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m_paw_finegrid.F90
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m_paw_finegrid.F90
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!!****m* ABINIT/m_paw_finegrid
!! NAME
!! m_paw_finegrid
!!
!! FUNCTION
!! This module contains a set of routines to compute various quantities
!! on the fine grid around a given atom.
!!
!! COPYRIGHT
!! Copyright (C) 2013-2022 ABINIT group (MT,FJ)
!! This file is distributed under the terms of the
!! GNU General Public License, see ~abinit/COPYING
!! or http://www.gnu.org/copyleft/gpl.txt .
!!
!! NOTES
!! FOR DEVELOPPERS: in order to preserve the portability of libPAW library,
!! please consult ~abinit/src/??_libpaw/libpaw-coding-rules.txt
!!
!! SOURCE
#include "libpaw.h"
MODULE m_paw_finegrid
USE_DEFS
USE_MSG_HANDLING
USE_MEMORY_PROFILING
use m_pawtab, only : pawtab_type
use m_paw_sphharm, only : initylmr
use m_paw_numeric, only : paw_jbessel,paw_splint,paw_uniform_splfit,paw_sort_dp
implicit none
private
!public procedures.
public :: pawgylm ! g_l(r-R)*Y_lm(r-R) (and derivatives)
public :: pawgylmg ! Fourier transform of g_l(r-R)*Y_lm(r-R), plane-waves case
public :: pawrfgd_fft ! r-R, plane-waves case
public :: pawrfgd_wvl ! r-R, wavelets case
public :: pawexpiqr ! exp(i.q.(r-R))
!declarations for the whole module (were needed to replace the statement functions)
!MG: Why this? Global variables are powerful but extremely DANGEROUS
integer,private,save :: lambda
real(dp),private,save :: pi_over_rshp,sigma
real(dp),private,save,allocatable :: alpha(:,:),qq(:,:)
!!***
CONTAINS
!===========================================================
!!***
!----------------------------------------------------------------------
!!****f* m_paw_finegrid/pawgylm
!! NAME
!! pawgylm
!!
!! FUNCTION
!! Compute g_l(r-R)*Y_lm(r-R) (and derivatives) on the fine (rectangular) grid
!! around one atom (g_l=radial shape function).
!! R is the position of the atom
!!
!! INPUTS
!! lm_size=number of lm components to be calculated
!! nfgd= number of (fine grid) FFT points in the paw sphere around current atom
!! optgr0= 1 if g_l(r-R)*Y_lm(r-R) are computed
!! optgr1= 1 if first derivatives of g_l(r-R)*Y_lm(r-R) are computed
!! optgr2= 1 if second derivatives of g_l(r-R)*Y_lm(r-R) are computed
!! pawtab <type(pawtab_type)>=paw tabulated starting data for current atom
!! rfgd(3,nfgd)= coordinates of r-R on the fine rect. grid around current atom
!!
!! OUTPUT
!! if (optgr0==1)
!! gylm(nfgd,lm_size)= g_l(r-R)*Y_lm(r-R) around current atom
!! if (optgr1==1)
!! gylmgr(3,nfgd,lm_size)= derivatives of g_l(r-R)*Y_lm(r-R) wrt cart. coordinates
!! if (optgr2==1)
!! gylmgr2(6,nfgd,lm_size)= second derivatives of g_l(r-R)*Y_lm(r-R) wrt cart. coordinates
!!
!! SOURCE
subroutine pawgylm(gylm,gylmgr,gylmgr2,lm_size,nfgd,optgr0,optgr1,optgr2,pawtab,rfgd)
!Arguments ---------------------------------------------
!scalars
integer,intent(in) :: lm_size,nfgd,optgr0,optgr1,optgr2
type(pawtab_type),intent(in) :: pawtab
!arrays
real(dp),intent(in) :: rfgd(:,:)
real(dp),intent(out) :: gylm(nfgd,optgr0*lm_size)
real(dp),intent(out) :: gylmgr(3,nfgd,optgr1*lm_size)
real(dp),intent(out) :: gylmgr2(6,nfgd,optgr2*lm_size)
!Local variables ------------------------------
!scalars
integer :: ic,ilm,izero,l_size,ll,normchoice,option,shape_type
real(dp) :: arg
real(dp) :: jbes1,jbes2,jbesp1,jbesp2,jbespp1,jbespp2,rcut
real(dp) :: splfact
logical :: compute_gr0,compute_gr1,compute_gr2
character(len=500) :: msg
!arrays
integer,allocatable :: isort(:)
real(dp),parameter :: ffact(1:9)=(/1._dp,3._dp,15._dp,105._dp,945._dp,10395._dp,&
& 135135._dp,2027025._dp,34459425._dp/)
real(dp),parameter :: toldev=tol3
real(dp) :: ss(3)
real(dp),allocatable :: cc(:,:),d2gfact(:,:),d2shpfuncnum(:,:),dgfact(:,:)
real(dp),allocatable :: dshpfuncnum(:,:),gfact(:,:)
real(dp),allocatable :: rnrm(:),rnrm_inv(:),rnrm_sort(:)
real(dp),allocatable :: shpfuncnum(:,:),work(:),ylmr(:,:),ylmrgr(:,:,:)
! *************************************************************************
if (optgr0==0.and.optgr1==0.and.optgr2==0) return
if (nfgd==0) return
!Compatibility test
!==========================================================
if (size(rfgd)/=3*nfgd) then
msg='rfgd array must be allocated at rfgd(3,nfgd)!'
LIBPAW_BUG(msg)
end if
if (pawtab%lcut_size>9) then
msg='l_size>10 forbidden!'
LIBPAW_BUG(msg)
end if
if (pawtab%shape_type==1.and.pawtab%shape_lambda<2) then
msg='Exponent lambda of gaussian shape function must be > 1!'
LIBPAW_ERROR(msg)
end if
!Initializations
!==========================================================
!Options for computation
compute_gr0=(optgr0==1.or.optgr1==1.or.optgr2==1)
compute_gr1=(optgr1==1.or.optgr2==1)
compute_gr2=(optgr2==1)
l_size=pawtab%lcut_size
!Norms of vectors around the atom
LIBPAW_ALLOCATE(rnrm,(nfgd))
izero=-1
do ic=1,nfgd
rnrm(ic)=sqrt(rfgd(1,ic)**2+rfgd(2,ic)**2+rfgd(3,ic)**2)
if (rnrm(ic)<=tol10) izero=ic ! Has to be consistent with initylmr !!
end do
!Initializations
if (optgr0==1) gylm=zero
if (optgr1==1) gylmgr=zero
if (optgr2==1) gylmgr2=zero
!Some definitions concerning shape function g_l(r)
shape_type=pawtab%shape_type
sigma=pawtab%shape_sigma;lambda=pawtab%shape_lambda
pi_over_rshp=pi/pawtab%rshp
rcut=tol12+pawtab%rshp
if (shape_type==3) then
LIBPAW_ALLOCATE(alpha,(2,l_size))
LIBPAW_ALLOCATE(qq,(2,l_size))
do ll=1,l_size
alpha(1:2,ll)=pawtab%shape_alpha(1:2,ll)
qq(1:2,ll)=pawtab%shape_q(1:2,ll)
end do
end if
!If needed, sort selected radii by increasing norm
if (shape_type==-1) then
LIBPAW_ALLOCATE(isort,(nfgd))
LIBPAW_ALLOCATE(rnrm_sort,(nfgd))
do ic=1,nfgd
isort(ic)=ic
end do
rnrm_sort(1:nfgd)=rnrm(1:nfgd)
call paw_sort_dp(nfgd,rnrm_sort,isort,tol16)
end if
!If shape function is "numeric", spline it onto selected radii
if (shape_type==-1) then
LIBPAW_ALLOCATE(work,(nfgd))
if (compute_gr0) then
LIBPAW_ALLOCATE(shpfuncnum,(nfgd,l_size))
do ll=1,l_size
call paw_splint(pawtab%mesh_size,pawtab%rad_for_spline,pawtab%shapefunc(:,ll),&
& pawtab%dshpfunc(:,ll,2),nfgd,rnrm_sort,work)
do ic=1,nfgd
shpfuncnum(isort(ic),ll)=work(ic)
end do
end do
end if
if(compute_gr1) then
LIBPAW_ALLOCATE(dshpfuncnum,(nfgd,l_size))
do ll=1,l_size
call paw_splint(pawtab%mesh_size,pawtab%rad_for_spline,pawtab%dshpfunc(:,ll,1),&
& pawtab%dshpfunc(:,ll,3),nfgd,rnrm_sort,work)
do ic=1,nfgd
dshpfuncnum(isort(ic),ll)=work(ic)
end do
end do
end if
if(compute_gr2) then
LIBPAW_ALLOCATE(d2shpfuncnum,(nfgd,l_size))
do ll=1,l_size
call paw_splint(pawtab%mesh_size,pawtab%rad_for_spline,pawtab%dshpfunc(:,ll,2),&
& pawtab%dshpfunc(:,ll,4),nfgd,rnrm_sort,work)
do ic=1,nfgd
d2shpfuncnum(isort(ic),ll)=work(ic)
end do
end do
end if
LIBPAW_DEALLOCATE(work)
end if
if (shape_type==-1) then
LIBPAW_DEALLOCATE(isort)
LIBPAW_DEALLOCATE(rnrm_sort)
end if
!If needed, compute limits at r=0 of shape function and derivatives
if (izero>0) then
LIBPAW_ALLOCATE(cc,(3,min(l_size,3)))
cc=zero
if (shape_type==-1) then
splfact=(pawtab%rad_for_spline(4)-pawtab%rad_for_spline(1))&
& /(pawtab%rad_for_spline(3)-pawtab%rad_for_spline(2))
end if
do ll=1,min(l_size,3)
! cc(2,l) is g_prime(0)
if (optgr0==1.or.optgr1==1.or.optgr2==1) then
if (shape_type==-1) then
ss(1:3)=pawtab%shapefunc(2:4,ll)/pawtab%rad_for_spline(2:4)**(ll-1)
cc(1,ll)=ss(3)+(ss(1)-ss(2))*splfact
else if (shape_type==1.or.shape_type==2) then
cc(1,ll)=one
else if (shape_type==3) then
cc(1,ll)=(alpha(1,ll)*qq(1,ll)**(ll-1) &
& +alpha(2,ll)*qq(2,ll)**(ll-1))/ffact(ll)
end if
cc(1,ll)=cc(1,ll)*pawtab%gnorm(ll)
end if
! cc(2,l) is g_prime(0)
if (optgr1==1.or.optgr2==1) then
if (shape_type==-1) then
ss(1:3)=(ss(1:3)-cc(1,ll))/pawtab%rad_for_spline(2:4)
cc(2,ll)=ss(3)+(ss(1)-ss(2))*splfact
else if (shape_type==1.and.lambda==1) then
cc(2,ll)=-one/sigma
else
cc(2,ll)=zero
end if
cc(2,ll)=cc(2,ll)*pawtab%gnorm(ll)
end if
! cc(3,l) is g_prime_prime(0)
if (optgr2==1) then
if (shape_type==-1) then
ss(1:3)=(ss(1:3)-cc(2,ll))/pawtab%rad_for_spline(2:4)
cc(3,ll)=two*(ss(3)+(ss(1)-ss(2))*splfact)
else if (shape_type==1) then
if (lambda==1) cc(3,ll)=one/sigma**2
if (lambda==2) cc(3,ll)=-two/sigma**2
if (lambda >2) cc(3,ll)=zero
else if (shape_type==2) then
cc(3,ll)=-(two/three)*pi_over_rshp**2
else if (shape_type==3) then
cc(3,ll)=-(alpha(1,ll)*qq(1,ll)**(ll+1) &
& +alpha(2,ll)*qq(2,ll)**(ll+1))/ffact(ll+1)
end if
cc(3,ll)=cc(3,ll)*pawtab%gnorm(ll)
end if
end do
end if
!Y_lm(r-R) calculation
!==========================================================
normchoice=1 ; option=max(optgr0,2*optgr1,3*optgr2)
if(compute_gr0) then
LIBPAW_ALLOCATE(ylmr,(l_size**2,nfgd))
end if
if(compute_gr1.and.(.not.compute_gr2)) then
LIBPAW_ALLOCATE(ylmrgr,(3,l_size**2,nfgd))
end if
if(compute_gr2) then
LIBPAW_ALLOCATE(ylmrgr,(9,l_size**2,nfgd))
end if
if (compute_gr0.and.(.not.compute_gr1).and.(.not.compute_gr2)) then
call initylmr(l_size,normchoice,nfgd,rnrm,option,rfgd,ylmr)
else
call initylmr(l_size,normchoice,nfgd,rnrm,option,rfgd,ylmr,ylmrgr)
end if
!gl(r) and derivatives calculation for l>=0
!==========================================================
!Compute gl(r), gl_prime(r)/r and (gl_prime_prime(r)-gl_prime(r)/r)/r**2
if (compute_gr0) then
LIBPAW_BOUND2_ALLOCATE(gfact,BOUNDS(1,nfgd),BOUNDS(0,l_size-1))
gfact(:,:)=zero
end if
if (compute_gr1) then
LIBPAW_BOUND2_ALLOCATE(dgfact,BOUNDS(1,nfgd),BOUNDS(0,l_size-1))
dgfact(:,:)=zero
end if
if (compute_gr2) then
LIBPAW_BOUND2_ALLOCATE(d2gfact,BOUNDS(1,nfgd),BOUNDS(0,l_size-1))
d2gfact(:,:)=zero
end if
if(compute_gr1) then
LIBPAW_ALLOCATE(rnrm_inv,(nfgd))
do ic=1,nfgd
if (ic/=izero) rnrm_inv(ic)=one/rnrm(ic)
end do
if (izero>0) rnrm_inv(izero)=zero
end if
!----- type -1 -----
if (shape_type==-1) then
if (compute_gr0) then
do ll=0,l_size-1
do ic=1,nfgd
if (rnrm(ic)<=rcut) then
gfact(ic,ll)=shpfuncnum(ic,ll+1)
end if
end do
end do
end if
if (compute_gr1) then
do ll=0,l_size-1
do ic=1,nfgd
if (rnrm(ic)<=rcut) then
dgfact(ic,ll)=dshpfuncnum(ic,ll+1)*rnrm_inv(ic)
end if
end do
end do
end if
if(compute_gr2) then
do ll=0,l_size-1
do ic=1,nfgd
if (rnrm(ic)<=rcut) then
d2gfact(ic,ll)=(d2shpfuncnum(ic,ll+1)-dgfact(ic,ll))*rnrm_inv(ic)**2
end if
end do
end do
end if
! ----- type 1 or 2 -----
else if (shape_type==1.or.shape_type==2) then
! FIRST COMPUTE FACTORS FOR l=0
if (optgr0==1.and.optgr1==0.and.optgr2==0) then
if (shape_type==1) then
do ic=1,nfgd
arg=rnrm(ic)
if (arg<toldev) then
gfact(ic,0)=shapefunc1_0(arg)
else if (arg<=rcut) then
gfact(ic,0)=shapefunc1(arg)
end if
end do
else ! shape_type==2
do ic=1,nfgd
arg=rnrm(ic)
if (arg<toldev) then
gfact(ic,0)=shapefunc2_0(arg)
else if (arg<=rcut) then
gfact(ic,0)=shapefunc2(arg)
end if
end do
end if
else if (optgr1==1.and.optgr2==0) then
if (shape_type==1) then
do ic=1,nfgd
arg=rnrm(ic)
if (arg<toldev) then
gfact(ic,0)=shapefunc1_0(arg)
if (lambda==2) then
dgfact(ic,0)=dshpfunc1_ovr_0_2(arg)
else ! lambda>2
dgfact(ic,0)=dshpfunc1_ovr_0(arg)
end if
else if (arg<=rcut) then
gfact(ic,0)=shapefunc1(arg)
dgfact(ic,0)=dshpfunc1(arg)*rnrm_inv(ic)
end if
end do
else ! shape_type==2
do ic=1,nfgd
arg=rnrm(ic)
if (arg<toldev) then
gfact(ic,0)=shapefunc2_0(arg)
dgfact(ic,0)=dshpfunc2_ovr_0(arg)
else if (arg<=rcut) then
gfact(ic,0)=shapefunc2(arg)
dgfact(ic,0)=dshpfunc2(arg)*rnrm_inv(ic)
end if
end do
end if
else if (optgr2==1) then
if (shape_type==1) then
do ic=1,nfgd
arg=rnrm(ic)
if (arg<toldev) then
gfact(ic,0)=shapefunc1_0(arg)
if (lambda==2) then
dgfact(ic,0)=dshpfunc1_ovr_0_2(arg)
d2gfact(ic,0)=d2shpfunc1_ovr2_0_2(arg)
else if (lambda==3) then
dgfact(ic,0)=dshpfunc1_ovr_0(arg)
if (ic/=izero) then
d2gfact(ic,0)=d2shpfunc1_ovr2_0_3(arg)
else
d2gfact(ic,0)=zero ! Diverging case
end if
else if (lambda==4) then
dgfact(ic,0)=dshpfunc1_ovr_0(arg)
d2gfact(ic,0)=d2shpfunc1_ovr2_0_4(arg)
else ! lambda>4
dgfact(ic,0)=dshpfunc1_ovr_0(arg)
d2gfact(ic,0)=d2shpfunc1_ovr2_0(arg)
end if
else if (arg<=rcut) then
gfact(ic,0)=shapefunc1(arg)
dgfact(ic,0)=dshpfunc1(arg)*rnrm_inv(ic)
d2gfact(ic,0)=(d2shpfunc1(arg)-dgfact(ic,0))*rnrm_inv(ic)**2
end if
end do
else ! shape_type==2
do ic=1,nfgd
arg=rnrm(ic)
if (arg<toldev) then
gfact(ic,0)=shapefunc2_0(arg)
dgfact(ic,0)=dshpfunc2_ovr_0(arg)
d2gfact(ic,0)=d2shpfunc2_ovr2_0(arg)
else if (arg<=rcut) then
gfact(ic,0)=shapefunc2(arg)
dgfact(ic,0)=dshpfunc2(arg)*rnrm_inv(ic)
d2gfact(ic,0)=(d2shpfunc2(arg)-dgfact(ic,0))*rnrm_inv(ic)**2
end if
end do
end if
end if
! THEN COMPUTE FACTORS FOR l>0 (from l=0)
if (compute_gr0) then
if (l_size>1) then
do ll=1,l_size-1
do ic=1,nfgd
gfact(ic,ll)=pawtab%gnorm(ll+1)*gfact(ic,0)*rnrm(ic)**ll
end do
end do
end if
end if
if (compute_gr1) then
if (l_size>1) then
do ic=1,nfgd
dgfact(ic,1)=pawtab%gnorm(2)*(gfact(ic,0)*rnrm_inv(ic)+dgfact(ic,0)*rnrm(ic))
end do
end if
if (l_size>2) then
do ic=1,nfgd
dgfact(ic,2)=pawtab%gnorm(3)*(two*gfact(ic,0)+dgfact(ic,0)*rnrm(ic)**2)
end do
end if
if (l_size>3) then
do ll=3,l_size-1
do ic=1,nfgd
dgfact(ic,ll)=pawtab%gnorm(ll+1) &
& *(dble(ll)*gfact(ic,0)*rnrm(ic)**(ll-2)+dgfact(ic,0)*rnrm(ic)**ll)
end do
end do
end if
end if
if (compute_gr2) then
if (l_size>1) then
do ic=1,nfgd
d2gfact(ic,1)=pawtab%gnorm(2) &
& *(-gfact(ic,0)*rnrm_inv(ic)**3+two*dgfact(ic,0)*rnrm_inv(ic)+d2gfact(ic,0)*rnrm(ic))
end do
end if
if (l_size>2) then
do ic=1,nfgd
d2gfact(ic,2)=pawtab%gnorm(3)*(four*dgfact(ic,0)+d2gfact(ic,0)*rnrm(ic)**2)
end do
end if
if (l_size>3) then
do ic=1,nfgd
d2gfact(ic,3)=pawtab%gnorm(4) &
& *(three*gfact(ic,0)*rnrm_inv(ic)+6._dp*dgfact(ic,0)*rnrm(ic)+ d2gfact(ic,0)*rnrm(ic)**3)
end do
end if
if (l_size>4) then
do ic=1,nfgd
d2gfact(ic,4)=pawtab%gnorm(5) &
& *(8._dp*gfact(ic,0)+8._dp*dgfact(ic,0)*rnrm(ic)**2+d2gfact(ic,0)*rnrm(ic)**4)
end do
end if
if (l_size>5) then
do ll=5,l_size-1
do ic=1,nfgd
d2gfact(ic,ll)=pawtab%gnorm(ll+1) &
& *(dble(ll*(ll-2))*gfact(ic,0)*rnrm(ic)**(ll-4) &
& +dble(2*ll)*dgfact(ic,0)*rnrm(ic)**(ll-2) &
& +d2gfact(ic,0)*rnrm(ic)**ll)
end do
end do
end if
end if
if (compute_gr0) gfact(:,0)=gfact(:,0)*pawtab%gnorm(1)
if (compute_gr1) dgfact(:,0)=dgfact(:,0)*pawtab%gnorm(1)
if (compute_gr2) d2gfact(:,0)=d2gfact(:,0)*pawtab%gnorm(1)
! ----- type 3 -----
else if (shape_type==3) then
if (optgr0==1.and.optgr1==0.and.optgr2==0) then
do ll=0,l_size-1
do ic=1,nfgd
arg=rnrm(ic)
if (arg<=rcut) then
call paw_jbessel(jbes1,jbesp1,jbespp1,ll,0,qq(1,1+ll)*arg)
call paw_jbessel(jbes2,jbesp2,jbespp2,ll,0,qq(2,1+ll)*arg)
gfact(ic,ll)=shapefunc3(jbes1,jbes2,ll)
end if
end do
end do
else if (optgr1==1.and.optgr2==0) then
do ll=0,l_size-1
do ic=1,nfgd
arg=rnrm(ic)
if (arg<=rcut) then
call paw_jbessel(jbes1,jbesp1,jbespp1,ll,1,qq(1,1+ll)*arg)
call paw_jbessel(jbes2,jbesp2,jbespp2,ll,1,qq(2,1+ll)*arg)
gfact(ic,ll)=shapefunc3(jbes1,jbes2,ll)
dgfact(ic,ll)=dshpfunc3(jbesp1,jbesp2,ll)*rnrm_inv(ic)
end if
end do
end do
if (izero>0.and.l_size>=1) dgfact(izero,0)=-(alpha(1,1)*qq(1,1)+alpha(2,1)*qq(2,1))/three
if (izero>0.and.l_size>=3) dgfact(izero,2)=two/15._dp*(alpha(1,1)*qq(1,1)+alpha(2,1)*qq(2,1))
! Note: for l=1, dgfact is diverging - d2gfact is diverging for l<4
else if (optgr2==1) then
do ll=0,l_size-1
do ic=1,nfgd
arg=rnrm(ic)
if (arg<=rcut) then
call paw_jbessel(jbes1,jbesp1,jbespp1,ll,2,qq(1,1+ll)*arg)
call paw_jbessel(jbes2,jbesp2,jbespp2,ll,2,qq(2,1+ll)*arg)
gfact(ic,ll)=shapefunc3(jbes1,jbes2,ll)
dgfact(ic,ll)=dshpfunc3(jbesp1,jbesp2,ll)*rnrm_inv(ic)
d2gfact(ic,ll)=(d2shpfunc3(jbespp1,jbespp2,ll)-dgfact(ic,ll))*rnrm_inv(ic)**2
end if
end do
end do
if (izero>0.and.l_size>=1) dgfact(izero,0)=-(alpha(1,1)*qq(1,1)+alpha(2,1)*qq(2,1))/three
if (izero>0.and.l_size>=3) dgfact(izero,2)=two/15._dp*(alpha(1,1)*qq(1,1)+alpha(2,1)*qq(2,1))
! Note: for l=1, dgfact is diverging - d2gfact is diverging for l<4
end if
end if
!g_l(r-R)*Y_lm(r-R) calculation
!==========================================================
if (optgr0==1) then
do ll=0,l_size-1
do ilm=ll**2+1,min((ll+1)**2,lm_size)
do ic=1,nfgd
gylm(ic,ilm)=gfact(ic,ll)*ylmr(ilm,ic)
end do
end do
end do
! Special value at r-R=0 (supposing shapefunc(r)->C.r**l when r->0)
if (izero>0) then
gylm(izero,1:lm_size)=zero
if (lm_size>=1) gylm(izero,1)=ylmr(1,izero)*cc(1,1)
end if
end if
!d/dr{g_l(r-R)*Y_lm(r-R)} calculation
!==========================================================
if(optgr1==1) then
do ll=0,l_size-1
do ilm=ll**2+1,min((ll+1)**2,lm_size)
do ic=1,nfgd
gylmgr(1:3,ic,ilm)=gfact(ic,ll)*ylmrgr(1:3,ilm,ic)&
& +dgfact(ic,ll)*rfgd(1:3,ic)*ylmr(ilm,ic)
end do
end do
end do
! Special values at r-R=0 (supposing shapefunc(r)->C.r**l when r->0)
if (izero>0) then
gylmgr(1:3,izero,1:lm_size)=zero
if (lm_size>=1) then
arg=cc(2,1)/sqrt(four_pi)
gylmgr(1:3,izero,1)=arg
end if
if (lm_size>=2) then
arg=cc(1,2)*sqrt(three/four_pi)
gylmgr(2,izero,2)=arg
if (lm_size>=3) gylmgr(3,izero,3)=arg
if (lm_size>=4) gylmgr(1,izero,4)=arg
end if
end if
end if
!d2/dridrj{g_l(r-R)*Y_lm(r-R)} calculation
!==========================================================
if(optgr2==1) then
do ll=0,l_size-1
do ilm=ll**2+1,min((ll+1)**2,lm_size)
do ic=1,nfgd
gylmgr2(1,ic,ilm)=gfact(ic,ll)*ylmrgr(4,ilm,ic) &
& +dgfact(ic,ll)*(ylmr(ilm,ic)+two*rfgd(1,ic)*ylmrgr(1,ilm,ic)) &
& +d2gfact(ic,ll)*ylmr(ilm,ic)*rfgd(1,ic)*rfgd(1,ic)
gylmgr2(2,ic,ilm)=gfact(ic,ll)*ylmrgr(5,ilm,ic) &
& +dgfact(ic,ll)*(ylmr(ilm,ic)+two*rfgd(2,ic)*ylmrgr(2,ilm,ic)) &
& +d2gfact(ic,ll)*ylmr(ilm,ic)*rfgd(2,ic)*rfgd(2,ic)
gylmgr2(3,ic,ilm)=gfact(ic,ll)*ylmrgr(6,ilm,ic) &
& +dgfact(ic,ll)*(ylmr(ilm,ic)+two*rfgd(3,ic)*ylmrgr(3,ilm,ic)) &
& +d2gfact(ic,ll)*ylmr(ilm,ic)*rfgd(3,ic)*rfgd(3,ic)
gylmgr2(4,ic,ilm)=gfact(ic,ll)*ylmrgr(7,ilm,ic) &
& +dgfact(ic,ll)*(rfgd(3,ic)*ylmrgr(2,ilm,ic)+rfgd(2,ic)*ylmrgr(3,ilm,ic)) &
& +d2gfact(ic,ll)*ylmr(ilm,ic)*rfgd(3,ic)*rfgd(2,ic)
gylmgr2(5,ic,ilm)=gfact(ic,ll)*ylmrgr(8,ilm,ic) &
& +dgfact(ic,ll)*(rfgd(3,ic)*ylmrgr(1,ilm,ic)+rfgd(1,ic)*ylmrgr(3,ilm,ic)) &
& +d2gfact(ic,ll)*ylmr(ilm,ic)*rfgd(3,ic)*rfgd(1,ic)
gylmgr2(6,ic,ilm)=gfact(ic,ll)*ylmrgr(9,ilm,ic) &
& +dgfact(ic,ll)*(rfgd(1,ic)*ylmrgr(2,ilm,ic)+rfgd(2,ic)*ylmrgr(1,ilm,ic)) &
& +d2gfact(ic,ll)*ylmr(ilm,ic)*rfgd(1,ic)*rfgd(2,ic)
end do
end do
end do
! Special values at r-R=0 (supposing shapefunc(r)->C.r**l when r->0)
if (izero>0) then
gylmgr2(1:6,izero,1:lm_size)=zero
if (lm_size>=1) then
arg=cc(3,1)/sqrt(four_pi)
gylmgr2(1:3,izero,1)=arg
end if
if (lm_size>=2) then
arg=cc(2,2)*sqrt(three/four_pi)
gylmgr2(2,izero,2)=two*arg
gylmgr2(4,izero,2)= arg
if (lm_size>=3) then
gylmgr2(1,izero,3)=two*arg
gylmgr2(3,izero,3)=two*arg
end if
if (lm_size>=4) then
gylmgr2(5,izero,4)=arg
gylmgr2(6,izero,4)=arg
end if
end if
if (lm_size>=5) then
arg=cc(1,3)*sqrt(15._dp/four_pi)
gylmgr2(6,izero,5)=arg
if (lm_size>=6) gylmgr2(4,izero,6)=arg
if (lm_size>=7) then
gylmgr2(1,izero,7)= -arg/sqrt3
gylmgr2(2,izero,7)= -arg/sqrt3
gylmgr2(3,izero,7)=two*arg/sqrt3
end if
if (lm_size>=8) gylmgr2(5,izero,8)=arg
if (lm_size>=9) then
gylmgr2(1,izero,9)= arg
gylmgr2(2,izero,9)=-arg
end if
end if
end if
end if
!Memory deallocation
!==========================================================
LIBPAW_DEALLOCATE(rnrm)
if (allocated(cc)) then
LIBPAW_DEALLOCATE(cc)
end if
if (compute_gr0) then
LIBPAW_DEALLOCATE(gfact)
end if
if (compute_gr1) then
LIBPAW_DEALLOCATE(dgfact)
end if
if (compute_gr2) then
LIBPAW_DEALLOCATE(d2gfact)
end if
if (compute_gr1) then
LIBPAW_DEALLOCATE(rnrm_inv)
end if
if (shape_type==3) then
LIBPAW_DEALLOCATE(alpha)
LIBPAW_DEALLOCATE(qq)
end if
if (compute_gr0) then
LIBPAW_DEALLOCATE(ylmr)
end if
if (compute_gr1) then
LIBPAW_DEALLOCATE(ylmrgr)
end if
if (shape_type==-1) then
if (compute_gr0) then
LIBPAW_DEALLOCATE(shpfuncnum)
end if
if (compute_gr1) then
LIBPAW_DEALLOCATE(dshpfuncnum)
end if
if (compute_gr2) then
LIBPAW_DEALLOCATE(d2shpfuncnum)
end if
end if
! -----------------------------------------------------------------
!Small functions related to analytical expression of shape function
CONTAINS
!!***
! ------------------------------------------------
!!****f* m_paw_finegrid/shapefunc1
! shapefunc1 is g(x) (gaussian)
function shapefunc1(arg)
real(dp) :: shapefunc1
real(dp),intent(in) :: arg
shapefunc1=exp(-(arg/sigma)**lambda)
end function shapefunc1
!!***
! ------------------------------------------------
!!****f* m_paw_finegrid/shapefunc1_0
! shapefunc1_0 is g(x) (gaussian) for small x
function shapefunc1_0(arg)
real(dp) :: shapefunc1_0
real(dp),intent(in) :: arg
shapefunc1_0=one-(arg/sigma)**lambda+half*(arg/sigma)**(2*lambda)-(arg/sigma)**(3*lambda)/6._dp
end function shapefunc1_0
!!***
! ------------------------------------------------
!!****f* m_paw_finegrid/shapefunc2
! shapefunc2 is g(x) (sinc2)
function shapefunc2(arg)
real(dp) :: shapefunc2
real(dp),intent(in) :: arg
shapefunc2=(sin(pi_over_rshp*arg)/(pi_over_rshp*arg))**2
end function shapefunc2
!!***
! ------------------------------------------------
!!****f* m_paw_finegrid/shapefunc2_0
! shapefunc2_0 is g(x) (sinc2) for small x
function shapefunc2_0(arg)
real(dp) :: shapefunc2_0
real(dp),intent(in) :: arg
shapefunc2_0=one-(pi_over_rshp*arg)**2/three+two*(pi_over_rshp*arg)**4/45._dp
end function shapefunc2_0
!!***
! ------------------------------------------------
!!****f* m_paw_finegrid/shapefunc3
! shapefunc3 is g(x) (Bessel)
function shapefunc3(jbes1,jbes2,argl)
integer,intent(in) :: argl
real(dp) :: shapefunc3
real(dp),intent(in) :: jbes1,jbes2
shapefunc3= alpha(1,1+argl)*jbes1+alpha(2,1+argl)*jbes2
end function shapefunc3
!!***
! ------------------------------------------------
!!****f* m_paw_finegrid/dshpfunc1
! dshpfunc1(x) is g_prime(x) (gaussian)
function dshpfunc1(arg)
real(dp) :: dshpfunc1
real(dp),intent(in) :: arg
dshpfunc1=-lambda/sigma*(arg/sigma)**(lambda-1)*exp(-(arg/sigma)**lambda)
end function dshpfunc1
!!***
! ------------------------------------------------
!!****f* m_paw_finegrid/dshpfunc1_ovr_0
! dshpfunc1_ovr_0(x) is g_prime(x)/x (gaussian) for small x and lambda>2
function dshpfunc1_ovr_0(arg)
real(dp) :: dshpfunc1_ovr_0
real(dp),intent(in) :: arg
dshpfunc1_ovr_0=-lambda/sigma**2*((arg/sigma)**(lambda-2)-(arg/sigma)**(2*lambda-2))
end function dshpfunc1_ovr_0
!!***
! ------------------------------------------------
!!****f* m_paw_finegrid/dshpfunc1_ovr_0_2
! dshpfunc1_ovr_0_2(x) is g_prime(x)/x (gaussian) for small x and lambda=2
function dshpfunc1_ovr_0_2(arg)
real(dp) :: dshpfunc1_ovr_0_2
real(dp),intent(in) :: arg
dshpfunc1_ovr_0_2=-two/sigma**2*(one-(arg/sigma)**2+half*(arg/sigma)**4)
end function dshpfunc1_ovr_0_2
!!***
! ------------------------------------------------
!!****f* m_paw_finegrid/dshpfunc2
! dshpfunc2(x) is g_prime(x) (sinc2)
function dshpfunc2(arg)
real(dp) :: dshpfunc2
real(dp),intent(in) :: arg
dshpfunc2=two*pi_over_rshp*sin(pi_over_rshp*arg)/(pi_over_rshp*arg)**3&
& *(pi_over_rshp*arg*cos(pi_over_rshp*arg)-sin(pi_over_rshp*arg))
end function dshpfunc2
!!***
! ------------------------------------------------
!!****f* m_paw_finegrid/dshpfunc2_ovr_0
! dshpfunc2_ovr_0(x) is g_prime(x)/x (sinc2) for small x
function dshpfunc2_ovr_0(arg)
real(dp) :: dshpfunc2_ovr_0
real(dp),intent(in) :: arg
dshpfunc2_ovr_0=-two*pi_over_rshp**2/3._dp+8._dp*pi_over_rshp**4*arg**2/45._dp
end function dshpfunc2_ovr_0
!!***
! ------------------------------------------------
!!****f* m_paw_finegrid/dshpfunc3
! dshpfunc3(x) is g_prime(x) (Bessel)
function dshpfunc3(jbesp1,jbesp2,argl)
integer :: argl
real(dp) :: dshpfunc3
real(dp),intent(in) :: jbesp1,jbesp2
dshpfunc3= alpha(1,1+argl)*qq(1,1+argl)*jbesp1 &
& +alpha(2,1+argl)*qq(2,1+argl)*jbesp2
end function dshpfunc3
!!***
! ------------------------------------------------
!!****f* m_paw_finegrid/d2shpfunc1
! d2shpfunc1(x) is g_prime_prime(x) (gaussian)
function d2shpfunc1(arg)
real(dp) :: d2shpfunc1
real(dp),intent(in) :: arg
d2shpfunc1=lambda/(sigma**2)*(lambda*(arg/sigma)**(2*lambda-2) &
& -(lambda-1)*(arg/sigma)**(lambda-2))*exp(-(arg/sigma)**lambda)
end function d2shpfunc1
!!***
! ------------------------------------------------
!!****f* m_paw_finegrid/d2shpfunc1_ovr2_0
! d2shpfunc1_ovr2_0(x) is (g_prime_prime(x)-g_prime(x)/x)/x**2 (gaussian) for small x and lambda>4
function d2shpfunc1_ovr2_0(arg)
real(dp) :: d2shpfunc1_ovr2_0
real(dp),intent(in) :: arg
d2shpfunc1_ovr2_0=-lambda/(sigma**4)*((lambda-2)*(arg/sigma)**(lambda-4) &
& -(lambda-1)*two*(arg/sigma)**(2*lambda-4))
end function d2shpfunc1_ovr2_0
!!***
! ------------------------------------------------
!!****f* m_paw_finegrid/d2shpfunc1_ovr2_0_2
! d2shpfunc1_ovr2_0_2(x) is (g_prime_prime(x)-g_prime(x)/x)/x**2 (gaussian) for small x and lambda==2
function d2shpfunc1_ovr2_0_2(arg)
real(dp) :: d2shpfunc1_ovr2_0_2
real(dp),intent(in) :: arg
d2shpfunc1_ovr2_0_2=four/(sigma**4)*(one-(arg/sigma)**2)
end function d2shpfunc1_ovr2_0_2
!!***
! ------------------------------------------------
!!****f* m_paw_finegrid/d2shpfunc1_ovr2_0_3
! d2shpfunc1_ovr2_0_3(x) is (g_prime_prime(x)-g_prime(x)/x)/x**2 (gaussian) for small x and lambda==3
function d2shpfunc1_ovr2_0_3(arg)
real(dp) :: d2shpfunc1_ovr2_0_3
real(dp),intent(in) :: arg
d2shpfunc1_ovr2_0_3=-three/arg/sigma**3+12._dp*arg**2/sigma**6-half*21._dp*arg**5/sigma**9
end function d2shpfunc1_ovr2_0_3
!!***
! ------------------------------------------------
!!****f* m_paw_finegrid/d2shpfunc1_ovr2_0_4
! d2shpfunc1_ovr2_0_4(x) is (g_prime_prime(x)-g_prime(x)/x)/x**2 (gaussian) for small x and lambda==4
function d2shpfunc1_ovr2_0_4(arg)
real(dp) :: d2shpfunc1_ovr2_0_4
real(dp),intent(in) :: arg
d2shpfunc1_ovr2_0_4=-8._dp/(sigma**4)*(one-three*(arg/sigma)**4)
end function d2shpfunc1_ovr2_0_4
!!***
! ------------------------------------------------
!!****f* m_paw_finegrid/d2shpfunc2
! d2shpfunc2(x) is g_prime_prime(x) (sinc2)
function d2shpfunc2(arg)
real(dp) :: d2shpfunc2
real(dp),intent(in) :: arg
d2shpfunc2=two/(pi_over_rshp**2*arg**4)* &
& (pi_over_rshp**2*arg**2*(cos(pi_over_rshp*arg))**2 &
& +(three-pi_over_rshp**2*arg**2)*(sin(pi_over_rshp*arg))**2 &
& -four*pi_over_rshp*arg*cos(pi_over_rshp*arg)*sin(pi_over_rshp*arg))
end function d2shpfunc2
!!***
! ------------------------------------------------
!!****f* m_paw_finegrid/d2shpfunc2_ovr2_0
! d2shpfunc2_ovr2_0(x) is (g_prime_prime(x)-g_prime(x)/x)/x**2 (sinc2) for small x
function d2shpfunc2_ovr2_0(arg)
real(dp) :: d2shpfunc2_ovr2_0
real(dp),intent(in) :: arg
d2shpfunc2_ovr2_0=16._dp/45._dp*pi_over_rshp**4-8._dp/105._dp*pi_over_rshp**6*arg**2 &
& +41._dp/6300._dp*pi_over_rshp**8*arg**4
end function d2shpfunc2_ovr2_0
!!***
! ------------------------------------------------
!!****f* m_paw_finegrid/d2shpfunc3
! d2shpfunc3(x) is g_prime_prime(x) (Bessel)
function d2shpfunc3(jbespp1,jbespp2,argl)
integer,intent(in) :: argl
real(dp) :: d2shpfunc3
real(dp),intent(in) :: jbespp1,jbespp2
d2shpfunc3= alpha(1,1+argl)*(qq(1,1+argl)**2)*jbespp1 &
& +alpha(2,1+argl)*(qq(2,1+argl)**2)*jbespp2
end function d2shpfunc3
! ------------------------------------------------
end subroutine pawgylm
!!***
!----------------------------------------------------------------------
!!****f* m_paw_finegrid/pawgylmg
!! NAME
!! pawgylmg
!!
!! FUNCTION
!! PAW: Compute Fourier transform of each g_l(r).Y_lm(r) function
!!
!! INPUTS
!! gprimd(3,3)=dimensional reciprocal space primitive translations
!! kg(3,npw)=integer coordinates of planewaves in basis sphere for this k point.
!! kpg(npw,nkpg)= (k+G) components (only if useylm=1)
!! kpt(3)=reduced coordinates of k point
!! lmax=1+max. value of l angular momentum
!! nkpg=second dimension of kpg_k (0 if useylm=0)
!! npw=number of planewaves in basis sphere
!! ntypat=number of types of atoms
!! pawtab(ntypat) <type(pawtab_type)>=paw tabulated starting data
!! ylm(npw,lmax**2)=real spherical harmonics for each G and k point
!!
!! OUTPUT
!! gylmg(npw,lmax**2,ntypat)=Fourier transform of each g_l(r).Y_lm(r) function
!!
!! SOURCE
subroutine pawgylmg(gprimd,gylmg,kg,kpg,kpt,lmax,nkpg,npw,ntypat,pawtab,ylm)
!Arguments ------------------------------------
!scalars
integer,intent(in) :: lmax,nkpg,npw,ntypat
!arrays
integer,intent(in) :: kg(3,npw)
real(dp),intent(in) :: gprimd(3,3),kpg(npw,nkpg),kpt(3)
real(dp),intent(in) :: ylm(npw,lmax**2)
type(pawtab_type),intent(in) :: pawtab(ntypat)
real(dp),intent(out) :: gylmg(npw,lmax**2,ntypat)
!Local variables-------------------------------
!scalars
integer :: ig,ilm,itypat,ll,l0,mm,mqgrid
real(dp) :: kpg1,kpg2,kpg3,kpgc1,kpgc2,kpgc3
!arrays
real(dp),allocatable :: glg(:),qgrid(:),kpgnorm(:),shpf(:,:),work(:)
! *************************************************************************
!Get |k+G|:
LIBPAW_ALLOCATE(kpgnorm,(npw))
if (nkpg<3) then
do ig=1,npw
kpg1=kpt(1)+dble(kg(1,ig));kpg2=kpt(2)+dble(kg(2,ig));kpg3=kpt(3)+dble(kg(3,ig))
kpgc1=kpg1*gprimd(1,1)+kpg2*gprimd(1,2)+kpg3*gprimd(1,3)
kpgc2=kpg1*gprimd(2,1)+kpg2*gprimd(2,2)+kpg3*gprimd(2,3)
kpgc3=kpg1*gprimd(3,1)+kpg2*gprimd(3,2)+kpg3*gprimd(3,3)
kpgnorm(ig)=sqrt(kpgc1*kpgc1+kpgc2*kpgc2+kpgc3*kpgc3)
end do
else
do ig=1,npw
kpgc1=kpg(ig,1)*gprimd(1,1)+kpg(ig,2)*gprimd(1,2)+kpg(ig,3)*gprimd(1,3)
kpgc2=kpg(ig,1)*gprimd(2,1)+kpg(ig,2)*gprimd(2,2)+kpg(ig,3)*gprimd(2,3)
kpgc3=kpg(ig,1)*gprimd(3,1)+kpg(ig,2)*gprimd(3,2)+kpg(ig,3)*gprimd(3,3)
kpgnorm(ig)=sqrt(kpgc1*kpgc1+kpgc2*kpgc2+kpgc3*kpgc3)
end do
end if
LIBPAW_ALLOCATE(glg,(npw))
LIBPAW_ALLOCATE(work,(npw))
!Loop over types of atoms
do itypat=1,ntypat