\subsubsection{The coupled mean-field and BCS equations}
+
+The stationary equations are obtained variationally. Variation with
+respect to the single-particle wave functions $\psi_{\alpha}$ yields
+the mean field equations~\cite{Gre05aB,Mar10aB}
+\begin{subequations}
+ \label{eq:mfeq}
+ \begin{equation}
+ \hat{h}\psi_{\alpha}=\varepsilon_\alpha\psi_{\alpha}\;,
+ \label{eq:statmfeq}
+ \end{equation}
+ where $\hat{h}$ is the mean-field Hamiltonian (\ref{eq:spham}) and
+ $\varepsilon_\alpha$ is the single-particle energy of state
+ $\alpha$. Simultaneous variation of $\psi_{\alpha}$ together with
+ the occupation amplitude $v_\alpha$ yields the
+ Hartree-Fock-Bogolyubov equations~\cite{Gre05aB,Rin80aB,Ben03aR}
+ which complicate Eq.~(\ref{eq:statmfeq}) by non-diagonal terms on
+ the right-hand-side~\cite{Rei97a}. We employ here the BCS
+ approximation which exploits time-reversal symmetry where each
+ single-particle state has a time reversed partner
+ $\psi_{\alpha}\leftrightarrow\psi_{\overline{\alpha}}$ as was
+ already implied in the pairing densities $\xi_q$ in
+ Eq.~(\ref{eq:rtjpair}).
+ Each pair of time-conjugate states is associated with an occupation
+ amplitude $v_\alpha$. These are determined by the BCS equation
+ $\left(\varepsilon_\alpha-\epsilon_{\mathrm{F},q_\alpha}\right)
+ \left(u_\alpha^2-v_\alpha^2\right) = \Delta_\alpha
+ u_\alpha^{\mbox{}}v_\alpha^{\mbox{}} $ which can be resolved to a
+ closed expression for the occupation amplitudes as
+ \begin{eqnarray}
+ v_\alpha^2
+ & =&
+ \frac{1}{2} \left( 1 -
+ \frac{\varepsilon_\alpha-\epsilon_{\mathrm{F},q_\alpha}}
+ {\sqrt{(\varepsilon_\alpha-\epsilon_{\mathrm{F},q_\alpha})^2
+ +\Delta_\alpha^2}}
+ \right)
+ \quad,
+ \label{eq:va}\\
+ \Delta_\alpha
+ &
+ =
+ &
+ \frac{1}{2}V_\mathrm{pair,q_\alpha}
+ \int \D^3r\,\psi^+_\alpha\psi_\alpha^{\mbox{}}
+ \xi^{\mbox{}}_{q_\alpha}
+ \left[1 -\frac{\rho}{\rho_{0,\mathrm{pair}}}\right]
+ \quad,
+ \\
+ \epsilon_{\mathrm{F},q}
+ &:&
+ \sum_{\alpha\in q}v_\alpha^2=N_q
+ \quad.
+ \label{eq:parnum}
+ \end{eqnarray}
+ $q_\alpha$ denotes the nucleon type to which state $\alpha$
+ belongs, ${\alpha\in q}$ means all states of type $q$, and $N_q$ is
+ the number of nucleons of type $q$ (identified as $N_p=Z$ and
+ $N_n=N$). The Fermi energies $\epsilon_{\mathrm{F},q_\alpha}$ serve
+ to regulate the average particle number to the required values
+ $N_q$.
+\end{subequations}
+Here, the space of pairing-active states is just the space of states
+actually included in the calculation. The results of BCS pairing
+depend slightly on the size of the active space
+\cite{Gre05aB,Rin80aB}. We recommend using about
+$$N_q+\frac{5}{3}N_q^{2/3}$$
+single-nucleon states, which comes closest to the dynamical pairing
+space of Ref.~\cite{Ben00a}.
+
+
+\subsubsection{Iterative solution}
+\label{sec:statiter}
+
+The coupled mean-field and BCS equations (\ref{eq:mfeq}) are solved
+iteratively. The wave functions are iterated with a gradient step which
+is accelerated by kinetic-energy damping~\cite{Rei82a,Blu92a}
+\begin{equation}
+ \psi_\alpha^{(n+1)}
+ =
+ \mathcal{O}\left\{
+ \psi_\alpha^{(n)}
+ -
+ \frac{\delta}{\hat{T} + E_0}
+ \left( \hat{h}^{(n)} -
+ \langle\psi_\alpha^{(n)}|\hat{h}^{(n)}|\psi_\alpha^{(n)}\rangle
+ \right)\psi_\alpha^{(n)}\right\}
+ \label{eq:dampstep}
+\end{equation}
+where $\hat{T}=\hat{p}^2/(2m)$ is the operator of kinetic energy,
+$\mathcal{O}$ means orthonormalization of the whole set of new wave
+functions, and the upper index indicates the iteration number. Note
+that this sort of kinetic-energy damping is particularly suited for
+the fast Fourier techniques that we use in the present code. The
+damped gradient step has two numerical parameters, the step size
+$\delta$ and the damping regulator $E_0$. The latter should be chosen
+typically of the order of the depth of the local potential $U_q$. In
+practice, we find $E_0=100$ MeV a safe choice. The step size is of
+order of $\delta=0.1...0.8$. Larger values yield faster iteration, but
+can run more easily into pathological conditions. The optimum
+choice depends somewhat on the Skyrme parametrization. Those with
+effective mass $m^*/m\approx 1$ allow larger $\delta$ values. Low
+$m^*/m$ may require a reduction in the step size.
+
+Keeping the value of {\tt x0dmp} constant can cause slow convergence, so that a
+method was developed to change {\tt x0dmp} during the iterations. The
+value from the input is regarded as the minimum allowed one and
+saved in {\tt x0dmpmin}. At the start of the iterations, however, {\tt
+ x0dmp} is multiplied by 3 to attempt a faster convergence.
+
+The change in {\tt x0dmp} is then implemented by comparing the HF
+energy as well as the fluctuations and
+to the previous values. If the energy decreases or one
+of the fluctuations decreases by a factor of less than $1-10^{-5}$,
+{\tt x0dmp} is increased by a factor 1.005 to further speed up
+convergence. If none of these conditions holds, it is assumed that the
+step was too large and {\tt x0dmp} is reduced by a factor 0.8, but is
+never allowed to fall below {\tt x0dmpmin}.
+
+This whole process is turned on only if the input variable {\tt
+ tvaryx\_0} in the namelist "static" is {\tt .TRUE.} The default
+value is {\tt .FALSE.}
+
+After each such wave function step, the BCS equations
+(\ref{eq:va}--\ref{eq:parnum}) are solved with
+$\varepsilon_\alpha=\langle\psi_\alpha|\hat{h}|\psi_\alpha\rangle$,
+the densities are updated (Eqs.~\ref{eq:rtjeven}-\ref{eq:rtjpair}),
+and new mean fields computed (\ref{eq:spham}-\ref{eq:spot}).
+This then provides the starting point for the next
+iteration. The process is continued until sufficient convergence is
+achieved. We consider as the convergence criterion the average energy
+variance, or fluctuation, of the single particle
+states
+\begin{subequations}
+ \label{eq:totvar}
+ \begin{eqnarray}
+ \overline{\Delta\varepsilon}
+ &=&
+ \sqrt{\frac{\sum_\alpha\Delta\varepsilon_\alpha^2}{\sum_\alpha 1}}
+ \quad,
+ \\
+ \Delta\varepsilon_\alpha^2
+ &=&
+ \langle\psi_\alpha|\hat{h}^2|\psi_\alpha\rangle
+ -
+ \varepsilon_\alpha^2
+ \quad,
+ \label{eq:spvar}
+ \\
+ \varepsilon_\alpha
+ &=&
+ \langle\psi_\alpha|\hat{h}|\psi_\alpha\rangle
+ \quad.
+ \label{eq:spenerg}
+ \end{eqnarray}
+\end{subequations}
+The single particle energy $\varepsilon_\alpha$ is defined here as an
+expectation value. It finally becomes an eigenvalue in
+Eq.~(\ref{eq:statmfeq}) if the iteration has converged to
+$\Delta\varepsilon_\alpha\approx 0$. Vanishing total variance
+$\overline{\Delta\varepsilon}$ signals that we have reached minimum
+energy, i.e. a solution of the mean-field plus BCS equations. One has
+to be aware, however, that this may be only a local minimum (isomeric
+state). It requires experience to judge whether one has found the
+absolute energy minimum. In case of doubt, one should redo a couple of
+static iterations from very different initial configurations.
+
+
+This raises the question of how to initialize the iteration. We take
+as a starting point the wave functions of the deformed harmonic
+oscillator (see point \ref{it:iniho} in Section~\ref{sec:init}). These
+are characterized by $\vec{n}=(n_x,n_y,n_z)$, the number of nodes in
+each direction. We stack the wave functions in order of increasing
+oscillator energy
+$\epsilon_\alpha^{(0)}=\hbar\omega_xn_x+\hbar\omega_yn_y+\hbar\omega_zn_z$
+and stop if the desired number of states is reached. The deformation of
+the initializing oscillator influences the initial state in two ways:
+first, through the deformation of the basis wave functions as such,
+and second, through the energy ordering of the $\epsilon_\alpha^{(0)}$
+and corresponding sequence of levels built. Variation of initial
+conditions means basically a variation of the oscillator deformation.
+For example, the iteration will most probably end up in a prolate
+minimum if the initial state was sufficiently prolate, and in an
+oblate minimum after an oblate initial state. It depends on the
+nucleus which one is the absolute minimum.
+
+\subsection{TDHF}
+
+\subsubsection{The time-dependent mean-field equations}
+
+The TDHF equations are determined from the variation of the action
+\begin{equation*}
+ S=\int \D t\left[E[\{\psi_\alpha\}]-
+ \sum_\alpha\langle\psi_\alpha|\I \partial_t|\psi_\alpha\rangle
+ \right]\;,
+\end{equation*}
+with respect to the wave functions $\psi_\alpha^+$ where the energy is
+given as in Eqs.~(\ref{eq:efundet}-\ref{eq:ehfc})~\cite{Mar10aB}.
+This yields the TDHF
+equation
+\begin{equation}
+ \I \partial_t\psi_{\alpha}=\hat{h}\psi_{\alpha}\;,
+ \label{eq:dynmfeq}
+\end{equation}
+where $\hat{h}$ is, again, the mean-field Hamiltonian
+(\ref{eq:spham}). For simplicity, we are not considering variation of
+the occupation amplitude in the time-dependent case. The occupation
+amplitudes obtained from static iteration are kept frozen during time
+evolution. For studies of mean-field flow at moderate
+excitations (heavy-ion collisions, giant resonances) this
+approximation is legitimate. However, a study of truly low energy
+dynamics in the range of a few MeV (soft vibrations, fission) requires
+a full time-dependent Hartree-Fock-Bogolyubov treatment and should not
+be undertaken with the present code.
+
+
+\subsubsection{Time development algorithm}
+\label{sec:timevol}
+
+
+The TDHF equation (\ref{eq:dynmfeq}) can be formally resolved into an
+integral equation as
+\begin{subequations}
+ \begin{eqnarray}
+ |\psi_\alpha(t\!+\!\Delta t)\rangle
+ &=&
+ \hat U(t,t\!+\!\Delta t)|\psi_\alpha(t)\rangle
+ \\
+ \hat U(t,t\!+\!\Delta t)
+ &=&
+ \hat{\mathcal{T}}\exp\left(-\frac{\I}{\hbar}\int_t^{t+\Delta t}
+ \hat h(t')\,\D t'\right)
+ \;,
+ \label{eq:timexp}
+ \end{eqnarray}
+\end{subequations}
+where $\hat U$ is the time-evolution operator and $\hat{\mathcal{T}}$
+the time-ordering operation. This time evolution is unitary, thus
+conserving orthonormalization of the single-particle wave functions,
+and it conserves the total energy (\ref{eq:efundet}) provided that there are
+no time-dependent external fields. To convert this involved
+operator into an efficiently computable but also sufficiently accurate
+form a predictor-corrector strategy is used:
+\begin{enumerate}
+\item In a first step (predictor), we determine the single-particle
+ Hamiltonian at midtime $\hat{h}(t\!+\!\Delta t/2)$.
+ To that end, a trial step by $\Delta t$
+ \begin{equation}
+ \tilde{\psi}_\alpha
+ =
+ \exp\left(-\tfrac{\I}{\hbar}
+ \hat h(t)\,\Delta t\right)\psi_\alpha(t)
+ \label{eq:midstep}
+ \end{equation}
+ is performed using the mean field $\hat{h}(t)$ at initial time $t$.
+ The density $\tilde\rho$ and similarly also other densities and
+ currents at $t+\Delta t$ are accumulated from the wave functions
+ $\tilde{\psi}_\alpha$, which can be discarded immediately, so that
+ it is not necessary to store a complete second set of wave
+ functions. They are used to compute an estimate for the densities at
+ half step $\rho_{\rm pre}=(\rho(t)+\tilde\rho)/2$ and analogously
+ for the other densities and currents. These are then used to
+ calculate the estimated Hamiltonian $\hat h_{\rm pre}$ at $t+\Delta
+ t/2$ according to Eqs.~(\ref{eq:spham}-\ref{eq:spot}).
+
+\item This is used to perform the corrector step to advance the wave
+ functions to the end of the time step (again with frozen
+ Hamiltonian, but now $\tilde{h}_\mathrm{pre}$)
+ \begin{equation}
+ \psi_\alpha(t\!+\!\Delta t)
+ =
+ \exp\left(-\tfrac{\I}{\hbar}{\hat h}_\mathrm{pre}\,\Delta t\right)
+ \psi_\alpha(t)
+ \;.
+ \label{eq:finstep}
+ \end{equation}
+\item In both cases the operator exponential is evaluated by a Taylor
+ series expansion up to order $m$:
+ \begin{equation}
+ \exp\left(-\tfrac{\I}{\hbar}\hat{h}\,\Delta t\right)\psi
+ \approx
+ \sum_{n=0}^m
+ \frac{(-\I\Delta t)^n}{\hbar^n\,n!}\hat h^n\psi\;,
+ \label{eq:timepower}
+ \end{equation}
+ where $\hat{h}$ is the actual mean field in step (\ref{eq:midstep}),
+ or (\ref{eq:finstep}) respectively. $\hat h^n\psi$ is computed
+ in straightforward manner by successive application of the mean
+ field Hamiltonian, i.e. $\hat
+ h^n\psi=\underbrace{\hat{h}(...(\hat{h}}_{n \mbox{\tiny
+ times}}\psi)...)$.
+\end{enumerate}
+The Taylor expansion spoils strict unitarity of the exponential
+$\exp\left(-\tfrac{\I}{\hbar}\hat{h}\,\Delta t\right)$ and energy
+conservation. We turn this flaw into an advantage and use norm
+conservation as well as energy conservation (if it applies) as
+counter-check of the quality of the step along the propagation. The
+reliability depends, of course, on a proper choice of the numerical
+parameters in this step which are the step size $\Delta t$ and the
+order of the Taylor expansion $m$. The step size is limited by the
+maximum possible kinetic energy and by the typical time scales of changes in
+the mean field $\hat{h}$. The maximum kinetic energy, in turn, depends
+on the grid spacing as $\propto \Delta x^{-2}$. A choice of $\Delta
+t=0.1-0.2$fm/c is applicable in connection with $\Delta x=0.7-1$/fm.
+For the order of Taylor expansion, one needs at least $m=4$. Although
+there are formal reasons for $m=4$ \cite{bertsch}, in practice
+$m>4$ may also be used, but choosing $m>6$ is not so efficient for
+the values of $\Delta t$ considered here.
+
+\subsubsection{Collective excitations}
+\label{sec:collex}
+
+Giant resonances are prominent excitation modes of nuclei. Best known
+is probably the isovector giant dipole resonance, but there are many more
+modes depending on isospin
+and angular momentum. The typical resonance energies lie in a region
+from 10 to 30 MeV where the present TDHF code with frozen occupation
+numbers is applicable because the relevant energy range lies far above
+the pairing gap (1--2 MeV). The generation of these modes is
+particularly simple within the present TDHF treatment. One first
+produces a stationary state as outlined in Section~\ref{sec:static}
+and then triggers the excitation by a time-dependent external
+field as described in Section~\ref{sec:external}. A broad pulse allows
+triggering particular excitation energies. An infinitely short pulse
+amounts to an instantaneous boost.
+
+
+The boost is a generic excitation of a system which gives the same
+weight to all frequencies. It is thus ideally suited for analyzing in
+an unbiased manner the excitation spectra of a system. This, in turn,
+allows a thorough spectral analysis. To obtain the spectral
+distribution of isovector dipole strength, one applies a boost with
+small strength $\eta$ and $F_q=\hat{D}\propto r^1Y_{10}\tau_z$ the
+isovector dipole operator. The Slater determinant $|\Phi(t)\rangle$ is
+propagated in TDHF for a sufficiently long time while recording the
+dipole moment $D(t)=\langle\Phi(t)|\hat{D}|\Phi(t)\rangle$. The dipole
+strength is finally extracted from the Fourier transform
+$\tilde{D}(\omega)$ as
+$S_D(\omega)=\Im\left\{\tilde{D}(\omega)\right\}/\eta$. Note that
+this is valid only in the linear case, i.~e., if the amplitude of the
+vibration if proportional to the boost velocity \cite{Cal97a,Rei07a}.
+This should be checked in the calculations.
+
+The straightforward Fourier transform leads to artifacts if the dipole
+signal has not fully died out at the end of the simulation time. In
+the general case, some filtering is necessary to suppress artifacts
+from cutting the signal at a certain final time~\cite{Pre92aB}. In
+practice, it is most convenient to use filtering in the time domain by
+damping the signal $D(t)$ towards the final time. A robust choice is
+\begin{equation}
+ D(t)
+ \quad\longrightarrow\quad
+ D_\mathrm{fil}(t)
+ =
+ D(t)\cos\left(\frac{\pi}{2}\frac{t}{t_\mathrm{final}}\right)^{n_\mathrm{fil}}
+\end{equation}
+where $t_\mathrm{final}$ is the final time of the simulation. This
+guarantees that the effective signal $D_\mathrm{fil}$ vanishes at the
+end of the interval. The $\cos^n$ profile switches off gently and
+leaves as much as possible from the relevant signal at early times.
+The parameter $n_\mathrm{fil}$ determines the strengths of filtering.
+Value of order of 4--6 are recommended to suppress the
+artifacts safely. For a detailed description of this spectral analysis see
+\cite{Cal97a}. For typical applications in nuclear physics see
+\cite{Rei07a}. It is to be noted that the code does not include this
+final step of spectral analysis. The time dependent signals are
+printed on the protocol files {\tt monopoles.res}, {\tt dipoles.res},
+and {\tt quadrupoles.res}. It is
+left to the user to perform the final steps towards a spectral
+distribution. A word is in order about $t_\mathrm{final}$. It
+determines the resolution of the spectral analysis. The corresponding
+energy bins are given by
+$\delta{E}_\mathrm{exc}=\hbar\pi/t_\mathrm{final}$. Windowing effectively
+reduces the time span in which relevant information is contained
+and roughly doubles the relevant $\delta{E}_\mathrm{exc}$. For
+example, to obtain a spectral resolution of 1 MeV, one needs to
+simulate up to about 1200 fm/c.
+
+
+Although excitation spectra are one of the most basic properties of the system, there
+are many other dynamical features of interest. The multipole signals
+in the time domain (printed in the protocol files) are as such
+interesting quantities. One can have, e.g., a look at cross-talk
+between the multipole channels. It is particularly interesting to
+study excitation dynamics for varying excitation strength $\eta$, from
+the regime of linear response (small $\eta$) deep into the non-linear
+regime. It is inefficient to perform a full three-dimensional TDHF
+calculation to obtain linear-regime excitation spectra for spherical nuclei. This is
+better done in a dedicated RPA calculation on a spherical basis (see,
+e.g.,~\cite{Rei92b}) for which there exist an overwhelming multitude
+of codes. The realm of TDHF calculations of nuclear excitations are
+spectra in deformed systems, stability analysis of exotic
+configurations, and in particular non-linear dynamics.
+
+There are many more details worth looking at. One may check the
+densities and currents to visualize the flow pattern associated with a
+mode. A most elaborate analysis deals with a phase-space picture of
+nuclear dynamics by virtue of the Wigner transformation~\cite{Loe11a}.
+The code allows saving all ingredients needed for such elaborate
+analysis in dedicated output files, see Section~\ref{sec:silo}. It is
+left to the user to work out the further steps of the analysis.
+
+A serious problem that can occur in collective excitation studies is
+the evaporation of particles at higher excitation energies. Some of
+the single-particle wave functions become unbound and move towards the
+boundaries for the computational box with a non-negligible part of
+their probability distributions. Since the code assumes periodic
+boundary conditions for the wave functions (for details see
+Sect~\ref{sec:bc}), effectively the particles reenter the box from the
+opposite side and can interact with the nucleus again, leading to a
+spurious revival of the excitation \cite{Nak05a,Rei06a}. This effect
+can be reduced in several ways; for a brief discussion see
+Sect.~\ref{sec:bc}.
+
+
+\subsubsection{Nuclear reactions}
+\label{sec:react}
+Collisions of nuclei are a prime application of nuclear TDHF. They
+were, in fact, the major motivation for its realization
+\cite{Bon78a,Dav85a}. The present code is designed to initialize such
+collision scenarios in a most flexible manner. We start by explaining
+the simplest case of a collision of two nuclei. First, we prepare the
+ground states of the two nuclei as explained in
+Section~\ref{sec:statiter}. The static solutions are centered around
+the origin $\vec{r}=0$ of their initial grid. The static wave
+functions $\psi_{\alpha,I}^{\mbox{(stat)}}$ where $I=1,2$ labels the
+two nuclei are shifted to new centers $\vec{R}_I$ where the distance
+$|\vec{R}_2-\vec{R}_1|$ should be sufficiently large that the nuclei
+have negligible overlap and negligible Coulomb distortion from the
+other nucleus (the latter condition usually only loosely fulfilled).
+The shifted wave functions
+$\psi_{\alpha,I}^{\mbox{(stat)}}(\vec{r}-\vec{R}_I,s)$ are obtained by
+interpolation on the grid. The interpolation is done by transforming
+to momentum space, applying translation factors, and going back. It is
+obvious that the collisions need a larger numerical box than the
+static Hartree-Fock calculations. Thus, we may compute the static wave
+functions on a smaller box since we are shifting and interpolating the
+result anyway for dynamical initialization.
+
+It should be noted that the nuclei may be placed at arbitrary
+positions in the new grid. It is highly recommended, however, to
+displace them by an integer number of grid points from the original
+position, since otherwise the interpolation may lead to some degree of
+excitation. In practice, if the center of mass in the static calculation
+was at the origin, the new position should be of the form ($m_x$ {\tt
+ dx}, $m_y$ {\tt dy}, $m_z$ {\tt dz}) with integer factors $m_x$
+etc. If for some reason this is not desired, accuracy can be restored
+by placing the nucleus {\em alone} in the larger grid and running this
+as input for a static calculation with a sufficient number of time steps
+before using the resulting wave functions as input for the dynamic one.
+
+At this point we have the
+nuclei resting at a safe distance. To set them in motion we need to
+give each nucleus a momentum $\vec{P}_1=-\vec{P}_2$ (note that the
+total momentum of the combined system still vanishes). Consequently,
+the initial configuration is given by the Slater state built from the
+shifted and boosted single-particle wave functions (see fragment
+initialization, point \ref{it:frag} in Section~\ref{sec:init})
+\begin{equation}
+\begin{array}{ll}
+ \psi_{\alpha,1}(\vec{r},s;t\!=\!0)
+ =e^{\I\vec{p}_1\cdot\vec{r}}
+ \psi_{\alpha,1}^{\mbox{(stat)}}(\vec{r}-\vec{R}_1,s),&
+ \vec{p}_1 =\frac{\vec{P}_1}{A_1}
+ \quad,
+ \\
+ \psi_{\alpha,2}(\vec{r},s;t\!=\!0)
+ = e^{\I\vec{p}_2\cdot\vec{r}}
+ \psi_{\alpha,2}^{\mbox{(stat)}}(\vec{r}-\vec{R}_2,s),&
+ \vec{p}_2=\frac{\vec{P}_2}{A_2}
+ \quad.
+ \label{eq:inittdhf}
+\end{array}
+\end{equation}
+The distance between the nuclei is large, but inevitably finite. This may
+induce minor violations of orthonormality. Thus the
+full set of wave functions (\ref{eq:inittdhf}) is orthonormalized as a final step of
+initial preparation.
+
+The occupation amplitudes $v_{\alpha,I}$ are taken over from the
+static solution and frozen along the dynamical evolution. In fact,
+most of the collision studies will be principally to explore
+the dynamical features in the regimes of fusion and inelastic
+collisions. It is then recommended to use the conceptually simplest
+and most robust strategy, namely to start from simple Slater states
+(not BCS states) for the two nuclei. This means that in most cases
+the static solution is calculated without pairing, fixing $v_\alpha=1$ and
+including just as many states as there are nucleons.
+
+The above example deals with two initial fragments. The code is more
+flexible than that. It allows an initial state composed from several
+fragments. The strategy remains the same as for the binary system. It
+is simply repeated for each new fragment.
+
+The time evolution is performed as outlined in Section
+\ref{sec:timevol}. It requires some effort to visualize the complex
+dynamics which emerges in collisions. A rough picture is, again,
+provided by the multipole moments. The quadrupole moment, e.g., can
+serve as a measure of stretching of the total system. Small values
+indicate a compound nucleus while asymptotically growing values signal
+fragmentation. One may want, particularly in case of collisions, more
+detailed pictures of the flow as, e.g., snapshots of the density of
+current distributions and, ultimately, a full phase space picture
+\cite{Loe11a}. Material for that can be output on demand, as detailed
+in Section \ref{sec:silo}. Again, we leave it to the user to extract
+the wanted information and to prepare it for visualization.
+
+In nuclear collisions at higher energies the emission of particles can
+also cause problems; in those cases one better uses absorbing boundary
+conditions. See Sect.~\ref{sec:bc} for details.
+
+\subsection{Observables}
+\label{sec:observables}
+
+It was already mentioned in Sections~\ref{sec:collex} and
+\ref{sec:react} which observables may be used to analyze nuclear
+dynamics. We here briefly summarize the observables computed
+and output in the code and indicate how further observables may be
+extracted. Basic features of the description by the Skyrme
+energy-density functional are, of course, energy and densities.
+
+\subsubsection{Multipole moments}
+\label{sec:mulmom}
+
+The gross features of the density distribution are well characterized
+by the multipole moments. The most important moment is the
+center of mass (c.m.)
+\begin{subequations}
+ \label{eq:mulmom}
+ \begin{equation}
+ \vec{R}^\mathrm{(type)}
+ =
+ \frac{\int \D^3r\,\vec{r}\,\rho^\mathrm{(type)}(\vec r)}{A}
+ \quad,
+ \label{eq:cmcart}
+ \end{equation}
+ where $A=\int \D^3r\,\rho(\vec r)$ is the total mass number and
+ ``type'' can refer to proton c.m.\ from $\rho_p$, neutron c.m.\
+ from $\rho_n$, isoscalar or total c.m.\ from the total density
+ $\rho=\rho_p+\rho_n\equiv\rho_{T=0}$, or isovector moment related to
+ the isovector density $\rho_{T=1}=\frac{N}{A}\rho_p-\frac{Z}{A}\rho_n$.
+ The
+ definition of $\vec{R}^\mathrm{(type)}$ directly employs the
+ Cartesian coordinate $r_i$. The same holds for the Cartesian
+ quadrupole tensor
+ \begin{equation}
+ \mathcal{Q}_{kl}^\mathrm{(type)}
+ =
+ \int \D^3r \left(3(r_k-R_k)(r_l-R_l)-\delta_{kl}\sum_i(r_i-R_i)^2\right)
+ \rho^\mathrm{(type)}(\vec r)
+ \quad,
+ \label{eq:Qcart}
+ \end{equation}
+ again for the various types as discussed above. The matrix
+ $\mathcal{Q}_{kl}$ is not invariant under rotations of the
+ coordinate frame. There is a preferred coordinate system: the system
+ of principle axes. It is obtained by diagonalizing
+ $\mathcal{Q}_{kl}$. The quadrupole matrix in the principle-axis
+ frame thus has only three non-vanishing entries $Q_{xx}$, $Q_{yy}$,
+ and $Q_{zz}$ together with the trace condition
+ $Q_{xx}+Q_{yy}+Q_{zz}=0$.
+
+For the quadrupole case the spherical moments defined as
+ \begin{equation}
+ Q_{2m}^\mathrm{(type)}
+ =
+ \int \D^3r\,r^2Y_{2m}\,\rho^\mathrm{(type)}(\vec{r}-\vec{R})\;,
+ \label{eq:spherQ}
+ \end{equation}
+ are also useful, where $r=|\vec{r}|$ and $Y_{2m}$ are the
+spherical harmonics. They are often expressed as dimensionless
+ quadrupole moments
+ \begin{equation}
+ a_m
+ =
+ \frac{4\pi}{5}
+ \frac{Q_{2m}}{AR^2}
+ \quad,
+ \label{eq:dimlessQ}
+ \end{equation}
+ with $R=r_0A^{1/3}$ a fixed radius derived from the total mass
+ number $A$.
+ This, again, could be defined for any ``type'', but is used, in
+ practice, mainly for the isoscalar moments.
+
+ The dimensionless moments have
+ the advantage of being free of an overall scale which was removed by
+ the denominator $AR^2$. They allow characterization of the
+ shape of the nucleus. However, the general $a_m$ are not invariant
+ under rotations of the coordinate frame. We obtain a unique
+ characterization by transforming to the principle-axis system. These
+ are defined by the conditions $a_{\pm 1}=0$ and $a_2=a_{-2}$. There
+ remain only two shape parameters $a_0$ and $a_2$. These are often reexpressed
+ as total deformation $\beta$ and triaxiality $\gamma$, often
+ called Bohr-Mottelson parameters, through
+ \begin{equation}
+ \beta
+ =
+ \sqrt{a_0^2+2a_2^2}
+ \quad,\quad
+ \gamma
+ =
+ \mbox{atan}\left(\frac{\sqrt{2}\,a_2}{a_0}\right)
+ \quad.
+ \label{eq:triax}
+ \end{equation}
+ Triaxiality $\gamma$ is handled like an angle. It can, in principle,
+ take all values between $0^o$ and $360^o$, but physically relevant
+ parameters stay in the $0\ldots60^\circ$ range. The other sectors
+ correspond to equivalent configurations~\cite{Gre05aB}.
+
+ The monopole moment just corresponds to the total particle number,
+ so to describe monopole vibrations usually the r.m.s.\ radii or
+ their squares are employed. The r.m.s. radii
+ are defined as
+ \begin{equation}
+ r_\mathrm{rms}^\mathrm{(type)}
+ =
+ \sqrt{\frac{\int \D^3r\,(\vec{r}-\vec{R})^2\,\rho^\mathrm{(type)}(\vec{r})}
+ {\int \D^3r\,\rho^\mathrm{(type)}(\vec{r})}}
+ \label{eq:rmsrad}
+ \end{equation}
+ where ``type'' can be proton, neutron, or total. The isovector
+ variant does not make sense here.
+
+ We supply printouts of all the above variants of multipole moments
+ to allow a most flexible analysis. For the reason given above the
+ r.m.s.\ radii are output in a file called {\tt monopolesfile}.
+\end{subequations}
+
+\subsubsection{Alternative way to evaluate the total energy}
+\label{sec:alterenerg}
+
+The key observable is the total energy $E_\mathrm{tot}$. It is computed as
+given in Eqs. (\ref{eq:efundet}-\ref{eq:ehfc}). More detailed energy observables are
+provided by the s.p.\ energies (\ref{eq:spenerg}). These can also be
+used to compute the total energy. The traditional HF scheme deals with
+pure two-body interactions and exploits that to simplify~\cite{Gre05aB}
+\begin{equation}
+ \label{eq:koopman}
+ E_\mathrm{tot,HF}
+ =
+ \frac{1}{2}\sum_\alpha\left(t_\alpha+\epsilon_\alpha\right)
+\end{equation}
+where $t_\alpha=\langle\psi_\alpha|\hat{T}|\psi_\alpha\rangle$ is the
+s.p.\ kinetic energy. This is possible because
+\begin{equation*}
+ \epsilon_\alpha=t_\alpha+u_\alpha
+ \quad,\quad
+ u_\alpha
+ =
+ \sum_{\beta}\left[v_{\alpha\beta\alpha\beta}-v_{\alpha\beta\beta\alpha}\right]
+ -\tfrac1{2} v_\alpha=\epsilon_\alpha-t_\alpha
+ \quad,
+\end{equation*}
+where $u_\alpha$ is the s.p.\ mean-field potential energy and $v$ the
+two-body interaction, and
+\begin{equation*}
+ E_\mathrm{tot,HF}
+ =
+ \sum_\alpha t_\alpha
+ +
+ \frac{1}{2}\sum_{\alpha\beta}
+ \left[v_{\alpha\beta\alpha\beta}-v_{\alpha\beta\beta\alpha}\right]
+ \quad.
+\end{equation*}
+The Skyrme force does not simply have this two-body structure. Still
+the total energy is very often computed along the strategy of
+Eq.~(\ref{eq:koopman}). However, the density dependence requires
+augmenting this recipe by a rearrangement energy which accounts for a
+contribution missing in the simple recipe (\ref{eq:koopman}). The
+extension to the Skyrme energy thus reads
+\begin{subequations}
+ \label{eq:koopman2}
+ \begin{eqnarray}
+ E_\mathrm{tot,HF}
+ &=&
+ \frac{1}{2}\sum_\alpha\left(t_\alpha+\epsilon_\alpha\right)
+ + E_{3,\rm corr} + E_\mathrm{C,corr}
+ \quad,
+ \\
+ E_{3,\rm corr}
+ & = &
+ \int \D^3r \frac{\alpha}{6} \rho^{\alpha}
+ \left[b_3\rho^2 - b'_3 (\rho_{p}^2+\rho_{n}^2) \right]
+ \quad,
+ \label{eq:e3corr}
+ \\
+ E_\mathrm{C,corr}
+ &=& \frac{1}{4} \left( \frac{3}{\pi} \right)^{1/3}
+ \int \D^3r\, \rho_{pr}^{4/3}
+ \quad.
+ \label{eq:ecorc}
+ \end{eqnarray}
+\end{subequations}
+In the code the total energy is computed both ways, from the
+straightforward Skyrme energy (\ref{eq:efundet}) as well as from the
+above recipe (\ref{eq:koopman2}). Numerically these values are close but not
+identical.
+
+\subsection{Two-body analysis}
+For binary collisions with two fragments in the final state, the
+properties of these and of the relative motion form an important set
+of observables. They are computed at the end of the calculation when a
+reasonable distance --- determined by the input variable {\tt rsep}
+--- between the fragments has been reached using the module {\tt
+ Twobody}. Some parts of the routines in this module are also called
+every time step in order to check whether separation has occurred and,
+in that case, what the distance between the fragments is in order to
+stop the calculation when the desired separation has been reached. At
+that point then the full two-body analysis is performed.
+
+Since the calculational grid cannot be made too large, the fragments
+are not completely decoupled as they would be if the point-particle
+approximation were valid. Instead there are Coulomb interactions
+between the higher multipoles. The code tries to treat this problem by
+a sophisticated calculation of the Coulomb interaction energy: it
+computes the Coulomb energy separately for each isolated fragment and
+then subtracts that from the Coulomb energy of the total system. This
+was shown to yield almost constant asymptotic relative kinetic
+energies for a broader range of separation distances. The typical
+behavior is illustrated in Fig.~\ref{fig:twobody}.
+
+If it is desired to investigate the dependence of the two-body results
+on the separation distance, simply restart the code with a larger
+value of {\tt rsep}
+
+\begin{figure}[h]
+\centerline{\includegraphics[width=\linewidth]{Twobody.eps}}
+\caption{\label{fig:twobody} Variation of the extrapolated relative
+ kinetic energy and scattering angle as the distance between the
+ fragments increases from 20 to 31~fm. The angular momenta also show
+ changes on the 1\% level. This sample calculation is for a collision
+ of ${}^{68}$Ni+${}^{208}$Pb at a center-of-mass energy of 290~MeV
+ and impact parameter $b=$4~fm.}
+\end{figure}
+
+The observables calculated are: mass and charge numbers of the
+fragments, final asymptotic kinetic energy, orbital angular momenta of
+the fragments (internal) and of relative motion, scattering angle (also
+from an extrapolation to infinite separation. For most of these the
+initial values are also printed.
+
+The two-body analysis assumes that orbital motion is in the
+$y$-$z$-plane. It uses the following steps:
+\begin{enumerate}
+\item Find out whether separation has occurred. The axis of largest
+ quadrupole moment is calculated and then it is checked whether the
+ total density along this axis shows a double-humped behavior with a
+ sufficient low minimum between. If so, a plane perpendicular to axis
+ at the lowest density is constructed to divide the computational grid
+ into two parts. This iterated a few times to get consistent results.
+\item Calculate masses, charges, centers of mass, total momenta,
+ angular momenta, and rotational frequency vectors for the two parts.
+\item The data from the initialization of the calculation is
+ read again to get the relative energy and impact parameter before
+ the collision. Then the scattering angle is calculated as the sum of
+ three contributions: deflections for the incoming and outgoing
+ Rutherford trajectories and rotation angle during the TDHF
+ calculation. {\bf Note: The simple summation of these three angles
+ may not work in special cases. The user should look at the final
+ direction the fragments are moving.}
+\end{enumerate}
+
+\subsection{Discretization}
+\subsubsection{Data types}
+In the module {\tt Params} a type {\tt db} is defined for 12-digit
+accuracy and on all present machines should amount to double
+precision, i.~e., {\tt REAL(db)} and {\tt COMPLEX(db)} are actually
+{\tt REAL(8)} and {\tt COMPLEX(8)} for most compilers. Keeping the symbolic type
+throughout of course makes the code more flexible for future hardware.
+Using single precision is not recommended.
+
+Since the external libraries are based on C or Fortran-77 coding,
+special care has to be taken in this respect. The FFTW3 library stores
+its plans in 8-byte integers, which in the modules {\tt Coulomb} and
+{\tt Fourier} are defined using the type {\tt C\_LONG} from the
+system-supplied module {\tt ISO\_C\_BINDING}. If this is not
+available, they may be defined as {\tt INTEGER(8)} or if that also
+causes problems, {\tt DOUBLE PRECISION}, which certainly corresponds
+to at least 8 bytes. For the {\tt LAPACK} routines, double-precision
+real and complex variables are necessary, which we also define using
+``{\tt db''}. If {\tt db} should ever be changed in such a way that
+{\tt REAL(db)} no longer corresponds to 8 bytes, different {\tt LAPACK}
+routines must be selected.
+
+Note that data conversion needs some care. If {\tt a} and {\tt b} are
+double precision, {\tt CMPLX(a,b)} is not; according to the standard
+it returns the default accuracy, which on many machines will still be
+single precision. Thus that for example in {\tt EXP(CMPLX(a,b))} the
+exponential would be evaluated in single precision. That is why in the
+code the expression {\tt CMPLX(a,b,db)} is used consistently; this is
+also safe for future changes in accuracy.
+
+The other conversion functions used are safe. {\tt AIMAG} is generic
+and {\tt REAL} reproduces the accuracy of (only) a {\em complex
+ argument}.
+
+\subsubsection{Grid definition}
+\label{sec:griddef}
+All wave functions and fields are defined on a three-dimensional
+regular Cartesian grid of {\tt nx} by {\tt ny} by {\tt nz} grid
+points. {\em {\tt nx}, {\tt ny}, and {\tt nz} must be even numbers}.
+The physical spacing between the points is given as {\tt dx}, {\tt
+ dy}, and {\tt dz} (in fm). In principle these could be different for
+the three directions, but since this will lead to a loss of accuracy
+it is highly recommended to give the same value to all of them. A
+typical range is 0.5--1.0~fm.
+
+The grid is automatically arranged in such a way that in each
+direction the same number of grid points are located on both sides of the
+origin. This means that the three-dimensional origin is in the center
+of a cubic cell and has the advantage that exact parity properties for
+the wave functions can be maintained. The coordinate values for e.~g.,
+the $x$-direction are thus:
+\begin{equation}\begin{split}
+ -\frac{{\tt nx}-1}{2}&{\tt dx},\quad -\frac{{\tt nx}-1}{2}{\tt
+ dx}+{\tt dx},\quad\ldots \quad -\frac{{\tt dx}}{2},\\
+ &+\frac{{\tt dx}}{2},\quad \ldots\quad \frac{{\tt nx}-1}{2}{\tt
+ dx}.
+\end{split}
+\end{equation}
+The corresponding values are available in the arrays {\tt x(nx)}, {\tt
+ y(ny)}, and {\tt z(nz)}.
+
+\subsubsection{Derivatives}
+\label{sec:deriv}
+
+The computation of the kinetic densities and currents and the
+application of the mean-field Hamiltonian require first and second
+derivatives at several places in the code. We define them in Fourier
+space. For simplicity, the strategy is explained here for one
+dimension. The generalization to 3D is obvious.
+
+The {\tt nx} discrete grid points $x_\nu$ in coordinate space are
+related to the same number of grid points $k_n$ in Fourier space
+(physically equivalent to momentum space) as
+\begin{subequations}
+\label{eq:FT}
+\begin{eqnarray}
+ x_\nu
+ &=&
+ \left(-\frac{{\tt nx}-1}{2}+\nu\right){\tt dx}
+ \quad,
+ \nu=1,...,{\tt nx}
+ \quad,
+\\ \label{eq:kvalues}
+ k_n &=&
+ (n-1) {\tt dk},\quad n=1,\ldots {\tt nx}/2 \quad,
+\\ \nonumber
+ k_n&=&(n-{\tt nx}-1)\,{\tt dk},\quad n={\tt nx}/2+1,\ldots,{\tt nx} \quad,
+\\
+ {\tt dk}
+ &=&
+ \frac{2\pi}{{\tt nx}\cdot{\tt dx}}
+ \quad.
+\end{eqnarray}
+Note the particular indexing for the $k$-values. In principle, the
+values $k_n=(n-1){\tt dk}$ for all $n$ are equivalent for the Fourier
+transform, but for the second half of this range the negative
+$k$-values should be chosen because of their smaller magnitude. For
+the Fourier expansion, $k=-{\tt dk}$ and $k=({\tt nx}-1){\tt dk}$ are
+equivalent because of periodicity in $k$-space.
+
+A function $f(x_\nu)$ in coordinate space is connected to a
+function $\tilde{f}(k_n)$ in Fourier space by
+\begin{eqnarray}
+ \tilde{f}(k_n)
+ &=&
+ \sum_{\nu=1}^{\tt nx}
+ \exp{\left(-\I k_nx_\nu\right)}f(x_\nu)
+ \quad,
+\label{eq:FTforward}\\
+ f(x_\nu)
+ &=&
+ \frac{1}{{\tt nx}}\sum_{n=1}^{\tt nx}
+ \exp{\left(\I k_nx_\nu\right)}\tilde{f}(k_n)
+\label{eq:FTbackward}
+\end{eqnarray}
+\end{subequations}
+This complex Fourier representation implies that the function $f$ is
+periodic with $f(x+{\tt dx}\cdot{\tt nx})=f(x)$. The appropriate
+integration scheme is the trapezoidal rule which complies with the above
+summations adding up all terms with equal weight.
+The derivatives of the exponential basis functions are
+\begin{equation}\label{eq:expbas}
+ \frac{\D^m}{\D x^m}\exp{(\I k_nx)}
+ =
+ (\I k_n)^m\exp{(\I k_nx)}
+ \quad.
+\end{equation}
+Computation of the $m$th derivative thus becomes a trivial
+multiplication by $(\I k_n)^m$ in Fourier space. Time critical
+derivatives are best evaluated in Fourier space using the fast Fourier
+transformation (FFT). To that end a forward transform
+(\ref{eq:FTforward}) is performed, then the values $\tilde{f}(k_n)$
+are multiplied by $(\I k_n)^m$ as given in
+Eq.~(\ref{eq:expbas}) and finally transformed $(\I
+k_n)^m\tilde{f}(k_n)$ back to coordinate space by the transformation
+(\ref{eq:FTbackward}). This strategy is coded in the subroutines
+\texttt{cdervx}, \texttt{cdervy}, and \texttt{cdervz} contained in
+module \texttt{Levels}. It is used for derivatives of wave functions
+provided the switch {\tt TFFT} is set.
+
+For coding purposes, it is often useful to perform derivatives as a
+matrix operation directly in coordinate space. The derivative matrices
+are built by evaluating the double summation of forward and backward
+transform ahead of time. For the $m$th derivative this reads
+\begin{equation*}\begin{split}
+ f^{(m)}(x_\nu)
+ &=
+ \frac{1}{\tt nx}
+ \sum_n\exp{(ik_nx_\nu)}(\I k_n)^m
+ \sum_{\nu'}\exp{(-\I k_nx_{\nu'})}f(x_{\nu'})
+\\
+ &=
+ \sum_{\nu'}\;
+ \underbrace{
+ \frac{1}{\tt nx}
+ \sum_n\exp{(\I k_nx_\nu)}(\I k_n)^m
+ \exp{(-\I k_nx_{\nu'})}}_{D^{(m)}_{\nu\nu'}}\;
+ f(x_{\nu'})
+ \quad.
+\end{split}
+\end{equation*}
+From here, the detailed handling depends on the order of
+derivative. The $k_n$ run over the values
+$k_n=0,
+\pm{\tt dk},\pm 2{\tt dk},...,({\tt nx}/2-1){\tt dk},+{\tt dk}\,{\tt
+ nx}/2$.
+Here the index ordering given in Eq.~(\ref{eq:kvalues}) does not matter
+as the index is summed over.
+Note that the first and the last value come alone while all others
+come in pairs of $\pm$ partners. These pairwise terms can be combined
+into a sine function for $n=1$ and a cosine for $n=2$. The derivative
+matrices thus read in detail
+\begin{subequations}
+\begin{align}
+\label{eq:derv1}
+ D^{(1)}_{\nu\nu'}
+ =&
+ -\frac{2{\tt dk}}{\tt nx}
+ \sum_{n=1}^{{\tt nx}/2-1}n\sin(k_n{\tt dx}(\nu\!-\!\nu'))
+\notag\\
+ &
+ -\frac{{\tt dk}}{\tt nx}
+ \frac{{\tt nx}}{2}\sin((\nu\!-\!\nu'){\tt dk}\,{\tt nx}/2)
+ \quad,
+\\
+\intertext{and}
+\label{eq:derv2}
+ D^{(2)}_{\nu\nu'}
+ =&
+ -\frac{2{\tt dk}^2}{\tt nx}
+ \sum_{n=1}^{{\tt nx}/2-1}n^2\cos(k_n{\tt dx}(\nu\!-\!\nu'))
+\notag\\
+ &
+ -\frac{{\tt dk}^2}{\tt nx}
+ \left(\frac{{\tt nx}}{2}\right)^2\cos((\nu\!-\!\nu'){\tt dk}\,{\tt nx}/2)
+ \quad.
+\end{align}
+\end{subequations}
+A word is in order about the first derivative. The upper point in the
+$k$-grid, ${\tt dk}\,{\tt nx}/2$, is ambiguous. Exploiting periodicity,
+it could be equally well $-{\tt dk}\,{\tt nx}/2$. In order, to deal
+with a $\pm k$ symmetric derivative we have anti-symmetrized this last
+point. The price for this is a slight violation of hermiticity which,
+however, should be very small as we anyway should not have significant
+wave-function contributions at the upper edge of the $k$-grid.
+
+The derivative matrices $D^{(m)}_{\nu\nu'}$ can be prepared ahead of
+time and are then at disposal for any derivative in the course of the
+program. Actually, the matrices for the first derivative are prepared
+in routine \texttt{sder} and for the second derivative in routine
+\texttt{sder2}, both contained in module \texttt{Grids}. These
+routines are applied to generate the derivative matrices
+\texttt{derv1x}, \texttt{derv1y}, \texttt{derv1z}, \texttt{derv2x},
+\texttt{derv2y}, and \texttt{derv2z}, for first and second derivatives
+in the $x$, $y$ and $z$ directions.
+
+The matrix formulation of the derivatives is used in the code in two
+ways: on the one hand, the derivatives of the real-valued densities,
+currents, and mean-field components are always calculated using these
+derivative matrices, because they are real and the Fourier transform
+method would require converting them to complex values (using special
+Fourier techniques for real arrays is in principle possible but has
+not been worked out yet). In addition the user can switch to using the
+matrix method everywhere, which may give a slight speed advantage for
+small grid dimensions.
+
+\subsubsection{Boundary conditions}\label{sec:bc}
+The code uses a periodic Fourier transform to calculate derivatives.
+This is valid only with {\em periodic boundary conditions}. Thus in
+principle the wave functions and potentials are assumed to be repeated
+periodically in each Cartesian direction. Because of the short range
+of the nuclear force, this is not a serious problem in most cases; at
+higher energies, however, as mentioned in Sect.~\ref{sec:collex}
+and~\ref{sec:react} the emission of low-density material from
+the nuclei can interfere with the dynamics in the neighboring box and
+cause problems in the conservation of energy and angular momentum; for
+a detailed discussion see~\cite{Guo08a}. This is aggravated by
+the fact that even with periodic boundary conditions periodicity is
+truly fulfilled only for the wave functions and mean-field
+components. Since the vector $\vec r$ itself is not periodic but
+jumps at the boundary, operators such as the orbital angular
+momentum are not periodic.
+
+Several ways to solve, or reduce, the problem have been brought up.
+The most obvious and conceptually simplest approach is to increase the
+size of the numerical box. This is, however, not an option in 3D
+calculations as the expense grows cubically with the box length. Very
+recently, a multi-grid method has been proposed \cite{deGio12} which
+renders the use of enlarged boxes feasible (although still at the
+boundaries of present days computer capabilities). Perfect removal of
+escaping particles is achieved by radiating or exact boundary
+conditions \cite{Bou97,Man98a,Pardi2013,pardi13} which, again,
+are not yet practicable in 3D calculations. Robust and efficient are
+absorbing boundary layers using an especially tailored imaginary
+potential \cite{Nak05a} or by applying a mask function during time
+evolution \cite{Kra92a}. The latter technique is particularly easy to
+implement and has been widely used in the past. Its robustness and
+efficiency allow developing advanced analyzing techniques on the grid
+as, e.g., the computation of kinetic-energy spectra and angular
+distributions of the emitted particles \cite{Rei06f}. Those who are
+not afraid of a little bit of coding can easily implement the
+mask-function technique for absorbing boundary conditions into the
+present code. A detailed description and discussion of this approach
+and the proper choice of numerical parameters is found in
+\cite{Rei06a}.
+
+On the other hand, for the Coulomb field with its long range periodicity
+would be clearly wrong. Details are given in Section~\ref{sec:coulomb}
+and in the module description.
+
+\subsubsection{Wave function storage}
+Module {\tt Levels} handles the single-particle wave functions and
+associated quantities.
+
+The principal array for the wave function is called {\tt psi}, which
+is of type {\tt COMPLEX(db)}. Its dimension is {\tt
+ (nx,ny,nz,2,nstloc)}, where the first three indices naturally refer
+to the spatial position. The 4th index corresponds to spin: index~1
+refers to spin up and~2 to spin down, quantization being along the
+$z$-direction.
+
+The last index numbers the wave functions. If the code is run on a
+single node, the value is {\tt nstmax}, the total number of
+single-particle wave functions. They are divided up into neutron and
+proton states, with the index range given by {\tt npmin} and {\tt
+ npsi}. The sub-ranges are:
+\begin{itemize}
+\item {\tt npmin(1)}$\ldots${\tt npsi(1)} : the neutron states,
+\item {\tt npmin(2)}$\ldots${\tt npsi(2)} : the proton states.
+\end{itemize}
+In the present code {\tt npmin(1)=1} and {\tt npsi(2)=nstmax}.
+
+\begin{figure}
+\centerline{\includegraphics[width=0.7\linewidth]{Wavefun.eps}}
+\caption{\label{fig:wavefun}
+Storage arrangement of the single-particle wave functions. On the left
+the case for single-processor or {\tt OpenMP} is shown, which for the
+case of distributed memory under {\tt MPI} is mapped to the individual
+processors as shown on the right. Note that each node will have its
+own values of {\tt nstloc} and {\tt globalindex}.
+}
+\end{figure}
+
+If the code is run in parallel (MPI) on several nodes, only {\tt nstloc}
+single-particle wave functions are stored on a given node, where {\tt
+ nstloc} may vary. Pointers are then defined to indicate the
+relationship between the local index and that in the global array of
+wave functions. For details see the section on parallelization; the
+general layout is given in Fig.~\ref{fig:wavefun}.
+
+There are a number of arrays containing the physical properties of the wave
+functions, such as the single-particle energy. The names start with
+{\tt sp\_} and they are defined in module {\tt Levels}. They are not
+split up in the parallel case, but on each node only the pertinent
+index positions are used.
+
+\subsubsection{Densities and currents}
+The various densities necessary for constructing the mean field are
+actually kept in separate arrays and can be output onto data files for
+later analysis (see subroutine {\tt write\_densities}). The
+dimensioning is {\tt (nx,ny,nz,2)} with the last index referring to
+isospin for scalar densities, so that {\tt rho(:,:,:,1)} is the
+neutron density and {\tt rho(:,:,:,2)} the proton density. For vector
+densities there is an additional index with values~1 to~3 for the
+Cartesian direction, thus {\tt sdens(nx,ny,nz,3,2)} containing the
+spin density in each direction for neutrons and protons.
+
+Since it is often not necessary to keep the neutron and proton
+contributions separate, subroutine {\tt write\_densities} has the option of
+adding them up before output.
+
+\subsubsection{The Coulomb solver}\label{sec:coulomb}
+The choice of boundary condition affects only the Coulomb
+potential, since the wave functions are always periodic.
+The logical input variable {\tt periodic} selects the desired case.
+The consequences are as follows:
+\begin{itemize}
+\item {\bf Periodic boundary condition:} the complete system is
+ assumed to be periodically repeated in all three spatial
+ directions, similar to a crystal structure. This may be useful,
+ e.~g., to follow a translating nucleus for a longer distance. The
+ Coulomb field is, however, influenced by the periodic repetitions of
+ the charge density (so that usually isolated boundary conditions
+ will be more relevant). The Coulomb potential can be simply
+ calculated via Fourier transformation. The folding integral in
+\begin{equation}
+V_C(\vec r)=e^2\int D^3 r'\frac{\rho_p(\vec r')}{|\vec r-\vec r'|}
+\end{equation}
+for the 3-dimensional Fourier transforms $\tilde\rho(\vec k)$ and
+$\tilde V_C(\vec k)$ leads to
+\begin{equation}
+\tilde V_C(\vec k)=\frac{4\pi e^2}{k^2}\,\tilde\rho(\vec k),
+\end{equation}
+where $\vec k$ is the vector with the components given in
+Eq,~\ref{eq:FT} and $4\pi/k^2$ is the Green's function. Transforming
+back into coordinate space then yields the solution $V_C(\vec
+r)$. Assuming $\tilde\rho(0)=0$, i.~e., vanishing total charge
+(corresponding to the Jellium approximation) the singularity at
+$\vec k=0$ is removed.
+
+\item For an {\bf isolated system} the boundary condition is that the
+ Coulomb potential should go to zero at infinite distance, which
+ clearly is a problem when the calculation is restricted to a finite
+ grid. We use the method of Eastwood and Brownrigg \cite{Eas79},
+ which produces high-quality solutions even close to the
+ boundary. The basic idea is to extend the computational mesh to
+ double its size in all three directions, filling the new region with
+ zero density. Then the Fourier transform method is applied to
+ calculate the potential throughout this bigger mesh, but the
+ solution is only used inside the original (physical) region.
+
+ The Green's function in this case, however, is no longer the simple
+ $4\pi/k^2$ for the infinite system, but has to be determined by
+ Fourier transforming the function $1/\vec r^2$ in the expanded grid. The
+ origin of $\vec r$ must be taken at the usual place in the physical
+ region.
+
+ Here the point $\vec r=0$ poses problems. It should reflect the
+ average of the Green's function over the numerical cell at the
+ origin with a typical density profile in that cell. We found best
+ results with a Gaussian density profile leading to a value of
+ \begin{equation}
+ \frac{2.84\times 3}{{\tt dx}+{\tt dy}+{\tt dz}}
+ \end{equation}
+ for the Green's function in that cell. {\bf Note that this is not an exact
+ result but obtained by optimizing the result compared to other
+ methods. It should work quite well if the grid spacings are close to
+ 1~fm, but for much smaller or much larger values and other density
+ profiles may need to be optimized. }
+
+ More details on the discretized form of these equations are given in
+ the description of the module {\tt Coulomb}.
+
+ The wave functions are assumed periodic also in this case, which
+ should not be a problem as long as the density distribution does not
+ get too close to the boundaries.
+
+\end{itemize}
+
+\subsection{Initialization}
+\label{sec:init}
+A particular strength of the code is its flexible initialization.
+There are essentially three types of initialization, which can be
+selected through the input variable {\tt nof}:
+
+\begin{enumerate}
+\item\label{it:iniho} {\em Harmonic oscillator:} {\tt nof=0}: this is
+ applicable only to static calculations. The initial wave functions
+ are generated from harmonic oscillator states with initial radii
+ {\tt radinx}, {\tt radiny}, and {\tt radinz} in the three
+ directions. It is advisable to choose the three radii different to
+ avoid being kept in a symmetric configuration for non-spherical
+ nuclei. Note that this is a very simple initialization and has some
+ defects; for example, the initial deformation is controlled more by
+ the occupation of the oscillator states than by the radius
+ parameters. This should eventually be replaced by, e.~g., Nilsson
+ wave functions.
+
+ For this case the type of nucleus is determined by the input numbers
+ {\tt nneut} and {\tt nprot} giving the number of neutrons and
+ protons, while {\tt npsi} can be used to add some unoccupied states
+ (this sometimes leads to faster convergence).
+
+\item\label{it:frag} {\em Fragment initialization:} {\tt nof>0}: wave
+ functions for a number {\tt nof} of fragments are read in and
+ positioned in the grid at certain positions. The wave functions are
+ read from files produced by the static code with the file names
+ given by the input {\tt filename}, they are positioned at
+ center-of-mass positions {\tt fcent} and given an initial velocity
+ controlled by {\tt fboost}. The code determines the number of wave
+ functions needed from these data files and also checks the agreement of
+ Skyrme force and grid used. This initialization is used, e.g., for
+ nuclear reactions (see Section~\ref{sec:react}).
+
+ The number of fragments read in is arbitrary, but there are two
+ special cases:
+ \begin{itemize}
+ \item for {\tt nof=1} a single fragment is read in. This can be
+ useful for initializing with static wave functions to study
+ collective vibrations in a nucleus using the TDHF mode.
+ \item for {\tt nof=2} a special initialization can be done where the
+ initial velocities are not given directly but computed from a
+ center-of-mass energy {\tt ecm} and an impact parameter {\tt b}.
+ \end{itemize}
+
+ Some consistency checks are made when initializing from fragment
+ files. For dynamic calculations the Skyrme force given in the input
+ must agree with that in all the fragment files. For static
+ calculations this is skipped, because it may be useful to start such
+ a calculation using data from a similar one with a different Skyrme force.
+
+\item {\em User initialization:} a user-supplied routine {\tt
+ user\_init} can be employed to set up the wave functions in any
+ desired way. The only condition is that the index ranges etc.\ are
+ set up correctly and the wave function array {\tt psi} is filled
+ with the proper values. It was found useful, e.~g., to use initial
+ Gaussians distributed in various geometric patterns for
+ $\alpha$-cluster studies.
+\end{enumerate}
+
+\subsection{Restarting a calculation}\label{sec:restart}
+Sometimes it is necessary to continue a calculation that was not run
+to the desired completion because of a machine failure or because the
+number of iterations or time steps was set too low. In such cases the
+last wave function file with name {\tt wffile}, which is generated at
+regular intervals of {\tt mrest} iterations or time steps, can be used
+to initialize a continuation. The program handles this in a simple fashion: if
+the logical variable {\tt trestart} is input as {\tt TRUE}, it sets up
+an initialization with one fragment (read from the initialization file)
+placed at the origin and with zero
+velocity. The only other modifications to the regular setup are then
+to take the initial iteration number and time from that file instead
+of starting at zero, as well as suppressing some unneeded
+initialization steps.
+
+Restarting works for both static and dynamic calculations. To continue
+a calculation that was stopped because the desired number of
+iterations or time steps was reached, a new limit for these should be
+provided in the input.
+
+This flexible restart makes it possible to use a different grid for
+the continuation in the sense that the grid spacings must agree, but
+the new grid can be larger than the old one.
+
+\subsection{Accuracy considerations}\label{sec:accuracy}
+The grid representation and solution methods introduced above depend
+on several numerical parameters. Their proper choice is crucial
+for the accuracy and speed of the calculations. In this Section, we
+want to briefly address the dependence on numerical parameters. An
+extensive discussion of grid representations and static iteration is
+found in~\cite{Blu92a}.
+
+\begin{figure}[h]
+\centerline{\includegraphics[width=\linewidth]{static-grid.eps}}
+\caption{\label{fig:static-grid} Binding energy (left) and
+ r.m.s. radius (right) of $^{16}$O computed for the force SkI3 drawn
+ as functions of grid spacing $\Delta x=\Delta y=\Delta z$. A
+ logarithmic scale us used for $\Delta x$. The number of grid points
+ has been chosen to keep the box size constant at $N_x\Delta x=24$
+ fm.}
+\end{figure}
+Figure \ref{fig:static-grid} shows the sensitivity with respect to the grid
+spacings $\Delta x$, $\Delta y$, $\Delta z$. The trend is the same for
+both observables, energy and radius: The results have very high
+quality and change very little up to $\Delta x=$ 0.75 fm. They quickly
+degrade above that spacing. But even at $\Delta x=$ 1 fm, we still
+find an acceptable quality which suffices for most applications,
+particularly for large scale explorations. If high accuracy matters,
+$\Delta x\approx$ 0.75 fm should be chosen; not much is gained
+by going to even finer gridding. This holds for
+ground states and moderate excitations. High excitations and fast
+collisions may require a finer mesh. Note that the maximum
+representable kinetic energy is
+$E_\mathrm{kin,max}=(\hbar^2/2m)(\pi/\Delta x)^2$, which amounts to
+about 200 MeV for $\Delta x=$ 1 fm. The actual energies of interest
+should stay far below this limit. It is an instructive exercise to
+study uniform center-of-mass motion at various velocities to explore the
+limits of a given representation.
+
+The number of grid points $N_x=N_y=N_z$ in the tests of Figure
+\ref{fig:static-grid} were chosen such that the box size was the same
+in all cases. The actual choice of $N_x$ depends sensitively on the
+system, its size and separation energy. As a rule of thumb, the
+density decreases asymptotically as
+$\rho\propto\exp{(-2\sqrt{2m\varepsilon_N}r/\hbar)}$ where
+$\varepsilon_N$ is the single particle energy of the least bound
+state. One should aim for at least $\rho<10^{-8}\,{\rm fm}^{-3}$ at
+the boundaries.
+
+\begin{figure}
+\centerline{\includegraphics[width=0.5\linewidth]{dynamic-grids.eps}}
+\caption{\label{fig:dynamic-grids} Time evolution of the quadrupole
+ momentum for two different grid spacings $\Delta x=\Delta y=\Delta
+ z$ and constant box size of 24 fm. The test case is $^{16}$O excited by
+ an instantaneous boost computed with the force SkI3.}
+\end{figure}
+Figure \ref{fig:dynamic-grids} explores the effect of grid spacing for
+dynamics. Two different spacings are compared for a quadrupole
+oscillation following an instantaneous quadrupole boost. Practically
+no difference can be seen for the ``safe choice'' $\Delta x=$ 0.75 fm
+and the robust choice $\Delta x=$ 1 fm. Dynamical applications,
+oscillations and collisions, are in general less demanding and can
+be performed very well with $\Delta x=$ 1 fm. This is pleasing as dynamical
+calculations are usually much more costly than purely static ones.
+
+There are two parameters regulating the static iteration according to
+Eq.~(\ref{eq:dampstep}), the damping energy {\tt e0inv} and the step
+size {\tt x0dmp}. {\tt e0inv} should correspond to the depth of
+the binding potential. The overall step size {\tt x0dmp} can be of
+order of one if {\tt e0inv} is well chosen. Nuclear binding is very
+similar all over the chart of nuclei. This allows to develop one safe
+choice for nearly all cases. We recommend {\tt e0inv}$\approx 100$ MeV
+together with {\tt x0dmp}$\approx 1/2$, reducing the latter slightly
+if convergence problems appear. Of course,
+ a few percent in iteration speed might be gained by fine-tuning these
+parameters for a given case, but this is not worth the effort unless
+large scale surveys for a given class of nuclei and forces are
+planned.
+
+\begin{figure}
+\centerline{\includegraphics*[width=0.95\linewidth]{dynamicSV-energ.eps}}
+\caption{\label{fig:dynamicSV-energ} Time evolution of the binding energy
+ of $^{16}$O after an excitation by a soft cos$^2$ pulse of width 20
+ fm/c computed for the force SV-bas~\cite{Klu09a} with a grid spacing
+ $\Delta x=0.75$ fm and box size of 24 fm. Results are shown for
+ different sizes of time step {\tt dt} and different order of Taylor
+ expansion {\tt m} of the exponential evolution. The left panel shows
+ the full evolution from ground state energy to the excited
+ energy. The right panels concentrate on the times after the pulse is
+ over in a narrower energy range relevant for this excitation.}
+\end{figure}
+
+The time stepping using the exponential propagator has the two parameters,
+step size {\tt dt} and order {\tt mxp}$=m$ of the Taylor expansion
+(\ref{eq:timepower}) of the exponential. Intuitively, one expects that
+small {\tt dt} and large {\tt mxp} improve the quality of the
+step. An efficient stepping scheme, however, looks for the largest {\tt
+ dt} and smallest {\tt mxp} which still provide acceptable and
+stable results. It is hard to give general rules as good working
+values for the parameters depend on all details of the actual calculation:
+gridding, nuclei involved, excitation energy, and kind of
+excitation.
+
+Figure \ref{fig:dynamicSV-energ} demonstrates the dependence of a
+typical dynamical evolution on these time-stepping parameters. We
+consider a time interval up to 1260 fm/c which is a long time for
+heavy-ion collisions and just sufficient for a spectral analysis of
+oscillations~\cite{Rei07a}. The excitation is done by a soft sin$^2$
+pulse of finite extension in time. The energy increases during the
+initial excitation phase, as can be seen from the left panel in the
+figure. After the external pulse is over, energy conservation holds,
+which is nicely seen at plotting resolution in the left
+panel. Normalization should be conserved at all times. Both
+conservation laws serve as tests for the time step. Norm is conserved
+up to at least six digits for all cases and times shown in Figure
+\ref{fig:dynamicSV-energ}. The energy is more critical. The right
+panels show the energy in a small window around the final energy after
+the excitation phase is over. The right lower panel shows a variation
+of the Taylor order $m$ for fixed time step. The most prominent effect
+is the sudden turn to catastrophic failure for $m=6$. In fact,
+propagation by approximate exponential evolution explodes sooner or
+later in all cases. The art is to extend the stable interval by
+a proper choice of the stepping parameters. It is plausible that the
+cases with $m>6$ maintain stability longer because the exponential is
+better approximated. It is surprising that $m=4$ is also stable over
+the whole time interval. There seem to be subtle cancellations of
+error going on. Considering the stable signals, we see very little
+differences between the cases. One may generally be happy with low
+$m$. It is mainly stability demands which could call for larger
+$m$. Note that this is not a generic result. Stability for a given
+test case should be checked once in a while and particularly before
+launching larger surveys.
+
+The right upper panel in Figure \ref{fig:dynamicSV-energ} shows
+results for different {\tt dt} (as we have seen, the $m$ values are
+not important as long as we achieve stable results). Here we see a
+clear dependence on the step size. The energies remain constant in the
+average. But there are energy fluctuations and these depend
+sensitively on {\tt dt}. Smaller {\tt dt} yields smaller fluctuations.
+As far as one can read off from the figure, the amplitude of the
+fluctuations shrink $\propto${\tt dt}$^2$. It depends on the intended
+analysis to which level of precision the time
+evolution should be driven. A value of {\tt dt}$\approx$ 0.4 fm/c will be acceptable in most
+cases because the average trend remains far smaller than the
+fluctuations. Here also it must be emphasized that this is not a
+generic number. Forces with lower effective mass (SV-bas has
+$m^*/m=0.9$) are more demanding and usually require smaller {\tt
+ dt}. On the other hand, running propagation without the spin-orbit term
+allows even larger time steps because the spin-orbit potential is the
+most critical piece in the mean-field Hamiltonian. The mix of
+$\mathbf{p}$ and $\mathbf{r}$ imposes high demands on the numerical
+representations. We again strongly recommend running a few
+tests when switching forces or excitation schemes.
+
+\section{Code structure}
+The code is completely modularized to provide as large a degree of
+encapsulation as possible in order to ease modification. Most modules
+read their operating parameters from an associated {\tt NAMELIST} and
+have their local initialization routines. In addition, a modern style
+of programming is used that employs a minimum number of local
+variables and streamlined array calculations that make the code lines
+very close to the physical equations being solved.
+
+Here we give a brief overview over the modules and their
+purpose. There is a comprehensive manual supplied with the electronic
+version that gives a detailed description of all modules. The source
+files containing the modules have the same names, but all in lower
+case, with an extension of {\tt .f90}. The higher-level modules are:
+
+\begin{description}
+\item{Main program}: It calls initialization routines, sets up the
+ initial wave functions using either harmonic oscillator states or
+ reading wave functions of static Hartree-Fock solutions from given
+ input files (module {\tt Fragments}). It then calls either {\tt
+ statichf} from module {\tt Static} or {\tt dynamichf} from module
+ {\tt Dynamic} to run the calculation.
+\item{Static}: This contains the code for the static iterations {\tt
+ statichf} and the subroutine {\tt sinfo} to generate output of the
+ results.
+\item{Dynamic}: runs the dynamic calculation in {\tt dynamichf} and
+ generates output in {\tt tinfo}. Also controls the inclusion of an
+ external excitation implemented in module {\tt External}.
+\item{Densities}: calculates the densities and current densities by
+ summing over the single-particle states.
+\item{Meanfield}: contains the central physics calculation: the
+ computation of the components of the mean field (subroutine {\tt
+ skyrme}) and the application of the single-particle Hamiltonian to
+ a wave function (subroutine {\tt hpsi}).
+\item{Coulomb}: calculation of the Coulomb potential.
+\item{Energies}: calculation of the total energies and its various
+ contributions.
+\item{External}: calculation of the action of an external potential or
+ initial collective boost of the wave functions.
+\item{Pairs}: Implementation of the pairing correlations in the BCS
+ approximation.
+\item{Moment}: calculation of moments and deformation parameters for
+ the bulk density.
+\item{Twobody}: attempts to divide up the system into two separated
+ nuclei and to calculate their properties and relative motion.
+\end{description}
+
+The lower-level supporting modules are:
+\begin{description}
+\item{Params}: general parameters used throughout the code.
+\item{Forces}: defines parameters of the Skyrme force and the pairing
+ interaction and constructs them according to input.
+\item{Grids}: defines everything associated with the numerical grid
+ and sets it up.
+\item{Levels}: definition of the single-particle wave functions and
+ elementary operations on them such as derivatives.
+\item{Fragments}: controls the reading of static wave functions from
+ precomputed data and setting them up in the grid.
+\item{Inout}: contains the subroutines for I/O of wave functions and
+ densities.
+\item{Trivial}: defines some very basic operations on wave functions
+ and densities.
+\item{Fourier}: sets up the transform plans for the {\tt FFTW3}
+ package to calculate Fourier transforms of wave functions and
+ densities.
+\item{Parallel}: This comes in two versions. The source file {\tt
+ parallel.f90} contains the routines to handle {\tt MPI} message
+ passing, while {\tt sequential.f90} sets up essentially dummy
+ replacements for sequential or {\tt OpenMP} mode.
+\item{User}: contains a sample user initialization code which can be
+ used as a template for more complicated setups.
+\end{description}
+
+\section{Parallelization}
+For both OpenMP~\cite{Cha08aB} and MPI~\cite{none12} the code can be
+run in parallel mode. Parallelization for the static mode works in
+OpenMP but not in MPI: the reason is in the orthogonalization step
+which is not easily amenable for distributed-memory parallel
+computation. This should be worked on in the future.
+
+MPI and OpenMP can be used jointly if there are computing nodes with
+multiple processors.
+
+In both cases parallelization is done over the wave functions. The
+code applies the time-development operator or the gradient iteration, which use the fixed set
+of mean-field components, to each wave function, and this can naturally be
+parallelized. Computing the mean fields and densities by summing up
+over single-particle wave functions is also easily parallelizable.
+
+The library {\tt FFTW3}~\cite{Fri05a} itself can also run on multiple
+processors in parallel. This can be used in addition to OpenMP or MPI,
+but was found to be helpful only in the sequential version of the
+code.
+
+\subsection{OpenMP}
+The application of the subroutine {\tt tstep} for propagating one wave
+function for one time step, and of {\tt add\_densities} for adding one
+wave function's contribution to the mean fields is done in
+parallel loops. The only complicating factor is that the densities,
+being accumulated in several subsets, must be kept separate using the
+{\tt REDUCTION(+)} clause of OpenMP. The summation cannot be done
+internally in {\tt add\_densities}, because for the half time step the
+wave functions are immediately discarded after adding their
+contribution to the densities to avoid having to store the full set at
+half time. Thus only the combined {\tt tstep}-{\tt add\_densities}
+loop should be parallelized.
+
+The OpenMP program version can be compiled using the appropriate
+compiler option. A separate {\tt Makefile.openmp} is provided which
+just contains the {\tt -fopenmp} option for the GNU compiler. The
+number of parallel threads is not set by the code: the user should set
+the environment variable {\tt OMP\_NUM\_THREADS} to the desired
+number.
+
+The user should also check compatibility of the locally installed {¸tt
+ LAPACK/BLAS} libraries with {\tt OpenMP}.
+
+\subsection{MPI}
+In principle MPI uses the same technique as OpenMP, parallelizing over
+wave functions. In this case, however, each node contains only a
+fraction of the wave functions. This has several consequences:
+\begin{enumerate}
+\item The time-stepping of the wave functions can be done
+ independently on each node, but requires that the densities are
+ broadcast to all nodes after each half or full time step by summing
+ up partial densities from the nodes in subroutine {\tt
+ collect\_densities}.
+\item The other calculation that uses wave functions directly is that
+ of the single-particle properties. These are calculated on each node
+ for the wave functions present on that node and then collected using
+ subroutine {\tt collect\_sp\_properties}.
+\item Only one node must be allowed to produce output. This is
+ regulated by choosing node \#0 and setting the flag {\tt } to
+ {\tt .TRUE.}\ on that node..
+\item The saving of the wave functions is done in the following way:
+ Node zero writes a header file containing the job information on
+ file {\tt wffile}, then each node writes a separate file {\tt
+ wwfile.001}, {\tt wwfile.002}, and so on up to the number of
+ nodes. This avoids having to collect the wave functions on one node.
+\item Using these parallel output files as fragment initialization or
+ restart files is handled so flexibly that they can be read into a
+ different nodal configuration or even a sequential run.
+\end{enumerate}
+
+The MPI version needs the appropriate compiler and linker calls for
+the system used. The sequential or OpenMP versions are obtained simply
+by linking with {\tt sequential.f90} instead of {\tt parallel.f90},
+which replaces the MPI calls with a set of dummy routines and sets up
+the descriptor arrays for the wave function allocation in a trivial
+way.
+
+The {\tt Makefile.mpi} shows the procedure; in practice systems differ
+considerably and the user should look up the compilation commands for
+his particular system.
+
+\section{Input description}
+All the input is through {\tt NAMELIST} and many variables have
+default values. {\em The {\tt NAMELIST}s should be in this file in the
+ order in which they are described here, any {\tt NAMELIST} not used
+ for a particular job may be omitted or left in the input file, in
+ which case it is ignored}. The input is from a file called {\tt
+ for005}, so the input data have to be produced with an editor. If the
+large output listing is to go into {\tt output}, the code should be
+run using, e.~g.,
+\begin{verbatim*}
+./sky3d.seq > output
+\end{verbatim*}
+The reason for not using redirected input is that in most MPI
+implementations for an input through ``{\tt < for005}''
+is passed only to
+node~0, while all nodes can read the same file in parallel using an
+explicit {\tt OPEN} statement.
+
+
+\subsection{Namelist {\tt files}}
+\label{sec:filenamein}
+This {\tt NAMELIST} contains names for the files used in the code.
+They are defined in module \texttt{Params} and are:
+\begin{description}
+\item[\texttt{wffile}]: file to contain the static single-particle
+ wave functions plus some additional data. This can be used for
+ fragment initialization or for restarting a job. Default: {\tt
+ 'none'}, i.~e., nothing is written.
+\item[\texttt{converfile}]: contains convergence information for the
+ static calculation. Default: {\tt conver.res}.
+\item[\texttt{monopolesfile}]: contains moment values of monopole
+ type. Default: {\tt monopoles.res}.
+\item[\texttt{dipolesfile}]: contains moment values of dipole type.
+ Default: {\tt dipoles.res}.
+\item[\texttt{quadrupolesfile}]: contains moment values of
+ quadrupole type. Default: {\tt quadrupoles.res}.
+\item[\texttt{momentafile}]: contains components of the total
+ momentum. Default: {\tt momenta.res}.
+\item[\texttt{energiesfile}]: energy data for time-dependent calculations.
+ Default: {\tt energies.res}.
+\item[\texttt{spinfile}]: time-dependent total, orbital, and spin
+ angular-momentum data as three-dimensional vectors.
+\item[\texttt{extfieldfile}]:
+time dependence of expectation value of the external field.
+\end{description}
+
+\subsection{Namelist {\tt force}}
+This defines the Skyrme force to be used. In most cases it should just
+use two input values:
+\begin{description}
+\item[{\tt name}]: the name of the force, referring to the predefined
+ forces in {\tt forces.data}.
+\item[{\tt pairing}]: the type of pairing, at present either {\tt
+ NONE} for no pairing, \texttt{VDI} for the volume-delta
+ pairing, or \texttt{DDDI} for density-dependent delta pairing.
+ The pairing parameters are included in the force definition. {\em Note
+ that the pairing type must be written in upper case.}
+\end{description}
+There is also the possibility for inputting a user-defined force; this
+is described in detail with module \texttt{Forces} in the online
+technical documentation.
+
+\subsection{Namelist {\tt main}}\texttt{main}
+This contains general variables applicable to both static and dynamic
+mode. They are mostly defined in module \texttt{Params}.
+\begin{description}
+\item[\texttt{tcoul}:] determines whether the Coulomb field should be
+ included. Default is {\tt true}.
+\item[\texttt{trestart}:] if {\tt true}, restarts the calculation from
+ {\tt wffile}. Default is {\tt false}.
+\item[\texttt{tfft}:] if {\tt true}, the derivatives of the wave
+ functions, but not of the densities, are done directly through FFT.
+ Otherwise matrix multiplication is used, but with the matrix also
+ obtained from FFT. Default is {\tt true}.
+\item[\texttt{mprint}:] control for printer output. If {\tt mprint} is
+ greater than zero, more detailed output is produced every {\tt
+ mprint} iterations or time steps on standard output.
+\item[\texttt{mplot}:] if {\tt mplot} is greater than zero, a printer plot is
+ produced and the densities are dumped every {\tt mplot} time steps
+ or iterations. Default is~0.
+\item{\texttt{mrest}:} if greater than zero, a {\tt wffile} is produced
+ every {\tt mrest} iteration or time step. Default is~0.
+\item[\texttt{writeselect}]: selects the output of densities by giving
+ a string of characters choosing them (see subroutine
+\texttt{write\_densities} for details. Default is {\tt 'r'}, i.~e., only
+the density is written.
+\item[\texttt{write\_isospin}]: determines whether the densities should
+ be output isospin-summed ({\tt false}) or separately for neutrons
+ and protons ({\tt true}). Default is {\tt false}.
+\item[\texttt{imode}]: selects a static {\tt imode=1} or dynamic {\tt
+ imode=2} calculation.
+\item[\texttt{nof}]: (number of fragments) selects the initialization.
+ {\tt nof=0}: initialization from harmonic oscillator, only for the
+ static case; {\tt nof<0}: user-defined initialization by subroutine
+ \texttt{init\_user} in module \texttt{User}; {\tt nof>0}:
+ initialization from fragment data as determined in {\tt NAMELIST}
+ {\tt fragments}.
+\item{\texttt{r0}:} nuclear radius parameter. The nuclear radius
+ $R=r_0A^{1/3}$ is used to compute the $\beta$ and $\gamma$
+ deformation parameters in subroutine \texttt{moments}. Units: fm,
+ default value 1.2~fm.
+\end{description}
+
+\subsection{Namelist {\tt grid}}\texttt{grid}
+This defines the properties of the numerical grid.
+\begin{description}
+\item[\texttt{nx}, \texttt{ny}, \texttt{nz}]: number of grid points in the
+ three Cartesian directions. They must be even numbers.
+\item[\texttt{dx}, \texttt{dy}, \texttt{dz}]: spacing between grid points in fm.
+ If only {\tt dx} is given in the input, all three grid spacings
+ become equal.
+ The grid positions are then set up to be symmetric with the
+ coordinate zero centrally between point number {\tt nx/2} and {\tt
+ nx/2+1}.
+\item[\texttt{periodic}]: chooses a periodic ({\tt true}) or isolated
+ ({\tt false}) system.
+\end{description}
+
+\subsection{Namelist {\tt static}}\texttt{static}
+These input variables control the static calculations.
+\begin{description}
+\item[\texttt{tdiag}:] if {\tt true}, there is a diagonalization of the
+ Hamiltonian during the later (after the 20th) static iterations. This
+ 20 is hard coded in {\tt static.f90}. Default is {\tt false}.
+\item[\texttt{tlarge}:] if {\tt true}, during the diagonalization the new
+ wave functions are temporarily written on disk to avoid doubling the
+ memory requirements. Default is {\tt false}.
+\item[\texttt{nneut}, \texttt{nprot}:] The numbers of neutrons and protons
+ in the nucleus. These are used for the harmonic-oscillator
+and user initialization.
+\item[\texttt{npsi}:] the numbers of neutron ({\tt npsi(1)}) and proton
+ ({\tt npsi(2)}) wave functions actually used including unfilled
+ orbitals. Again, useful only for harmonic-oscillator or user initialization.
+\item[\texttt{radinx}, \texttt{radiny}, \texttt{radinz}:] the radius parameters of the
+ harmonic oscillator in the three Cartesian directions, in fm.
+\item[\texttt{e0dmp}:] the damping parameter. For its use see subroutine
+ \texttt{setdmc}. The default value is 100~MeV.
+\item[\texttt{x0dmp}:] parameters controlling the relaxation. The
+ default value is 0.2. In special cases it may be desirable to
+ change this to accelerate convergence.
+\item[\texttt{tvaryx\_0}:] determines whether the variable {\tt x0dmp}
+ should be adjusted during the iterations.
+\item[\texttt{serr}:] this parameter is used for a convergence
+ check. If the sum of fluctuations in the single-particle energies,
+ {\tt sumflu} goes below this value, the calculation stops. A
+ typical value is {\tt 1.E-5}, but for heavier systems and with
+ pairing this may be too demanding.
+\end{description}
+
+\subsection{Namelist {\tt dynamic}}\texttt{dynamic}
+These are variables controlling the dynamic (TDHF) calculation.
+
+\begin{description}
+\item[\texttt{nt}]: number of time steps to be run.
+\item[\texttt{dt}]: the time step in fm/c. A standard value is of the
+ order of 0.2 to 0.3~fm/c, it depends somewhat on the value of {\tt
+ mxpact}. If the combination of these two is not good enough, the
+ calculation becomes unstable after some time, in the sense that the
+ norm of the wave functions and the energy drift off and can diverge
+ (see Sect.~\ref{sec:accuracy}).
+\item[\texttt{mxpact}]: the order of expansion for the exponential
+ time-development operator. The predictor (trial) step calculation
+ uses {\tt mxpact/2} as the order. For more information see
+ Sect.~\ref{sec:accuracy}.
+\item[\texttt{rsep}]: termination condition. If the final state in a
+ two-body reaction is also of two-body character, the calculation is
+ terminated as soon as the separation distance exceeds {\tt rsep} and
+ the full two-body analysis is performed.
+ Units: fm. No default. The purpose of this variable is to prevent
+ the calculation of continuing into meaningless configurations, like
+ crossing of the boundary.
+\item[\texttt{texternal}]: indicates that an external perturbing field is
+ used. In this case the namelist {\tt extern} must be
+ present. Default: {\tt false}.
+\end{description}
+
+\subsection{Namelist {\tt extern}}\texttt{extern}
+The variables read here describe the external field that is applied to
+get the nucleus into a collective vibration. Details can be found in
+the description of module \texttt{External}. It is read only if the
+parameter \texttt{texternal} read in namelist \texttt{dynamic} is
+true.
+\begin{description}
+\item[\texttt{ipulse}]: the type of pulse applied. For {\tt ipulse=0} the
+ wave function is multiplied with a phase factor that produces an
+ initial excitation. For {\tt ipulse=1} a Gaussian time dependence is
+ used, for {\tt ipulse=2} a $\cos^2$ one. Default: 0. Details are
+ given in Eqs.~(\ref{eq:extgauss}) and (\ref{eq:extcos}).
+\item[\texttt{isoext}]: isospin character of the excitation. If this is
+ zero, protons and neutrons are exited in the same way. For a value
+ of 1, they behave oppositely but with a coupling that leaves the
+ center-of-mass invariant. Default: 0.
+\item[\texttt{tau0}, \texttt{taut}]: time at which the excitation field
+ reaches its maximum, and width of the pulse. No defaults.
+\item[\texttt{omega}]: if this is nonzero, the time-dependence of the
+ external field gets an additional cosine factor with frequency {\tt
+ omega}.
+\item[\texttt{radext}, \texttt{widext}]: radius and width of a
+ Woods-Saxon-type cutoff factor in radius for the external field.
+ Defaults: 100~fm and 1~fm, which practically implies no damping.
+Definition in Eq.~(\ref{eq:extdamp}).
+\item[\texttt{amplq0}]: amplitude for quadrupole excitation of the
+ $Q_{20}$ type. Defined as usual with respect to the $z$-axis.
+\end{description}
+
+\subsection{Namelist {\tt fragments}}\texttt{fragments}
+The variables in this namelist control fragment initialization for the
+case of \texttt{nof}{\tt >0}. Most quantities are dimensioned for the fragments
+and we indicate this by index ``{\tt i}'' in the following.
+\begin{description}
+\item[\texttt{filename}{\tt (i)}]: the name of the file containing the wave
+ functions of fragment {\tt i}.
+\item[\texttt{fcent}{\tt (1:3,i)}]: initial position of fragment {\tt i} given
+ as three Cartesian coordinate values in fm. The position must be
+ such that the complete fragment grid fits inside the new
+ computational grid.
+\item[\texttt{fix\_boost}]: used only for the two-fragment case. if this
+ logical variable is {\tt TRUE}, the initial velocities are
+ calculated from the {\tt fboost} values; otherwise from the relative
+ motion quantities {\tt ecm} and {\tt b}.
+\item[\texttt{fboost}{\tt (1:3,i)}]: the initial boost of the fragment in the three
+ Cartesian directions. It is given as the total kinetic energy in
+ each direction in MeV, with the sign indicating positive or negative
+ direction. Thus {\tt SUM(ABS(fboost(:,i)))} is the total kinetic
+ energy of fragment {\tt i}.
+\item[\texttt{ecm}, \texttt{b}]: center-of-mass kinetic energy in MeV and
+ impact parameter in fm. Used only if {\tt fix\_boost} is {\tt FALSE}.
+ These are the values at infinite distance
+ and are corrected using Rutherford trajectories (assuming spherical
+ nuclei) for initialization at the finite distance given by
+ the {\tt fcent} coordinates.
+\end{description}
+
+\subsection{Namelist {\tt user}}\texttt{user}
+This namelist is read only if needed for user initialization (see
+module \texttt{User}). Its contents depend on the specific user
+initialization and the only thing to be said here is that it should
+appear last in the input file. Since the namelist is defined and used
+only in module {\tt User}, its name can also be changed arbitrarily,
+of course.
+
+\section{Output description}
+The code produces a number of output files containing various pieces
+of information.
+The bulky observables, such as densities or currents, are selectively
+output at certain time steps into special binary output files
+{\em nnnnnn}{\tt .tdd}, where {\em nnnnnn} indicates the iteration or
+time step number. These files can then be used for further analysis or
+converted to be used as input in visualization codes. Examples of this
+are found among the utility codes provided.
+
+The complete set of wave functions is saved at regular
+intervals of {\tt mrest} iterations or time steps. Because this leads
+to large storage requirements, only the last such file in a run is
+kept. It can be used for restarting the calculation or for inputting
+fragment wave functions for initializing another calculation.
+
+In {\tt MPI} mode the wave functions are distributed over several
+files, each containing only those present on a specific processor. An
+additional header file contains the remaining information and can be
+used to read the wave functions even on a different processor
+configuration.
+
+Aside from these binary files there are a number of text files. The
+{\tt *.res} files contain one line for each time step or iteration
+where output is triggered according to the value of \texttt{mprint}.
+There is an explanatory header line in these that has a leading '\#',
+so that is treated as a comment by {\tt gnuplot} --- thus {\tt
+ gnuplot} can be used immediately to plot the behavior of any one
+column of numbers, i.~e., the dependence of a physical quantity on
+iteration or time. In addition more complicated output is printed on
+standard output, which can be redirected into a file using shell
+redirection.
+
+The names of the output files can be adjusted using input variables as
+listed in Section~\ref{sec:filenamein}, so that the files are here
+denoted by the default names given there. The {\tt *.res} files are
+relatively small, so that no mechanism was implemented to suppress
+them.
+
+\subsection{File {\tt conver.res}}
+This is produced in \texttt{sinfo} only in the static calculation and its
+purpose is to give a quick impression of the convergence behavior. The
+numbers given in each line are the iteration count, the total energy
+in MeV, the relative change in energy from one iteration to the next,
+the average uncertainties in the single-particle energies {\tt efluct1} and
+{\tt efluct2}, the root-mean-square radius in fm and finally the
+deformation parameters $\beta$ and $\gamma$ (see Section
+\ref{sec:mulmom}). The latter give an impression as to where the
+nuclear shape is ending up.
+
+For the judging of convergence, the {\tt efluct} values are more
+important than the change in total energy, since the energy can remain
+constant while the wave functions still change considerably.
+
+\subsection{File {\tt monopoles.res}}
+At present this file is generated in subroutine
+\texttt{moment\_shortprint}, but only in the dynamic mode. It contains
+the time, the neutron, proton, and total root-mean-square radii, and
+the difference of neutron minus proton root-mean-square radii.
+
+\subsection{File {\tt dipoles.res}}
+This file is produced both in the static and dynamic calculations in
+subroutines \texttt{sinfo} and \texttt{tinfo}. It contains the iteration or
+time step number followed by the three components of the center of
+mass vector $\vec{R}$ and those of the difference of proton minus
+neutron center-of-mass vectors $\vec{R}^{(T=1)}$, both in fm, for the
+definition see Eq.~(\ref{eq:cmcart}). The first of these is useful as
+a check to see whether the center of mass drifts off during the
+calculation, while the second vector may be useful to look at proton
+vs. neutron vibrations.
+
+\subsection{File {\tt quadrupoles.res}}
+This is also generated in \texttt{moment\_shortprint} and thus only in
+dynamic mode. It contains the Cartesian quadrupole moments
+(\ref{eq:Qcart}) for neutrons, protons, and the full mass distribution
+followed by the expectation values of $x^2$, $y^2$, and $z^2$ for
+neutrons and protons, all in fm$^2$.
+
+\subsection{File {\tt energies.res}}
+This is the important monitoring file for the dynamic calculation. It
+is written in subroutine \texttt{tinfo} and each line contains the
+simulation time, the number of neutrons and protons in the system
+(these should be constant, so this is a stability check), the total
+energy (again, this should be conserved), the total kinetic energy,
+and finally the collective energy {\tt ecoll} separately for neutrons
+and protons. Units for the energies are
+all MeV.
+
+\subsection{Standard output}
+This contains all the additional information that in most cases is not
+needed directly for further processing in, e.~g., graphics programs. If it should
+be found necessary to utilize some data from this file, it is in most
+cases easy to use {\tt grep} or a scripting language like {\tt Perl}
+or {\tt Python} to extract the necessary data. Of course the code can
+also be modified to produce additional output files.
+
+The initial part of the output essentially echoes all the data from
+the {\tt NAMELIST}s in tabular form, to enable checking the
+correctness of input data. In the case of fragment initialization this
+is more involved and is discussed below for the dynamic case, since it
+is not so common for static calculations.
+
+In general the layout of the information is compact with sequences of
+``*'' characters to provide separation between input groups as some
+guidance for the eye.
+
+\subsubsection{Static calculation}
+The code first prints the current iteration number. Iteration ``0'' refers
+to the state before iterations are started, for the later iteration
+numbers, the information refers to the end of the iteration.
+
+The overview of the various energy contributions is printed: the
+first part is similar to what is in file {\tt conver.res}, while a
+second list shows the energies calculated from the density functional
+and split up for the various contributions.
+
+Next there is a simple printer plot of the density distribution in the
+$(x-z)$-plane. This is often quite helpful, since it shows what is
+going on in the calculation without the need to start a graphics
+program, which requires converting the data first.
+
+Next there is a listing of single-particle states. For each state this
+shows its parity, occupation probability {\tt wocc} (which is called
+$v^2$, as it is interesting mostly in the pairing case), the energy
+fluctuations \texttt{sp\_efluct1} and \texttt{sp\_efluct2}, the norm, the kinetic and total
+energies of the state, and finally the expectation values of the three
+components of the orbital and spin angular momentum, respectively.
+
+Finally a summary of some integrated quantities is given, separately for
+neutrons, protons, and all nucleons: the particle number, the
+root-mean-square radius, quadrupole moment, and the average of the
+coordinates squared, followed by the center-of-mass components.
+
+Then iterations continue and only one line is printed for each, as
+this may be quite slow and it is important to be able to check
+progress while the code is running. After \texttt{mprint} iterations
+the detailed information is repeated.
+
+\subsubsection{Dynamic calculation}
+After echoing the parameters for the dynamic calculation, the fragment
+definitions are given and all the resulting information is printed:
+the computed boost values in case of twobody initialization, the
+properties of the single-particle states read in, including which
+index in the fragment file is transferred to which index in the total
+set of wave functions.
+
+In case of an external field, the input data is also echoed in a
+detailed form.
+
+The time stepping starts and detailed output is produced every
+\texttt{mprint} steps at the end of the time step. Much of it is similar to
+the static case, so only the differences are pointed out.
+
+Because in the dynamic case the situation can have a general
+three-dimensional character, the full information on the quadrupole
+tensor (\ref{eq:Qcart}) is printed, separately for the neutron,
+proton, and total mass distributions. The three eigenvalues and
+associated normalized eigenvectors are given, followed by Cartesian
+and polar deformation parameters $a_{0}$, $a_{2}$, $\beta$, and
+$\gamma$, as defined in Section~\ref{sec:mulmom}.
+
+The separation of the two fragments and its time derivative is printed
+next {\em and repeated every time step}, as examining these quantities
+is meaningful only with more frequent sampling.
+
+The energy information is shortened by omitting the quantities not of
+interest in the dynamic case.
+{\em It should be kept in mind that the two-body
+ analysis is only valid if the reaction plane is the $(x-z)$-plane
+ and the results printed may not be useful if the physical situation
+ is not of two-body nature but the code does not recognize that.}
+
+The single-particle property list and the integrated quantities are as
+in the static case, but the energy fluctuations are omitted.
+
+The next time step is then indicated and the fragment separation data
+are printed for every time step until after \texttt{mprint} steps the
+full output recurs.
+
+\section{Utilities}
+A set of short programs is designed to help with further processing of
+output from the code. The currently available set is described here.
+
+Most of these routines contain a loop to input a file name from the
+terminal. If it is desired to do this in a loop over a set of files, a
+simple trick can be used: generate a list of file names using, e.~g.,
+
+{\tt ls -1 *.tdd > list}
+
+and then execute the program with ``list'' as input, e.~g.,
+
+{\tt ./fileinfo < list}
+
+For {\tt Tdhf2Silo} a script {\tt convert} is provided that handles
+this (see below). It can easily be adapted to the other utilities and
+must be stored in the same directory as the executable utility program
+itself in order for the {\tt dirname} command to work properly.
+
+\subsection{Fileinfo}
+This is a short program to print information about binary files
+generated by Sky3D. It takes the name of either a {\tt *.tdd} or a
+wave function file as input and prints out essentially all the
+information contained in the header. It can be compiled simply by
+executing
+
+{\tt gfortran -o fileinfo fileinfo.f90}
+
+\subsection{Inertia}
+This program calculates the tensor of inertia relative to the center
+of mass from the density distribution. It is intended as an example of
+an analysis code reading {\tt *.tdd} files and doing some computation,
+which can be used as a model for doing similar things. It illustrates
+looking for the desired field in the file and taking into account
+whether it is stored as a total density or isospin-separated. Being
+given a filename as input, it reads the density and calculates the
+inertia tensor to print all its 9~components. Compile it using
+
+{\tt gfortran -o Inertia Inertia.f90}
+
+\subsection{Cuts}
+This utility reads the density from a file {\em nnnnnn}{\tt .tdd} file and produces
+output files named {\em nnnnnn}{\tt rxy.tdd}, {\em nnnnnn}{\tt
+ rxz.tdd}, and {\em nnnnnn}{\tt ryz.tdd}, which contains
+two-dimensional cuts through the system in the $(x,y)$, $(x,z)$, and
+$(y,z)$ plane, respectively. The cuts are evaluated at the origin for
+the third coordinate by averaging the two neighboring planes.
+
+These data files are written in such a format that they can be read
+by {\tt gnuplot} for use in its commands for 2-dimensional plotting.
+
+This program is intended again as a template that can be modified for
+other applications.
+
+\subsection{Overlap}
+This is a code to calculate the overlap of two Slater determinants.
+Given the names of two wave function files (which must contain
+compatible data: dimensions, force, etc.) it reads the wave functions,
+generates the matrices of overlaps between one set and the other
+separately for neutrons and protons, and then calculates the
+determinant of each, which is the overlap between the two Slater
+determinants. It prints some summary information: distance between the
+centers of mass of each set, minimum and maximum diagonal elements,
+maximum absolute value of off-diagonal elements, and finally the
+overlaps for protons and neutrons as well as their product.
+
+This code uses subroutines from {\tt LINPACK} (stored at {\tt
+ NETLIB.ORG}), which are included in the file {\tt det.f} with
+appropriate copyright. It can be compiled using
+
+{\tt gfortran -o overlap overlap.f90 det.f}.
+
+\subsection{Tdhf2Silo}
+\label{sec:silo}
+This program is quite complicated. It reads a set of {\tt *.tdd} files
+and converts them into {\tt Silo} files. {\tt Silo} is a library for
+handling scientific datasets developed at Lawrence Livermore National
+Laboratory ({\tt https://wci.llnl.gov/codes/silo/index.html}). This is
+the most appropriate library to use in conjunction with the LLNL
+graphics visualization tool {\tt VisIt} ({\tt
+ https://wci.llnl.gov/codes/visit/home.html}), which was found to be
+highly suitable for plotting {\tt Sky3D} results and producing
+movies. It should be noted that a copy of the included file {\tt
+ silo.inc} is provided, which defines symbolic names for the various
+parameters used in the library calls. This file is from Silo version
+4.9. {\em Older versions of this file may cause problems as they use
+fixed-format Fortran style; the present version of {\tt silo.inc}
+should, however, also work with older
+versions of the library.}
+
+The conversion code is quite flexible in that it decides what
+to produce for the different field types: isospin-summed or not,
+vector or scalar. They are given appropriate names for {\tt Silo} with
+suffixes p and n for protons and neutrons, and x, y, z for the vector
+components. In the case of vector fields a variable containing the
+vector definition is also written so that the field can be plotted
+immediately as a vector field in {\tt VisIt}.
+
+If the user wants another dataset handling method, the code should be
+readily adaptable to other libraries. {\tt VisIt} itself has many ways
+of importing data, but of course there are also alternative 3D
+visualization systems.
+
+\section{Running the code}
+\subsection{Compilation and linking}
+To produce executable files the code comes with several {\tt
+ Makefile}s. The standard {\tt Makefile} produces a sequential code
+{\tt sky3d.seq}, the file {\tt Makefile.openmp} a parallel code using
+{\tt OpenMP}, while {\tt Makefile.mpi} should produce an {\tt MPI}
+distributed system code.
+
+The {\tt Makefile}s are written for the {\tt gfortran} compiler and
+the commands and options must be adapted if other compilers are used.
+{\em The user may also have to modify the library names and execution
+ of the code under MPI will require consulting the local
+ documentation or system administrator.}
+
+No attempt was made to select the compiler and linker options
+optimally for speed, since experience has shown that optimization at
+the cutting edge is highly time-dependent. Thus users should do some
+speed tests before embarking on major calculations.
+
+\subsection{External libraries needed}
+The {\tt LAPACK} library is used in the code to supply the routines
+{\tt ZHBEVD} and {\tt DSYEV}. {\tt LAPACK} should be installed in most
+scientific computing centers; if not, the files can be obtained from
+{\tt www.netlib.org} and just be added as additional source files to
+the code. Note that a complete set of routines called by these two
+subroutines must be downloaded.
+
+The other external routine library that is used is {\tt FFTW3}. Again,
+it will be preinstalled in most systems. If not, there are two
+possibilities:
+\begin{enumerate}
+\item Download the source code from {\tt www.fftw.org} and compile the
+ library yourself. In our experience this worked smoothly. The
+ generated library can be installed in a system library directory or
+ kept in a user account. In the latter case the use of {\tt -lfftw3}
+ in the makefiles does not work anymore and the full path name of the
+ library file must be given.
+\item Replace it by another FFT routine. This requires quite a bit of
+ work: {\tt FFTW} organizes its calls around ``plans'', which
+ describe a set of operations to be done on the three-dimensional
+ arrays. In \texttt{init\_fft} quite sophisticated plans are set up to,
+ for example, transform in the y-direction for all x- and
+ z-values. This means that all calls to subroutines beginning with
+ {\tt dfftw} have to be examined and possibly replaced by loops over
+ one-dimensional FFT transforms. This should be relatively
+ straightforward, but there are two more important points to
+ consider: 1)~normalization differs between FFT codes. For {\tt FFTW}
+ transformation followed by inverse transformation multiplies the
+ original data by {\tt nx*ny*nz} and this factor is taken into
+ account in several places. 2)~For the non-periodic case the Fourier
+ transform in the Coulomb solver uses doubled dimensions in all three
+ directions. Some FFT codes have an initialization that sets up the
+ transformation factors depending on the dimension; in such cases
+ the initialization may have to be repeated.
+\end{enumerate}
+
+\subsection{Running with {\tt OPENMP}}
+The {\tt OPENMP} version can be compiled using the file {\tt
+ Makefile.openmp}, which produces an executable {\tt sky3d.omp}. The
+main difference to the sequential makefile is the addition of an
+openmp compiler option. Since this depends on the compiler used, it
+may have to be modified. For {\tt gfortran} the option is {\tt
+ -fopenmp}, while for Intel Fortran it is simply {\tt -openmp}.
+
+For controlling the running of the code the user should set the
+environment variable {\tt OMP\_NUM\_THREADS} to the number of parallel
+threads to be used (usually the number of processors). In addition it
+may be necessary to set {\tt OMP\_STACKSIZE}. The two parallel loops
+in {\tt dynamichf} need to store all the density fields in parallel,
+and the second loop adds {\tt ps4} to that. Taking into account vector
+fields and isospin, a total of~24 three-dimensional {\tt COMPLEX(8)}
+fields need to be stored, amounting to {\tt nx*ny*nz*24*16} bytes,
+which is the stack size needed.
+
+\subsection{Running under {\tt MPI}}
+The situation for {\tt MPI} is a bit more complex than for {\tt
+ OPENMP}, so that the file {\tt Makefile.mpi} will almost certainly
+have to be modified. One crucial difference to the other makefiles is
+that the module {\tt Parallel} is now generated from the source file
+{\tt parallel.f90}. In addition the compilation commands have to be
+adapted; something like {\tt mpif90} will be needed but is
+installation dependent. In addition a command like {\tt mpirun} will
+be needed for execution; the user is advised to consult local
+documentation.
+
+\subsection{Required input}
+Here it is just summarized what input is needed for a static or dynamic
+calculation. A full description can be found with the documentation
+for the {\tt NAMELIST}s.
+
+\subsubsection{Static calculation}
+
+The {\tt NAMELIST}s needed are, in that order:
+\begin{quote}
+\texttt{files}, \texttt{force}, \texttt{main}, \texttt{grid},
+\texttt{static}.
+In addition, if initialization is from fragments, \texttt{fragments},
+and for user initialization possibly \texttt{user} (only if the user
+initialization requires input).
+\end{quote}
+
+\subsubsection{Dynamic calculation}
+The {\tt NAMELIST}s needed are, in that order:
+\begin{quote}
+\texttt{files}, \texttt{force}, \texttt{main}, \texttt{grid},
+\texttt{dynamic}, For external field excitation \texttt{extern}, and
+in all cases \texttt{fragments}.
+\end{quote}
+
+\subsection{Test cases}
+To allow checking the proper behavior of the code, we provide three
+test cases exercising different functions: a static calculation for
+the ground state of $^{16}$O, a dynamic calculation using an external
+excitation to stimulate a giant resonance in $^{16}$O, and finally a
+sample deep-inelastic collision of two $^{16}$O nuclei. The test cases
+directory contains a more detailed description of these cases and what
+to look for principally in the results. Note that since the
+calculations are quite large-scale, differences in roundoff errors may
+lead to the output not being quite identical to the samples provided.
+
+\section{Caveats concerning the code}
+The user should be aware of the limitations of the code in various
+respects but also note some less straightforward procedures for
+improving accuracy in certain cases.
+
+\subsection{Static calculations}
+Since the main use of the code is expected to be in time-dependent
+calculations, the static part is less highly developed. The omission
+of all symmetry restrictions, while very useful for innovative
+applications with time-dependence, can cause some problems in the
+static case.
+\begin{enumerate}
+\item The spin is not aligned along a fixed direction. Since for
+ even-even nuclei Kramers degeneracy operates, two degenerate levels
+ will mix in an uncontrolled way to produce an arbitrary spin
+ alignment. This can be remedied by diagonalizing the spin operators
+ in such a two-level subspace.
+\item The center of mass may move away from the origin during the
+ iterations. This is typically a very small effect and will be
+ corrected when the wave functions are placed at a given position in
+ the dynamic initialization. The calculation of observables, however,
+ should always use coordinates relative to the real center of mass.
+\item In very heavy nuclei sometimes even a rotation was seen, as the
+ reoccupation of high-lying levels can change the geometric
+ orientation. If this is a serious problem, constraints should be
+ introduced or another code used for the static calculation.
+\item The harmonic oscillator initialization can be quite deficient
+ for heavier nuclei. It is planned to develop a Nilsson-model
+ alternative; meanwhile in case of problems the use of wave functions
+ from axial or symmetry-restricting codes could be implemented by
+ generating a wave function file from such results. Since this will
+ involve interpolation, a number of static iterations should then be
+ performed using the present code to improve stability.
+\end{enumerate}
+
+\subsection{Dynamic calculations}
+Here it is important what the required accuracy will be. Most
+exploratory calculations will not pose high accuracy demands. There
+are several possible ways in which accuracy can be improved if needed:
+\begin{enumerate}
+\item The initial configuration may be improved. If the fragment
+ nuclei are not situated at grid points, one should run a number of
+ static iterations with each fragment situated alone in the new
+ grid.
+\item If the fragments are deformed, the energy estimate from the
+ Rutherford trajectory will not be reliable; in this case it is
+ recommended to run a number of dynamic iterations, observe the
+ change in relative distance, and then correct the boost energies to
+ match the correct velocity of relative motion.
+\end{enumerate}
+
+\section{Modifying the code}\label{sec:modify}
+Since the code was developed with a view for easy modification, in this
+Section we give some advice on how to add new things to it and how to
+run simulations.
+
+\subsection{Modifying the Skyrme force}
+The code comes with a quite large database of Skyrme force
+parametrizations together with appropriate pairing parameters built
+in. This is certainly useful to avoid mistakes in the input by having
+to indicate only the name of the force. Still there will be a need to
+add new forces and even forces with a different density functional to
+it, for which we suggest three different approaches.
+
+\subsubsection{Direct parameter input}
+If a specific parametrization is not expected to be a permanent
+addition to the code, for example if one or several parameters are
+varied to study the sensitivity of the results to specific parts of
+the density functional, the best way is to use the facility for giving
+the parameters directly in the input. This is triggered by using some
+force name that is not in the database, in which case the routine
+expects all parameters to be given in {\tt NAMELIST forces}.
+
+\subsubsection{Expanding the database}
+At present the database of forces is contained in the file {\tt
+ forces.data} as a long initialization statement. This has the
+advantage of readability and avoids having to make a database file
+accessible in every directory used for code applications.
+
+So a new force which will be used more permanently can be added simply
+by adding the appropriate lines in {\tt forces.data} and increasing
+the number assigned to {\tt nforce} accordingly. There is a slight
+danger that the total length of the list will exceed the 255 lines
+allowed by the Fortran standard; in this case either remove some
+outdated forces, remove the separation lines of all stars, or if there
+is still a problem, convert the initialization to {\tt DATA}
+statements initializing a smaller number of forces in each case.
+
+\subsubsection{Adding new physics to the density functional}
+This can of course require a lot more modifications to the code.
+Generally speaking, such a new term will appear not only in the
+density functional, but also leads to new contributions in the
+single-particle Hamiltonian and may require new types of densities and
+currents. In addition, it will involve new force parameters. So in
+general the following steps will be needed:
+\begin{enumerate}
+\item {\em Parameter definition:} The new parameter can be added to
+ the general {\tt Force} type definition in {\tt forces.f90}. This is
+ the most logical and systematic way, but we recommend it only if the
+ new physics is to be there permanently, since in this case the whole
+ database has to be updated to give a default value --- probably zero
+ --- to the new parameter for each existing force. The alternative is
+ to leave this parameter as a separate entity, which can still be a
+ module variable in {\tt Forces} and be input using the same
+ namelist. Derived parameters should also be calculated here.
+
+\item {\em New densities and currents:} everything that can be defined
+ directly as a sum over the occupied single-particle wave functions
+ should be defined and calculated in module {\tt Densities}. It
+ should be a module variable and allocated during initialization
+ similar to the densities already defined. Then the contributions of
+ the different derivatives of the wave function can be accumulated
+ separately as is done for the existing contributions.
+
+\item {\em Calculation of mean-field components:} subroutine {\tt
+ skyrme} in module {\tt Meanfield} is the place where the fields
+ appearing in the single-particle Hamiltonian are calculated. The
+ difference to module {\tt Densities} is that wave functions are not
+ involved directly, but only combinations and derivatives of the
+ densities and currents need to be evaluated. The new fields should
+ be defined and allocated and then calculated in subroutine {\tt
+ skyrme}. The handling of derivatives again can be imitated based
+ on the existing terms.
+
+\item {\em Single-particle Hamiltonian:} The additional contribution
+ to the single-particle Hamiltonian must be calculated in subroutine
+ {\tt hpsi} of module {\tt Meanfield}. Code has to be added to calculate
+ how the additional terms act on the input wave function {\tt pinn}
+ and the result has to be added to the output wave function {\tt
+ pout}. Again, for efficiency the spatial derivatives can be
+ handled in separate loops.
+
+\item {\em Contribution to the energy:} subroutine {\tt integ\_energy}
+ in module {\tt Energies} must include an additional contribution of
+ the new term, which in this case means simply computing the
+ expression for the energy functional. Probably it will be useful to
+ add this up in some new variable (defined as a module variable), so
+ that the contribution of the new term can be printed out together
+ with the other contributions in subroutines {\tt sinfo} of module
+ {\tt Static} or subroutine {\tt tinfo} of module {\tt Dynamic},
+ respectively.
+
+\item {\em Output of densities:} it may be desirable to write out the
+ new densities, currents, or mean-field contributions into the {\tt
+ *.tdd} files. To do that, subroutine {\tt write\_densities} in
+ module {\tt Inout} must be modified. A new letter to use for {\tt
+ writeselect} must be defined and selected in the {\tt SELECT CASE}
+ statement, and then depending on whether it is a scalar or a vector
+ field, {\tt write\_one\_density} or {\tt write\_vec\_density} is
+ called with a descriptive name given to the field.
+
+ The complication that occurs here is that these writing routines
+ assume isospin-dependent fields and output either isospin-separate
+ or isospin-summed fields depending on the value of {\tt
+ write\_isospin}. If the field has no isospin dependence, the
+ statements used to output {\tt wcoul} should be imitated.
+\end{enumerate}
+
+\subsection{Using constraints in the static calculation}
+There are many situations in which it is useful to solve the static
+Hartree-Fock equations with added constraints. The most well-known
+application is a quadrupole constraint, in which the expectation value
+$\langle\Psi|\hat H-\lambda\hat Q_{20}|\Psi\rangle$ is minimized to
+obtain deformed states of the nucleus. In the simplest case the Lagrange multiplier
+$\lambda$ may be kept fixed, in which case one has to accept the
+resulting quadrupole moment, or, alternatively, some iterative change
+in $\lambda$ is applied to make the solution converge to a desired
+value for $\langle\Psi|\hat Q_{20}|\Psi\rangle$.
+
+The various ways of introducing a constraint are discussed extensively
+in chapter 7.6 of \cite{Rin80aB}, so that here we only briefly discuss
+practical considerations for adding a constraint to Sky3D.
+
+Adding a constraint corresponds to including another potential term in
+the single-particle Hamiltonian, e.~g., for the quadrupole case
+\begin{equation}
+ \hat h\longrightarrow\hat h-\lambda (2z^2-x^2-y^2)
+\end{equation}
+This could be implemented essentially analogously to the use of an
+external exciting field in the time-dependent case. Construct a
+subroutine to compute this external potential and add it to the
+mean-field potential {\tt upot}. This should be applied every time the
+single-particle Hamiltonian is applied to a wave function in module
+{\tt Static}, i.~e., after each call to subroutine {\tt skyrme}. Then
+an additional subroutine should be written which is called at the end
+of each iteration to compute the expectation value of the constraining
+operator (if not already done elsewhere in the code) and adjust the
+value of $\lambda$ if necessary. The method for adjusting $\lambda$
+will depend on the specific constraint used \cite{Cus85a,Sta10a}.
+
+If the constraint is generated by some thing more complicated than a
+scalar potential, like the orbital and spin operators in the case of
+cranking, it will be more convenient to implement the constraint
+inside the subroutine {\tt hpsi} at those places where the appropriate
+derivatives are available.
+
+Since the numerical method used does not restrict deformation, care
+must be taken for unstable constraints like the quadrupole, which can
+go to large values at the edges of the computational box and may pull
+the wave functions there. If that becomes a problem, the operator
+should be damped suitably \cite{Rut95a}.
+
+\subsection{Analyzing the results in new ways}
+ The code provides quite a large number of physical observables in
+ its output files. For new applications it may be necessary to look
+ at additional ones. There are essentially two ways to implement
+ this:
+ \begin{enumerate}
+ \item If the new observable depends only on density and mean-field
+ components, the easiest way is to use the {\tt *.tdd} files, where
+ if necessary more fields can be output by modifying the subroutine
+ \texttt{write\_densities}. As an example for reading the {\tt *.tdd}
+ files, we provide a code calculating the tensor of inertia among
+ the utilities.
+ \item It becomes more complicated if the wave functions have to be
+ used. Here one possibility is to add a call to a user-written
+ routine at the beginning of the static or dynamic loops, which
+ then can use the array \texttt{psi} in any way desired. This may
+ not be a good option if the calculations are lengthy and the
+ analysis routine may have to be modified several times. In this
+ case it is better to generate a new file name for the {\tt wffile}
+ each time \texttt{write\_wavefunctions} is called, similar to the
+ way it is done in \texttt{write\_densities}, and to do the
+ analysis by reading the wave function files.
+ \end{enumerate}
+
+\section*{Acknowledgments}
+This work was supported by the Bundesministerium f\"ur Bildung und
+Forschung under contract No.\ 05P12RFFTG, by the UK STFC under grant
+number ST/J000051/1, and by the U.S. Department of Energy under grant
+No. DE-FG02-96ER40975 with Vanderbilt University.
+
+\bibliographystyle{elsarticle-num}
+\bibliography{Userguide}
+\end{document}
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+} def
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+%End of PostScript Level 2 code
+%
+/PatternBgnd {
+ TransparentPatterns {} {gsave 1 setgray fill grestore} ifelse
+} def
+%
+% Substitute for Level 2 pattern fill codes with
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+%
+/Level1PatternFill {
+/Pattern1 {0.250 Density} bind def
+/Pattern2 {0.500 Density} bind def
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+/Pattern5 {0.375 Density} bind def
+/Pattern6 {0.625 Density} bind def
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+} def
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+ get 1 eq {dup 2 get exch dup 3 get exch 6 get stringwidth pop -2 div
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+ show 2 index {aload pop M neg 3 -1 roll neg R pop pop} {pop pop pop
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+ forall} def
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diff --git a/Paper/v1.1/elsarticle-num.bst b/Paper/v1.1/elsarticle-num.bst
new file mode 100755
index 0000000000000000000000000000000000000000..24654f12a66a1f07ce0fbc6150da3ebdb173b815
--- /dev/null
+++ b/Paper/v1.1/elsarticle-num.bst
@@ -0,0 +1,1507 @@
+%%
+%% This is file `elsarticle-num.bst',
+%%
+%% Copyright 2007, 2008, 2009 Elsevier Ltd
+%%
+%% This file is part of the 'Elsarticle Bundle'.
+%% ---------------------------------------------
+%%
+%% It may be distributed under the conditions of the LaTeX Project Public
+%% License, either version 1.2 of this license or (at your option) any
+%% later version. The latest version of this license is in
+%% http://www.latex-project.org/lppl.txt
+%% and version 1.2 or later is part of all distributions of LaTeX
+%% version 1999/12/01 or later.
+%%
+%% The list of all files belonging to the 'Elsarticle Bundle' is
+%% given in the file `manifest.txt'.
+%%
+%%
+%% $Id: elsarticle-num.bst 19 2009-12-17 07:07:52Z rishi $
+%%
+%%
+%% ----------------------------------------
+%%
+ENTRY
+ { address
+ author
+ booktitle
+ chapter
+ edition
+ editor
+ howpublished
+ institution
+ journal
+ key
+ month
+ note
+ number
+ organization
+ pages
+ publisher
+ school
+ series
+ title
+ type
+ volume
+ year
+ eprint % urlbst
+ doi % urlbst
+ url % urlbst
+ lastchecked % urlbst
+ }
+ {}
+ { label }
+
+INTEGERS { output.state before.all mid.sentence after.sentence after.block }
+
+STRINGS { urlintro eprinturl eprintprefix doiprefix doiurl openinlinelink closeinlinelink } % urlbst...
+INTEGERS { hrefform inlinelinks makeinlinelink addeprints adddoiresolver }
+FUNCTION {init.urlbst.variables}
+{
+ "Available from: " 'urlintro := % prefix before URL
+ "http://arxiv.org/abs/" 'eprinturl := % prefix to make URL from eprint ref
+ "arXiv:" 'eprintprefix := % text prefix printed before eprint ref
+ "http://dx.doi.org/" 'doiurl := % prefix to make URL from DOI
+ "doi:" 'doiprefix := % text prefix printed before DOI ref
+ #1 'addeprints := % 0=no eprints; 1=include eprints
+ #1 'adddoiresolver := % 0=no DOI resolver; 1=include it
+ #2 'hrefform := % 0=no crossrefs; 1=hypertex xrefs; 2=hyperref refs
+ #1 'inlinelinks := % 0=URLs explicit; 1=URLs attached to titles
+ % the following are internal state variables, not config constants
+ #0 'makeinlinelink := % state variable managed by setup.inlinelink
+ "" 'openinlinelink := % ditto
+ "" 'closeinlinelink := % ditto
+}
+INTEGERS {
+ bracket.state
+ outside.brackets
+ open.brackets
+ within.brackets
+ close.brackets
+}
+FUNCTION {init.state.consts}
+{ #0 'outside.brackets := % urlbst
+ #1 'open.brackets :=
+ #2 'within.brackets :=
+ #3 'close.brackets :=
+
+ #0 'before.all :=
+ #1 'mid.sentence :=
+ #2 'after.sentence :=
+ #3 'after.block :=
+}
+
+STRINGS { s t }
+
+FUNCTION {output.nonnull.original}
+{ 's :=
+ output.state mid.sentence =
+ { ", " * write$ }
+ { output.state after.block =
+ { add.period$ write$
+ newline$
+ "\newblock " write$
+ }
+ { output.state before.all =
+ 'write$
+ { add.period$ " " * write$ }
+ if$
+ }
+ if$
+ mid.sentence 'output.state :=
+ }
+ if$
+ s
+}
+
+FUNCTION {setup.inlinelink}
+{ makeinlinelink
+ { hrefform #1 = % hypertex
+ { "\special {html:<a href=" quote$ * url * quote$ * "> }{" * 'openinlinelink :=
+ "\special {html:</a>}" 'closeinlinelink :=
+ }
+ { hrefform #2 = % hyperref
+ { "\href{" url * "}{" * 'openinlinelink :=
+ "}" 'closeinlinelink :=
+ }
+ 'skip$
+ if$ % hrefform #2 =
+ }
+ if$ % hrefform #1 =
+ #0 'makeinlinelink :=
+ }
+ 'skip$
+ if$ % makeinlinelink
+}
+FUNCTION {add.inlinelink}
+{ openinlinelink empty$
+ 'skip$
+ { openinlinelink swap$ * closeinlinelink *
+ "" 'openinlinelink :=
+ }
+ if$
+}
+FUNCTION {output.nonnull}
+{ % Save the thing we've been asked to output
+ 's :=
+ % If the bracket-state is close.brackets, then add a close-bracket to
+ % what is currently at the top of the stack, and set bracket.state
+ % to outside.brackets
+ bracket.state close.brackets =
+ { "]" *
+ outside.brackets 'bracket.state :=
+ }
+ 'skip$
+ if$
+ bracket.state outside.brackets =
+ { % We're outside all brackets -- this is the normal situation.
+ % Write out what's currently at the top of the stack, using the
+ % original output.nonnull function.
+ s
+ add.inlinelink
+ output.nonnull.original % invoke the original output.nonnull
+ }
+ { % Still in brackets. Add open-bracket or (continuation) comma, add the
+ % new text (in s) to the top of the stack, and move to the close-brackets
+ % state, ready for next time (unless inbrackets resets it). If we come
+ % into this branch, then output.state is carefully undisturbed.
+ bracket.state open.brackets =
+ { " [" * }
+ { ", " * } % bracket.state will be within.brackets
+ if$
+ s *
+ close.brackets 'bracket.state :=
+ }
+ if$
+}
+
+FUNCTION {inbrackets}
+{ bracket.state close.brackets =
+ { within.brackets 'bracket.state := } % reset the state: not open nor closed
+ { open.brackets 'bracket.state := }
+ if$
+}
+
+FUNCTION {format.lastchecked}
+{ lastchecked empty$
+ { "" }
+ { inbrackets "cited " lastchecked * }
+ if$
+}
+
+FUNCTION {output}
+{ duplicate$ empty$
+ 'pop$
+ 'output.nonnull
+ if$
+}
+
+FUNCTION {output.check}
+{ 't :=
+ duplicate$ empty$
+ { pop$ "empty " t * " in " * cite$ * warning$ }
+ 'output.nonnull
+ if$
+}
+
+FUNCTION {fin.entry.original}
+{ add.period$
+ write$
+ newline$
+}
+
+FUNCTION {new.block}
+{ output.state before.all =
+ 'skip$
+ { after.block 'output.state := }
+ if$
+}
+
+FUNCTION {new.sentence}
+{ output.state after.block =
+ 'skip$
+ { output.state before.all =
+ 'skip$
+ { after.sentence 'output.state := }
+ if$
+ }
+ if$
+}
+
+FUNCTION {add.blank}
+{ " " * before.all 'output.state :=
+}
+
+FUNCTION {date.block}
+{
+ add.blank
+}
+
+FUNCTION {not}
+{ { #0 }
+ { #1 }
+ if$
+}
+
+FUNCTION {and}
+{ 'skip$
+ { pop$ #0 }
+ if$
+}
+
+FUNCTION {or}
+{ { pop$ #1 }
+ 'skip$
+ if$
+}
+
+FUNCTION {new.block.checka}
+{ empty$
+ 'skip$
+ 'new.block
+ if$
+}
+
+FUNCTION {new.block.checkb}
+{ empty$
+ swap$ empty$
+ and
+ 'skip$
+ 'new.block
+ if$
+}
+
+FUNCTION {new.sentence.checka}
+{ empty$
+ 'skip$
+ 'new.sentence
+ if$
+}
+
+FUNCTION {new.sentence.checkb}
+{ empty$
+ swap$ empty$
+ and
+ 'skip$
+ 'new.sentence
+ if$
+}
+
+FUNCTION {field.or.null}
+{ duplicate$ empty$
+ { pop$ "" }
+ 'skip$
+ if$
+}
+
+FUNCTION {emphasize}
+{ skip$ }
+
+FUNCTION {capitalize}
+{ "u" change.case$ "t" change.case$ }
+
+FUNCTION {space.word}
+{ " " swap$ * " " * }
+
+ % Here are the language-specific definitions for explicit words.
+ % Each function has a name bbl.xxx where xxx is the English word.
+ % The language selected here is ENGLISH
+FUNCTION {bbl.and}
+{ "and"}
+
+FUNCTION {bbl.etal}
+{ "et~al." }
+
+FUNCTION {bbl.editors}
+{ "Eds." }
+
+FUNCTION {bbl.editor}
+{ "Ed." }
+
+FUNCTION {bbl.edby}
+{ "edited by" }
+
+FUNCTION {bbl.edition}
+{ "Edition" }
+
+FUNCTION {bbl.volume}
+{ "Vol." }
+
+FUNCTION {bbl.of}
+{ "of" }
+
+FUNCTION {bbl.number}
+{ "no." }
+
+FUNCTION {bbl.nr}
+{ "no." }
+
+FUNCTION {bbl.in}
+{ "in" }
+
+FUNCTION {bbl.pages}
+{ "pp." }
+
+FUNCTION {bbl.page}
+{ "p." }
+
+FUNCTION {bbl.chapter}
+{ "Ch." }
+
+FUNCTION {bbl.techrep}
+{ "Tech. Rep." }
+
+FUNCTION {bbl.mthesis}
+{ "Master's thesis" }
+
+FUNCTION {bbl.phdthesis}
+{ "Ph.D. thesis" }
+
+FUNCTION {bbl.first}
+{ "1st" }
+
+FUNCTION {bbl.second}
+{ "2nd" }
+
+FUNCTION {bbl.third}
+{ "3rd" }
+
+FUNCTION {bbl.fourth}
+{ "4th" }
+
+FUNCTION {bbl.fifth}
+{ "5th" }
+
+FUNCTION {bbl.st}
+{ "st" }
+
+FUNCTION {bbl.nd}
+{ "nd" }
+
+FUNCTION {bbl.rd}
+{ "rd" }
+
+FUNCTION {bbl.th}
+{ "th" }
+
+MACRO {jan} {"Jan."}
+
+MACRO {feb} {"Feb."}
+
+MACRO {mar} {"Mar."}
+
+MACRO {apr} {"Apr."}
+
+MACRO {may} {"May"}
+
+MACRO {jun} {"Jun."}
+
+MACRO {jul} {"Jul."}
+
+MACRO {aug} {"Aug."}
+
+MACRO {sep} {"Sep."}
+
+MACRO {oct} {"Oct."}
+
+MACRO {nov} {"Nov."}
+
+MACRO {dec} {"Dec."}
+
+FUNCTION {eng.ord}
+{ duplicate$ "1" swap$ *
+ #-2 #1 substring$ "1" =
+ { bbl.th * }
+ { duplicate$ #-1 #1 substring$
+ duplicate$ "1" =
+ { pop$ bbl.st * }
+ { duplicate$ "2" =
+ { pop$ bbl.nd * }
+ { "3" =
+ { bbl.rd * }
+ { bbl.th * }
+ if$
+ }
+ if$
+ }
+ if$
+ }
+ if$
+}
+
+MACRO {acmcs} {"ACM Comput. Surv."}
+
+MACRO {acta} {"Acta Inf."}
+
+MACRO {cacm} {"Commun. ACM"}
+
+MACRO {ibmjrd} {"IBM J. Res. Dev."}
+
+MACRO {ibmsj} {"IBM Syst.~J."}
+
+MACRO {ieeese} {"IEEE Trans. Softw. Eng."}
+
+MACRO {ieeetc} {"IEEE Trans. Comput."}
+
+MACRO {ieeetcad}
+ {"IEEE Trans. Comput.-Aided Design Integrated Circuits"}
+
+MACRO {ipl} {"Inf. Process. Lett."}
+
+MACRO {jacm} {"J.~ACM"}
+
+MACRO {jcss} {"J.~Comput. Syst. Sci."}
+
+MACRO {scp} {"Sci. Comput. Programming"}
+
+MACRO {sicomp} {"SIAM J. Comput."}
+
+MACRO {tocs} {"ACM Trans. Comput. Syst."}
+
+MACRO {tods} {"ACM Trans. Database Syst."}
+
+MACRO {tog} {"ACM Trans. Gr."}
+
+MACRO {toms} {"ACM Trans. Math. Softw."}
+
+MACRO {toois} {"ACM Trans. Office Inf. Syst."}
+
+MACRO {toplas} {"ACM Trans. Prog. Lang. Syst."}
+
+MACRO {tcs} {"Theoretical Comput. Sci."}
+
+FUNCTION {write.url}
+{ url empty$
+ { skip$ }
+ { "\newline\urlprefix\url{" url * "}" * write$ newline$ }
+ if$
+}
+
+INTEGERS { nameptr namesleft numnames }
+
+FUNCTION {format.names}
+{ 's :=
+ #1 'nameptr :=
+ s num.names$ 'numnames :=
+ numnames 'namesleft :=
+ { namesleft #0 > }
+ { s nameptr
+ "{f.~}{vv~}{ll}{, jj}" format.name$
+ 't :=
+ nameptr #1 >
+ {
+ namesleft #1 >
+ { ", " * t * }
+ {
+ "," *
+ s nameptr "{ll}" format.name$ duplicate$ "others" =
+ { 't := }
+ { pop$ }
+ if$
+ t "others" =
+ {
+ " " * bbl.etal *
+ }
+ { " " * t * }
+ if$
+ }
+ if$
+ }
+ 't
+ if$
+ nameptr #1 + 'nameptr :=
+ namesleft #1 - 'namesleft :=
+ }
+ while$
+}
+FUNCTION {format.names.ed}
+{ format.names }
+FUNCTION {format.authors}
+{ author empty$
+ { "" }
+ { author format.names }
+ if$
+}
+
+FUNCTION {format.editors}
+{ editor empty$
+ { "" }
+ { editor format.names
+ editor num.names$ #1 >
+ { " (" * bbl.editors * ")" * }
+ { " (" * bbl.editor * ")" * }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.in.editors}
+{ editor empty$
+ { "" }
+ { editor format.names.ed
+ editor num.names$ #1 >
+ { " (" * bbl.editors * ")" * }
+ { " (" * bbl.editor * ")" * }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.note}
+{
+ note empty$
+ { "" }
+ { note #1 #1 substring$
+ duplicate$ "{" =
+ 'skip$
+ { output.state mid.sentence =
+ { "l" }
+ { "u" }
+ if$
+ change.case$
+ }
+ if$
+ note #2 global.max$ substring$ *
+ }
+ if$
+}
+
+FUNCTION {format.title}
+{ title empty$
+ { "" }
+ { title "t" change.case$
+ }
+ if$
+}
+
+FUNCTION {output.bibitem.original}
+{ newline$
+ "\bibitem{" write$
+ cite$ write$
+ "}" write$
+ newline$
+ ""
+ before.all 'output.state :=
+}
+
+FUNCTION {n.dashify}
+{
+ 't :=
+ ""
+ { t empty$ not }
+ { t #1 #1 substring$ "-" =
+ { t #1 #2 substring$ "--" = not
+ { "--" *
+ t #2 global.max$ substring$ 't :=
+ }
+ { { t #1 #1 substring$ "-" = }
+ { "-" *
+ t #2 global.max$ substring$ 't :=
+ }
+ while$
+ }
+ if$
+ }
+ { t #1 #1 substring$ *
+ t #2 global.max$ substring$ 't :=
+ }
+ if$
+ }
+ while$
+}
+
+FUNCTION {word.in}
+{ bbl.in
+ ":" *
+ " " * }
+
+FUNCTION {format.date}
+{ year empty$
+ { month empty$
+ { "" }
+ { "there's a month but no year in " cite$ * warning$
+ month
+ }
+ if$
+ }
+ { month empty$
+ 'year
+ { month " " * year * }
+ if$
+ }
+ if$
+ duplicate$ empty$
+ 'skip$
+ {
+ before.all 'output.state :=
+ " (" swap$ * ")" *
+ }
+ if$
+}
+
+FUNCTION{format.year}
+{ year duplicate$ empty$
+ { "empty year in " cite$ * warning$ pop$ "" }
+ { "(" swap$ * ")" * }
+ if$
+}
+
+FUNCTION {format.btitle}
+{ title
+}
+
+FUNCTION {tie.or.space.connect}
+{ duplicate$ text.length$ #3 <
+ { "~" }
+ { " " }
+ if$
+ swap$ * *
+}
+
+FUNCTION {either.or.check}
+{ empty$
+ 'pop$
+ { "can't use both " swap$ * " fields in " * cite$ * warning$ }
+ if$
+}
+
+FUNCTION {format.bvolume}
+{ volume empty$
+ { "" }
+ { bbl.volume volume tie.or.space.connect
+ series empty$
+ 'skip$
+ { bbl.of space.word * series emphasize * }
+ if$
+ "volume and number" number either.or.check
+ }
+ if$
+}
+
+FUNCTION {format.number.series}
+{ volume empty$
+ { number empty$
+ { series field.or.null }
+ { output.state mid.sentence =
+ { bbl.number }
+ { bbl.number capitalize }
+ if$
+ number tie.or.space.connect
+ series empty$
+ { "there's a number but no series in " cite$ * warning$ }
+ { bbl.in space.word * series * }
+ if$
+ }
+ if$
+ }
+ { "" }
+ if$
+}
+
+FUNCTION {is.num}
+{ chr.to.int$
+ duplicate$ "0" chr.to.int$ < not
+ swap$ "9" chr.to.int$ > not and
+}
+
+FUNCTION {extract.num}
+{ duplicate$ 't :=
+ "" 's :=
+ { t empty$ not }
+ { t #1 #1 substring$
+ t #2 global.max$ substring$ 't :=
+ duplicate$ is.num
+ { s swap$ * 's := }
+ { pop$ "" 't := }
+ if$
+ }
+ while$
+ s empty$
+ 'skip$
+ { pop$ s }
+ if$
+}
+
+FUNCTION {convert.edition}
+{ edition extract.num "l" change.case$ 's :=
+ s "first" = s "1" = or
+ { bbl.first 't := }
+ { s "second" = s "2" = or
+ { bbl.second 't := }
+ { s "third" = s "3" = or
+ { bbl.third 't := }
+ { s "fourth" = s "4" = or
+ { bbl.fourth 't := }
+ { s "fifth" = s "5" = or
+ { bbl.fifth 't := }
+ { s #1 #1 substring$ is.num
+ { s eng.ord 't := }
+ { edition 't := }
+ if$
+ }
+ if$
+ }
+ if$
+ }
+ if$
+ }
+ if$
+ }
+ if$
+ t
+}
+
+FUNCTION {format.edition}
+{ edition empty$
+ { "" }
+ { output.state mid.sentence =
+ { convert.edition "l" change.case$ " " * bbl.edition * }
+ { convert.edition "t" change.case$ " " * bbl.edition * }
+ if$
+ }
+ if$
+}
+
+INTEGERS { multiresult }
+
+FUNCTION {multi.page.check}
+{ 't :=
+ #0 'multiresult :=
+ { multiresult not
+ t empty$ not
+ and
+ }
+ { t #1 #1 substring$
+ duplicate$ "-" =
+ swap$ duplicate$ "," =
+ swap$ "+" =
+ or or
+ { #1 'multiresult := }
+ { t #2 global.max$ substring$ 't := }
+ if$
+ }
+ while$
+ multiresult
+}
+
+FUNCTION {format.pages}
+{ pages empty$
+ { "" }
+ { pages multi.page.check
+ { bbl.pages pages n.dashify tie.or.space.connect }
+ { bbl.page pages tie.or.space.connect }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.journal.pages}
+{ pages empty$
+ 'skip$
+ { duplicate$ empty$
+ { pop$ format.pages }
+ {
+ " " *
+ format.year * " " *
+ pages n.dashify *
+ }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.vol.num.pages}
+{
+ % volume field.or.null
+ " "
+ volume empty$
+ { pop$ "" }
+ { volume * }
+ if$
+ number empty$
+ 'skip$
+ {
+ "~(" number * ")" * *
+ volume empty$
+ { "there's a number but no volume in " cite$ * warning$ }
+ 'skip$
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.chapter.pages}
+{ chapter empty$
+ { "" }
+ { type empty$
+ { bbl.chapter }
+ { type "l" change.case$ }
+ if$
+ chapter tie.or.space.connect
+ }
+ if$
+}
+
+FUNCTION {format.in.ed.booktitle}
+{ booktitle empty$
+ { "" }
+ { editor empty$
+ { word.in booktitle * }
+ { word.in format.in.editors * ", " *
+ booktitle * }
+ if$
+ }
+ if$
+}
+
+FUNCTION {empty.misc.check}
+{ author empty$ title empty$ howpublished empty$
+ month empty$ year empty$ note empty$
+ and and and and and
+ { "all relevant fields are empty in " cite$ * warning$ }
+ 'skip$
+ if$
+}
+
+FUNCTION {format.thesis.type}
+{ type empty$
+ 'skip$
+ { pop$
+ type "t" change.case$
+ }
+ if$
+}
+
+FUNCTION {format.tr.number}
+{ type empty$
+ { bbl.techrep }
+ 'type
+ if$
+ number empty$
+ { "t" change.case$ }
+ { number tie.or.space.connect }
+ if$
+}
+
+FUNCTION {format.article.crossref}
+{
+ key empty$
+ { journal empty$
+ { "need key or journal for " cite$ * " to crossref " * crossref *
+ warning$
+ ""
+ }
+ { word.in journal emphasize * }
+ if$
+ }
+ { word.in key * " " *}
+ if$
+ " \cite{" * crossref * "}" *
+}
+
+FUNCTION {format.crossref.editor}
+{ editor #1 "{vv~}{ll}" format.name$
+ editor num.names$ duplicate$
+ #2 >
+ { pop$
+ " " * bbl.etal *
+ }
+ { #2 <
+ 'skip$
+ { editor #2 "{ff }{vv }{ll}{ jj}" format.name$ "others" =
+ {
+ " " * bbl.etal *
+ }
+ { bbl.and space.word * editor #2 "{vv~}{ll}" format.name$
+ * }
+ if$
+ }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.book.crossref}
+{ volume empty$
+ { "empty volume in " cite$ * "'s crossref of " * crossref * warning$
+ word.in
+ }
+ { bbl.volume volume tie.or.space.connect
+ bbl.of space.word *
+ }
+ if$
+ editor empty$
+ editor field.or.null author field.or.null =
+ or
+ { key empty$
+ { series empty$
+ { "need editor, key, or series for " cite$ * " to crossref " *
+ crossref * warning$
+ "" *
+ }
+ { series emphasize * }
+ if$
+ }
+ { key * }
+ if$
+ }
+ { format.crossref.editor * }
+ if$
+ " \cite{" * crossref * "}" *
+}
+
+FUNCTION {format.incoll.inproc.crossref}
+{
+ editor empty$
+ editor field.or.null author field.or.null =
+ or
+ { key empty$
+ { booktitle empty$
+ { "need editor, key, or booktitle for " cite$ * " to crossref " *
+ crossref * warning$
+ ""
+ }
+ { word.in booktitle * }
+ if$
+ }
+ { word.in key * " " *}
+ if$
+ }
+ { word.in format.crossref.editor * " " *}
+ if$
+ " \cite{" * crossref * "}" *
+}
+
+FUNCTION {format.org.or.pub}
+{ 't :=
+ ""
+ year empty$
+ { "empty year in " cite$ * warning$ }
+ 'skip$
+ if$
+ address empty$ t empty$ and
+ year empty$ and
+ 'skip$
+ {
+ t empty$
+ { address empty$
+ 'skip$
+ { address * }
+ if$
+ }
+ { t *
+ address empty$
+ 'skip$
+ { ", " * address * }
+ if$
+ }
+ if$
+ year empty$
+ 'skip$
+ { t empty$ address empty$ and
+ 'skip$
+ { ", " * }
+ if$
+ year *
+ }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.publisher.address}
+{ publisher empty$
+ { "empty publisher in " cite$ * warning$
+ ""
+ }
+ { publisher }
+ if$
+ format.org.or.pub
+}
+
+FUNCTION {format.organization.address}
+{ organization empty$
+ { "" }
+ { organization }
+ if$
+ format.org.or.pub
+}
+
+FUNCTION {make.href.null}
+{
+ pop$
+}
+FUNCTION {make.href.hypertex}
+{
+ "\special {html:<a href=" quote$ *
+ swap$ * quote$ * "> }" * swap$ *
+ "\special {html:</a>}" *
+}
+FUNCTION {make.href.hyperref}
+{
+ "\href {" swap$ * "} {\path{" * swap$ * "}}" *
+}
+FUNCTION {make.href}
+{ hrefform #2 =
+ 'make.href.hyperref % hrefform = 2
+ { hrefform #1 =
+ 'make.href.hypertex % hrefform = 1
+ 'make.href.null % hrefform = 0 (or anything else)
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.url}
+{ inlinelinks #1 = url empty$ or
+ { "" }
+ { hrefform #1 =
+ { % special case -- add HyperTeX specials
+ urlintro "\url{" url * "}" * url make.href.hypertex * }
+ { urlintro "\url{" * url * "}" * }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.eprint}
+{ eprint empty$
+ { "" }
+ { eprintprefix eprint * eprinturl eprint * make.href }
+ if$
+}
+
+FUNCTION {format.doi}
+{ doi empty$
+ { "" }
+ { doiprefix doi * doiurl doi * make.href }
+ if$
+}
+
+FUNCTION {output.url}
+{ url empty$
+ 'skip$
+ { new.block
+ format.url output
+ format.lastchecked output
+ }
+ if$
+}
+
+FUNCTION {output.web.refs}
+{
+ new.block
+ output.url
+ addeprints eprint empty$ not and
+ { format.eprint output.nonnull }
+ 'skip$
+ if$
+ adddoiresolver doi empty$ not and
+ { format.doi output.nonnull }
+ 'skip$
+ if$
+}
+
+FUNCTION {output.bibitem}
+{ outside.brackets 'bracket.state :=
+ output.bibitem.original
+ inlinelinks url empty$ not and
+ { #1 'makeinlinelink := }
+ { #0 'makeinlinelink := }
+ if$
+}
+
+FUNCTION {fin.entry}
+{ output.web.refs % urlbst
+ makeinlinelink % ooops, it appears we didn't have a title for inlinelink
+ { setup.inlinelink % add some artificial link text here, as a fallback
+ "[link]" output.nonnull }
+ 'skip$
+ if$
+ bracket.state close.brackets = % urlbst
+ { "]" * }
+ 'skip$
+ if$
+ fin.entry.original
+}
+
+FUNCTION {webpage}
+{ output.bibitem
+ author empty$
+ { editor empty$
+ 'skip$ % author and editor both optional
+ { format.editors output.nonnull }
+ if$
+ }
+ { editor empty$
+ { format.authors output.nonnull }
+ { "can't use both author and editor fields in " cite$ * warning$ }
+ if$
+ }
+ if$
+ new.block
+ title empty$ 'skip$ 'setup.inlinelink if$
+ format.title "title" output.check
+ inbrackets "online" output
+ new.block
+ year empty$
+ 'skip$
+ { format.date "year" output.check }
+ if$
+ % We don't need to output the URL details ('lastchecked' and 'url'),
+ % because fin.entry does that for us, using output.web.refs. The only
+ % reason we would want to put them here is if we were to decide that
+ % they should go in front of the rather miscellaneous information in 'note'.
+ new.block
+ note output
+ fin.entry
+}
+
+FUNCTION {article}
+{ output.bibitem
+ format.authors "author" output.check
+ title empty$ 'skip$ 'setup.inlinelink if$ % urlbst
+ format.title "title" output.check
+ crossref missing$
+ { journal
+ "journal" output.check
+ % add.blank
+ before.all 'output.state :=
+ format.vol.num.pages output
+ }
+ { format.article.crossref output.nonnull
+ format.pages output
+ }
+ if$
+ format.journal.pages
+ format.note output
+ fin.entry
+ write.url
+}
+
+FUNCTION {book}
+{ output.bibitem
+ author empty$
+ { format.editors "author and editor" output.check
+ }
+ { format.authors output.nonnull
+ crossref missing$
+ { "author and editor" editor either.or.check }
+ 'skip$
+ if$
+ }
+ if$
+ title empty$ 'skip$ 'setup.inlinelink if$ % urlbst
+ format.btitle "title" output.check
+ crossref missing$
+ { format.edition output
+ format.bvolume output
+ format.number.series output
+ format.publisher.address output
+ }
+ {
+ format.book.crossref output.nonnull
+ }
+ if$
+ format.note output
+ fin.entry
+ write.url
+}
+
+FUNCTION {booklet}
+{ output.bibitem
+ format.authors output
+ title empty$ 'skip$ 'setup.inlinelink if$ % urlbst
+ format.title "title" output.check
+ howpublished output
+ address output
+ format.note output
+ format.date output
+ fin.entry
+ write.url
+}
+
+FUNCTION {inbook}
+{ output.bibitem
+ author empty$
+ { format.editors "author and editor" output.check
+ }
+ { format.authors output.nonnull
+ crossref missing$
+ { "author and editor" editor either.or.check }
+ 'skip$
+ if$
+ }
+ if$
+ title empty$ 'skip$ 'setup.inlinelink if$ % urlbst
+ format.btitle "title" output.check
+ crossref missing$
+ {
+ format.edition output
+ format.bvolume output
+ format.number.series output
+ format.publisher.address output
+ format.chapter.pages "chapter and pages" output.check
+ }
+ {
+ format.chapter.pages "chapter and pages" output.check
+ format.book.crossref output.nonnull
+ }
+ if$
+ format.pages "pages" output.check
+ format.note output
+ fin.entry
+ write.url
+}
+
+FUNCTION {incollection}
+{ output.bibitem
+ format.authors "author" output.check
+ title empty$ 'skip$ 'setup.inlinelink if$ % urlbst
+ format.title "title" output.check
+ crossref missing$
+ { format.in.ed.booktitle "booktitle" output.check
+ format.edition output
+ format.bvolume output
+ format.number.series output
+ format.publisher.address output
+ format.chapter.pages output
+ }
+ { format.incoll.inproc.crossref output.nonnull
+ format.chapter.pages output
+ }
+ if$
+ format.pages "pages" output.check
+ format.note output
+ fin.entry
+ write.url
+}
+
+FUNCTION {inproceedings}
+{ output.bibitem
+ format.authors "author" output.check
+ title empty$ 'skip$ 'setup.inlinelink if$ % urlbst
+ format.title "title" output.check
+ crossref missing$
+ { format.in.ed.booktitle "booktitle" output.check
+ format.edition output
+ format.bvolume output
+ format.number.series output
+ publisher empty$
+ { format.organization.address output }
+ { organization output
+ format.publisher.address output
+ }
+ if$
+ }
+ { format.incoll.inproc.crossref output.nonnull
+ }
+ if$
+ format.pages "pages" output.check
+ format.note output
+ fin.entry
+ write.url
+}
+
+FUNCTION {conference} { inproceedings }
+
+FUNCTION {manual}
+{ output.bibitem
+ author empty$
+ { organization empty$
+ 'skip$
+ { organization output.nonnull
+ address output
+ }
+ if$
+ }
+ { format.authors output.nonnull }
+ if$
+ title empty$ 'skip$ 'setup.inlinelink if$ % urlbst
+ format.btitle "title" output.check
+ author empty$
+ { organization empty$
+ {
+ address output
+ }
+ 'skip$
+ if$
+ }
+ {
+ organization output
+ address output
+ }
+ if$
+ format.edition output
+ format.note output
+ format.date output
+ fin.entry
+ write.url
+}
+
+FUNCTION {mastersthesis}
+{ output.bibitem
+ format.authors "author" output.check
+ title empty$ 'skip$ 'setup.inlinelink if$ % urlbst
+ format.title "title" output.check
+ bbl.mthesis format.thesis.type output.nonnull
+ school "school" output.check
+ address output
+ format.note output
+ format.date "year" output.check
+ fin.entry
+ write.url
+}
+
+FUNCTION {misc}
+{ output.bibitem
+ format.authors output
+ title empty$ 'skip$ 'setup.inlinelink if$ % urlbst
+ format.title output
+ howpublished output
+ format.note output
+ format.date output
+ fin.entry
+ write.url
+ empty.misc.check
+}
+
+FUNCTION {phdthesis}
+{ output.bibitem
+ format.authors "author" output.check
+ title empty$ 'skip$ 'setup.inlinelink if$ % urlbst
+ format.title "title" output.check
+ bbl.phdthesis format.thesis.type output.nonnull
+ school "school" output.check
+ address output
+ format.note output
+ format.date "year" output.check
+ fin.entry
+ write.url
+}
+
+FUNCTION {proceedings}
+{ output.bibitem
+ editor empty$
+ { organization output }
+ { format.editors output.nonnull }
+ if$
+ title empty$ 'skip$ 'setup.inlinelink if$ % urlbst
+ format.btitle "title" output.check
+ format.bvolume output
+ format.number.series output
+ editor empty$
+ { publisher empty$
+ 'skip$
+ {
+ format.publisher.address output
+ }
+ if$
+ }
+ { publisher empty$
+ {
+ format.organization.address output }
+ {
+ organization output
+ format.publisher.address output
+ }
+ if$
+ }
+ if$
+ format.note output
+ fin.entry
+ write.url
+}
+
+FUNCTION {techreport}
+{ output.bibitem
+ format.authors "author" output.check
+ title empty$ 'skip$ 'setup.inlinelink if$ % urlbst
+ format.title "title" output.check
+ format.tr.number output.nonnull
+ institution "institution" output.check
+ address output
+ format.note output
+ format.date "year" output.check
+ fin.entry
+ write.url
+}
+
+FUNCTION {unpublished}
+{ output.bibitem
+ format.authors "author" output.check
+ title empty$ 'skip$ 'setup.inlinelink if$ % urlbst
+ format.title "title" output.check
+ format.note "note" output.check
+ format.date output
+ fin.entry
+ write.url
+}
+
+FUNCTION {default.type} { misc }
+
+READ
+
+STRINGS { longest.label }
+
+INTEGERS { number.label longest.label.width }
+
+FUNCTION {initialize.longest.label}
+{ "" 'longest.label :=
+ #1 'number.label :=
+ #0 'longest.label.width :=
+}
+
+FUNCTION {longest.label.pass}
+{ number.label int.to.str$ 'label :=
+ number.label #1 + 'number.label :=
+ label width$ longest.label.width >
+ { label 'longest.label :=
+ label width$ 'longest.label.width :=
+ }
+ 'skip$
+ if$
+}
+
+EXECUTE {initialize.longest.label}
+
+ITERATE {longest.label.pass}
+
+FUNCTION {begin.bib}
+{ preamble$ empty$
+ 'skip$
+ { preamble$ write$ newline$ }
+ if$
+ "\begin{thebibliography}{" longest.label * "}" *
+ write$ newline$
+ "\expandafter\ifx\csname url\endcsname\relax"
+ write$ newline$
+ " \def\url#1{\texttt{#1}}\fi"
+ write$ newline$
+ "\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi"
+ write$ newline$
+ "\expandafter\ifx\csname href\endcsname\relax"
+ write$ newline$
+ " \def\href#1#2{#2} \def\path#1{#1}\fi"
+ write$ newline$
+}
+
+EXECUTE {begin.bib}
+
+EXECUTE {init.urlbst.variables}
+EXECUTE {init.state.consts}
+
+ITERATE {call.type$}
+
+FUNCTION {end.bib}
+{ newline$
+ "\end{thebibliography}" write$ newline$
+}
+
+EXECUTE {end.bib}
+%% End of customized bst file
+%%
+%% End of file `elsarticle-num.bst'.
+
+
diff --git a/Paper/v1.1/elsarticle.cls b/Paper/v1.1/elsarticle.cls
new file mode 100755
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+%%
+%% This is file `elsarticle.cls',
+%% generated with the docstrip utility.
+%%
+%% The original source files were:
+%%
+%% elsarticle.dtx (with options: `class')
+%%
+%% Copyright 2007, 2008, 2009 Elsevier Ltd
+%%
+%% This file is part of the 'Elsarticle Bundle'.
+%% -------------------------------------------
+%%
+%% It may be distributed under the conditions of the LaTeX Project Public
+%% License, either version 1.2 of this license or (at your option) any
+%% later version. The latest version of this license is in
+%% http://www.latex-project.org/lppl.txt
+%% and version 1.2 or later is part of all distributions of LaTeX
+%% version 1999/12/01 or later.
+%%
+%% The list of all files belonging to the 'Elsarticle Bundle' is
+%% given in the file `manifest.txt'.
+%%
+%%
+%% $Id: elsarticle.dtx 19 2009-12-17 07:07:52Z rishi $
+%%
+ \def\RCSfile{elsarticle}%
+ \def\RCSversion{1.20b}%
+ \def\RCSdate{2009/09/17}%
+ \def\@shortjnl{\relax}
+ \def\@journal{Elsevier Ltd} \def\@company{Elsevier Ltd}
+ \def\@issn{000-0000}
+ \def\@shortjid{elsarticle}
+\NeedsTeXFormat{LaTeX2e}[1995/12/01]
+\ProvidesClass{\@shortjid}[\RCSdate, \RCSversion: \@journal]
+\def\ABD{\AtBeginDocument}
+\newif\ifpreprint \preprintfalse
+\newif\iflongmktitle \longmktitlefalse
+
+\def\@blstr{1}
+\newdimen\@bls
+\@bls=\baselineskip
+
+\def\@finalWarning{%
+ *****************************************************\MessageBreak
+ This document is typeset in the CRC style which\MessageBreak
+ is not suitable for submission.\MessageBreak
+ \MessageBreak
+ Please typeset again using 'preprint' option\MessageBreak
+ for creating PDF suitable for submission.\MessageBreak
+ ******************************************************\MessageBreak
+}
+
+\DeclareOption{preprint}{\global\preprinttrue
+ \gdef\@blstr{1}\xdef\jtype{0}%
+ \AtBeginDocument{\@twosidefalse\@mparswitchfalse}}
+\DeclareOption{final}{\gdef\@blstr{1}\global\preprintfalse}
+\DeclareOption{review}{\global\preprinttrue\gdef\@blstr{1.5}}
+\DeclareOption{authoryear}{\xdef\@biboptions{round,authoryear}}
+\DeclareOption{number}{\xdef\@biboptions{numbers}}
+\DeclareOption{numbers}{\xdef\@biboptions{numbers}}
+\DeclareOption{longtitle}{\global\longmktitletrue}
+\DeclareOption{5p}{\xdef\jtype{5}\global\preprintfalse
+ \ExecuteOptions{twocolumn}}
+ \def\jtype{0}
+\DeclareOption{3p}{\xdef\jtype{3}\global\preprintfalse}
+\DeclareOption{1p}{\xdef\jtype{1}\global\preprintfalse
+ \AtBeginDocument{\@twocolumnfalse}}
+\DeclareOption{times}{\IfFileExists{txfonts.sty}%
+ {\AtEndOfClass{\RequirePackage{txfonts}%
+ \gdef\ttdefault{cmtt}%
+ \let\iint\relax
+ \let\iiint\relax
+ \let\iiiint\relax
+ \let\idotsint\relax
+ \let\openbox\relax}}{\RequirePackage{times}}}
+\ExecuteOptions{a4paper,10pt,oneside,onecolumn,number,preprint}
+\DeclareOption*{\PassOptionsToClass{\CurrentOption}{article}}
+\ProcessOptions
+\LoadClass{article}
+\RequirePackage{graphicx}
+\let\comma\@empty
+\let\tnotesep\@empty
+\def\title#1{\gdef\@title{#1}}
+\let\@title\@empty
+
+\def\elsLabel#1{\@bsphack\protected@write\@auxout{}%
+ {\string\Newlabel{#1}{\@currentlabel}}\@esphack}
+\def\Newlabel#1#2{\expandafter\xdef\csname X@#1\endcsname{#2}}
+
+\def\elsRef#1{\@ifundefined{X@#1}{0}{\csname X@#1\endcsname}%
+}
+
+\def\tnotemark[#1]{\textsuperscript{\@for\@@tmark:=#1\do{%
+ \edef\tnotenum{\@ifundefined{X@\@@tmark}{1}{\elsRef{\@@tmark}}}%
+ \ifcase\tnotenum\or\ding{73}\or,\ding{73}\ding{73}\fi}}%
+}
+\let\@tnotemark\@empty
+
+\let\@tnotes\@empty
+\RequirePackage{pifont}
+\newcounter{tnote}
+\def\tnotetext[#1]#2{\g@addto@macro\@tnotes{%
+ \refstepcounter{tnote}\elsLabel{#1}%
+ \def\thefootnote{\ifcase\c@tnote\or\ding{73}\or\ding{73}\ding{73}\fi}%
+ \footnotetext{#2}}}
+
+\let\@nonumnotes\@empty
+\def\nonumnote#1{\g@addto@macro\@nonumnotes{%
+ \let\thefootnote\relax\footnotetext{#1}}}
+
+\newcounter{fnote}
+\def\fnmark[#1]{\let\comma\@empty
+ \def\@fnmark{\@for\@@fnmark:=#1\do{%
+ \edef\fnotenum{\@ifundefined{X@\@@fnmark}{1}{\elsRef{\@@fnmark}}}%
+ \unskip\comma\fnotenum\let\comma,}}%
+}
+
+\let\@fnotes\@empty\let\@fnmark\@empty
+\def\fntext[#1]#2{\g@addto@macro\@fnotes{%
+ \refstepcounter{fnote}\elsLabel{#1}%
+ \def\thefootnote{\thefnote}%
+ \global\setcounter{footnote}{\thefnote}%
+ \footnotetext{#2}}}
+
+\def\cormark[#1]{\edef\cnotenum{\elsRef{#1}}%
+ \unskip\textsuperscript{\sep\ifcase\cnotenum\or
+ $\ast$\or$\ast\ast$\fi\hspace{-1pt}}\let\sep=,}
+
+\let\@cormark\@empty
+\let\@cornotes\@empty
+\newcounter{cnote}
+\def\cortext[#1]#2{\g@addto@macro\@cornotes{%
+ \refstepcounter{cnote}\elsLabel{#1}%
+ \def\thefootnote{\ifcase\thecnote\or$\ast$\or
+ $\ast\ast$\fi}%
+ \footnotetext{#2}}}
+
+\let\@corref\@empty
+\def\corref#1{\edef\cnotenum{\elsRef{#1}}%
+ \edef\@corref{\ifcase\cnotenum\or
+ $\ast$\or$\ast\ast$\fi\hskip-1pt}}
+
+\def\fnref#1{\fnmark[#1]}
+\def\tnoteref#1{\tnotemark[#1]}
+
+\def\resetTitleCounters{\c@cnote=0
+ \c@fnote=0 \c@tnote=0 \c@footnote=0}
+
+\let\eadsep\@empty
+\let\@elseads\@empty
+\let\@elsuads\@empty
+\let\@cormark\@empty
+\def\hashchar{\expandafter\@gobble\string\~}
+\def\underscorechar{\expandafter\@gobble\string\_}
+\def\lbracechar{\expandafter\@gobble\string\{}
+\def\rbracechar{\expandafter\@gobble\string\}}
+
+\def\ead{\@ifnextchar[{\@uad}{\@ead}}
+\gdef\@ead#1{\bgroup\def\_{\string\underscorechar\space}%
+ \def\{{\string\lbracechar\space}%
+ \def~{\hashchar\space}%
+ \def\}{\string\rbracechar\space}%
+ \edef\tmp{\the\@eadauthor}
+ \immediate\write\@auxout{\string\emailauthor
+ {#1}{\expandafter\strip@prefix\meaning\tmp}}%
+ \egroup
+}
+\newcounter{ead}
+\gdef\emailauthor#1#2{\stepcounter{ead}%
+ \g@addto@macro\@elseads{\raggedright%
+ \let\corref\@gobble
+ \eadsep\texttt{#1} (#2)\def\eadsep{\unskip,\space}}%
+}
+\gdef\@uad[#1]#2{\bgroup
+ \def~{\string\hashchar\space}%
+ \def\_{\string\underscorechar\space}%
+ \edef\tmp{\the\@eadauthor}
+ \immediate\write\@auxout{\string\urlauthor
+ {#2}{\expandafter\strip@prefix\meaning\tmp}}%
+ \egroup
+}
+\def\urlauthor#1#2{\g@addto@macro\@elsuads{\let\corref\@gobble%
+ \raggedright\eadsep\texttt{#1}\space(#2)%
+ \def\eadsep{\unskip,\space}}%
+}
+
+\def\elsauthors{}
+\def\pprinttitle{}
+\let\authorsep\@empty
+\let\sep\@empty
+\newcounter{author}
+\def\author{\@ifnextchar[{\@@author}{\@author}}
+
+\newtoks\@eadauthor
+\def\@@author[#1]#2{\g@addto@macro\elsauthors{%
+ \def\baselinestretch{1}%
+ \authorsep#2\unskip\textsuperscript{%#1%
+ \@for\@@affmark:=#1\do{%
+ \edef\affnum{\@ifundefined{X@\@@affmark}{1}{\elsRef{\@@affmark}}}%
+ \unskip\sep\affnum\let\sep=,}%
+ \ifx\@fnmark\@empty\else\unskip\sep\@fnmark\let\sep=,\fi
+ \ifx\@corref\@empty\else\unskip\sep\@corref\let\sep=,\fi
+ }%
+ \def\authorsep{\unskip,\space}%
+ \global\let\sep\@empty\global\let\@corref\@empty
+ \global\let\@fnmark\@empty}%
+ \@eadauthor={#2}
+}
+
+\def\@author#1{\g@addto@macro\elsauthors{\normalsize%
+ \def\baselinestretch{1}%
+ \upshape\authorsep#1\unskip\textsuperscript{%
+ \ifx\@fnmark\@empty\else\unskip\sep\@fnmark\let\sep=,\fi
+ \ifx\@corref\@empty\else\unskip\sep\@corref\let\sep=,\fi
+ }%
+ \def\authorsep{\unskip,\space}%
+ \global\let\@fnmark\@empty
+ \global\let\@corref\@empty
+ \global\let\sep\@empty}%
+ \@eadauthor={#1}
+}
+
+\def\elsaddress{}
+\def\addsep{\par\vskip6pt}
+\def\address{\@ifnextchar[{\@@address}{\@address}}
+
+\def\@alph#1{%
+ \ifcase#1\or a\or b\or c\or d\or e\or f\or g\or h\or i\or j\or k\or
+ l\or m\or n\or o\or p\or q\or r\or s\or t\or u\or v\or w\or x\or
+ y\or z%
+ \or aa\or ab\or ac\or ad\or ae\or af\or ag\or ah\or ai\or aj\or
+ ak\or al\or am\or an\or ao\or ap\or aq\or ar\or as\or at\or au\or
+ av\or aw\or ax\or ay\or az%
+ \or ba\or bb\or bc\or bd\or be\or bf\or bg\or bh\or bi\or bj\or
+ bk\or bl\or bm\or bn\or bo\or bp\or bq\or br\or bs\or bt\or bu\or
+ bv\or bw\or bx\or by\or bz%
+ \or ca\or cb\or cc\or cd\or ce\or cf\or cg\or ch\or ci\or cj\or
+ ck\or cl\or cm\or cn\or co\or cp\or cq\or cr\or cs\or ct\or cu\or
+ cv\or cw\or cx\or cy\or cz%
+ \or da\or db\or dc\or dd\or de\or df\or dg\or dh\or di\or dj\or
+ dk\or dl\or dm\or dn\or do\or dp\or dq\or dr\or ds\or dt\or du\or
+ dv\or dw\or dx\or dy\or dz%
+ \or ea\or eb\or ec\or ed\or ee\or ef\or eg\or eh\or ei\or ej\or
+ ek\or el\or em\or en\or eo\or ep\or eq\or er\or es\or et\or eu\or
+ ev\or ew\or ex\or ey\or ez%
+ \or fa\or fb\or fc\or fd\or fe\or ff\or fg\or fh\or fi\or fj\or
+ fk\or fl\or fm\or fn\or fo\or fp\or fq\or fr\or fs\or ft\or fu\or
+ fv\or fw\or fx\or fy\or fz%
+ \or ga\or gb\or gc\or gd\or ge\or gf\or gg\or gh\or gi\or gj\or
+ gk\or gl\or gm\or gn\or go\or gp\or gq\or gr\or gs\or gt\or gu\or
+ gv\or gw\or gx\or gy\or gz%
+ \else\@ctrerr\fi}
+
+\newcounter{affn}
+\renewcommand\theaffn{\alph{affn}}
+
+\long\def\@@address[#1]#2{\g@addto@macro\elsaddress{%
+ \def\baselinestretch{1}%
+ \refstepcounter{affn}
+ \xdef\@currentlabel{\theaffn}
+ \elsLabel{#1}%
+ \textsuperscript{\theaffn}#2\par}}
+
+\long\def\@address#1{\g@addto@macro\elsauthors{%
+ \def\baselinestretch{1}%
+ \addsep\footnotesize\itshape#1\def\addsep{\par\vskip6pt}%
+ \def\authorsep{\par\vskip8pt}}}
+
+\newbox\absbox
+\renewenvironment{abstract}{\global\setbox\absbox=\vbox\bgroup
+ \hsize=\textwidth\def\baselinestretch{1}%
+ \noindent\unskip\textbf{Abstract}
+ \par\medskip\noindent\unskip\ignorespaces}
+ {\egroup}
+
+\newbox\keybox
+\def\keyword{%
+ \def\sep{\unskip, }%
+ \def\MSC{\@ifnextchar[{\@MSC}{\@MSC[2000]}}
+ \def\@MSC[##1]{\par\leavevmode\hbox {\it ##1~MSC:\space}}%
+ \def\PACS{\par\leavevmode\hbox {\it PACS:\space}}%
+ \def\JEL{\par\leavevmode\hbox {\it JEL:\space}}%
+ \global\setbox\keybox=\vbox\bgroup\hsize=\textwidth
+ \normalsize\normalfont\def\baselinestretch{1}
+ \parskip\z@
+ \noindent\textit{Keywords: }
+ \raggedright % Keywords are not justified.
+ \ignorespaces}
+\def\endkeyword{\par \egroup}
+
+\newdimen\Columnwidth
+\Columnwidth=\columnwidth
+
+\def\printFirstPageNotes{%
+ \iflongmktitle
+ \let\columnwidth=\textwidth\fi
+ \ifx\@tnotes\@empty\else\@tnotes\fi
+ \ifx\@nonumnotes\@empty\else\@nonumnotes\fi
+ \ifx\@cornotes\@empty\else\@cornotes\fi
+ \ifx\@elseads\@empty\relax\else
+ \let\thefootnote\relax
+ \footnotetext{\ifnum\theead=1\relax
+ \textit{Email address:\space}\else
+ \textit{Email addresses:\space}\fi
+ \@elseads}\fi
+ \ifx\@elsuads\@empty\relax\else
+ \let\thefootnote\relax
+ \footnotetext{\textit{URL:\space}%
+ \@elsuads}\fi
+ \ifx\@fnotes\@empty\else\@fnotes\fi
+ \iflongmktitle\if@twocolumn
+ \let\columnwidth=\Columnwidth\fi\fi
+}
+
+\long\def\pprintMaketitle{\clearpage
+ \iflongmktitle\if@twocolumn\let\columnwidth=\textwidth\fi\fi
+ \resetTitleCounters
+ \def\baselinestretch{1}%
+ \printFirstPageNotes
+ \begin{center}%
+ \thispagestyle{pprintTitle}%
+ \def\baselinestretch{1}%
+ \Large\@title\par\vskip18pt
+ \normalsize\elsauthors\par\vskip10pt
+ \footnotesize\itshape\elsaddress\par\vskip36pt
+ \hrule\vskip12pt
+ \ifvoid\absbox\else\unvbox\absbox\par\vskip10pt\fi
+ \ifvoid\keybox\else\unvbox\keybox\par\vskip10pt\fi
+ \hrule\vskip12pt
+ \end{center}%
+ \gdef\thefootnote{\arabic{footnote}}%
+ }
+
+\def\printWarning{%
+ \mbox{}\par\vfill\par\bgroup
+ \fboxsep12pt\fboxrule1pt
+ \hspace*{.18\textwidth}
+ \fcolorbox{gray50}{gray10}{\box\warnbox}
+ \egroup\par\vfill\thispagestyle{empty}
+ \setcounter{page}{0}
+ \clearpage}
+
+\long\def\finalMaketitle{%
+ \resetTitleCounters
+ \def\baselinestretch{1}%
+ \MaketitleBox
+ \thispagestyle{pprintTitle}%
+ \gdef\thefootnote{\arabic{footnote}}%
+ }
+
+\long\def\MaketitleBox{%
+ \resetTitleCounters
+ \def\baselinestretch{1}%
+ \begin{center}%
+ \def\baselinestretch{1}%
+ \Large\@title\par\vskip18pt
+ \normalsize\elsauthors\par\vskip10pt
+ \footnotesize\itshape\elsaddress\par\vskip36pt
+ \hrule\vskip12pt
+ \ifvoid\absbox\else\unvbox\absbox\par\vskip10pt\fi
+ \ifvoid\keybox\else\unvbox\keybox\par\vskip10pt\fi
+ \hrule\vskip12pt
+ \end{center}%
+ }
+
+\def\FNtext#1{\par\bgroup\footnotesize#1\egroup}
+\newdimen\space@left
+\def\alarm#1{\typeout{******************************}%
+ \typeout{#1}%
+ \typeout{******************************}%
+}
+\long\def\getSpaceLeft{%\global\@twocolumnfalse%
+ \global\setbox0=\vbox{\hsize=\textwidth\MaketitleBox}%
+ \global\setbox1=\vbox{\hsize=\textwidth
+ \let\footnotetext\FNtext
+ \printFirstPageNotes}%
+ \xdef\noteheight{\the\ht1}%
+ \xdef\titleheight{\the\ht0}%
+ \@tempdima=\vsize
+ \advance\@tempdima-\noteheight
+ \advance\@tempdima-1\baselineskip
+}
+
+ \skip\footins=24pt
+
+\newbox\els@boxa
+\newbox\els@boxb
+
+\ifpreprint
+ \def\maketitle{\pprintMaketitle}
+ \else
+ \ifnum\jtype=1
+ \def\maketitle{%
+ \iflongmktitle\getSpaceLeft
+ \global\setbox\els@boxa=\vsplit0 to \@tempdima
+ \box\els@boxa\par\resetTitleCounters
+ \thispagestyle{pprintTitle}%
+ \printFirstPageNotes
+ \box0%
+ \else
+ \finalMaketitle\printFirstPageNotes
+ \fi
+ \gdef\thefootnote{\arabic{footnote}}}%
+ \else
+ \ifnum\jtype=5
+ \def\maketitle{%
+ \iflongmktitle\getSpaceLeft
+ \global\setbox\els@boxa=\vsplit0 to \@tempdima
+ \box\els@boxa\par\resetTitleCounters
+ \thispagestyle{pprintTitle}%
+ \printFirstPageNotes
+ \twocolumn[\box0]%\printFirstPageNotes
+ \else
+ \twocolumn[\finalMaketitle]\printFirstPageNotes
+ \fi
+ \gdef\thefootnote{\arabic{footnote}}}
+ \else
+ \if@twocolumn
+ \def\maketitle{%
+ \iflongmktitle\getSpaceLeft
+ \global\setbox\els@boxa=\vsplit0 to \@tempdima
+ \box\els@boxa\par\resetTitleCounters
+ \thispagestyle{pprintTitle}%
+ \printFirstPageNotes
+ \twocolumn[\box0]%
+ \else
+ \twocolumn[\finalMaketitle]\printFirstPageNotes
+ \fi
+ \gdef\thefootnote{\arabic{footnote}}}%
+ \else
+ \def\maketitle{%
+ \iflongmktitle\getSpaceLeft
+ \global\setbox\els@boxa=\vsplit0 to \@tempdima
+ \box\els@boxa\par\resetTitleCounters
+ \thispagestyle{pprintTitle}%
+ \printFirstPageNotes
+ \box0%
+ \else
+ \finalMaketitle\printFirstPageNotes
+ \fi
+ \gdef\thefootnote{\arabic{footnote}}}%
+ \fi
+ \fi
+ \fi
+\fi
+\def\ps@pprintTitle{%
+ \let\@oddhead\@empty
+ \let\@evenhead\@empty
+ \def\@oddfoot{\footnotesize\itshape
+ Preprint submitted to \ifx\@journal\@empty Elsevier
+ \else\@journal\fi\hfill\today}%
+ \let\@evenfoot\@oddfoot}
+\def\@seccntDot{.}
+\def\@seccntformat#1{\csname the#1\endcsname\@seccntDot\hskip 0.5em}
+
+\renewcommand\section{\@startsection {section}{1}{\z@}%
+ {18\p@ \@plus 6\p@ \@minus 3\p@}%
+ {9\p@ \@plus 6\p@ \@minus 3\p@}%
+ {\normalsize\bfseries\boldmath}}
+\renewcommand\subsection{\@startsection{subsection}{2}{\z@}%
+ {12\p@ \@plus 6\p@ \@minus 3\p@}%
+ {3\p@ \@plus 6\p@ \@minus 3\p@}%
+ {\normalfont\normalsize\itshape}}
+\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}%
+ {12\p@ \@plus 6\p@ \@minus 3\p@}%
+ {\p@}%
+ {\normalfont\normalsize\itshape}}
+
+\def\paragraph{\secdef{\els@aparagraph}{\els@bparagraph}}
+\def\els@aparagraph[#1]#2{\elsparagraph[#1]{#2.}}
+\def\els@bparagraph#1{\elsparagraph*{#1.}}
+
+\newcommand\elsparagraph{\@startsection{paragraph}{4}{0\z@}%
+ {10\p@ \@plus 6\p@ \@minus 3\p@}%
+ {-6\p@}%
+ {\normalfont\itshape}}
+\newdimen\leftMargin
+\leftMargin=2em
+\newtoks\@enLab %\newtoks\@enfont
+\def\@enQmark{?}
+\def\@enLabel#1#2{%
+ \edef\@enThe{\noexpand#1{\@enumctr}}%
+ \@enLab\expandafter{\the\@enLab\csname the\@enumctr\endcsname}%
+ \@enloop}
+\def\@enSpace{\afterassignment\@enSp@ce\let\@tempa= }
+\def\@enSp@ce{\@enLab\expandafter{\the\@enLab\space}\@enloop}
+\def\@enGroup#1{\@enLab\expandafter{\the\@enLab{#1}}\@enloop}
+\def\@enOther#1{\@enLab\expandafter{\the\@enLab#1}\@enloop}
+\def\@enloop{\futurelet\@entemp\@enloop@}
+\def\@enloop@{%
+ \ifx A\@entemp \def\@tempa{\@enLabel\Alph }\else
+ \ifx a\@entemp \def\@tempa{\@enLabel\alph }\else
+ \ifx i\@entemp \def\@tempa{\@enLabel\roman }\else
+ \ifx I\@entemp \def\@tempa{\@enLabel\Roman }\else
+ \ifx 1\@entemp \def\@tempa{\@enLabel\arabic}\else
+ \ifx \@sptoken\@entemp \let\@tempa\@enSpace \else
+ \ifx \bgroup\@entemp \let\@tempa\@enGroup \else
+ \ifx \@enum@\@entemp \let\@tempa\@gobble \else
+ \let\@tempa\@enOther
+ \fi\fi\fi\fi\fi\fi\fi\fi
+ \@tempa}
+\newlength{\@sep} \newlength{\@@sep}
+\setlength{\@sep}{.5\baselineskip plus.2\baselineskip
+ minus.2\baselineskip}
+\setlength{\@@sep}{.1\baselineskip plus.01\baselineskip
+ minus.05\baselineskip}
+\providecommand{\sfbc}{\rmfamily\upshape}
+\providecommand{\sfn}{\rmfamily\upshape}
+\def\@enfont{\ifnum \@enumdepth >1\let\@nxt\sfn \else\let\@nxt\sfbc \fi\@nxt}
+\def\enumerate{%
+ \ifnum \@enumdepth >3 \@toodeep\else
+ \advance\@enumdepth \@ne
+ \edef\@enumctr{enum\romannumeral\the\@enumdepth}\fi
+ \@ifnextchar[{\@@enum@}{\@enum@}}
+\def\@@enum@[#1]{%
+ \@enLab{}\let\@enThe\@enQmark
+ \@enloop#1\@enum@
+ \ifx\@enThe\@enQmark\@warning{The counter will not be printed.%
+ ^^J\space\@spaces\@spaces\@spaces The label is: \the\@enLab}\fi
+ \expandafter\edef\csname label\@enumctr\endcsname{\the\@enLab}%
+ \expandafter\let\csname the\@enumctr\endcsname\@enThe
+ \csname c@\@enumctr\endcsname7
+ \expandafter\settowidth
+ \csname leftmargin\romannumeral\@enumdepth\endcsname
+ {\the\@enLab\hskip\labelsep}%
+ \@enum@}
+\def\@enum@{\list{{\@enfont\csname label\@enumctr\endcsname}}%
+ {\usecounter{\@enumctr}\def\makelabel##1{\hss\llap{##1}}%
+ \ifnum \@enumdepth>1\setlength{\topsep}{\@@sep}\else
+ \setlength{\topsep}{\@sep}\fi
+ \ifnum \@enumdepth>1\setlength{\itemsep}{0pt plus1pt minus1pt}%
+ \else \setlength{\itemsep}{\@@sep}\fi
+ %\setlength\leftmargin{\leftMargin}%%%{1.8em}
+ \setlength{\parsep}{0pt plus1pt minus1pt}%
+ \setlength{\parskip}{0pt plus1pt minus1pt}
+ }}
+
+\def\endenumerate{\par\ifnum \@enumdepth >1\addvspace{\@@sep}\else
+ \addvspace{\@sep}\fi \endlist}
+
+\def\sitem{\@noitemargtrue\@item[\@itemlabel *]}
+
+\def\itemize{\@ifnextchar[{\@Itemize}{\@Itemize[]}}
+
+\def\@Itemize[#1]{\def\next{#1}%
+ \ifnum \@itemdepth >\thr@@\@toodeep\else
+ \advance\@itemdepth\@ne
+ \ifx\next\@empty\else\expandafter\def\csname
+ labelitem\romannumeral\the\@itemdepth\endcsname{#1}\fi%
+ \edef\@itemitem{labelitem\romannumeral\the\@itemdepth}%
+ \expandafter\list\csname\@itemitem\endcsname
+ {\def\makelabel##1{\hss\llap{##1}}}%
+ \fi}
+\def\newdefinition#1{%
+ \@ifnextchar[{\@odfn{#1}}{\@ndfn{#1}}}%]
+\def\@ndfn#1#2{%
+ \@ifnextchar[{\@xndfn{#1}{#2}}{\@yndfn{#1}{#2}}}
+\def\@xndfn#1#2[#3]{%
+ \expandafter\@ifdefinable\csname #1\endcsname
+ {\@definecounter{#1}\@newctr{#1}[#3]%
+ \expandafter\xdef\csname the#1\endcsname{%
+ \expandafter\noexpand\csname the#3\endcsname \@dfncountersep
+ \@dfncounter{#1}}%
+ \global\@namedef{#1}{\@dfn{#1}{#2}}%
+ \global\@namedef{end#1}{\@enddefinition}}}
+\def\@yndfn#1#2{%
+ \expandafter\@ifdefinable\csname #1\endcsname
+ {\@definecounter{#1}%
+ \expandafter\xdef\csname the#1\endcsname{\@dfncounter{#1}}%
+ \global\@namedef{#1}{\@dfn{#1}{#2}}%
+ \global\@namedef{end#1}{\@enddefinition}}}
+\def\@odfn#1[#2]#3{%
+ \@ifundefined{c@#2}{\@nocounterr{#2}}%
+ {\expandafter\@ifdefinable\csname #1\endcsname
+ {\global\@namedef{the#1}{\@nameuse{the#2}}
+ \global\@namedef{#1}{\@dfn{#2}{#3}}%
+ \global\@namedef{end#1}{\@enddefinition}}}}
+\def\@dfn#1#2{%
+ \refstepcounter{#1}%
+ \@ifnextchar[{\@ydfn{#1}{#2}}{\@xdfn{#1}{#2}}}
+\def\@xdfn#1#2{%
+ \@begindefinition{#2}{\csname the#1\endcsname}\ignorespaces}
+\def\@ydfn#1#2[#3]{%
+ \@opargbegindefinition{#2}{\csname the#1\endcsname}{#3}\ignorespaces}
+\def\@dfncounter#1{\noexpand\arabic{#1}}
+\def\@dfncountersep{.}
+\def\@begindefinition#1#2{\trivlist
+ \item[\hskip\labelsep{\bfseries #1\ #2.}]\upshape}
+\def\@opargbegindefinition#1#2#3{\trivlist
+ \item[\hskip\labelsep{\bfseries #1\ #2\ (#3).}]\upshape}
+\def\@enddefinition{\endtrivlist}
+
+\def\@begintheorem#1#2{\trivlist
+ \let\baselinestretch\@blstr
+ \item[\hskip \labelsep{\bfseries #1\ #2.}]\itshape}
+\def\@opargbegintheorem#1#2#3{\trivlist
+ \let\baselinestretch\@blstr
+ \item[\hskip \labelsep{\bfseries #1\ #2\ (#3).}]\itshape}
+
+\def\newproof#1{%
+ \@ifnextchar[{\@oprf{#1}}{\@nprf{#1}}}
+\def\@nprf#1#2{%
+ \@ifnextchar[{\@xnprf{#1}{#2}}{\@ynprf{#1}{#2}}}
+\def\@xnprf#1#2[#3]{%
+ \expandafter\@ifdefinable\csname #1\endcsname
+ {\@definecounter{#1}\@newctr{#1}[#3]%
+ \expandafter\xdef\csname the#1\endcsname{%
+ \expandafter\noexpand\csname the#3\endcsname \@prfcountersep
+ \@prfcounter{#1}}%
+ \global\@namedef{#1}{\@prf{#1}{#2}}%
+ \global\@namedef{end#1}{\@endproof}}}
+\def\@ynprf#1#2{%
+ \expandafter\@ifdefinable\csname #1\endcsname
+ {\@definecounter{#1}%
+ \expandafter\xdef\csname the#1\endcsname{\@prfcounter{#1}}%
+ \global\@namedef{#1}{\@prf{#1}{#2}}%
+ \global\@namedef{end#1}{\@endproof}}}
+\def\@oprf#1[#2]#3{%
+ \@ifundefined{c@#2}{\@nocounterr{#2}}%
+ {\expandafter\@ifdefinable\csname #1\endcsname
+ {\global\@namedef{the#1}{\@nameuse{the#2}}%
+ \global\@namedef{#1}{\@prf{#2}{#3}}%
+ \global\@namedef{end#1}{\@endproof}}}}
+\def\@prf#1#2{%
+ \refstepcounter{#1}%
+ \@ifnextchar[{\@yprf{#1}{#2}}{\@xprf{#1}{#2}}}
+\def\@xprf#1#2{%
+ \@beginproof{#2}{\csname the#1\endcsname}\ignorespaces}
+\def\@yprf#1#2[#3]{%
+ \@opargbeginproof{#2}{\csname the#1\endcsname}{#3}\ignorespaces}
+\def\@prfcounter#1{\noexpand\arabic{#1}}
+\def\@prfcountersep{.}
+\def\@beginproof#1#2{\trivlist\let\baselinestretch\@blstr
+ \item[\hskip \labelsep{\scshape #1.}]\rmfamily}
+\def\@opargbeginproof#1#2#3{\trivlist\let\baselinestretch\@blstr
+ \item[\hskip \labelsep{\scshape #1\ (#3).}]\rmfamily}
+\def\@endproof{\endtrivlist}
+\newcommand*{\qed}{\hbox{}\hfill$\Box$}
+
+\@ifundefined{@biboptions}{\xdef\@biboptions{numbers}}{}
+\InputIfFileExists{\jobname.spl}{}{}
+\RequirePackage[\@biboptions]{natbib}
+
+\newwrite\splwrite
+\immediate\openout\splwrite=\jobname.spl
+\def\biboptions#1{\def\next{#1}\immediate\write\splwrite{%
+ \string\g@addto@macro\string\@biboptions{%
+ ,\expandafter\strip@prefix\meaning\next}}}
+
+\let\baselinestretch=\@blstr
+
+\ifnum\jtype=1
+ \RequirePackage{geometry}
+ \geometry{twoside,
+ paperwidth=210mm,
+ paperheight=297mm,
+ textheight=562pt,
+ textwidth=384pt,
+ centering,
+ headheight=50pt,
+ headsep=12pt,
+ footskip=12pt,
+ footnotesep=24pt plus 2pt minus 12pt,
+ }
+ \global\let\bibfont=\footnotesize
+ \global\bibsep=0pt
+ \if@twocolumn\global\@twocolumnfalse\fi
+\else\ifnum\jtype=3
+ \RequirePackage{geometry}
+ \geometry{twoside,
+ paperwidth=210mm,
+ paperheight=297mm,
+ textheight=622pt,
+ textwidth=468pt,
+ centering,
+ headheight=50pt,
+ headsep=12pt,
+ footskip=18pt,
+ footnotesep=24pt plus 2pt minus 12pt,
+ columnsep=2pc
+ }
+ \global\let\bibfont=\footnotesize
+ \global\bibsep=0pt
+ \if@twocolumn\input{fleqn.clo}\fi
+\else\ifnum\jtype=5
+ \RequirePackage{geometry}
+ \geometry{twoside,
+ paperwidth=210mm,
+ paperheight=297mm,
+ textheight=682pt,
+ textwidth=522pt,
+ centering,
+ headheight=50pt,
+ headsep=12pt,
+ footskip=18pt,
+ footnotesep=24pt plus 2pt minus 12pt,
+ columnsep=18pt
+ }%
+ \global\let\bibfont=\footnotesize
+ \global\bibsep=0pt
+ \input{fleqn.clo}
+ \global\@twocolumntrue
+%%
+%% End of option '5p'
+%%
+\fi\fi\fi
+\def\journal#1{\gdef\@journal{#1}}
+ \let\@journal\@empty
+\newenvironment{frontmatter}{}{\maketitle}
+
+\long\def\@makecaption#1#2{%
+ \vskip\abovecaptionskip\footnotesize
+ \sbox\@tempboxa{#1: #2}%
+ \ifdim \wd\@tempboxa >\hsize
+ #1: #2\par
+ \else
+ \global \@minipagefalse
+ \hb@xt@\hsize{\hfil\box\@tempboxa\hfil}%
+ \fi
+ \vskip\belowcaptionskip}
+
+\AtBeginDocument{\@ifpackageloaded{hyperref}
+ {\def\@linkcolor{blue}
+ \def\@anchorcolor{blue}
+ \def\@citecolor{blue}
+ \def\@filecolor{blue}
+ \def\@urlcolor{blue}
+ \def\@menucolor{blue}
+ \def\@pagecolor{blue}
+\begingroup
+ \@makeother\`%
+ \@makeother\=%
+ \edef\x{%
+ \edef\noexpand\x{%
+ \endgroup
+ \noexpand\toks@{%
+ \catcode 96=\noexpand\the\catcode`\noexpand\`\relax
+ \catcode 61=\noexpand\the\catcode`\noexpand\=\relax
+ }%
+ }%
+ \noexpand\x
+ }%
+\x
+\@makeother\`
+\@makeother\=
+}{}}
+%%
+\def\appendixname{Appendix }
+\renewcommand\appendix{\par
+ \setcounter{section}{0}%
+ \setcounter{subsection}{0}%
+ \setcounter{equation}{0}
+ \gdef\thefigure{\@Alph\c@section.\arabic{figure}}%
+ \gdef\thetable{\@Alph\c@section.\arabic{table}}%
+ \gdef\thesection{\appendixname~\@Alph\c@section}%
+ \@addtoreset{equation}{section}%
+ \gdef\theequation{\@Alph\c@section.\arabic{equation}}%
+ \addtocontents{toc}{\string\let\string\numberline\string\tmptocnumberline}{}{}
+}
+
+%%%% \numberline width calculation for appendix.
+\newdimen\appnamewidth
+\def\tmptocnumberline#1{%
+ \setbox0=\hbox{\appendixname}
+ \appnamewidth=\wd0
+ \addtolength\appnamewidth{2.5pc}
+ \hb@xt@\appnamewidth{#1\hfill}
+}
+
+%% Added for work with amsrefs.sty
+
+\@ifpackageloaded{amsrefs}%
+ {}
+ {%\let\bibsection\relax%
+ \AtBeginDocument{\def\cites@b#1#2,#3{%
+ \begingroup[%
+ \toks@{\InnerCite{#2}#1}%
+ \ifx\@empty#3\@xp\@gobble\fi
+ \cites@c#3%
+}}}
+%%
+%% Added for avoiding clash with cleveref.sty
+
+\@ifpackageloaded{cleveref}%
+ {}
+ {\def\tnotetext[#1]#2{\g@addto@macro\@tnotes{%
+ \refstepcounter{tnote}%
+ \immediate\write\@auxout{\string\Newlabel{#1}{\thetnote}}
+ \def\thefootnote{\ifcase\c@tnote\or\ding{73}\or\ding{73}\ding{73}\fi}%
+ \footnotetext{#2}}}
+%%%
+ \def\fntext[#1]#2{\g@addto@macro\@fnotes{%
+ \refstepcounter{fnote}%
+ \immediate\write\@auxout{\string\Newlabel{#1}{\thefnote}}
+ \def\thefootnote{\thefnote}%
+ \global\setcounter{footnote}{\thefnote}%
+ \footnotetext{#2}}}
+%%%
+ \def\cortext[#1]#2{\g@addto@macro\@cornotes{%
+ \refstepcounter{cnote}%
+ \immediate\write\@auxout{\string\Newlabel{#1}{\thecnote}}
+ \def\thefootnote{\ifcase\thecnote\or$\ast$\or
+ $\ast\ast$\fi}%
+ \footnotetext{#2}}}
+}
+\endinput
+%%
+%% End of file `elsarticle.cls'.
diff --git a/Paper/v1.1/epj.bst b/Paper/v1.1/epj.bst
new file mode 100755
index 0000000000000000000000000000000000000000..92ead5a826daa1592694ce425736e528ae104c07
--- /dev/null
+++ b/Paper/v1.1/epj.bst
@@ -0,0 +1,2376 @@
+%%
+%% This is file ``epj.bst'' derived from ``apsrev'',
+%% generated with the docstrip utility.
+%%
+%% The original source files were:
+%%
+%% merlin.mbs (with options: `head,ay,nat,seq-no,nm-init,ed-au,nmlm,x10,x0,m10,m0,nmft,nmft-def,fnm-def,nmfted,lab,lab-def,keyxyr,yr-par,note-yr,tit-it,jtit-x,jttl-rm,thtit-x,trtit-x,vol-bf,vnum-x,volp-com,jpg-1,num-xser,jnm-x,pub-date,pre-pub,isbn,issn,edby,blk-com,pp,ed,abr,ednx,ord,jabr,revdata,eprint,url,url-blk,misc-simple,bibinfo,nfss,{}')
+%% physjour.mbs (with options: `ay,nat,seq-no,nm-init,ed-au,nmlm,x10,x0,m10,m0,nmft,nmft-def,fnm-def,nmfted,lab,lab-def,keyxyr,yr-par,note-yr,tit-it,jtit-x,jttl-rm,thtit-x,trtit-x,vol-bf,vnum-x,volp-com,jpg-1,num-xser,jnm-x,pub-date,pre-pub,isbn,issn,edby,blk-com,pp,ed,abr,ednx,ord,jabr,revdata,eprint,url,url-blk,misc-simple,bibinfo,nfss,{}')
+%% geojour.mbs (with options: `ay,nat,seq-no,nm-init,ed-au,nmlm,x10,x0,m10,m0,nmft,nmft-def,fnm-def,nmfted,lab,lab-def,keyxyr,yr-par,note-yr,tit-it,jtit-x,jttl-rm,thtit-x,trtit-x,vol-bf,vnum-x,volp-com,jpg-1,num-xser,jnm-x,pub-date,pre-pub,isbn,issn,edby,blk-com,pp,ed,abr,ednx,ord,jabr,revdata,eprint,url,url-blk,misc-simple,bibinfo,nfss,{}')
+%% photjour.mbs (with options: `ay,nat,seq-no,nm-init,ed-au,nmlm,x10,x0,m10,m0,nmft,nmft-def,fnm-def,nmfted,lab,lab-def,keyxyr,yr-par,note-yr,tit-it,jtit-x,jttl-rm,thtit-x,trtit-x,vol-bf,vnum-x,volp-com,jpg-1,num-xser,jnm-x,pub-date,pre-pub,isbn,issn,edby,blk-com,pp,ed,abr,ednx,ord,jabr,revdata,eprint,url,url-blk,misc-simple,bibinfo,nfss,{}')
+%% merlin.mbs (with options: `tail,ay,nat,seq-no,nm-init,ed-au,nmlm,x10,x0,m10,m0,nmft,nmft-def,fnm-def,nmfted,lab,lab-def,keyxyr,yr-par,note-yr,tit-it,jtit-x,jttl-rm,thtit-x,trtit-x,vol-bf,vnum-x,volp-com,jpg-1,num-xser,jnm-x,pub-date,pre-pub,isbn,issn,edby,blk-com,pp,ed,abr,ednx,ord,jabr,revdata,eprint,url,url-blk,misc-simple,bibinfo,nfss,{}')
+%% ----------------------------------------
+%% *** Bibliographic Style for The European Physical Journal ***
+%%
+%% Copyright 2005 (EDP Sciences)
+ % ===============================================================
+ % IMPORTANT NOTICE:
+ % This bibliographic style (bst) file has been generated from one or
+ % more master bibliographic style (mbs) files, listed above.
+ %
+ % This generated file can be redistributed and/or modified under the terms
+ % of the LaTeX Project Public License Distributed from CTAN
+ % archives in directory macros/latex/base/lppl.txt; either
+ % version 1 of the License, or any later version.
+ % ===============================================================
+ % Name and version information of the main mbs file:
+ % \ProvidesFile{merlin.mbs}[2000/05/04 4.01 (PWD, AO, DPC)]
+ % For use with BibTeX version 0.99a or later
+ %-------------------------------------------------------------------
+ % This bibliography style file is intended for texts in ENGLISH
+ % This is an author-year citation style bibliography. As such, it is
+ % non-standard LaTeX, and requires a special package file to function properly.
+ % Such a package is natbib.sty by Patrick W. Daly
+ % The form of the \bibitem entries is
+ % \bibitem[Jones et al.(1990)]{key}...
+ % \bibitem[Jones et al.(1990)Jones, Baker, and Smith]{key}...
+ % The essential feature is that the label (the part in brackets) consists
+ % of the author names, as they should appear in the citation, with the year
+ % in parentheses following. There must be no space before the opening
+ % parenthesis!
+ % With natbib v5.3, a full list of authors may also follow the year.
+ % In natbib.sty, it is possible to define the type of enclosures that is
+ % really wanted (brackets or parentheses), but in either case, there must
+ % be parentheses in the label.
+ % The \cite command functions as follows:
+ % \citet{key} ==>> Jones et al. (1990)
+ % \citet*{key} ==>> Jones, Baker, and Smith (1990)
+ % \citep{key} ==>> (Jones et al., 1990)
+ % \citep*{key} ==>> (Jones, Baker, and Smith, 1990)
+ % \citep[chap. 2]{key} ==>> (Jones et al., 1990, chap. 2)
+ % \citep[e.g.][]{key} ==>> (e.g. Jones et al., 1990)
+ % \citep[e.g.][p. 32]{key} ==>> (e.g. Jones et al., p. 32)
+ % \citeauthor{key} ==>> Jones et al.
+ % \citeauthor*{key} ==>> Jones, Baker, and Smith
+ % \citeyear{key} ==>> 1990
+ %---------------------------------------------------------------------
+
+ENTRY
+ { address
+ archive
+ author
+ booktitle
+ chapter
+ collaboration
+ edition
+ editor
+ eid
+ eprint
+ howpublished
+ institution
+ isbn
+ issn
+ journal
+ key
+ month
+ note
+ number
+ numpages
+ organization
+ pages
+ publisher
+ school
+ series
+ title
+ type
+ url
+ volume
+ year
+ }
+ {}
+ { label extra.label sort.label short.list }
+INTEGERS { output.state before.all mid.sentence after.sentence after.block }
+FUNCTION {init.state.consts}
+{ #0 'before.all :=
+ #1 'mid.sentence :=
+ #2 'after.sentence :=
+ #3 'after.block :=
+}
+STRINGS { s t}
+FUNCTION {output.nonnull}
+{ 's :=
+ output.state mid.sentence =
+ { ", " * write$ }
+ { output.state after.block =
+ { add.period$ write$
+ newline$
+ "\newblock " write$
+ }
+ { output.state before.all =
+ 'write$
+ { add.period$ " " * write$ }
+ if$
+ }
+ if$
+ mid.sentence 'output.state :=
+ }
+ if$
+ s
+}
+FUNCTION {output}
+{ duplicate$ empty$
+ 'pop$
+ 'output.nonnull
+ if$
+}
+FUNCTION {output.check}
+{ 't :=
+ duplicate$ empty$
+ { pop$ "empty " t * " in " * cite$ * warning$ }
+ 'output.nonnull
+ if$
+}
+FUNCTION {fin.entry}
+{ write$
+ newline$
+}
+
+FUNCTION {new.block}
+{ output.state before.all =
+ 'skip$
+ { after.block 'output.state := }
+ if$
+}
+FUNCTION {new.sentence}
+{ output.state after.block =
+ 'skip$
+ { output.state before.all =
+ 'skip$
+ { after.sentence 'output.state := }
+ if$
+ }
+ if$
+}
+FUNCTION {add.blank}
+{ " " * before.all 'output.state :=
+}
+
+FUNCTION {date.block}
+{
+ skip$
+}
+
+FUNCTION {not}
+{ { #0 }
+ { #1 }
+ if$
+}
+FUNCTION {and}
+{ 'skip$
+ { pop$ #0 }
+ if$
+}
+FUNCTION {or}
+{ { pop$ #1 }
+ 'skip$
+ if$
+}
+FUNCTION {new.block.checkb}
+{ empty$
+ swap$ empty$
+ and
+ 'skip$
+ 'new.block
+ if$
+}
+FUNCTION {field.or.null}
+{ duplicate$ empty$
+ { pop$ "" }
+ 'skip$
+ if$
+}
+FUNCTION {emphasize}
+{ duplicate$ empty$
+ { pop$ "" }
+ { "\emph{" swap$ * "}" * }
+ if$
+}
+FUNCTION {bolden}
+{ duplicate$ empty$
+ { pop$ "" }
+ { "\textbf{" swap$ * "}" * }
+ if$
+}
+FUNCTION {bib.name.font}
+{ duplicate$ empty$
+ { pop$ "" }
+ { "" swap$ * "" * }
+ if$
+}
+FUNCTION {bib.fname.font}
+{ duplicate$ empty$
+ { pop$ "" }
+ { "" swap$ * "" * }
+ if$
+}
+FUNCTION {cite.name.font}
+{ duplicate$ empty$
+ { pop$ "" }
+ { "{" swap$ * "}" * }
+ if$
+}
+FUNCTION {tie.or.space.prefix}
+{ duplicate$ text.length$ #3 <
+ { "~" }
+ { " " }
+ if$
+ swap$
+}
+
+FUNCTION {capitalize}
+{ "u" change.case$ "t" change.case$ }
+
+FUNCTION {space.word}
+{ " " swap$ * " " * }
+ % Here are the language-specific definitions for explicit words.
+ % Each function has a name bbl.xxx where xxx is the English word.
+ % The language selected here is ENGLISH
+FUNCTION {bbl.and}
+{ ","}
+
+FUNCTION {bbl.etal}
+{ "et~al." }
+
+FUNCTION {bbl.editors}
+{ "eds." }
+
+FUNCTION {bbl.editor}
+{ "ed." }
+
+FUNCTION {bbl.edby}
+{ "edited by" }
+
+FUNCTION {bbl.edition}
+{ "edn." }
+
+FUNCTION {bbl.volume}
+{ "Vol." }
+
+FUNCTION {bbl.of}
+{ "of" }
+
+FUNCTION {bbl.number}
+{ "Number" }
+
+FUNCTION {bbl.nr}
+{ "no." }
+
+FUNCTION {bbl.in}
+{ "in" }
+
+FUNCTION {bbl.pages}
+{ "pp." }
+
+FUNCTION {bbl.page}
+{ "p." }
+
+FUNCTION {bbl.eidpp}
+{ "" }
+
+FUNCTION {bbl.chapter}
+{ "chap." }
+
+FUNCTION {bbl.techrep}
+{ "Tech. Rep." }
+
+FUNCTION {bbl.mthesis}
+{ "Master's thesis" }
+
+FUNCTION {bbl.phdthesis}
+{ "Ph.D. thesis" }
+
+FUNCTION {bbl.first}
+{ "1st" }
+
+FUNCTION {bbl.second}
+{ "2nd" }
+
+FUNCTION {bbl.third}
+{ "3rd" }
+
+FUNCTION {bbl.fourth}
+{ "4th" }
+
+FUNCTION {bbl.fifth}
+{ "5th" }
+
+FUNCTION {bbl.st}
+{ "st" }
+
+FUNCTION {bbl.nd}
+{ "nd" }
+
+FUNCTION {bbl.rd}
+{ "rd" }
+
+FUNCTION {bbl.th}
+{ "th" }
+
+MACRO {jan} {"Jan."}
+
+MACRO {feb} {"Feb."}
+
+MACRO {mar} {"Mar."}
+
+MACRO {apr} {"Apr."}
+
+MACRO {may} {"May"}
+
+MACRO {jun} {"Jun."}
+
+MACRO {jul} {"Jul."}
+
+MACRO {aug} {"Aug."}
+
+MACRO {sep} {"Sep."}
+
+MACRO {oct} {"Oct."}
+
+MACRO {nov} {"Nov."}
+
+MACRO {dec} {"Dec."}
+
+FUNCTION {eng.ord}
+{ duplicate$ "1" swap$ *
+ #-2 #1 substring$ "1" =
+ { bbl.th * }
+ { duplicate$ #-1 #1 substring$
+ duplicate$ "1" =
+ { pop$ bbl.st * }
+ { duplicate$ "2" =
+ { pop$ bbl.nd * }
+ { "3" =
+ { bbl.rd * }
+ { bbl.th * }
+ if$
+ }
+ if$
+ }
+ if$
+ }
+ if$
+}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ %-------------------------------------------------------------------
+
+ % Begin module:
+
+
+ % \ProvidesFile{physjour.mbs}[1999/10/05 2.1 (PWD)]
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
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+
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+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+MACRO {aa}{"Astron. \& Astrophys."}
+
+
+MACRO {aasup}{"Astron. \& Astrophys. Suppl. Ser."}
+
+
+MACRO {aj} {"Astron. J."}
+
+
+MACRO {aph} {"Acta Phys."}
+
+MACRO {advp} {"Adv. Phys."}
+
+MACRO {ajp} {"Amer. J. Phys."}
+
+MACRO {ajm} {"Amer. J. Math."}
+
+MACRO {amsci} {"Amer. Sci."}
+
+MACRO {anofd} {"Ann. Fluid Dyn."}
+
+MACRO {am} {"Ann. Math."}
+
+MACRO {ap} {"Ann. Phys. (NY)"}
+
+MACRO {adp} {"Ann. Phys. (Leipzig)"}
+
+MACRO {ao} {"Appl. Opt."}
+
+MACRO {apl} {"Appl. Phys. Lett."}
+
+MACRO {app} {"Astroparticle Phys."}
+
+MACRO {apj} {"Astrophys. J."}
+
+
+MACRO {apjsup} {"Astrophys. J. Suppl."}
+
+
+MACRO {apss} {"Astrophys. Space Sci."}
+
+
+MACRO {araa} {"Ann. Rev. Astron. Astrophys."}
+
+
+MACRO {baas} {"Bull. Amer. Astron. Soc."}
+
+
+MACRO {baps} {"Bull. Amer. Phys. Soc."}
+
+MACRO {cmp} {"Comm. Math. Phys."}
+
+MACRO {cpam} {"Commun. Pure Appl. Math."}
+
+MACRO {cppcf} {"Comm. Plasma Phys. \& Controlled Fusion"}
+
+MACRO {cpc} {"Comp. Phys. Comm."}
+
+MACRO {cqg} {"Class. Quant. Grav."}
+
+MACRO {cra} {"C. R. Acad. Sci. A"}
+
+MACRO {fed} {"Fusion Eng. \& Design"}
+
+MACRO {ft} {"Fusion Tech."}
+
+MACRO {grg} {"Gen. Relativ. Gravit."}
+
+MACRO {ieeens} {"IEEE Trans. Nucl. Sci."}
+
+MACRO {ieeeps} {"IEEE Trans. Plasma Sci."}
+
+MACRO {ijimw} {"Interntl. J. Infrared \& Millimeter Waves"}
+
+MACRO {ip} {"Infrared Phys."}
+
+MACRO {irp} {"Infrared Phys."}
+
+MACRO {jap} {"J. Appl. Phys."}
+
+MACRO {jasa} {"J. Acoust. Soc. America"}
+
+MACRO {jcp} {"J. Comp. Phys."}
+
+MACRO {jetp} {"Sov. Phys.--JETP"}
+
+MACRO {jfe} {"J. Fusion Energy"}
+
+MACRO {jfm} {"J. Fluid Mech."}
+
+MACRO {jmp} {"J. Math. Phys."}
+
+MACRO {jne} {"J. Nucl. Energy"}
+
+MACRO {jnec} {"J. Nucl. Energy, C: Plasma Phys., Accelerators, Thermonucl. Res."}
+
+MACRO {jnm} {"J. Nucl. Mat."}
+
+MACRO {jpc} {"J. Phys. Chem."}
+
+MACRO {jpp} {"J. Plasma Phys."}
+
+MACRO {jpsj} {"J. Phys. Soc. Jpn"}
+
+MACRO {jsi} {"J. Sci. Instrum."}
+
+MACRO {jvst} {"J. Vac. Sci. \& Tech."}
+
+MACRO {nat} {"Nature"}
+
+MACRO {nature} {"Nature"}
+
+
+
+MACRO {nedf} {"Nucl. Eng. \& Design/Fusion"}
+
+MACRO {nf} {"Nucl. Fusion"}
+
+MACRO {nim} {"Nucl. Inst. \& Meth."}
+
+MACRO {nimpr} {"Nucl. Inst. \& Meth. in Phys. Res."}
+
+MACRO {np} {"Nucl. Phys."}
+
+MACRO {npb} {"Nucl. Phys. B"}
+
+MACRO {nt/f} {"Nucl. Tech./Fusion"}
+
+MACRO {npbpc} {"Nucl. Phys. B (Proc. Suppl.)"}
+
+MACRO {inc} {"Nuovo Cimento"}
+
+MACRO {nc} {"Nuovo Cimento"}
+
+MACRO {pf} {"Phys. Fluids"}
+
+MACRO {pfa} {"Phys. Fluids A: Fluid Dyn."}
+
+MACRO {pfb} {"Phys. Fluids B: Plasma Phys."}
+
+MACRO {pl} {"Phys. Lett."}
+
+MACRO {pla} {"Phys. Lett. A"}
+
+MACRO {plb} {"Phys. Lett. B"}
+
+MACRO {prep} {"Phys. Rep."}
+
+MACRO {pnas} {"Proc. Nat. Acad. Sci. USA"}
+
+MACRO {pp} {"Phys. Plasmas"}
+
+MACRO {ppcf} {"Plasma Phys. \& Controlled Fusion"}
+
+MACRO {phitrsl} {"Philos. Trans. Roy. Soc. London"}
+
+MACRO {prl} {"Phys. Rev. Lett."}
+
+MACRO {pr} {"Phys. Rev."}
+
+MACRO {physrev} {"Phys. Rev."}
+
+MACRO {pra} {"Phys. Rev. A"}
+
+MACRO {prb} {"Phys. Rev. B"}
+
+MACRO {prc} {"Phys. Rev. C"}
+
+MACRO {prd} {"Phys. Rev. D"}
+
+MACRO {pre} {"Phys. Rev. E"}
+
+MACRO {ps} {"Phys. Scripta"}
+
+MACRO {procrsl} {"Proc. Roy. Soc. London"}
+
+MACRO {rmp} {"Rev. Mod. Phys."}
+
+MACRO {rsi} {"Rev. Sci. Inst."}
+
+MACRO {science} {"Science"}
+
+
+MACRO {sciam} {"Sci. Am."}
+
+MACRO {sam} {"Stud. Appl. Math."}
+
+MACRO {sjpp} {"Sov. J. Plasma Phys."}
+
+MACRO {spd} {"Sov. Phys.--Doklady"}
+
+MACRO {sptp} {"Sov. Phys.--Tech. Phys."}
+
+MACRO {spu} {"Sov. Phys.--Uspeki"}
+
+MACRO {st} {"Sky and Telesc."}
+
+
+
+
+
+ % End module: physjour.mbs
+
+
+
+ %-------------------------------------------------------------------
+ % Begin module:
+ % \ProvidesFile{geojour.mbs}[1999/11/16 2.0g (PWD)]
+MACRO {aisr} {"Adv. Space Res."}
+MACRO {ag} {"Ann. Geophys."}
+MACRO {anigeo} {"Ann. Geofis."}
+MACRO {angl} {"Ann. Glaciol."}
+MACRO {andmet} {"Ann. d. Meteor."}
+MACRO {andgeo} {"Ann. d. Geophys."}
+MACRO {andphy} {"Ann. Phys.-Paris"}
+MACRO {afmgb} {"Arch. Meteor. Geophys. Bioklimatol."}
+MACRO {atph} {"Atm\'osphera"}
+MACRO {aao} {"Atmos. Ocean"}
+MACRO {ass}{"Astrophys. Space Sci."}
+MACRO {atenv} {"Atmos. Environ."}
+MACRO {aujag} {"Aust. J. Agr. Res."}
+MACRO {aumet} {"Aust. Meteorol. Mag."}
+MACRO {blmet} {"Bound.-Lay. Meteorol."}
+MACRO {bams} {"Bull. Amer. Meteorol. Soc."}
+MACRO {cch} {"Clim. Change"}
+MACRO {cdyn} {"Clim. Dynam."}
+MACRO {cbul} {"Climatol. Bull."}
+MACRO {cap} {"Contrib. Atmos. Phys."}
+MACRO {dsr} {"Deep-Sea Res."}
+MACRO {dhz} {"Dtsch. Hydrogr. Z."}
+MACRO {dao} {"Dynam. Atmos. Oceans"}
+MACRO {eco} {"Ecology"}
+MACRO {empl}{"Earth, Moon and Planets"}
+MACRO {envres} {"Environ. Res."}
+MACRO {envst} {"Environ. Sci. Technol."}
+MACRO {ecms} {"Estuarine Coastal Mar. Sci."}
+MACRO {expa}{"Exper. Astron."}
+MACRO {geoint} {"Geofis. Int."}
+MACRO {geopub} {"Geofys. Publ."}
+MACRO {geogeo} {"Geol. Geofiz."}
+MACRO {gafd} {"Geophys. Astrophys. Fluid Dyn."}
+MACRO {gfd} {"Geophys. Fluid Dyn."}
+MACRO {geomag} {"Geophys. Mag."}
+MACRO {georl} {"Geophys. Res. Lett."}
+MACRO {grl} {"Geophys. Res. Lett."}
+MACRO {ga} {"Geophysica"}
+MACRO {gs} {"Geophysics"}
+MACRO {ieeetap} {"IEEE Trans. Antenn. Propag."}
+MACRO {ijawp} {"Int. J. Air Water Pollut."}
+MACRO {ijc} {"Int. J. Climatol."}
+MACRO {ijrs} {"Int. J. Remote Sens."}
+MACRO {jam} {"J. Appl. Meteorol."}
+MACRO {jaot} {"J. Atmos. Ocean. Technol."}
+MACRO {jatp} {"J. Atmos. Terr. Phys."}
+MACRO {jce} {"J. Climate"}
+MACRO {jcam} {"J. Climate Appl. Meteor."}
+MACRO {jcm} {"J. Climate Meteor."}
+MACRO {jcy} {"J. Climatol."}
+MACRO {jgr} {"J. Geophys. Res."}
+MACRO {jga} {"J. Glaciol."}
+MACRO {jh} {"J. Hydrol."}
+MACRO {jmr} {"J. Mar. Res."}
+MACRO {jmrj} {"J. Meteor. Res. Japan"}
+MACRO {jm} {"J. Meteor."}
+MACRO {jpo} {"J. Phys. Oceanogr."}
+MACRO {jra} {"J. Rech. Atmos."}
+MACRO {jaes} {"J. Aeronaut. Sci."}
+MACRO {japca} {"J. Air Pollut. Control Assoc."}
+MACRO {jas} {"J. Atmos. Sci."}
+MACRO {jmts} {"J. Mar. Technol. Soc."}
+MACRO {jmsj} {"J. Meteorol. Soc. Japan"}
+MACRO {josj} {"J. Oceanogr. Soc. Japan"}
+MACRO {jwm} {"J. Wea. Mod."}
+MACRO {lao} {"Limnol. Oceanogr."}
+MACRO {mwl} {"Mar. Wea. Log"}
+MACRO {mau} {"Mausam"}
+MACRO {meteor} {"``Meteor'' Forschungsergeb."}
+MACRO {map} {"Meteorol. Atmos. Phys."}
+MACRO {metmag} {"Meteor. Mag."}
+MACRO {metmon} {"Meteor. Monogr."}
+MACRO {metrun} {"Meteor. Rundsch."}
+MACRO {metzeit} {"Meteor. Z."}
+MACRO {metgid} {"Meteor. Gidrol."}
+MACRO {mwr} {"Mon. Weather Rev."}
+MACRO {nwd} {"Natl. Weather Dig."}
+MACRO {nzjmfr} {"New Zeal. J. Mar. Freshwater Res."}
+MACRO {npg} {"Nonlin. Proc. Geophys."}
+MACRO {om} {"Oceanogr. Meteorol."}
+MACRO {ocac} {"Oceanol. Acta"}
+MACRO {oceanus} {"Oceanus"}
+MACRO {paleoc} {"Paleoceanography"}
+MACRO {pce} {"Phys. Chem. Earth"}
+MACRO {pmg} {"Pap. Meteor. Geophys."}
+MACRO {ppom} {"Pap. Phys. Oceanogr. Meteor."}
+MACRO {physzeit} {"Phys. Z."}
+MACRO {pps} {"Planet. Space Sci."}
+MACRO {pss} {"Planet. Space Sci."}
+MACRO {pag} {"Pure Appl. Geophys."}
+MACRO {qjrms} {"Quart. J. Roy. Meteorol. Soc."}
+MACRO {quatres} {"Quat. Res."}
+MACRO {rsci} {"Radio Sci."}
+MACRO {rse} {"Remote Sens. Environ."}
+MACRO {rgeo} {"Rev. Geophys."}
+MACRO {rgsp} {"Rev. Geophys. Space Phys."}
+MACRO {rdgeo} {"Rev. Geofis."}
+MACRO {revmeta} {"Rev. Meteorol."}
+MACRO {sgp}{"Surveys in Geophys."}
+MACRO {sp} {"Solar Phys."}
+MACRO {ssr} {"Space Sci. Rev."}
+MACRO {tellus} {"Tellus"}
+MACRO {tac} {"Theor. Appl. Climatol."}
+MACRO {tagu} {"Trans. Am. Geophys. Union (EOS)"}
+MACRO {wrr} {"Water Resour. Res."}
+MACRO {weather} {"Weather"}
+MACRO {wafc} {"Weather Forecast."}
+MACRO {ww} {"Weatherwise"}
+MACRO {wmob} {"WMO Bull."}
+MACRO {zeitmet} {"Z. Meteorol."}
+ % End module: geojour.mbs
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ %-------------------------------------------------------------------
+
+ % Begin module:
+
+
+ % \ProvidesFile{photjour.mbs}[1999/02/24 2.0b (PWD)]
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+MACRO {appopt} {"Appl. Opt."}
+
+MACRO {bell} {"Bell Syst. Tech. J."}
+
+MACRO {ell} {"Electron. Lett."}
+
+MACRO {jasp} {"J. Appl. Spectr."}
+
+MACRO {jqe} {"IEEE J. Quantum Electron."}
+
+MACRO {jlwt} {"J. Lightwave Technol."}
+
+MACRO {jmo} {"J. Mod. Opt."}
+
+MACRO {josa} {"J. Opt. Soc. America"}
+
+MACRO {josaa} {"J. Opt. Soc. Amer.~A"}
+
+MACRO {josab} {"J. Opt. Soc. Amer.~B"}
+
+MACRO {jdp} {"J. Phys. (Paris)"}
+
+MACRO {oc} {"Opt. Commun."}
+
+MACRO {ol} {"Opt. Lett."}
+
+MACRO {phtl} {"IEEE Photon. Technol. Lett."}
+
+MACRO {pspie} {"Proc. Soc. Photo-Opt. Instrum. Eng."}
+
+MACRO {sse} {"Solid-State Electron."}
+
+MACRO {sjot} {"Sov. J. Opt. Technol."}
+
+MACRO {sjqe} {"Sov. J. Quantum Electron."}
+
+MACRO {sleb} {"Sov. Phys.--Leb. Inst. Rep."}
+
+MACRO {stph} {"Sov. Phys.--Techn. Phys."}
+
+MACRO {stphl} {"Sov. Techn. Phys. Lett."}
+
+MACRO {vr} {"Vision Res."}
+
+MACRO {zph} {"Z. f. Physik"}
+
+MACRO {zphb} {"Z. f. Physik~B"}
+
+MACRO {zphd} {"Z. f. Physik~D"}
+
+MACRO {CLEO} {"CLEO"}
+
+MACRO {ASSL} {"Adv. Sol.-State Lasers"}
+
+MACRO {OSA} {"OSA"}
+
+
+
+
+ % End module: photjour.mbs
+
+
+
+%% Copyright 1994-2000 Patrick W Daly
+MACRO {acmcs} {"ACM Comput. Surv."}
+
+MACRO {acta} {"Acta Inf."}
+
+MACRO {cacm} {"Commun. ACM"}
+
+MACRO {ibmjrd} {"IBM J. Res. Dev."}
+
+MACRO {ibmsj} {"IBM Syst.~J."}
+
+MACRO {ieeese} {"IEEE Trans. Software Eng."}
+
+MACRO {ieeetc} {"IEEE Trans. Comput."}
+
+MACRO {ieeetcad}
+ {"IEEE Trans. Comput. Aid. Des."}
+
+MACRO {ipl} {"Inf. Process. Lett."}
+
+MACRO {jacm} {"J.~ACM"}
+
+MACRO {jcss} {"J.~Comput. Syst. Sci."}
+
+MACRO {scp} {"Sci. Comput. Program."}
+
+MACRO {sicomp} {"SIAM J. Comput."}
+
+MACRO {tocs} {"ACM Trans. Comput. Syst."}
+
+MACRO {tods} {"ACM Trans. Database Syst."}
+
+MACRO {tog} {"ACM Trans. Graphic."}
+
+MACRO {toms} {"ACM Trans. Math. Software"}
+
+MACRO {toois} {"ACM Trans. Office Inf. Syst."}
+
+MACRO {toplas} {"ACM Trans. Progr. Lang. Syst."}
+
+MACRO {tcs} {"Theor. Comput. Sci."}
+
+FUNCTION {bibinfo.check}
+{ swap$
+ duplicate$ missing$
+ {
+ pop$ pop$
+ ""
+ }
+ { duplicate$ empty$
+ {
+ swap$ pop$
+ }
+ { swap$
+ "" * swap$ * "" *
+ }
+ if$
+ }
+ if$
+}
+FUNCTION {bibinfo.warn}
+{ swap$
+ duplicate$ missing$
+ {
+ swap$ "missing " swap$ * " in " * cite$ * warning$ pop$
+ ""
+ }
+ { duplicate$ empty$
+ {
+ swap$ "empty " swap$ * " in " * cite$ * warning$
+ }
+ { swap$
+ "" * swap$ * "" *
+ }
+ if$
+ }
+ if$
+}
+FUNCTION {format.eprint}
+{ eprint duplicate$ empty$
+ 'skip$
+ { "\texttt"
+ archive empty$
+ 'skip$
+ { "[" * archive * "]" * }
+ if$
+ "{" * swap$ * "}" *
+ }
+ if$
+}
+FUNCTION {format.url}
+{ url empty$
+ { "" }
+ { "\texttt{" url * "}" * }
+ if$
+}
+
+STRINGS { bibinfo}
+INTEGERS { nameptr namesleft numnames }
+
+FUNCTION {format.names}
+{ 'bibinfo :=
+ duplicate$ empty$ 'skip$ {
+ 's :=
+ "" 't :=
+ #1 'nameptr :=
+ s num.names$ 'numnames :=
+ numnames 'namesleft :=
+ { namesleft #0 > }
+ { s nameptr
+ "{f{.}.}" format.name$ duplicate$ empty$ 'skip$
+ { tie.or.space.prefix bib.fname.font swap$ * }
+ if$
+ s nameptr
+ "{vv~}{ll}" format.name$ bib.name.font *
+ s nameptr
+ "{jj}" format.name$ duplicate$ empty$ 'skip$
+ { bib.fname.font ", " swap$ * }
+ if$
+ *
+ bibinfo bibinfo.check
+ 't :=
+ nameptr #1 >
+ {
+ nameptr #0
+ #10 +
+ #1 + =
+ numnames #0
+ #10 +
+ > and
+ { "others" 't :=
+ #1 'namesleft := }
+ 'skip$
+ if$
+ namesleft #1 >
+ { ", " * t * }
+ {
+ numnames #2 >
+ { "" * }
+ 'skip$
+ if$
+ s nameptr "{ll}" format.name$ duplicate$ "others" =
+ { 't := }
+ { pop$ }
+ if$
+ t "others" =
+ {
+ " " * bbl.etal bib.name.font *
+ }
+ {
+ bbl.and
+ bib.name.font
+ " " * * t *
+ }
+ if$
+ }
+ if$
+ }
+ 't
+ if$
+ nameptr #1 + 'nameptr :=
+ namesleft #1 - 'namesleft :=
+ }
+ while$
+ } if$
+}
+FUNCTION {format.names.ed}
+{
+ format.names
+}
+FUNCTION {format.key}
+{ empty$
+ { key field.or.null }
+ { "" }
+ if$
+}
+
+FUNCTION {format.authors}
+{ author "" format.names
+ duplicate$ empty$ 'skip$
+ { collaboration "" bibinfo.check
+ duplicate$ empty$ 'skip$
+ { " (" swap$ * ")" * }
+ if$
+ *
+ }
+ if$
+}
+FUNCTION {get.bbl.editor}
+{ editor num.names$ #1 > 'bbl.editors 'bbl.editor if$ }
+
+FUNCTION {format.editors}
+{ editor "" format.names duplicate$ empty$ 'skip$
+ {
+ "," *
+ " " *
+ get.bbl.editor
+ *
+ }
+ if$
+}
+FUNCTION {format.isbn}
+{ isbn "" bibinfo.check
+ duplicate$ empty$ 'skip$
+ {
+ "ISBN " swap$ *
+ }
+ if$
+}
+
+FUNCTION {format.issn}
+{ issn "" bibinfo.check
+ duplicate$ empty$ 'skip$
+ {
+ "ISSN " swap$ *
+ }
+ if$
+}
+
+FUNCTION {format.note}
+{
+ note empty$
+ { "" }
+ { note #1 #1 substring$
+ duplicate$ "{" =
+ 'skip$
+ { output.state mid.sentence =
+ { "l" }
+ { "u" }
+ if$
+ change.case$
+ }
+ if$
+ note #2 global.max$ substring$ * "" bibinfo.check
+ }
+ if$
+}
+
+FUNCTION {format.title}
+{ title
+ duplicate$ empty$
+ { skip$ }
+ { "t" change.case$ }
+ if$
+ "" bibinfo.check
+ duplicate$ empty$ 'skip$
+ {
+ emphasize
+ }
+ if$
+}
+
+FUNCTION {format.full.names}
+{'s :=
+ "" 't :=
+ #1 'nameptr :=
+ s num.names$ 'numnames :=
+ numnames 'namesleft :=
+ { namesleft #0 > }
+ { s nameptr
+ "{vv~}{ll}" format.name$
+ 't :=
+ nameptr #1 >
+ {
+ nameptr #0
+ #10 +
+ #1 + =
+ numnames #0
+ #10 +
+ > and
+ { "others" 't :=
+ #1 'namesleft := }
+ 'skip$
+ if$
+ namesleft #1 >
+ { ", " * t * }
+ {
+ s nameptr "{ll}" format.name$ duplicate$ "others" =
+ { 't := }
+ { pop$ }
+ if$
+ t "others" =
+ {
+ " " * bbl.etal *
+ cite.name.font
+ }
+ {
+ numnames #2 >
+ { "," * }
+ 'skip$
+ if$
+ bbl.and
+ space.word * t *
+ }
+ if$
+ }
+ if$
+ }
+ 't
+ if$
+ nameptr #1 + 'nameptr :=
+ namesleft #1 - 'namesleft :=
+ }
+ while$
+ t "others" =
+ 'skip$
+ { cite.name.font }
+ if$
+}
+
+FUNCTION {author.editor.key.full}
+{ author empty$
+ { editor empty$
+ { key empty$
+ { cite$ #1 #3 substring$ }
+ 'key
+ if$
+ }
+ { editor format.full.names }
+ if$
+ }
+ { author format.full.names }
+ if$
+}
+
+FUNCTION {author.key.full}
+{ author empty$
+ { key empty$
+ { cite$ #1 #3 substring$ }
+ 'key
+ if$
+ }
+ { author format.full.names }
+ if$
+}
+
+FUNCTION {editor.key.full}
+{ editor empty$
+ { key empty$
+ { cite$ #1 #3 substring$ }
+ 'key
+ if$
+ }
+ { editor format.full.names }
+ if$
+}
+
+FUNCTION {make.full.names}
+{ type$ "book" =
+ type$ "inbook" =
+ or
+ 'author.editor.key.full
+ { type$ "proceedings" =
+ 'editor.key.full
+ 'author.key.full
+ if$
+ }
+ if$
+}
+
+%%%%%%%%%%%%%%%%%% EDP Sciences %%%%%%%%%%%%%%%%%%%%
+%FUNCTION {output.bibitem}
+%{ newline$
+% "\bibitem[{" write$
+% label write$
+% ")" make.full.names duplicate$ short.list =
+% { pop$ }
+% { * }
+% if$
+% "}]{" * write$
+% cite$ write$
+% "}" write$
+% newline$
+% ""
+% before.all 'output.state :=
+%}
+%
+FUNCTION {output.bibitem}
+{ newline$
+ "\bibitem" "{" * write$
+ cite$ write$
+ "}" write$
+ newline$
+ ""
+ before.all 'output.state :=
+}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+FUNCTION {n.dashify}
+{
+ 't :=
+ ""
+ { t empty$ not }
+ { t #1 #1 substring$ "-" =
+ { t #1 #2 substring$ "--" = not
+ { "--" *
+ t #2 global.max$ substring$ 't :=
+ }
+ { { t #1 #1 substring$ "-" = }
+ { "-" *
+ t #2 global.max$ substring$ 't :=
+ }
+ while$
+ }
+ if$
+ }
+ { t #1 #1 substring$ *
+ t #2 global.max$ substring$ 't :=
+ }
+ if$
+ }
+ while$
+}
+
+FUNCTION {word.in}
+{ bbl.in
+ " " * }
+
+FUNCTION {format.date}
+{ year "" bibinfo.check duplicate$ empty$
+ {
+ "empty year in " cite$ * "; set to ????" * warning$
+ pop$ "????"
+ }
+ 'skip$
+ if$
+ extra.label *
+ before.all 'output.state :=
+ " (" swap$ * ")" *
+}
+FUNCTION {format.date.misc}
+{
+ year empty$ 'skip$ {
+ format.date "year" output.check
+ } if$
+}
+FUNCTION {format.btitle}
+{ title "" bibinfo.check
+ duplicate$ empty$ 'skip$
+ {
+ emphasize
+ }
+ if$
+}
+FUNCTION {either.or.check}
+{ empty$
+ 'pop$
+ { "can't use both " swap$ * " fields in " * cite$ * warning$ }
+ if$
+}
+FUNCTION {format.bvolume}
+{ volume empty$
+ { "" }
+ { bbl.volume volume tie.or.space.prefix
+ "" bibinfo.check * *
+ series "" bibinfo.check
+ duplicate$ empty$ 'pop$
+ { swap$ bbl.of space.word * swap$
+ emphasize * }
+ if$
+ "volume and number" number either.or.check
+ }
+ if$
+}
+FUNCTION {format.number.series}
+{ volume empty$
+ { number empty$
+ { series field.or.null }
+ { series empty$
+ { number "" bibinfo.check
+ }
+ { output.state mid.sentence =
+ { bbl.number }
+ { bbl.number capitalize }
+ if$
+ number tie.or.space.prefix "" bibinfo.check * *
+ bbl.in space.word *
+ series "" bibinfo.check *
+ }
+ if$
+ }
+ if$
+ }
+ { "" }
+ if$
+}
+FUNCTION {is.num}
+{ chr.to.int$
+ duplicate$ "0" chr.to.int$ < not
+ swap$ "9" chr.to.int$ > not and
+}
+
+FUNCTION {extract.num}
+{ duplicate$ 't :=
+ "" 's :=
+ { t empty$ not }
+ { t #1 #1 substring$
+ t #2 global.max$ substring$ 't :=
+ duplicate$ is.num
+ { s swap$ * 's := }
+ { pop$ "" 't := }
+ if$
+ }
+ while$
+ s empty$
+ 'skip$
+ { pop$ s }
+ if$
+}
+
+FUNCTION {convert.edition}
+{ extract.num "l" change.case$ 's :=
+ s "first" = s "1" = or
+ { bbl.first 't := }
+ { s "second" = s "2" = or
+ { bbl.second 't := }
+ { s "third" = s "3" = or
+ { bbl.third 't := }
+ { s "fourth" = s "4" = or
+ { bbl.fourth 't := }
+ { s "fifth" = s "5" = or
+ { bbl.fifth 't := }
+ { s #1 #1 substring$ is.num
+ { s eng.ord 't := }
+ { edition 't := }
+ if$
+ }
+ if$
+ }
+ if$
+ }
+ if$
+ }
+ if$
+ }
+ if$
+ t
+}
+
+FUNCTION {format.edition}
+{ edition duplicate$ empty$ 'skip$
+ {
+ convert.edition
+ output.state mid.sentence =
+ { "l" }
+ { "t" }
+ if$ change.case$
+ "" bibinfo.check
+ "~" * bbl.edition *
+ }
+ if$
+}
+INTEGERS { multiresult }
+FUNCTION {multi.page.check}
+{ 't :=
+ #0 'multiresult :=
+ { multiresult not
+ t empty$ not
+ and
+ }
+ { t #1 #1 substring$
+ duplicate$ "-" =
+ swap$ duplicate$ "," =
+ swap$ "+" =
+ or or
+ { #1 'multiresult := }
+ { t #2 global.max$ substring$ 't := }
+ if$
+ }
+ while$
+ multiresult
+}
+FUNCTION {format.pages}
+{ pages duplicate$ empty$ 'skip$
+ { duplicate$ multi.page.check
+ {
+ bbl.pages swap$
+ n.dashify
+ }
+ {
+ bbl.page swap$
+ }
+ if$
+ tie.or.space.prefix
+ "" bibinfo.check
+ * *
+ }
+ if$
+}
+FUNCTION {first.page}
+{ 't :=
+ ""
+ { t empty$ not t #1 #1 substring$ "-" = not and }
+ { t #1 #1 substring$ *
+ t #2 global.max$ substring$ 't :=
+ }
+ while$
+}
+
+FUNCTION {format.journal.pages}
+{ pages duplicate$ empty$ 'pop$
+ { swap$ duplicate$ empty$
+ { pop$ pop$ format.pages }
+ {
+ ", " *
+ swap$
+ first.page
+ "" bibinfo.check
+ *
+ }
+ if$
+ }
+ if$
+}
+FUNCTION {format.journal.eid}
+{ eid "" bibinfo.check
+ duplicate$ empty$ 'skip$
+ { swap$ duplicate$ empty$ 'skip$
+ {
+ ", " *
+ }
+ if$
+ swap$
+ numpages empty$ 'skip$
+ { bbl.eidpp numpages tie.or.space.prefix
+ "" bibinfo.check * *
+ " (" swap$ * ")" * *
+ }
+ if$
+ }
+ if$ *
+}
+FUNCTION {format.vol.num.pages}
+{ volume field.or.null
+ bolden
+ number empty$
+ 'skip$
+ { "(" number * ")" * *
+ volume empty$
+ { "there's a number but no volume in " cite$ * warning$ }
+ 'skip$
+ if$
+ }
+ if$
+ eid empty$
+ { format.journal.pages }
+ { format.journal.eid }
+ if$
+}
+
+FUNCTION {format.chapter.pages}
+{ chapter empty$
+ 'format.pages
+ { type empty$
+ { bbl.chapter }
+ { type "l" change.case$
+ "" bibinfo.check
+ }
+ if$
+ chapter tie.or.space.prefix
+ "" bibinfo.check
+ * *
+ pages empty$
+ 'skip$
+ { ", " * format.pages * }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.booktitle}
+{
+ booktitle "" bibinfo.check
+ emphasize
+}
+FUNCTION {format.in.ed.booktitle}
+{ format.booktitle duplicate$ empty$ 'skip$
+ {
+ editor "" format.names.ed duplicate$ empty$ 'pop$
+ {
+ bbl.edby
+ " " * swap$ *
+ swap$
+ "," *
+ " " * swap$
+ * }
+ if$
+ word.in swap$ *
+ }
+ if$
+}
+FUNCTION {format.thesis.type}
+{ type duplicate$ empty$
+ 'pop$
+ { swap$ pop$
+ "t" change.case$ "" bibinfo.check
+ }
+ if$
+}
+FUNCTION {format.tr.number}
+{ number "" bibinfo.check
+ type duplicate$ empty$
+ { pop$ bbl.techrep }
+ 'skip$
+ if$
+ "" bibinfo.check
+ swap$ duplicate$ empty$
+ { pop$ "t" change.case$ }
+ { tie.or.space.prefix * * }
+ if$
+}
+FUNCTION {format.article.crossref}
+{
+ word.in
+ " \cite{" * crossref * "}" *
+}
+FUNCTION {format.book.crossref}
+{ volume duplicate$ empty$
+ { "empty volume in " cite$ * "'s crossref of " * crossref * warning$
+ pop$ word.in
+ }
+ { bbl.volume
+ swap$ tie.or.space.prefix "" bibinfo.check * * bbl.of space.word *
+ }
+ if$
+ " \cite{" * crossref * "}" *
+}
+FUNCTION {format.incoll.inproc.crossref}
+{
+ word.in
+ " \cite{" * crossref * "}" *
+}
+FUNCTION {format.org.or.pub}
+{ 't :=
+ ""
+ add.blank "(" *
+ t empty$
+ { address "" bibinfo.check *
+ }
+ { t *
+ address empty$
+ 'skip$
+ { ", " * address "" bibinfo.check *
+ }
+ if$
+ }
+ if$
+ year duplicate$ empty$
+ { "empty year in " cite$ * "; set to ????" * warning$
+ pop$ "????" }
+ { "" bibinfo.check
+ extra.label * }
+ if$
+ t empty$ address empty$ and
+ { * }
+ { ", " swap$ * * }
+ if$
+ ")" *
+}
+FUNCTION {format.publisher.address}
+{ publisher "" bibinfo.warn format.org.or.pub
+}
+
+FUNCTION {format.organization.address}
+{ organization "" bibinfo.check
+format.org.or.pub
+}
+
+FUNCTION {article}
+{ output.bibitem
+ format.authors "author" output.check
+ author format.key output
+ crossref missing$
+ {
+ ""
+ journal
+ duplicate$ empty$
+ { pop$ pop$ } {
+ "" bibinfo.check
+ * "journal" output.check
+ add.blank
+ } if$
+ format.vol.num.pages output
+ format.date "year" output.check
+ }
+ { format.article.crossref output.nonnull
+ format.pages output
+ }
+ if$
+ format.issn output
+ format.note output
+ format.eprint output
+ format.url output
+ fin.entry
+}
+FUNCTION {book}
+{ output.bibitem
+ author empty$
+ { format.editors "author and editor" output.check
+ editor format.key output
+ }
+ { format.authors output.nonnull
+ crossref missing$
+ { "author and editor" editor either.or.check }
+ 'skip$
+ if$
+ }
+ if$
+ format.btitle "title" output.check
+ crossref missing$
+ { format.bvolume output
+ format.number.series output
+ format.edition output
+ format.publisher.address output
+ }
+ {
+ format.book.crossref output.nonnull
+ format.date "year" output.check
+ }
+ if$
+ format.isbn output
+ format.note output
+ format.eprint output
+ format.url output
+ fin.entry
+}
+FUNCTION {booklet}
+{ output.bibitem
+ format.authors output
+ author format.key output
+ format.title "title" output.check
+ howpublished "" bibinfo.check output
+ address "" bibinfo.check output
+ format.date "year" output.check
+ format.isbn output
+ format.note output
+ format.eprint output
+ format.url output
+ fin.entry
+}
+
+FUNCTION {inbook}
+{ output.bibitem
+ author empty$
+ { format.editors "author and editor" output.check
+ editor format.key output
+ }
+ { format.authors output.nonnull
+ crossref missing$
+ { "author and editor" editor either.or.check }
+ 'skip$
+ if$
+ }
+ if$
+ format.btitle "title" output.check
+ crossref missing$
+ {
+ format.publisher.address output
+ format.bvolume output
+ format.chapter.pages "chapter and pages" output.check
+ format.number.series output
+ }
+ {
+ format.chapter.pages "chapter and pages" output.check
+ format.book.crossref output.nonnull
+ format.date "year" output.check
+ }
+ if$
+ format.edition output
+ crossref missing$
+ { format.isbn output }
+ 'skip$
+ if$
+ format.note output
+ format.eprint output
+ format.url output
+ fin.entry
+}
+
+FUNCTION {incollection}
+{ output.bibitem
+ format.authors "author" output.check
+ author format.key output
+ crossref missing$
+ { format.in.ed.booktitle "booktitle" output.check
+ format.publisher.address output
+ format.bvolume output
+ format.number.series output
+ format.chapter.pages output
+ format.edition output
+ format.isbn output
+ }
+ { format.incoll.inproc.crossref output.nonnull
+ format.chapter.pages output
+ }
+ if$
+ format.note output
+ format.eprint output
+ format.url output
+ fin.entry
+}
+FUNCTION {inproceedings}
+{ output.bibitem
+ format.authors "author" output.check
+ author format.key output
+ format.btitle "title" output.check
+ crossref missing$
+ { format.in.ed.booktitle "booktitle" output.check
+ publisher empty$
+ { format.organization.address output }
+ { organization "" bibinfo.check output
+ format.publisher.address output
+ }
+ if$
+ format.bvolume output
+ format.number.series output
+ format.pages output
+ format.isbn output
+ format.issn output
+ }
+ { format.incoll.inproc.crossref output.nonnull
+ format.pages output
+ }
+ if$
+ format.note output
+ format.eprint output
+ format.url output
+ fin.entry
+}
+FUNCTION {conference} { inproceedings }
+FUNCTION {manual}
+{ output.bibitem
+ format.authors output
+ author format.key output
+ format.btitle "title" output.check
+ organization "" bibinfo.check output
+ address "" bibinfo.check output
+ format.edition output
+ format.date "year" output.check
+ format.note output
+ format.eprint output
+ format.url output
+ fin.entry
+}
+
+FUNCTION {mastersthesis}
+{ output.bibitem
+ format.authors "author" output.check
+ author format.key output
+ bbl.mthesis format.thesis.type output.nonnull
+ school "" bibinfo.warn output
+ address "" bibinfo.check output
+ format.date "year" output.check
+ format.note output
+ format.eprint output
+ format.url output
+ fin.entry
+}
+
+FUNCTION {misc}
+{ output.bibitem
+ format.authors output
+ author format.key output
+ format.title output
+ howpublished "" bibinfo.check output
+ format.date.misc
+ format.note output
+ format.eprint output
+ format.url output
+ fin.entry
+}
+FUNCTION {phdthesis}
+{ output.bibitem
+ format.authors "author" output.check
+ author format.key output
+ bbl.phdthesis format.thesis.type output.nonnull
+ school "" bibinfo.warn output
+ address "" bibinfo.check output
+ format.date "year" output.check
+ format.note output
+ format.eprint output
+ format.url output
+ fin.entry
+}
+
+FUNCTION {proceedings}
+{ output.bibitem
+ format.editors output
+ editor format.key output
+ format.btitle "title" output.check
+ format.bvolume output
+ format.number.series output
+ publisher empty$
+ { format.organization.address output }
+ { organization "" bibinfo.check output
+ format.publisher.address output
+ }
+ if$
+ format.isbn output
+ format.issn output
+ format.note output
+ format.eprint output
+ format.url output
+ fin.entry
+}
+
+FUNCTION {techreport}
+{ output.bibitem
+ format.authors "author" output.check
+ author format.key output
+ format.tr.number output.nonnull
+ institution "" bibinfo.warn output
+ address "" bibinfo.check output
+ format.date "year" output.check
+ format.note output
+ format.eprint output
+ format.url output
+ fin.entry
+}
+
+FUNCTION {unpublished}
+{ output.bibitem
+ format.authors "author" output.check
+ author format.key output
+ format.date.misc
+ format.note "note" output.check
+ format.eprint output
+ format.url output
+ fin.entry
+}
+
+FUNCTION {default.type} { misc }
+READ
+FUNCTION {sortify}
+{ purify$
+ "l" change.case$
+}
+INTEGERS { len }
+FUNCTION {chop.word}
+{ 's :=
+ 'len :=
+ s #1 len substring$ =
+ { s len #1 + global.max$ substring$ }
+ 's
+ if$
+}
+FUNCTION {format.lab.names}
+{ 's :=
+ "" 't :=
+ s #1 "{vv~}{ll}" format.name$
+ s num.names$ duplicate$
+ #2 >
+ { pop$
+ " " * bbl.etal *
+ cite.name.font
+ "others" 't :=
+ }
+ { #2 <
+ 'skip$
+ { s #2 "{ff }{vv }{ll}{ jj}" format.name$ "others" =
+ {
+ " " * bbl.etal *
+ cite.name.font
+ "others" 't :=
+ }
+ { bbl.and space.word * s #2 "{vv~}{ll}" format.name$
+ * }
+ if$
+ }
+ if$
+ }
+ if$
+ t "others" =
+ 'skip$
+ { cite.name.font }
+ if$
+}
+
+FUNCTION {author.key.label}
+{ author empty$
+ { key empty$
+ { cite$ #1 #3 substring$ }
+ 'key
+ if$
+ }
+ { author format.lab.names }
+ if$
+}
+
+FUNCTION {author.editor.key.label}
+{ author empty$
+ { editor empty$
+ { key empty$
+ { cite$ #1 #3 substring$ }
+ 'key
+ if$
+ }
+ { editor format.lab.names }
+ if$
+ }
+ { author format.lab.names }
+ if$
+}
+
+FUNCTION {editor.key.label}
+{ editor empty$
+ { key empty$
+ { cite$ #1 #3 substring$ }
+ 'key
+ if$
+ }
+ { editor format.lab.names }
+ if$
+}
+
+FUNCTION {calc.short.authors}
+{ type$ "book" =
+ type$ "inbook" =
+ or
+ 'author.editor.key.label
+ { type$ "proceedings" =
+ 'editor.key.label
+ 'author.key.label
+ if$
+ }
+ if$
+ 'short.list :=
+}
+
+FUNCTION {calc.label}
+{ calc.short.authors
+ short.list
+ "("
+ *
+ year duplicate$ empty$
+ short.list key field.or.null = or
+ { pop$ "" }
+ 'skip$
+ if$
+ *
+ 'label :=
+}
+
+FUNCTION {sort.format.names}
+{ 's :=
+ #1 'nameptr :=
+ ""
+ s num.names$ 'numnames :=
+ numnames 'namesleft :=
+ { namesleft #0 > }
+ { s nameptr
+ "{vv{ } }{ll{ }}{ f{ }}{ jj{ }}"
+ format.name$ 't :=
+ nameptr #1 >
+ {
+ nameptr #0
+ #10 +
+ #1 + =
+ numnames #0
+ #10 +
+ > and
+ { "others" 't :=
+ #1 'namesleft := }
+ 'skip$
+ if$
+ " " *
+ namesleft #1 = t "others" = and
+ { "zzzzz" * }
+ { t sortify * }
+ if$
+ }
+ { t sortify * }
+ if$
+ nameptr #1 + 'nameptr :=
+ namesleft #1 - 'namesleft :=
+ }
+ while$
+}
+
+FUNCTION {sort.format.title}
+{ 't :=
+ "A " #2
+ "An " #3
+ "The " #4 t chop.word
+ chop.word
+ chop.word
+ sortify
+ #1 global.max$ substring$
+}
+FUNCTION {author.sort}
+{ author empty$
+ { key empty$
+ { "to sort, need author or key in " cite$ * warning$
+ ""
+ }
+ { key sortify }
+ if$
+ }
+ { author sort.format.names }
+ if$
+}
+FUNCTION {author.editor.sort}
+{ author empty$
+ { editor empty$
+ { key empty$
+ { "to sort, need author, editor, or key in " cite$ * warning$
+ ""
+ }
+ { key sortify }
+ if$
+ }
+ { editor sort.format.names }
+ if$
+ }
+ { author sort.format.names }
+ if$
+}
+FUNCTION {editor.sort}
+{ editor empty$
+ { key empty$
+ { "to sort, need editor or key in " cite$ * warning$
+ ""
+ }
+ { key sortify }
+ if$
+ }
+ { editor sort.format.names }
+ if$
+}
+INTEGERS { seq.num }
+FUNCTION {init.seq}
+{ #0 'seq.num :=}
+EXECUTE {init.seq}
+FUNCTION {int.to.fix}
+{ "000000000" swap$ int.to.str$ *
+ #-1 #10 substring$
+}
+FUNCTION {presort}
+{ calc.label
+ label sortify
+ " "
+ *
+ seq.num #1 + 'seq.num :=
+ seq.num int.to.fix
+ 'sort.label :=
+ sort.label
+ *
+ " "
+ *
+ title field.or.null
+ sort.format.title
+ *
+ #1 entry.max$ substring$
+ 'sort.key$ :=
+}
+
+ITERATE {presort}
+SORT
+STRINGS { last.label next.extra }
+INTEGERS { last.extra.num number.label }
+FUNCTION {initialize.extra.label.stuff}
+{ #0 int.to.chr$ 'last.label :=
+ "" 'next.extra :=
+ #0 'last.extra.num :=
+ #0 'number.label :=
+}
+FUNCTION {forward.pass}
+{ last.label label =
+ { last.extra.num #1 + 'last.extra.num :=
+ last.extra.num int.to.chr$ 'extra.label :=
+ }
+ { "a" chr.to.int$ 'last.extra.num :=
+ "" 'extra.label :=
+ label 'last.label :=
+ }
+ if$
+ number.label #1 + 'number.label :=
+}
+FUNCTION {reverse.pass}
+{ next.extra "" =
+ { "" 'extra.label := }
+ 'skip$
+ if$
+ extra.label 'next.extra :=
+ extra.label
+ duplicate$ empty$
+ 'skip$
+ { "" * }
+ if$
+ 'extra.label :=
+ label extra.label * 'label :=
+}
+EXECUTE {initialize.extra.label.stuff}
+ITERATE {forward.pass}
+REVERSE {reverse.pass}
+FUNCTION {bib.sort.order}
+{ sort.label
+ " "
+ *
+ year field.or.null sortify
+ *
+ " "
+ *
+ title field.or.null
+ sort.format.title
+ *
+ #1 entry.max$ substring$
+ 'sort.key$ :=
+}
+ITERATE {bib.sort.order}
+SORT
+FUNCTION {begin.bib}
+{ preamble$ empty$
+ 'skip$
+ { preamble$ write$ newline$ }
+ if$
+ "\begin{thebibliography}{" number.label int.to.str$ * "}" *
+ write$ newline$
+}
+EXECUTE {begin.bib}
+EXECUTE {init.state.consts}
+ITERATE {call.type$}
+FUNCTION {end.bib}
+{ newline$
+ "\end{thebibliography}" write$ newline$
+}
+EXECUTE {end.bib}
+%% End of customized bst file
+%%
+%% End of file `apsrev.bst'.
diff --git a/Paper/v1.1/template.tex b/Paper/v1.1/template.tex
new file mode 100644
index 0000000000000000000000000000000000000000..a4541cace6f3d934ecec7f769b7d8b46cc757e66
--- /dev/null
+++ b/Paper/v1.1/template.tex
@@ -0,0 +1,277 @@
+%% This template can be used to write a paper for
+%% Computer Physics Communications using LaTeX.
+%% For authors who want to write a computer program description,
+%% an example Program Summary is included that only has to be
+%% completed and which will give the correct layout in the
+%% preprint and the journal.
+%% The `elsarticle' style is used and more information on this style
+%% can be found at
+%% http://www.elsevier.com/wps/find/authorsview.authors/elsarticle.
+%%
+%%
+\documentclass[preprint,12pt]{elsarticle}
+
+%% Use the option review to obtain double line spacing
+%% \documentclass[preprint,review,12pt]{elsarticle}
+
+%% Use the options 1p,twocolumn; 3p; 3p,twocolumn; 5p; or 5p,twocolumn
+%% for a journal layout:
+%% \documentclass[final,1p,times]{elsarticle}
+%% \documentclass[final,1p,times,twocolumn]{elsarticle}
+%% \documentclass[final,3p,times]{elsarticle}
+%% \documentclass[final,3p,times,twocolumn]{elsarticle}
+%% \documentclass[final,5p,times]{elsarticle}
+%% \documentclass[final,5p,times,twocolumn]{elsarticle}
+
+%% if you use PostScript figures in your article
+%% use the graphics package for simple commands
+%% \usepackage{graphics}
+%% or use the graphicx package for more complicated commands
+%% \usepackage{graphicx}
+%% or use the epsfig package if you prefer to use the old commands
+%% \usepackage{epsfig}
+
+%% The amssymb package provides various useful mathematical symbols
+\usepackage{amssymb}
+%% The amsthm package provides extended theorem environments
+%% \usepackage{amsthm}
+
+%% The lineno packages adds line numbers. Start line numbering with
+%% \begin{linenumbers}, end it with \end{linenumbers}. Or switch it on
+%% for the whole article with \linenumbers after \end{frontmatter}.
+%% \usepackage{lineno}
+
+%% natbib.sty is loaded by default. However, natbib options can be
+%% provided with \biboptions{...} command. Following options are
+%% valid:
+
+%% round - round parentheses are used (default)
+%% square - square brackets are used [option]
+%% curly - curly braces are used {option}
+%% angle - angle brackets are used <option>
+%% semicolon - multiple citations separated by semi-colon
+%% colon - same as semicolon, an earlier confusion
+%% comma - separated by comma
+%% numbers- selects numerical citations
+%% super - numerical citations as superscripts
+%% sort - sorts multiple citations according to order in ref. list
+%% sort&compress - like sort, but also compresses numerical citations
+%% compress - compresses without sorting
+%%
+%% \biboptions{comma,round}
+
+% \biboptions{}
+
+%% This list environment is used for the references in the
+%% Program Summary
+%%
+\newcounter{bla}
+\newenvironment{refnummer}{%
+\list{[\arabic{bla}]}%
+{\usecounter{bla}%
+ \setlength{\itemindent}{0pt}%
+ \setlength{\topsep}{0pt}%
+ \setlength{\itemsep}{0pt}%
+ \setlength{\labelsep}{2pt}%
+ \setlength{\listparindent}{0pt}%
+ \settowidth{\labelwidth}{[9]}%
+ \setlength{\leftmargin}{\labelwidth}%
+ \addtolength{\leftmargin}{\labelsep}%
+ \setlength{\rightmargin}{0pt}}}
+ {\endlist}
+
+\journal{Computer Physics Communications}
+
+\begin{document}
+
+\begin{frontmatter}
+
+%% Title, authors and addresses
+
+%% use the tnoteref command within \title for footnotes;
+%% use the tnotetext command for the associated footnote;
+%% use the fnref command within \author or \address for footnotes;
+%% use the fntext command for the associated footnote;
+%% use the corref command within \author for corresponding author footnotes;
+%% use the cortext command for the associated footnote;
+%% use the ead command for the email address,
+%% and the form \ead[url] for the home page:
+%%
+%% \title{Title\tnoteref{label1}}
+%% \tnotetext[label1]{}
+%% \author{Name\corref{cor1}\fnref{label2}}
+%% \ead{email address}
+%% \ead[url]{home page}
+%% \fntext[label2]{}
+%% \cortext[cor1]{}
+%% \address{Address\fnref{label3}}
+%% \fntext[label3]{}
+
+\title{A \LaTeX{} template for CPC Computer Physics Descriptions}
+
+%% use optional labels to link authors explicitly to addresses:
+%% \author[label1,label2]{<author name>}
+%% \address[label1]{<address>}
+%% \address[label2]{<address>}
+
+\author[a]{First Author\corref{author}}
+\author[a,b]{Second Author}
+\author[b]{Third Author}
+
+\cortext[author] {Corresponding author.\\\textit{E-mail address:} firstAuthor@somewhere.edu}
+\address[a]{First Address}
+\address[b]{Second Address}
+
+\begin{abstract}
+%% Text of abstract
+A submitted program is expected to be of benefit to other physicists or physical chemists, or be an exemplar of good programming practice, or illustrate new or novel programming techniques which are of importance to some branch of computational physics or physical chemistry.
+
+Acceptable program descriptions can take different forms. The following Long Write-Up structure is a suggested structure but it is not obligatory. Actual structure will depend on the length of the program, the extent to which the algorithms or software have already been described in literature, and the detail provided in the user manual.
+
+Your manuscript and figure sources should be submitted through the Elsevier Editorial System (EES) by using the online submission tool at \\
+http://www.ees.elsevier.com/cpc.
+
+In addition to the manuscript you must supply: the program source code; job control scripts, where applicable; a README file giving the names and a brief description of all the files that make up the package and clear instructions on the installation and execution of the program; sample input and output data for at least one comprehensive test run; and, where appropriate, a user manual. These should be uploaded to EES during the submission process.
+
+\end{abstract}
+
+\begin{keyword}
+%% keywords here, in the form: keyword \sep keyword
+keyword1; keyword2; keyword3; etc.
+
+\end{keyword}
+
+\end{frontmatter}
+
+%%
+%% Start line numbering here if you want
+%%
+% \linenumbers
+
+% Computer program descriptions should contain the following
+% PROGRAM SUMMARY.
+
+{\bf PROGRAM SUMMARY/NEW VERSION PROGRAM SUMMARY}
+ %Delete as appropriate.
+
+\begin{small}
+\noindent
+{\em Manuscript Title:} \\
+{\em Authors:} \\
+{\em Program Title:} \\
+{\em Journal Reference:} \\
+ %Leave blank, supplied by Elsevier.
+{\em Catalogue identifier:} \\
+ %Leave blank, supplied by Elsevier.
+{\em Licensing provisions:} \\
+ %enter "none" if CPC non-profit use license is sufficient.
+{\em Programming language:} \\
+{\em Computer:} \\
+ %Computer(s) for which program has been designed.
+{\em Operating system:} \\
+ %Operating system(s) for which program has been designed.
+{\em RAM:} bytes \\
+ %RAM in bytes required to execute program with typical data.
+{\em Number of processors used:} \\
+ %If more than one processor.
+{\em Supplementary material:} \\
+ % Fill in if necessary, otherwise leave out.
+{\em Keywords:} Keyword one, Keyword two, Keyword three, etc. \\
+ % Please give some freely chosen keywords that we can use in a
+ % cumulative keyword index.
+{\em Classification:} \\
+ %Classify using CPC Program Library Subject Index, see (
+ % http://cpc.cs.qub.ac.uk/subjectIndex/SUBJECT_index.html)
+ %e.g. 4.4 Feynman diagrams, 5 Computer Algebra.
+{\em External routines/libraries:} \\
+ % Fill in if necessary, otherwise leave out.
+{\em Subprograms used:} \\
+ %Fill in if necessary, otherwise leave out.
+{\em Catalogue identifier of previous version:}* \\
+ %Only required for a New Version summary, otherwise leave out.
+{\em Journal reference of previous version:}* \\
+ %Only required for a New Version summary, otherwise leave out.
+{\em Does the new version supersede the previous version?:}* \\
+ %Only required for a New Version summary, otherwise leave out.
+{\em Nature of problem:}\\
+ %Describe the nature of the problem here.
+ \\
+{\em Solution method:}\\
+ %Describe the method solution here.
+ \\
+{\em Reasons for the new version:}*\\
+ %Only required for a New Version summary, otherwise leave out.
+ \\
+{\em Summary of revisions:}*\\
+ %Only required for a New Version summary, otherwise leave out.
+ \\
+{\em Restrictions:}\\
+ %Describe any restrictions on the complexity of the problem here.
+ \\
+{\em Unusual features:}\\
+ %Describe any unusual features of the program/problem here.
+ \\
+{\em Additional comments:}\\
+ %Provide any additional comments here.
+ \\
+{\em Running time:}\\
+ %Give an indication of the typical running time here.
+ \\
+
+\begin{thebibliography}{0}
+\bibitem{1}Reference 1 % This list should only contain those items referenced in the
+\bibitem{2}Reference 2 % Program Summary section.
+\bibitem{3}Reference 3 % Type references in text as [1], [2], etc.
+ % This list is different from the bibliography at the end of
+ % the Long Write-Up.
+\end{thebibliography}
+* Items marked with an asterisk are only required for new versions
+of programs previously published in the CPC Program Library.\\
+\end{small}
+
+
+%% main text
+\section{}
+\label{}
+
+%% The Appendices part is started with the command \appendix;
+%% appendix sections are then done as normal sections
+%% \appendix
+
+%% \section{}
+%% \label{}
+
+%% References
+%%
+%% Following citation commands can be used in the body text:
+%% Usage of \cite is as follows:
+%% \cite{key} ==>> [#]
+%% \cite[chap. 2]{key} ==>> [#, chap. 2]
+%%
+
+%% References with bibTeX database:
+
+\bibliographystyle{elsarticle-num}
+\bibliography{<your-bib-database>}
+
+%% Authors are advised to submit their bibtex database files. They are
+%% requested to list a bibtex style file in the manuscript if they do
+%% not want to use elsarticle-num.bst.
+
+%% References without bibTeX database:
+
+% \begin{thebibliography}{00}
+
+%% \bibitem must have the following form:
+%% \bibitem{key}...
+%%
+
+% \bibitem{}
+
+% \end{thebibliography}
+
+
+\end{document}
+
+%%
+%% End of file
diff --git a/Paper/v1.1/update.bib b/Paper/v1.1/update.bib
new file mode 100644
index 0000000000000000000000000000000000000000..6f01086029ed72e9b10729d127e2d62caa7bdfd5
--- /dev/null
+++ b/Paper/v1.1/update.bib
@@ -0,0 +1,11 @@
+@InCollection{HPV:VisIt,
+author = {Hank Childs and Eric Brugger and Brad Whitlock and Jeremy Meredith and Sean Ahern and David Pugmire and Kathleen Biagas and Mark Miller and Cyrus Harrison and Gunther H. Weber and Hari Krishnan and Thomas Fogal and Allen Sanderson and Christoph Garth and E. Wes Bethel and David Camp and Oliver R\"{u}bel and Marc Durant and Jean M. Favre and Paul Navr\'{a}til},
+title = {{VisIt: An End-User Tool For Visualizing and Analyzing Very Large Data}},
+year = "2012",
+pages = "357-372",
+month = "Oct",
+booktitle = {{High Performance Visualization--Enabling Extreme-Scale Scientific Insight}},
+}
+For graphics ouput
+with ViSiT \cite{HPV:VisIt} some suitable scientific database software
+should be used such as the Silo library {\tt https://wci.llnl.gov/simulation/computer-codes/silo}.\\
\ No newline at end of file
diff --git a/Paper/v1.1/update.tex b/Paper/v1.1/update.tex
new file mode 100755
index 0000000000000000000000000000000000000000..d1ce13be7e63619a016a1b33eefb8841e708d923
--- /dev/null
+++ b/Paper/v1.1/update.tex
@@ -0,0 +1,465 @@
+%% This template can be used to write a paper for
+%% Computer Physics Communications using LaTeX.
+%% For authors who want to write a computer program description,
+%% an example Program Summary is included that only has to be
+%% completed and which will give the correct layout in the
+%% preprint and the journal.
+%% The `elsarticle' style is used and more information on this style
+%% can be found at
+%% http://www.elsevier.com/wps/find/authorsview.authors/elsarticle.
+%%
+%%
+\documentclass[final,1p,twocolumn]{elsarticle}
+
+%% Use the option review to obtain double line spacing
+%% \documentclass[preprint,review,12pt]{elsarticle}
+
+%% Use the options 1p,twocolumn; 3p; 3p,twocolumn; 5p; or 5p,twocolumn
+%% for a journal layout:
+%% \documentclass[final,1p,times]{elsarticle}
+%% \documentclass[final,1p,times,twocolumn]{elsarticle}
+%% \documentclass[final,3p,times]{elsarticle}
+%% \documentclass[final,3p,times,twocolumn]{elsarticle}
+%% \documentclass[final,5p,times]{elsarticle}
+%% \documentclass[final,5p,times,twocolumn]{elsarticle}
+
+%% if you use PostScript figures in your article
+%% use the graphics package for simple commands
+%% \usepackage{graphics}
+%% or use the graphicx package for more complicated commands
+%% \usepackage{graphicx}
+%% or use the epsfig package if you prefer to use the old commands
+%% \usepackage{epsfig}
+
+%% The amssymb package provides various useful mathematical symbols
+\usepackage{amssymb}
+\usepackage[usenames]{color}
+%% The amsthm package provides extended theorem environments
+%% \usepackage{amsthm}
+
+%% The lineno packages adds line numbers. Start line numbering with
+%% \begin{linenumbers}, end it with \end{linenumbers}. Or switch it on
+%% for the whole article with \linenumbers after \end{frontmatter}.
+%% \usepackage{lineno}
+
+%% natbib.sty is loaded by default. However, natbib options can be
+%% provided with \biboptions{...} command. Following options are
+%% valid:
+
+%% round - round parentheses are used (default)
+%% square - square brackets are used [option]
+%% curly - curly braces are used {option}
+%% angle - angle brackets are used <option>
+%% semicolon - multiple citations separated by semi-colon
+%% colon - same as semicolon, an earlier confusion
+%% comma - separated by comma
+%% numbers- selects numerical citations
+%% super - numerical citations as superscripts
+%% sort - sorts multiple citations according to order in ref. list
+%% sort&compress - like sort, but also compresses numerical citations
+%% compress - compresses without sorting
+%%
+%% \biboptions{comma,round}
+
+% \biboptions{}
+
+%% This list environment is used for the references in the
+%% Program Summary
+%%
+\newcounter{bla}
+\newenvironment{refnummer}{%
+\list{[\arabic{bla}]}%
+{\usecounter{bla}%
+ \setlength{\itemindent}{0pt}%
+ \setlength{\topsep}{0pt}%
+ \setlength{\itemsep}{0pt}%
+ \setlength{\labelsep}{2pt}%
+ \setlength{\listparindent}{0pt}%
+ \settowidth{\labelwidth}{[9]}%
+ \setlength{\leftmargin}{\labelwidth}%
+ \addtolength{\leftmargin}{\labelsep}%
+ \setlength{\rightmargin}{0pt}}}
+ {\endlist}
+
+\journal{Computer Physics Communications}
+
+\begin{document}
+
+\begin{frontmatter}
+
+%% Title, authors and addresses
+
+%% use the tnoteref command within \title for footnotes;
+%% use the tnotetext command for the associated footnote;
+%% use the fnref command within \author or \address for footnotes;
+%% use the fntext command for the associated footnote;
+%% use the corref command within \author for corresponding author footnotes;
+%% use the cortext command for the associated footnote;
+%% use the ead command for the email address,
+%% and the form \ead[url] for the home page:
+%%
+%% \title{Title\tnoteref{label1}}
+%% \tnotetext[label1]{}
+%% \author{Name\corref{cor1}\fnref{label2}}
+%% \ead{email address}
+%% \ead[url]{home page}
+%% \fntext[label2]{}
+%% \cortext[cor1]{}
+%% \address{Address\fnref{label3}}
+%% \fntext[label3]{}
+
+\title{The TDHF Code Sky3D Version 1.1}
+
+\author{B.~Schuetrumpf\corref{gsi} }\ead{b.schuetrumpf@gsi.de}
+\address{National Superconducting Cyclotron Laboratory\\
+Michigan State University, East Lansing, Michigan 48824, USA}
+
+\author{P.-G.~Reinhard}\ead{Paul-Gerhard.Reinhard@physik.uni-erlangen.de}
+\address{Institut f\"ur Theoretische Physik II, Universit\"at
+ Erlangen-N\"urnberg, \\Staudtstrasse 7, 91058 Erlangen, Germany}
+
+\author{P.~D.~Stevenson}\ead{p.stevenson@surrey.ac.uk}
+\address{Department of Physics, University of Surrey, Guildford,
+ Surrey, GU2 7XH, United Kingdom}
+
+\author{A. S. Umar}\ead{sait.a.umar@Vanderbilt.Edu}
+\address{Department of Physics and Astronomy, Vanderbilt University, \\Nashville, Tennessee 37235, USA}
+
+\author{J.~A.~Maruhn\corref{author}} \ead{maruhn@th.physik.uni-frankfurt.de}
+\address{Institut f\"ur Theoretische Physik, Goethe-Universit\"at,
+ Max-von-Laue-Str. 1, \\60438 Frankfurt am Main, Germany}
+
+\cortext[gsi]{Present address: Gesellschaft f\"ur
+ Schwerionenforschung, 64291 Darmstadt, Germany}
+\cortext[author]{Corresponding author.}
+
+\begin{abstract}
+ The nuclear mean-field model based on Skyrme forces or related
+ density functionals has found widespread application to the
+ description of nuclear ground states, collective vibrational
+ excitations, and heavy-ion collisions. The code Sky3D solves the
+ static or dynamic equations on a three-dimensional Cartesian mesh
+ with isolated or periodic boundary conditions and no further
+ symmetry assumptions. Pairing can be included in the BCS
+ approximation for the static case. The code is implemented with a
+ view to allow easy modifications for including additional physics or
+ special analysis of the results.
+\end{abstract}
+
+\begin{keyword}
+Hartree-Fock; BCS; Density-functional theory; Skyrme energy functional; Giant Resonances;
+Heavy-Ion collisions.
+\end{keyword}
+\end{frontmatter}
+
+{\bf NEW VERSION PROGRAM SUMMARY}
+ %Delete as appropriate.
+{\bf PROGRAM SUMMARY}
+
+\begin{small}
+\noindent
+{\em Title:} The TDHF Code Sky3D Version 1.1\\
+{\em Authors:} B.~Schuetrumpf, P.-G.~Reinhard, P.~D.~Stevenson,
+A.~S.~Umar, and J.~A.~Maruhn, \\
+{\em Program Title:} Sky3D\\
+{\em Journal Reference:} \\
+ %Leave blank, supplied by Elsevier.
+{\em Catalogue identifier:} \\
+ %Leave blank, supplied by Elsevier.
+{\em Licensing provisions:} none\\
+ %enter "none" if CPC non-profit use license is sufficient.
+{\em Programming language:} Fortran 90. The {\tt OpenMP} version
+requires a relatively recent compiler; it was found to work using {\tt
+ gfortran 4.6.2} or later and the Intel compiler version 12 or later.\\
+{\em Computer:} All computers with a Fortran compiler supporting at
+least Fortran 90.\\
+ %Computer(s) for which program has been designed.
+{\em Operating system:} All operating systems with such a
+compiler. Some of the Makefiles and scripts depend on a Unix-like
+system and need modification under Windows.\\
+ %Operating system(s) for which program has been designed.
+{\em RAM:} 1~GB\\
+ %RAM in bytes required to execute program with typical data.
+{\em Number of processors used:} no built-in limit, parallelization
+using both OpenMP and MPI.\\
+ %If more than one processor.
+{\em Supplementary material:} Extensive documentation and a number of
+utility programs to analyze the results and prepare them for graphics
+output using the Silo library ({\tt
+ http://wci.llnl.gov/simulation/computer-codes/silo}) for use in VisIT
+\cite{HPV:VisIt} or Paraview ({\tt https://www.paraview.org}). The
+code can serve as a template for interfacing to other database or graphics systems. \\
+{\em Keywords:} Nuclear theory, Mean-field models, Nuclear reactions\\
+ % Please give some freely chosen keywords that we can use in a
+ % cumulative keyword index.
+{\em Classification:} 17.16 Theoretical Methods - General, 17.22
+Hartree-Fock Calculations, 17.23 Fission and Fusion Processes\\
+ %Classify using CPC Program Library Subject Index, see (
+ % http://cpc.cs.qub.ac.uk/subjectIndex/SUBJECT_index.html)
+ %e.g. 4.4 Feynman diagrams, 5 Computer Algebra.
+{\em External routines/libraries:} LAPACK, FFTW3.
+\\
+{\em Catalogue identifier of previous version:} \\
+ AESW\_v1\_0.\\
+ %Only required for a New Version summary, otherwise leave out.
+{\em Journal reference of previous version:} \\
+J.~A.~Maruhn, P.-G.~Reinhard, P.~D.~Stevenson, and A.~S.~Umar, "The
+TDHF Code Sky3D", Comp. Phys. Comm. {\bf 185}, 2195 (2014).\\
+ %Only required for a New Version summary, otherwise leave out.
+{\em Does the new version supersede the previous version?:} \\
+Yes.\\
+
+{\em Nature of problem:} The time-dependent Hartree-Fock equations can
+be used to simulate nuclear vibrations and collisions between nuclei
+for low energies. This code implements the equations based on a Skyrme
+energy functional and also allows the determination of the
+ground-state structure of nuclei through the static version of the
+equations. For the case of vibrations the principal aim is to
+calculate the excitation spectra by Fourier-analyzing the time
+dependence of suitable observables. In collisions, the formation of a
+neck between nuclei, the dissipation of energy from collective motion,
+processes like charge transfer and the approach to fusion are of
+principal interest.\\
+ %Describe the nature of the problem here.
+\\
+{\em Solution method:} The nucleonic wave function spinors are
+represented on a three-dimensional Cartesian mesh with no further
+symmetry restrictions. The boundary conditions are always periodic for
+the wave functions, while the Coulomb potential can also be calculated
+for an isolated charge distribution. All spatial derivatives are
+evaluated using the finite Fourier transform method. The code solves
+the static Hartree-Fock equations with a damped gradient iteration
+method and the time-dependent Hartree-Fock equations with an expansion
+of the time-development operator. Any number of initial nuclei can be
+placed into the mesh in with arbitrary positions and
+initial velocities.\\
+
+{\em Reasons for the new version:}\\
+A few bugs were fixed and a number of enhancements added concerning
+faster convergence, better stability, and more sophisticated analysis
+of some results.
+
+{\em Summary of revisions:}\\
+The following is a brief summary. A more complete documentation can be
+found as {\tt update.pdf} in the {\tt Documentation} subdirectory.
+
+{\bf New documentation:} It was decided to switch the documentation to using the {\tt
+ Doxygen} system (available from {\tt www.doxygen.org}), which can
+ generate the documentation in a variety of formats. To generate the
+ documentation, go into the {\tt Doc-doxygen} subdirectory and
+ execute {\tt make html}, {\tt make latex}, or {\tt make all} to
+ produce the corresponding version or both of them.
+
+{\color{red}
+ The documentation inserted into the source files accounts for most
+ of the formal changes in them. The general documentation is also
+ updated and present as ``Documentation.pdf''.
+}
+
+ {\bf Bug fixes:}
+\begin{enumerate}
+\item In the force database {\tt forces.data} two digits were
+ interchanged in the definition of SLy4d, leading to wrong results
+ for that force.
+
+\item If a restart is done for a two-body collision, the code changed
+ the number of fragments to {\tt nof}=1. The restart is then
+ initialized like a single-nucleus case with {\tt nof}=1. But two-body
+ analysis was activated only for {\tt nof}>1 such that it was absent
+ after restart.
+
+{\color{red}
+\item In the time-dependent mode, the wave functions were only save at
+ intervals of {\tt mprint} and {\tt mrest}, respectively. If a
+ calculation stops because of reaching the final distance or
+ fulfilling the convergence criterion, this may lead to a loss of
+ information, so that now both are done also in this event before the
+ job finishes.
+}
+
+\item The external field parameters were calculated directly from the
+ input in {\tt getin\_external}. Since this is called before the
+ fragment initialization is done, coefficients depending on proton or
+ neutron number will not be calculated correctly. For this reason,
+ the calculation of these coefficients is separated into a new
+ routine {\tt init\_external}, which is called directly before the
+ dynamic calculation starts.
+\end{enumerate}
+
+{\bf Array allocation:} It turned out that having the working arrays
+as automatic variables could cause problems, as they are allocated on
+the stack and the proper stack size must be calculated. Therefore in
+all cases where a larger array is concerned, it is now changed to {\tt
+ ALLOCATABLE} and allocated and deallocated as necessary.
+
+{\bf Elimination of ``guru'' mode of FFTW3} While the guru mode as
+defined in the FFTW3 package (see {\tt fftw.org}) offers an elegant
+formulation of complicated multidimensional transforms, it is not
+implemented in some support libraries like the Intel\textregistered\
+MKL. There is not much loss in speed when this is replaced by
+standard transforms with some explicit loops added where
+necessary. This affects the wave function transforms in the $y$ and
+$z$ direction.
+
+{\bf Enhancement of the makefile} In the previous version there were
+several versions of the makefile, which had to be edited by hand to
+use different compilers. This was reformulated using a more flexible
+file with various targets predefined. Thus to generate the executable
+code, it is sufficient to execute ``{\tt make} {\em target} `` in the
+{\tt Code} subdirectory, where {\em target} is one of the following:
+
+\begin{itemize}
+\item {\tt seq} : simple sequential version with the gfortran compiler.
+\item {\tt ifort}, {\tt ifort\_seq} : sequential version using the Intel compiler.
+\item {\tt omp} and {\tt ifort\_omp} produce the OpenMP version for
+ the gfortran and Intel compiler, respectively.
+\item {\tt mpi} : MPI version, which uses the compiler mpif90.
+\item {\tt mpi-omp} : MPI version also using OpenMP on each node.
+\item {\tt debug}, {\tt seq\_debug}, {\tt omp\_debug}, {\tt
+ mpi\_debug} : enable debugging mode for these cases. The first
+ three use the gfortran compiler.
+\item {\tt clean} : removes the generated object and module files.
+\item {\tt clean-exec} : same as {\tt clean} but removes the
+ executable files as well.
+\end{itemize}
+
+The generated executable files are called {\tt sky3d.seq}, {\tt
+ sky3d.ifort.seq}, {\tt sky3d.mpi}, {\tt sky3d.omp}, {\tt
+ sky3d.ifort.omp}, and {\tt sky3d.mpi-omp}, which should be
+self-explanatory. Thus several versions may be kept in the code
+directory, but a {\tt make clean} should be done before producing a
+new version to make sure the object and module files are correct.
+
+{\bf Skyrme-force compatibility for static restart:} the code normally
+checks that the Skyrme forces for all the input wave functions
+agree. It may be useful, however, to initialize a static calculation
+from results for a different Skyrme force. Therefore the consistency
+check was eliminated for the static case.
+
+{\bf Acceleration of the static calculations:} The basic parameters
+for the static iterations are (see Eq.~12 of the original paper) $x_0$
+(variable {\tt x0dmp}), which determines the size of the gradient
+step, and $E_0$ (variable {\tt e0dmp}) for the energy damping. These
+were read in and never changed throughout the calculation, except
+possibly through a restart. This can cause slow convergence, so that a
+method was developed to change {\tt x0dmp} during the iterations. The
+value from the input is now regarded as the minimum allowed one and
+saved in {\tt x0dmpmin}. At the start of the iterations, however, {\tt
+ x0dmp} is multiplied by 3 to attempt a faster convergence.
+
+The change in {\tt x0dmp} is then implemented by comparing the HF
+energy {\tt ehf} and the fluctuations {\tt efluct1} and {\tt efluct2}
+to the previous values saved in the variables {\tt ehfprev}, {\tt
+ efluct1prev}, and {\tt efluct2prev}. If the energy decreases or one
+of the fluctuations decreases by a factor of less than $1-10^{-5}$,
+{\tt x0dmp} is increased by a factor 1.005 to further speed up
+convergence. If none of these conditions holds, it is assumed that the
+step was too large and {\tt x0dmp} is reduced by a factor 0.8, but is
+never allowed to fall below {\tt x0dmpmin}.
+
+This whole process is turned on only if the input variable {\tt
+ tvaryx\_0} in the namelist "static" is {\tt .TRUE.} The default
+value is {\tt .FALSE.}
+
+A speedup up to a factor of 3 has been observed.
+
+{\bf External field expectation value} This value, which is printed in
+the file {\tt external.res}, was calculated from the spatial field
+including the (time-independent) amplitude {\tt amplq0}. The temporal
+Fourier transform then becomes quadratic in the amplitude, as the
+fluctuations in the density also grow linearly in {\tt amplq0}
+(provided the perturbation is not strong enough to take it into the
+nonlinear regime). This may be confusing and we therefore divided the
+expectation value by this factor. Note that if the external field is
+composed of a mixture of different multipoles (not coded presently),
+an overall scale factor should instead be used.
+
+{\color{red}
+{\bf Enhanced two-body analysis:} the analysis of the final two-body
+quantities after breakup included directly in the code was very
+simplified and actually it was superfluous to do this so frequently.
+This is replaced by a much more thorough analyis, including
+determination of the internal angular momenta of the fragments and of
+a quite accurate Coulomb energy. It is done only when the final
+separation is reached, while a simple determination of whether the
+fragments have separated and, if so, what their distance is, is
+performed every time step.
+}
+
+ {\bf Diagonalization} In the original program the diagonalization of
+ the Hamiltonian in the subroutine {\tt diagstep} was carried out
+ employing an eigenvalue decomposition using the LAPACK routine {\tt
+ ZHBEVD} which is optimized for banded matrices. This routine is
+ replaced in the update by the routine {\tt ZHEEVD} which is optimized
+ for general hermitian matrices. This change should result in a
+ moderate speed up for very large calculations. Furthermore the array
+ {\tt unitary}, previously a {\tt nstmax}$\times${\tt nstmax} array
+ has been reduced to a {\tt nlin}$\times${\tt nlin} array, where {\tt
+ nlin} is the number of wave functions of either neutrons or
+ protons. This array is now used as input and output for {\tt ZHEEVD}.
+
+{\bf New formulation of the spin-orbit term:}
+The action of the spin-orbit term has been corrected to comply with a
+strictly variational form. Starting from the spin-orbit energy
+\begin{equation}
+ E_{ls}
+ =
+ t_{ls}\int d^3r\,\vec{J}\cdot\nabla\rho
+ \quad,
+\end{equation}
+we obtain by variation with respect to the s.p. wavefunction $\psi^*$
+the spin-orbit term in the mean field in the symmetrized form
+\begin{equation}
+ \hat{h}_{ls}\psi
+ =
+ \frac{\mathrm{i}}{2}\left(
+ \vec{W}\cdot(\vec{\sigma}\times\nabla)\psi
+ +
+ \vec{\sigma}\cdot(\nabla\times(\vec{W}\psi)
+ \right)
+ \label{eq:hls-symm}
+\end{equation}
+where $\vec{W}=t_{ls}\nabla\rho$. In the previous version of the
+code, this term was simplified by applying the product rule for the
+$\nabla$ operator yielding
+\begin{eqnarray}
+ \frac{\mathrm{i}}{2}\left(
+ \vec{W}\cdot(\vec{\sigma}\times\nabla)\psi
+ +
+ \vec{\sigma}\cdot(\nabla\times(\vec{W}\psi)
+ \right)
+ &=&
+ \mathrm{i}\vec{W}\cdot(\vec{\sigma}\times\nabla)\psi
+ \quad.
+\end{eqnarray}
+Closer inspection reveals that the product rule is not perfectly
+fulfilled if the $\nabla$ operator is evaluated with finite Fourier
+transformation as inevitably done in the grid representation of the
+code. It turned out that this slight mismatch can accumulate to
+instabilities in TDHF runs over long times. Thus the variationally
+correct form (\ref{eq:hls-symm}) has been implemented now, although it
+leads to slightly longer running times.\\
+\\
+{\em Restrictions:} The reliability of the mean-field approximation is
+limited by the absence of hard nucleon-nucleon collisions. This limits
+the scope of applications to collision energies about a few MeV per
+nucleon above the Coulomb barrier and to relatively short interaction
+times. Similarly, some of the missing time-odd terms in the implementation
+of the Skyrme interaction may restrict the applications to even-even
+nuclei.\\
+\\
+{\em Unusual features:}\\
+The possibility of periodic boundary conditions and the highly
+flexible initialization make the code also suitable for astrophysical
+nuclear-matter applications.\\
+ \\
+{\em Running time:} The running time depends strongly on the size of
+the grid, the number of nucleons, and the duration of the
+collision. For a single-processor PC-type computer it can vary betweeneen
+a few minutes and weeks.\\
+ %Give an indication of the typical running time here.
+ \\
+\end{small}
+\bibliographystyle{elsarticle-num}
+\bibliography{update}
+\end{document}