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|
!
! Copyright (C) 2002 FPMD group
! This file is distributed under the terms of the
! GNU General Public License. See the file `License'
! in the root directory of the present distribution,
! or http://www.gnu.org/copyleft/gpl.txt .
!
!
! ==================================================================
SUBROUTINE LSD_PADE(RHO,ETA,EC,VCA,VCB)
! ==--------------------------------------------------------------==
! == PADE APPROXIMATION ==
! ==--------------------------------------------------------------==
USE kinds, ONLY: DP
IMPLICIT NONE
! arguments
REAL(DP) :: RHO,ETA,EC,VCA,VCB
! locals
REAL(DP) :: RS,FS,DFS,DFSA,DFSB,A0P,A1P,A2P,A3P,B1P,B2P,B3P,B4P
REAL(DP) :: TOP,DTOP,TOPX,BOT,DBOT,BOTX,VC,DX
REAL(DP), PARAMETER :: A0=.4581652932831429d0, A1=2.217058676663745d0, &
A2=0.7405551735357053d0, A3=0.01968227878617998d0
REAL(DP), PARAMETER :: B1=1.0D0, B2=4.504130959426697d0, &
B3=1.110667363742916d0, B4=0.02359291751427506d0
REAL(DP), PARAMETER :: DA0=.119086804055547D0, DA1=.6157402568883345d0, &
DA2=.1574201515892867d0, DA3=.003532336663397157d0
REAL(DP), PARAMETER :: DB1=0.0d0, DB2=.2673612973836267d0, &
DB3=.2052004607777787d0, DB4=.004200005045691381d0
REAL(DP), PARAMETER :: RSFAC=.6203504908994000d0, FSFAC=1.92366105093153617d0
! ==--------------------------------------------------------------==
RS=RSFAC*RHO**(-1.d0/3.d0)
FS=FSFAC*((1.d0+ETA)**(4.d0/3.d0)+(1.d0-ETA)**(4.d0/3.d0)-2.d0)
DFS=FSFAC*4.d0/3.d0* ((1.d0+ETA)**(1.d0/3.d0)-(1.d0-ETA)**(1.d0/3.d0))
DFSA=DFS*(1.d0-ETA)
DFSB=DFS*(-1.d0-ETA)
A0P=A0+FS*DA0
A1P=A1+FS*DA1
A2P=A2+FS*DA2
A3P=A3+FS*DA3
B1P=B1+FS*DB1
B2P=B2+FS*DB2
B3P=B3+FS*DB3
B4P=B4+FS*DB4
TOP=A0P+RS*(A1P+RS*(A2P+RS*A3P))
DTOP=A1P+RS*(2.d0*A2P+RS*3.d0*A3P)
TOPX=DA0+RS*(DA1+RS*(DA2+RS*DA3))
BOT=RS*(B1P+RS*(B2P+RS*(B3P+RS*B4P)))
DBOT=B1P+RS*(2.d0*B2P+RS*(3.d0*B3P+RS*4.d0*B4P))
BOTX=RS*(DB1+RS*(DB2+RS*(DB3+RS*DB4)))
EC=-TOP/BOT
VC=EC+RS*(DTOP/BOT-TOP*DBOT/(BOT*BOT))/3.d0
DX=-(TOPX/BOT-TOP*BOTX/(BOT*BOT))
VCA=VC+DX*DFSA
VCB=VC+DX*DFSB
! ==--------------------------------------------------------------==
RETURN
END SUBROUTINE LSD_PADE
!______________________________________________________________________
subroutine ggablyp4(nnr,nspin,gradr,rhor,exc)
! _________________________________________________________________
! becke-lee-yang-parr gga
!
! exchange: becke, pra 38, 3098 (1988) but derived from
! pw91 exchange formula given in prb 48, 14944 (1993)
! by setting "b3" and "b4" to 0.0
! correlation: miehlich et al., cpl 157, 200 (1989)
! method by ja white & dm bird, prb 50, 4954 (1994)
!
! spin-polarized version by andras stirling 10/1998,
! using original gga program of alfredo pasquarello 22/09/1994
! and spin-unpolarized blyp routine of olivier parisel and
! alfredo pasquarello (02/1997)
!
USE kinds, ONLY : DP
USE constants, ONLY: pi, fpi
!
implicit none
! input
integer nspin, nnr
real(DP) gradr(nnr,3,nspin), rhor(nnr,nspin)
! output
! on output: rhor contains the exchange-correlation potential
real(DP) exc
! local
integer isdw, isup, isign, ir
!
real(DP) abo, agdr, agdr2, agr, agr2, agur, agur2, arodw, &
arodw2, aroe, aroe2, aroup, aroup2, ax
real(DP) byagdr, byagr, byagur, cden, cf, cl1, cl11, cl2, &
cl21, cl22, cl23, cl24, cl25, cl26, cl27, clyp, csum
real(DP) dddn, dexcdg, dexcdgd, dexcdgu, df1d, df1u, df2d, &
df2u, dfd, dfnum1d, dfnum1u, dfnum2d, dfnum2u, dfs, dfu, &
dfxdd, dfxdg, dfxdgd, dfxdgu, dfxdu, dilta, dilta119, dl1dn, &
dl1dnd, dl1dnu, dl2dd, dl2dg, dl2dgd, dl2dgu, dl2dn, &
dl2dnd, dl2dnd1, dl2dnu, dl2dnu1, dl2do, dlt, dodn, &
disign, dwsign, dys, dysd, dysu
real(DP) ex, excupdt, exd, exu, fac1, fac2, factor1, factor2, &
fx, fxd, fxden, fxdend, fxdenu, fxnum, fxnumd, fxnumu, fxu
real(DP) gkf, gkfd, gkfu, grdx, grdy, grdz, grux, gruy, gruz, &
grx, gry, grz
real(DP) omiga, pd, pi2, pider2, piexch, pu
real(DP) rhodw, rhoup, roe, roedth, roeth, roeuth, rometh
real(DP) s, s2, sd, sd2, sddw, sdup, su, su2, sysl, sysld, syslu
real(DP) t113, upsign, usign
real(DP) x1124, x113, x118, x13, x143, x19, x23, x43, &
x4718, x53, x672, x718, x772, x83
real(DP) ys, ysd, ysl, ysld, yslu, ysr, ysrd, ysru, ysu
!===========================================================================
real(DP) bb1, bb2, bb5, aa, bb, cc, dd, delt, eps
parameter(bb1=0.19644797d0,bb2=0.2742931d0,bb5=7.79555418d0, &
aa=0.04918d0, &
bb=0.132d0,cc=0.2533d0,dd=0.349d0,delt=1.0d-12,eps=1.0d-14)
!
!
x13=1.0d0/3.0d0
x19=1.0d0/9.0d0
x23=2.0d0/3.0d0
x43=4.0d0/3.0d0
x53=5.0d0/3.0d0
x83=8.0d0/3.0d0
x113=11.0d0/3.0d0
x4718=47.0d0/18.0d0
x718=7.0d0/18.0d0
x118=1.0d0/18.0d0
x1124=11.0d0/24.0d0
x143=14.0d0/3.0d0
x772=7.0d0/72.0d0
x672=6.0d0/72.0d0
!
! _________________________________________________________________
! derived parameters from pi
!
pi2=pi*pi
ax=-0.75d0*(3.0d0/pi)**x13
piexch=-0.75d0/pi
pider2=(3.0d0*pi2)**x13
cf=0.3d0*pider2*pider2
! _________________________________________________________________
! other parameters
!
t113=2.0d0**x113
!
rhodw=0.0d0
grdx=0.0d0
grdy=0.0d0
grdz=0.0d0
!
fac1=1.0d0
! _________________________________________________________________
! main loop
!
isup=1
isdw=2
do ir=1,nnr
rhoup=rhor(ir,isup)
grux=gradr(ir,1,isup)
gruy=gradr(ir,2,isup)
gruz=gradr(ir,3,isup)
if(nspin.eq.2) then
rhodw=rhor(ir,isdw)
grdx=gradr(ir,1,isdw)
grdy=gradr(ir,2,isdw)
grdz=gradr(ir,3,isdw)
else
rhodw=0.0d0
grdx =0.0d0
grdy =0.0d0
grdz =0.0d0
endif
roe=rhoup+rhodw
if(roe.eq.0.0) goto 100
aroup=abs(rhoup)
arodw=abs(rhodw)
aroe=abs(roe)
grx=grux + grdx
gry=gruy + grdy
grz=gruz + grdz
agur2=grux*grux+gruy*gruy+gruz*gruz
agur=sqrt(agur2)
agdr2=grdx*grdx+grdy*grdy+grdz*grdz
agdr=sqrt(agdr2)
agr2=grx*grx+gry*gry+grz*grz
agr=sqrt(agr2)
roeth=aroe**x13
rometh=1.0d0/roeth
gkf=pider2*roeth
sd=1.0d0/(2.0d0*gkf*aroe)
s=agr*sd
s2=s*s
! _________________________________________________________________
! exchange
!
if(nspin.eq.1) then
!
!
ysr=sqrt(1.0d0+bb5*bb5*s2)
ys=bb5*s+ysr
ysl=log(ys)*bb1
sysl=s*ysl
fxnum=1.0d0+sysl+bb2*s2
fxden=1.0d0/(1.0d0+sysl)
fx=fxnum*fxden
!
ex=ax*fx*roeth*aroe
!
! ### potential contribution ###
!
dys=bb5*(1.0d0+bb5*s/ysr)/ys
dfs=-fxnum*(ysl+bb1*s*dys)*fxden*fxden &
& +(ysl+bb1*s*dys+2.0d0*s*bb2)*fxden
dfxdu=(ax*roeth*x43)*(fx-dfs*s)
dfxdg=ax*roeth*dfs*sd
!
! ### end of potential contribution ###
!
else
!
roeuth=(2.0d0*aroup)**x13
roedth=(2.0d0*arodw)**x13
gkfu=pider2*roeuth*aroup
gkfd=pider2*roedth*arodw
upsign=sign(1.d0,gkfu-eps)
dwsign=sign(1.d0,gkfd-eps)
factor1=0.5d0*(1+upsign)/(gkfu+(1-upsign)*eps)
fac1=gkfu*factor1
factor2=0.5d0*(1+dwsign)/(gkfd+(1-dwsign)*eps)
fac2=gkfd*factor2
sdup=1.0d0/2.0d0*factor1
sddw=1.0d0/2.0d0*factor2
su=agur*sdup
su2=su*su
sd=agdr*sddw
sd2=sd*sd
!
ysru=sqrt(1.0d0+bb5*bb5*su2)
ysu=bb5*su+ysru
yslu=log(ysu)*bb1
syslu=su*yslu
fxnumu=1.0d0+syslu+bb2*su2
fxdenu=1.0d0/(1.0d0+syslu)
fxu=fxnumu*fxdenu
exu=piexch*2.0d0*gkfu*fxu*fac1
!
ysrd=sqrt(1.0d0+bb5*bb5*sd2)
ysd=bb5*sd+ysrd
ysld=log(ysd)*bb1
sysld=sd*ysld
fxnumd=1.0d0+sysld+bb2*sd2
fxdend=1.0d0/(1.0d0+sysld)
fxd=fxnumd*fxdend
exd=piexch*2.0d0*gkfd*fxd*fac2
!
ex=0.5d0*(exu+exd)
!
! ### potential contribution ###
!
dysu=bb5*(1.0d0+bb5*su/ysru)/ysu
pu=2.0d0*su*bb2
dfnum1u=yslu+bb1*su*dysu+pu
df1u=dfnum1u*fxdenu
dfnum2u=fxnumu*(yslu+bb1*su*dysu)
df2u=dfnum2u*fxdenu*fxdenu
dfu=df1u-df2u
dfxdu=ax*roeuth*x43*1.0d0*(fxu-dfu*su)*fac1
dfxdgu=ax*aroup*roeuth*dfu*sdup*fac1
!
dysd=bb5*(1.0d0+bb5*sd/ysrd)/ysd
pd=2.0d0*sd*bb2
dfnum1d=ysld+bb1*sd*dysd+pd
df1d=dfnum1d*fxdend
dfnum2d=fxnumd*(ysld+bb1*sd*dysd)
df2d=dfnum2d*fxdend*fxdend
dfd=df1d-df2d
dfxdd=ax*roedth*x43*1.0d0*(fxd-dfd*sd)*fac2
dfxdgd=ax*arodw*roedth*dfd*sddw*fac2
!
! ### end of potential contribution ###
!
endif
! _________________________________________________________________
! correlation lyp(aroe,aroup,arodw,agr,agur,agdr)
!
cden=1.0d0+dd*rometh
cl1=-aa/cden
!
omiga=exp(-cc*rometh)/cden/aroe**x113
dilta=rometh*(cc+dd/cden)
aroe2=aroe*aroe
abo=aa*bb*omiga
!
dodn=x13*omiga/aroe*(dilta-11.0d0)
dddn=x13*(dd*dd*aroe**(-x53)/cden/cden-dilta/aroe)
!
if(nspin.eq.1) then
!
cl1=cl1*aroe
!
cl21=4.0d0*cf*aroe**x83
cl22=(x4718-x718*dilta)*agr2
cl23=(2.5d0-x118*dilta)*agr2/2.0d0
cl24=(dilta-11.0d0)/9.0d0*agr2/4.0d0
cl25=x1124*agr2
!
cl2=-abo*aroe2*(0.25d0*(cl21+cl22-cl23-cl24)-cl25)
!
! ### potential contribution ###
!
dl1dnu=-aa*(1/cden+x13*dd*rometh/cden/cden)
!
dlt=x672+2.0d0*x772*dilta
dl2dn=-abo*aroe*(cf*x143*aroe**x83-dlt*agr2)
dl2do=cl2/omiga
dl2dd=abo*aroe2*x772*agr2
dl2dnu=dl2dn+dl2do*dodn+dl2dd*dddn
!
dl2dg=abo*aroe2*agr*dlt
!
! ### end of potential contribution ###
!
else
!
cl11=cl1*4.0d0/aroe
cl1=cl11*aroup*arodw
!
aroup2=aroup*aroup
arodw2=arodw*arodw
!
cl21=t113*cf*(aroup**x83+arodw**x83)
cl22=(x4718-x718*dilta)*agr2
cl23=(2.5d0-x118*dilta)*(agur2+agdr2)
dilta119=(dilta-11.0d0)/9.0d0
cl24=dilta119/aroe*(aroup*agur2+arodw*agdr2)
cl25=x23*aroe2*agr2
cl26=(x23*aroe2-aroup2)*agdr2
cl27=(x23*aroe2-arodw2)*agur2
!
csum=cl21+cl22-cl23-cl24
cl2=-abo*(aroup*arodw*csum-cl25+cl26+cl27)
!
! ### potential contribution ###
!
! *** cl1 has changed its form! ***
!
dl1dn=cl1/aroe*(x13*dd/cden*rometh-1.0d0)
dl1dnu=dl1dn+cl11*arodw
dl1dnd=dl1dn+cl11*aroup
!
dl2dnu1=arodw*csum+ &
& arodw*aroup*(t113*cf*x83*aroup**x53- &
& dilta119*arodw/aroe2*(agur2-agdr2))-x43*aroe*agr2+ &
& x23*agdr2*(2.0d0*arodw-aroup)+x43*aroe*agur2
dl2dnd1=aroup*csum+ &
& aroup*arodw*(t113*cf*x83*arodw**x53+ &
& dilta119*aroup/aroe2*(agur2-agdr2))-x43*aroe*agr2+ &
& x23*agur2*(2.0d0*aroup-arodw)+x43*aroe*agdr2
!
dl2do=cl2/omiga
dl2dd=-abo*aroup*arodw* &
& (-x718*agr2+x118*(agur2+agdr2)- &
& x19*(aroup*agur2+arodw*agdr2)/aroe)
!
dl2dnu=-abo*dl2dnu1+dl2do*dodn+dl2dd*dddn
dl2dnd=-abo*dl2dnd1+dl2do*dodn+dl2dd*dddn
!
dl2dg=-abo* &
& (aroup*arodw*2.0d0*(x4718-x718*dilta)*agr- &
& x43*aroe2*agr)
dl2dgu=-2.0d0*abo*agur*((x118*dilta-2.5d0- &
& dilta119*aroup/aroe)*aroup*arodw &
& +x23*aroe2-arodw2)
dl2dgd=-2.0d0*abo*agdr*((x118*dilta-2.5d0- &
& dilta119*arodw/aroe)*aroup*arodw &
& +x23*aroe2-aroup2)
!
endif
!
clyp=cl1+cl2
! _________________________________________________________________
! updating of xc-energy
!
excupdt=ex+clyp
!
exc=exc+excupdt
!
! _________________________________________________________________
! first part xc-potential construction
!
!
rhor(ir,isup)=dfxdu+(dl1dnu+dl2dnu)*fac1
isign=sign(1.d0,agr-delt)
byagr=0.5d0*(1+isign)/(agr+(1-isign)*delt)
!
if(nspin.eq.1) then
!
dexcdg=(dfxdg*aroe+dl2dg)*byagr
gradr(ir,1,isup)=grx*dexcdg
gradr(ir,2,isup)=gry*dexcdg
gradr(ir,3,isup)=grz*dexcdg
!
else
!
rhor(ir,isdw)=dfxdd+(dl1dnd+dl2dnd)*fac2
!
usign =sign(1.d0,agur-delt)
disign=sign(1.d0,agdr-delt)
byagur=0.5d0*(1+ usign)/(agur+(1- usign)*delt)
byagdr=0.5d0*(1+disign)/(agdr+(1-disign)*delt)
!
dexcdgu=(dfxdgu+dl2dgu)*byagur
dexcdgd=(dfxdgd+dl2dgd)*byagdr
dexcdg=dl2dg*byagr
!
gradr(ir,1,isup)=(dexcdgu*grux+dexcdg*grx)*fac1
gradr(ir,2,isup)=(dexcdgu*gruy+dexcdg*gry)*fac1
gradr(ir,3,isup)=(dexcdgu*gruz+dexcdg*grz)*fac1
gradr(ir,1,isdw)=(dexcdgd*grdx+dexcdg*grx)*fac2
gradr(ir,2,isdw)=(dexcdgd*grdy+dexcdg*gry)*fac2
gradr(ir,3,isdw)=(dexcdgd*grdz+dexcdg*grz)*fac2
!
endif
!
100 continue
end do
!
return
end subroutine ggablyp4
!
!______________________________________________________________________
subroutine ggapbe(nnr,nspin,gradr,rhor,excrho)
! _________________________________________________________________
! Perdew-Burke-Ernzerhof gga
! Perdew, et al. PRL 77, 3865, 1996
!
USE kinds, ONLY: DP
use constants, only: pi, fpi
!
implicit none
! input
integer nspin, nnr
real(DP) gradr(nnr,3,nspin), rhor(nnr,nspin)
! output: excrho: exc * rho ; E_xc = \int excrho(r) d_r
! output: rhor: contains the exchange-correlation potential
real(DP) excrho
! local
integer ir, icar, iss, isup, isdw, nspinx
real(DP) lim1, lim2
parameter ( lim1=1.d-8, lim2=1.d-8, nspinx=2 )
real(DP) zet, arho(nspinx), grad(3,nspinx), agrad(nspinx), &
arhotot, gradtot(3), agradtot, &
scl, scl1, wrkup, wrkdw, &
exrho(nspinx), dexdrho(nspinx), dexdg(nspinx), &
ecrho, decdrho(nspinx), decdg
!
! main loop
!
isup=1
isdw=2
do ir=1,nnr
!
arho(isup) = abs(rhor(ir,isup))
arhotot = arho(isup)
zet = 0.d0
do icar = 1, 3
grad(icar,isup) = gradr(ir,icar,isup)
gradtot(icar) = gradr(ir,icar,isup)
enddo
!
if (nspin.eq.2) then
arho(isdw) = abs(rhor(ir,isdw))
arhotot = abs(rhor(ir,isup)+rhor(ir,isdw))
do icar = 1, 3
grad(icar,isdw) = gradr(ir,icar,isdw)
gradtot(icar) = gradr(ir,icar,isup)+gradr(ir,icar,isdw)
enddo
zet = (rhor(ir,isup) - rhor(ir,isdw)) / arhotot
if (zet.ge. 1.d0) zet = 1.d0
if (zet.le.-1.d0) zet = -1.d0
endif
!
do iss = 1, nspin
agrad(iss) = sqrt( grad(1,iss)*grad(1,iss) + &
& grad(2,iss)*grad(2,iss) + &
& grad(3,iss)*grad(3,iss) )
agradtot = sqrt( gradtot(1)*gradtot(1) + &
& gradtot(2)*gradtot(2) + &
& gradtot(3)*gradtot(3) )
enddo
!
! _________________________________________________________________
! First it calculates the energy density excrho
! exrho: exchange term
! ecrho: correlation term
!
if ( nspin.eq.2 ) then
scl = 2.d0
scl1 = 0.5d0
else
scl = 1.d0
scl1 = 1.d0
endif
do iss = 1, nspin
if ( arho(iss).gt.lim1) then
call exchpbe( scl*arho(iss), scl*agrad(iss), &
& exrho(iss),dexdrho(iss),dexdg(iss))
excrho = excrho + scl1*exrho(iss)
else
dexdrho(iss) = 0.d0
dexdg(iss) = 0.d0
endif
enddo
if ( arhotot.gt.lim1) then
call ecorpbe( arhotot, agradtot, zet, ecrho, &
& decdrho(1), decdrho(2), decdg, nspin )
excrho = excrho + ecrho
else
decdrho(isup) = 0.d0
decdrho(isdw) = 0.d0
decdg = 0.d0
endif
! _________________________________________________________________
! Now it calculates the potential and writes it in rhor
! it uses the following variables:
! dexdrho = d ( ex*rho ) / d (rho)
! decdrho = d ( ec*rho ) / d (rho)
! dexdg = (d ( ex*rho ) / d (grad(rho)_i)) * agrad / grad_i
! decdg = (d ( ec*rho ) / d (grad(rho)_i)) * agrad / grad_i
! gradr here is used as a working array
!
! _________________________________________________________________
! first part of the xc-potential : D(rho*exc)/D(rho)
!
do iss = 1, nspin
rhor(ir,iss) = dexdrho(iss) + decdrho(iss)
enddo
!
! gradr = D(rho*exc)/D(|grad rho|) * (grad rho) / |grad rho|
!
do iss = 1, nspin
do icar = 1,3
wrkup =0.d0
wrkdw =0.d0
if (agrad(iss).gt.lim2) &
& wrkup = dexdg(iss)*grad(icar,iss)/agrad(iss)
if (agradtot.gt.lim2) &
& wrkdw = decdg*gradtot(icar)/agradtot
gradr(ir,icar,iss) = wrkup + wrkdw
enddo
enddo
!
end do
!
return
end subroutine ggapbe
!
!______________________________________________________________________
subroutine exchpbe(rho,agrad,ex,dexdrho,dexdg)
! _________________________________________________________________
!
! Perdew-Burke-Ernzerhof gga, Exchange term:
! Calculates the exchange energy density and the two functional derivative
! that will be used to calculate the potential
!
USE kinds, ONLY: DP
implicit none
! input
! input rho: charge density
! input agrad: abs(grad rho)
real(DP) rho, agrad
! ouput
! output ex: Ex[rho,grad_rho] = \int ex dr
! output dexdrho: d ex / d rho
! output dexdg: d ex / d grad_rho(i) = dexdg*grad_rho(i)/abs(grad_rho)
real(DP) ex, dexdrho, dexdg
! local
real(DP) thrd, thrd4, pi32td, ax, al, um, uk, ul
parameter(thrd=.33333333333333333333d0,thrd4=4.d0/3.d0)
parameter(pi32td=3.09366772628014d0) ! pi32td=(3.d0*pi*pi)**0.333d0
parameter(al=0.161620459673995d0) ! al=1.0/(2.0*(pi32)**0.333d0)
parameter(ax=-0.738558766382022405884230032680836d0)
parameter(um=0.2195149727645171d0,uk=0.8040d0,ul=um/uk)
!
real(DP) rhothrd, exunif, dexunif, kf, s, s2, p0, fxpbe, fs
!----------------------------------------------------------------------
! construct LDA exchange energy density
!
rhothrd = rho**thrd
dexunif = ax*rhothrd
exunif = rho*dexunif
!----------------------------------------------------------------------
! construct PBE enhancement factor
!
kf = pi32td*rhothrd
s = agrad/(2.d0*kf*rho)
s2 = s*s
p0 = 1.d0 + ul*s2
fxpbe = 1.d0 + uk - uk/p0
ex = exunif*fxpbe
!----------------------------------------------------------------------
! now calculates the potential terms
!
! fs=(1/s)*d fxPBE/ ds
!
fs=2.d0*uk*ul/(p0*p0)
dexdrho = dexunif*thrd4*(fxpbe-s2*fs)
dexdg = ax*al*s*fs
!
return
end subroutine exchpbe
!----------------------------------------------------------------------
subroutine ecorpbe(rho,agrad,zet,ectot,decup,decdn,decdg,nspin)
! -----------------------------------------------------------------
!
! Adapted from the Official PBE correlation code. K. Burke, May 14, 1996.
!
! input: rho = rho_up + rho_down; total charge density
! input: agrad = abs( grad(rho) )
! input: zet = (rho_up-rho_down)/rho
! input: nspin
! output: ectot = ec*rho ---correlation energy density---
! output: decup = d ( ec*rho ) / d (rho_up)
! output: decdn = d ( ec*rho ) / d (rho_down)
! output: decdg = (d ( ec*rho ) / d (grad(rho)_i)) * agrad / grad_i
!----------------------------------------------------------------------
!----------------------------------------------------------------------
! References:
! [a] J.P.~Perdew, K.~Burke, and M.~Ernzerhof,
! {\sl Generalized gradient approximation made simple}, sub.
! to Phys. Rev.Lett. May 1996.
! [b] J. P. Perdew, K. Burke, and Y. Wang, {\sl Real-space cutoff
! construction of a generalized gradient approximation: The PW91
! density functional}, submitted to Phys. Rev. B, Feb. 1996.
! [c] J. P. Perdew and Y. Wang, Phys. Rev. B {\bf 45}, 13244 (1992).
!----------------------------------------------------------------------
!----------------------------------------------------------------------
USE kinds, ONLY: DP
USE constants, ONLY: pi
implicit none
real(DP) rho, agrad, zet, ectot, decup, decdn, decdg
integer nspin
real(DP) pi32, alpha, thrd, thrdm, thrd2, sixthm, thrd4, &
gam, fzz, gamma, bet, delt, eta
! thrd*=various multiples of 1/3
! numbers for use in LSD energy spin-interpolation formula, [c](9).
! gam= 2^(4/3)-2
! fzz=f''(0)= 8/(9*gam)
! numbers for construction of PBE
! gamma=(1-log(2))/pi^2
! bet=coefficient in gradient expansion for correlation, [a](4).
! eta=small number to stop d phi/ dzeta from blowing up at
! |zeta|=1.
parameter(pi32=29.608813203268075856503472999628d0)
parameter(alpha=1.91915829267751300662482032624669d0)
parameter(thrd=1.d0/3.d0,thrdm=-thrd,thrd2=2.d0*thrd)
parameter(sixthm=thrdm/2.d0)
parameter(thrd4=4.d0*thrd)
parameter(gam=0.5198420997897463295344212145565d0)
parameter(fzz=8.d0/(9.d0*gam))
parameter(gamma=0.03109069086965489503494086371273d0)
parameter(bet=0.06672455060314922d0,delt=bet/gamma)
parameter(eta=1.d-12)
real(DP) g, fk, rs, sk, twoksg, t
real(DP) rtrs, eu, eurs, ep, eprs, alfm, alfrsm, z4, f, ec
real(DP) ecrs, fz, eczet, comm, vcup, vcdn, g3, pon, b, b2, t2, t4
real(DP) q4, q5, h, g4, t6, rsthrd, gz, fac
real(DP) bg, bec, q8, q9, hb, hrs, hz, ht, pref
!----------------------------------------------------------------------
if (nspin.eq.1) then
g=1.d0
else
g=((1.d0+zet)**thrd2+(1.d0-zet)**thrd2)*0.5d0
endif
fk=(pi32*rho)**thrd
rs=alpha/fk
sk=sqrt(4.d0*fk/pi)
twoksg=2.d0*sk*g
t=agrad/(twoksg*rho)
!----------------------------------------------------------------------
!----------------------------------------------------------------------
! find LSD energy contributions, using [c](10) and Table I[c].
! eu=unpolarized LSD correlation energy
! eurs=deu/drs
! ep=fully polarized LSD correlation energy
! eprs=dep/drs
! alfm=-spin stiffness, [c](3).
! alfrsm=-dalpha/drs
! f=spin-scaling factor from [c](9).
! construct ec, using [c](8)
rtrs=dsqrt(rs)
call gcor2(0.0310907d0,0.21370d0,7.5957d0,3.5876d0,1.6382d0, &
& 0.49294d0,rtrs,eu,eurs)
if (nspin.eq.2) then
call gcor2(0.01554535d0,0.20548d0,14.1189d0,6.1977d0,3.3662d0, &
& 0.62517d0,rtrs,ep,eprs)
call gcor2(0.0168869d0,0.11125d0,10.357d0,3.6231d0,0.88026d0, &
& 0.49671d0,rtrs,alfm,alfrsm)
z4 = zet**4
f=((1.d0+zet)**thrd4+(1.d0-zet)**thrd4-2.d0)/gam
ec = eu*(1.d0-f*z4)+ep*f*z4-alfm*f*(1.d0-z4)/fzz
!----------------------------------------------------------------------
!----------------------------------------------------------------------
! LSD potential from [c](A1)
! ecrs = dec/drs [c](A2)
! eczet=dec/dzeta [c](A3)
! fz = df/dzeta [c](A4)
ecrs = eurs*(1.d0-f*z4)+eprs*f*z4-alfrsm*f*(1.d0-z4)/fzz
fz = thrd4*((1.d0+zet)**thrd-(1.d0-zet)**thrd)/gam
eczet = 4.d0*(zet**3)*f*(ep-eu+alfm/fzz)+fz*(z4*ep-z4*eu &
& -(1.d0-z4)*alfm/fzz)
comm = ec -rs*ecrs/3.d0-zet*eczet
vcup = comm + eczet
vcdn = comm - eczet
else
ecrs = eurs
ec = eu
vcup = ec -rs*ecrs/3.d0
endif
!----------------------------------------------------------------------
!----------------------------------------------------------------------
! PBE correlation energy
! g=phi(zeta), given after [a](3)
! delt=bet/gamma
! b=a of [a](8)
! g=((1.d0+zet)**thrd2+(1.d0-zet)**thrd2)/2.d0
g3 = g**3
pon=-ec/(g3*gamma)
b = delt/(dexp(pon)-1.d0)
b2 = b*b
t2 = t*t
t4 = t2*t2
q4 = 1.d0+b*t2
q5 = 1.d0+b*t2+b2*t4
h = g3*(bet/delt)*dlog(1.d0+delt*Q4*t2/Q5)
ectot = rho*(ec + h)
!----------------------------------------------------------------------
!----------------------------------------------------------------------
! energy done. Now the potential, using appendix e of [b].
t6 = t4*t2
rsthrd = rs/3.d0
fac = delt/b+1.d0
bec = b2*fac/(bet*g3)
q8 = q5*q5+delt*q4*q5*t2
q9 = 1.d0+2.d0*b*t2
hb = -bet*g3*b*t6*(2.d0+b*t2)/q8
hrs = -rsthrd*hb*bec*ecrs
ht = 2.d0*bet*g3*q9/q8
comm = h+hrs-7.d0*t2*ht/6.d0
if (nspin.eq.2) then
g4 = g3*g
bg = -3.d0*b2*ec*fac/(bet*g4)
gz=(((1.d0+zet)**2+eta)**sixthm- &
& ((1.d0-zet)**2+eta)**sixthm)/3.d0
hz = 3.d0*gz*h/g + hb*(bg*gz+bec*eczet)
pref = hz-gz*t2*ht/g
decup = vcup + comm + pref*( 1.d0 - zet)
decdn = vcdn + comm + pref*( -1.d0 - zet)
else
decup = vcup + comm
endif
decdg = t*ht/twoksg
!
return
end subroutine ecorpbe
!______________________________________________________________________
subroutine gcor2(a,a1,b1,b2,b3,b4,rtrs,gg,ggrs)
! _________________________________________________________________
! slimmed down version of GCOR used in PW91 routines, to interpolate
! LSD correlation energy, as given by (10) of
! J. P. Perdew and Y. Wang, Phys. Rev. B {\bf 45}, 13244 (1992).
! K. Burke, May 11, 1996.
!
USE kinds, ONLY : DP
implicit none
real(DP) a, a1, b1, b2, b3, b4, rtrs, gg, ggrs
real(DP) q0, q1, q2, q3
!
q0 = -2.d0*a*(1.d0+a1*rtrs*rtrs)
q1 = 2.d0*a*rtrs*(b1+rtrs*(b2+rtrs*(b3+b4*rtrs)))
q2 = dlog(1.d0+1.d0/q1)
gg = q0*q2
q3 = a*(b1/rtrs+2.d0*b2+rtrs*(3.d0*b3+4.d0*b4*rtrs))
ggrs = -2.d0*a*a1*q2-q0*q3/(q1*(1.d0+q1))
!
return
end subroutine gcor2
!
!______________________________________________________________________
subroutine ggapw(nnr,nspin,gradr,rhor,exc)
! _________________________________________________________________
! perdew-wang gga (PW91)
!
USE kinds, ONLY: DP
use constants, only: pi, fpi
!
implicit none
! input
integer nspin, nnr
real(DP) gradr(nnr,3,nspin), rhor(nnr,nspin)
! output
real(DP) exc
! local
integer isup, isdw, ir
real(DP) rhoup, rhodw, roe, aroe, rs, zeta
real(DP) grxu, gryu, grzu, grhou, grxd, gryd, grzd, grhod, grho
real(DP) ex, ec,vc, sc, v1x, v2x, v1c, v2c
real(DP) ecrs, eczeta
real(DP) exup, vcup, v1xup, v2xup, v1cup
real(DP) exdw, vcdw, v1xdw, v2xdw, v1cdw
real(DP), parameter:: pi34 = 0.75d0/pi, third = 1.d0/3.d0, &
small = 1.d-10
!
! _________________________________________________________________
! main loop
!
isup=1
isdw=2
exc=0.0d0
do ir=1,nnr
rhoup=rhor(ir,isup)
if(nspin.eq.2) then
rhodw=rhor(ir,isdw)
else
rhodw=0.0d0
end if
roe=rhoup+rhodw
aroe=abs(roe)
if (aroe.lt.small) then
rhor(ir,isup) =0.0d0
gradr(ir,1,isup)=0.0d0
gradr(ir,2,isup)=0.0d0
gradr(ir,3,isup)=0.0d0
if(nspin.eq.2) then
rhor(ir,isdw) =0.0d0
gradr(ir,1,isdw)=0.0d0
gradr(ir,2,isdw)=0.0d0
gradr(ir,3,isdw)=0.0d0
end if
go to 100
end if
grxu =gradr(ir,1,isup)
gryu =gradr(ir,2,isup)
grzu =gradr(ir,3,isup)
grhou=sqrt(grxu**2+gryu**2+grzu**2)
if(nspin.eq.2) then
grxd =gradr(ir,1,isdw)
gryd =gradr(ir,2,isdw)
grzd =gradr(ir,3,isdw)
grhod=sqrt(grxd**2+gryd**2+grzd**2)
else
grxd =0.0d0
gryd =0.0d0
grzd =0.0d0
grhod=0.0d0
endif
grho=sqrt((grxu+grxd)**2+(gryu+gryd)**2+(grzu+grzd)**2)
!
rs=(pi34/aroe)**third
if (nspin.eq.1) then
call exchpw91(aroe,grho,ex,v1x,v2x)
call pwlda(rs,ec,vc,ecrs)
call corpw91ns(rs,grho,ec,ecrs,sc,v1c,v2c)
exc = exc + roe*(ex+ec) + sc
rhor(ir,isup) = vc + v1x + v1c
!
! gradr = D(rho*exc)/D(|grad rho|) * (grad rho) / |grad rho|
!
gradr(ir,1,isup)=grxu*(v2x+v2c)
gradr(ir,2,isup)=gryu*(v2x+v2c)
gradr(ir,3,isup)=grzu*(v2x+v2c)
else
zeta=(rhoup-rhodw)/aroe
zeta=min(zeta, 1.d0)
zeta=max(zeta,-1.d0)
call exchpw91(2.d0*abs(rhoup),2.0d0*grhou,exup,v1xup,v2xup)
call exchpw91(2.d0*abs(rhodw),2.0d0*grhod,exdw,v1xdw,v2xdw)
call pwlsd(rs,zeta,ec,vcup,vcdw,ecrs,eczeta)
call corpw91(rs,zeta,grho,ec,ecrs,eczeta,sc,v1cup,v1cdw,v2c)
rhor(ir,isup) = vcup + v1xup + v1cup
rhor(ir,isdw) = vcdw + v1xdw + v1cdw
exc = exc+roe*(0.5d0*((1.d0+zeta)*exup+(1.d0-zeta)*exdw)+ec) &
+ sc
!
! gradr = D(rho*exc)/D(|grad rho|) * (grad rho) / |grad rho|
!
gradr(ir,1,isup)=grxu*(2.0d0*v2xup+v2c)+grxd*v2c
gradr(ir,2,isup)=gryu*(2.0d0*v2xup+v2c)+gryd*v2c
gradr(ir,3,isup)=grzu*(2.0d0*v2xup+v2c)+grzd*v2c
gradr(ir,1,isdw)=grxd*(2.0d0*v2xdw+v2c)+grxu*v2c
gradr(ir,2,isdw)=gryd*(2.0d0*v2xdw+v2c)+gryu*v2c
gradr(ir,3,isdw)=grzd*(2.0d0*v2xdw+v2c)+grzu*v2c
end if
100 continue
end do
!
return
end subroutine ggapw
!
!----------------------------------------------------------------------
subroutine exchpw91(rho,grho,ex,v1x,v2x)
!----------------------------------------------------------------------
!
! PW91 exchange for a spin-unpolarized electronic system
! Modified from the "official" PBE code of Perdew, Burke et al.
! input rho : density
! input grho: abs(grad rho)
! output: exchange energy per electron (ex) and potentials
! v1x = d(rho*exc)/drho
! v2x = d(rho*exc)/d|grho| * (1/|grho|)
!
USE kinds, ONLY : DP
USE constants, ONLY : pi
implicit none
! input
real(DP) rho, grho
! output
real(DP) ex, v1x, v2x
! local
real(DP) ex0, kf, s, s2, s4, f, fs, p0,p1,p2,p3,p4,p5,p6,p7
! parameters
real(DP) a1, a2, a3, a4, a, b1, bx, pi34, thrd, thrd4
parameter(a1=0.19645d0,a2=0.27430d0,a=7.7956d0,a4=100.d0)
! for becke exchange, set a3=b1=0
parameter(a3=0.15084d0,b1=0.004d0)
! pi34=3/(4pi) , bx=(3pi^2)^(1/3)
parameter(pi34=0.75d0/pi, bx=3.093667726d0, thrd=0.333333333333d0, &
thrd4=4.d0*thrd)
!
if (rho.lt.1.d-10) then
ex =0.0d0
v1x=0.0d0
v2x=0.0d0
end if
!
! kf=k_Fermi, ex0=Slater exchange energy
!
kf = bx*(rho**thrd)
ex0=-pi34*kf
if (grho.lt.1.d-10) then
ex =ex0
v1x=ex0*thrd4
v2x=0.0d0
end if
s = grho/(2.d0*kf*rho)
s2 = s*s
s4 = s2*s2
p0 = 1.d0/sqrt(1.d0+a*a*s2)
p1 = log(a*s+1.d0/p0)
p2 = exp(-a4*s2)
p3 = 1.d0/(1.d0+a1*s*p1+b1*s4)
p4 = 1.d0+a1*s*p1+(a2-a3*p2)*s2
! f is the enhancement factor
f = p3*p4
ex = ex0*f
! energy done. now the potential:
p5 = b1*s2-(a2-a3*p2)
p6 = a1*s*(p1+a*s*p0)
p7 = 2.d0*(a2-a3*p2)+2.d0*a3*a4*s2*p2-4.d0*b1*s2*f
! fs = (1/s) dF(s)/ds
fs = p3*(p3*p5*p6+p7)
v1x = ex0*thrd4*(f-s2*fs)
v2x = 0.5d0*ex0/kf*s*fs/grho
!
return
end subroutine exchpw91
!
!----------------------------------------------------------------------
subroutine corpw91ns(rs,grho,ec,ecrs,h,v1c,v2c)
!----------------------------------------------------------------------
!
! PW91 correlation (gradient correction term) - no spin case
! Modified from the "official" PBE code of Perdew, Burke et al.
!
! input rs: seitz radius
! input zeta: relative spin polarization
! input grho: abs(grad rho)
! input ec: Perdew-Wang correlation energy
! input ecrs: d(rho*ec)/d r_s
! output h : nonlocal part of correlation energy per electron
! output v1c: nonlocal parts of correlation potential
! v1c = d(rho*exc)/drho
! v2c = d(rho*exc)/d|grho|*(1/|grho|)
!
USE kinds, ONLY : DP
USE constants, ONLY : pi
implicit none
! input
real(DP) rs, grho, ec, ecrs
! output
real(DP) h, v1c, v2c
! local
real(DP) rho, t, ks, bet, delt, pon, b, b2, t2, t4, t6
real(DP) q4, q5, q6, q7, q8, q9, r0, r1, r2, r3, r4, rs2, rs3
real(DP) ccrs, rsthrd, fac, bec, coeff, cc
real(DP) h0, h0b, h0rs, h0t, h1, h1t, h1rs, hrs, ht
! parameters
real(DP) nu, cc0, cx, alf, c1, c2, c3, c4, c5, c6, a4, ax, pi34
parameter(nu=15.75592d0,cc0=0.004235d0,cx=-0.001667212d0)
parameter(c1=0.002568d0,c2=0.023266d0,c3=7.389d-6,c4=8.723d0)
parameter(c5=0.472d0,c6=7.389d-2,a4=100.d0, alf=0.09d0)
! ax=(4*1.9191583/pi)^(1/2), where k_F=1.9191583/r_s, k_s=boh*r_s^(1/2)
parameter(ax=1.5631853d0, pi34 = 0.75d0/pi)
!
!
rs2 = rs*rs
rs3 = rs2*rs
rho=pi34/rs3
! k_s=(4k_F/pi)^(1/2)
ks=ax/sqrt(rs)
! t=abs(grad rho)/(rho*2.*ks)
t=grho/(2.d0*rho*ks)
bet = nu*cc0
delt = 2.d0*alf/bet
pon = -delt*ec/bet
b = delt/(exp(pon)-1.d0)
b2 = b*b
t2 = t*t
t4 = t2*t2
t6 = t4*t2
q4 = 1.d0+b*t2
q5 = 1.d0+b*t2+b2*t4
q6 = c1+c2*rs+c3*rs2
q7 = 1.d0+c4*rs+c5*rs2+c6*rs3
cc = -cx + q6/q7
r0 = 0.663436444d0*rs
r1 = a4*r0
coeff = cc-cc0-3.d0*cx/7.d0
r2 = nu*coeff
r3 = exp(-r1*t2)
h0 = (bet/delt)*log(1.d0+delt*q4*t2/q5)
h1 = r3*r2*t2
h = (h0+h1)*rho
! energy done. now the potential:
ccrs = (c2+2.d0*c3*rs)/q7 - q6*(c4+2.d0*c5*rs+3.d0*c6*rs2)/q7**2
rsthrd = rs/3.d0
r4 = rsthrd*ccrs/coeff
fac = delt/b+1.d0
bec = b2*fac/bet
q8 = q5*q5+delt*q4*q5*t2
q9 = 1.d0+2.d0*b*t2
h0b = -bet*b*t6*(2.d0+b*t2)/q8
h0rs = -rsthrd*h0b*bec*ecrs
h0t = 2.d0*bet*q9/q8
h1rs = r3*r2*t2*(-r4+r1*t2/3.d0)
h1t = 2.d0*r3*r2*(1.d0-r1*t2)
hrs = h0rs+h1rs
ht = h0t+h1t
v1c = h0+h1+hrs-7.d0*t2*ht/6.d0
v2c = t*ht/(2.d0*ks*grho)
!
return
end subroutine corpw91ns
!
!----------------------------------------------------------------------
subroutine corpw91(rs,zeta,grho,ec,ecrs,eczeta,h,v1cup,v1cdn,v2c)
!----------------------------------------------------------------------
!
! PW91 correlation (gradient correction term)
! Modified from the "official" PBE code of Perdew, Burke et al.
!
! input rs: seitz radius
! input zeta: relative spin polarization
! input grho: abs(grad rho)
! input ec: Perdew-Wang correlation energy
! input ecrs: d(rho*ec)/d r_s ?
! input eczeta: d(rho*ec)/d zeta ?
! output h: nonlocal part of correlation energy per electron
! output v1cup,v1cdn: nonlocal parts of correlation potentials
! v1c** = d(rho*exc)/drho (up and down components)
! v2c = d(rho*exc)/d|grho|*(1/|grho|) (same for up and down)
!
USE kinds, ONLY : DP
USE constants, ONLY : pi
implicit none
! input
real(DP) rs, zeta, grho, ec, ecrs, eczeta
! output
real(DP) h, v1cup, v1cdn, v2c
! local
real(DP) rho, g, t, ks, gz, bet, delt, g3, g4, pon, b, b2, t2, t4, t6
real(DP) q4, q5, q6, q7, q8, q9, r0, r1, r2, r3, r4, rs2, rs3
real(DP) ccrs, rsthrd, fac, bg, bec, coeff, cc
real(DP) h0, h0b, h0rs, h0z, h0t, h1, h1t, h1rs, h1z
real(DP) hz, hrs, ht, comm, pref
! parameters
real(DP) nu, cc0, cx, alf, c1, c2, c3, c4, c5, c6, a4
real(DP) thrdm, thrd2, ax, eta, pi34
parameter(nu=15.75592d0,cc0=0.004235d0,cx=-0.001667212d0)
parameter(c1=0.002568d0,c2=0.023266d0,c3=7.389d-6,c4=8.723d0)
parameter(c5=0.472d0,c6=7.389d-2,a4=100.d0, alf=0.09d0)
parameter(thrdm=-0.333333333333d0,thrd2=0.666666666667d0)
! ax=(4*1.9191583/pi)^(1/2), where k_F=1.9191583/r_s, k_s=boh*r_s^(1/2)
parameter(ax=1.5631853d0, eta=1.d-12, pi34 = 0.75d0/pi )
!
!
if (grho.lt.1.d-10) then
h=0.0d0
v1cup=0.0d0
v1cdn=0.0d0
v2c=0.0d0
end if
rs2 = rs*rs
rs3 = rs2*rs
rho=pi34/rs3
g=((1.d0+zeta)**thrd2+(1.d0-zeta)**thrd2)/2.d0
! k_s=(4k_F/pi)^(1/2)
ks=ax/sqrt(rs)
! t=abs(grad rho)/(rho*2.*ks*g)
t=grho/(2.d0*rho*g*ks)
bet = nu*cc0
delt = 2.d0*alf/bet
g3 = g**3
g4 = g3*g
pon = -delt*ec/(g3*bet)
b = delt/(exp(pon)-1.d0)
b2 = b*b
t2 = t*t
t4 = t2*t2
t6 = t4*t2
q4 = 1.d0+b*t2
q5 = 1.d0+b*t2+b2*t4
q6 = c1+c2*rs+c3*rs2
q7 = 1.d0+c4*rs+c5*rs2+c6*rs3
cc = -cx + q6/q7
r0 = 0.663436444d0*rs
r1 = a4*r0*g4
coeff = cc-cc0-3.d0*cx/7.d0
r2 = nu*coeff*g3
r3 = dexp(-r1*t2)
h0 = g3*(bet/delt)*log(1.d0+delt*q4*t2/q5)
h1 = r3*r2*t2
h = (h0+h1)*rho
! energy done. now the potential:
ccrs = (c2+2.d0*c3*rs)/q7 - q6*(c4+2.d0*c5*rs+3.d0*c6*rs2)/q7**2
rsthrd = rs/3.d0
r4 = rsthrd*ccrs/coeff
! eta is a small quantity that avoids trouble if zeta=+1 or -1
gz = ((1.d0+zeta+eta)**thrdm - (1.d0-zeta+eta)**thrdm)/3.d0
fac = delt/b+1.d0
bg = -3.d0*b2*ec*fac/(bet*g4)
bec = b2*fac/(bet*g3)
q8 = q5*q5+delt*q4*q5*t2
q9 = 1.d0+2.d0*b*t2
h0b = -bet*g3*b*t6*(2.d0+b*t2)/q8
h0rs = -rsthrd*h0b*bec*ecrs
h0z = 3.d0*gz*h0/g + h0b*(bg*gz+bec*eczeta)
h0t = 2.d0*bet*g3*q9/q8
h1rs = r3*r2*t2*(-r4+r1*t2/3.d0)
h1z = gz*r3*r2*t2*(3.d0-4.d0*r1*t2)/g
h1t = 2.d0*r3*r2*(1.d0-r1*t2)
hrs = h0rs+h1rs
ht = h0t+h1t
hz = h0z+h1z
comm = h0+h1+hrs-7.d0*t2*ht/6.d0
pref = hz-gz*t2*ht/g
comm = comm-pref*zeta
v1cup = comm + pref
v1cdn = comm - pref
v2c = t*ht/(2.d0*ks*g*grho)
!
return
end subroutine corpw91
!----------------------------------------------------------------------
subroutine pwlda(rs,ec,vc,ecrs)
!----------------------------------------------------------------------
!
! uniform-gas, spin-unpolarised correlation of perdew and wang 1991
! input: rs seitz radius
! output: ec correlation energy per electron
! vc potential
! ecrs derivatives of ec wrt rs
!
USE kinds, ONLY : DP
implicit none
! input
real(DP) rs
! output
real(DP) ec, vc, ecrs
! local
real(DP) q0, rs12, q1, q2, q3
! parameters
real(DP) a, a1, b1, b2, b3, b4
parameter(a =0.0310907d0, a1=0.21370d0, b1=7.5957d0, &
b2=3.5876d0, b3=1.6382d0, b4=0.49294d0)
!
q0 = -2.d0*a*(1.d0+a1*rs)
rs12 = sqrt(rs)
q1 = 2.d0*a*rs12*(b1+rs12*(b2+rs12*(b3+b4*rs12)))
q2 = log(1.d0+1.d0/q1)
ec = q0*q2
q3 = a*(b1/rs12+2.d0*b2+3.d0*b3*rs12+2.d0*b4*2.d0*rs)
ecrs = -2.d0*a*a1*q2-q0*q3/(q1**2+q1)
vc = ec - rs*ecrs/3.d0
!
return
end subroutine pwlda
!----------------------------------------------------------------------
subroutine pwlsd(rs,zeta,ec,vcup,vcdn,ecrs,eczeta)
!----------------------------------------------------------------------
!
! uniform-gas correlation of perdew and wang 1991
! Modified from the "official" PBE code of Perdew, Burke et al.
! input: seitz radius (rs), relative spin polarization (zeta)
! output: correlation energy per electron (ec)
! up- and down-spin potentials (vcup,vcdn)
! derivatives of ec wrt rs (ecrs) & zeta (eczeta)
!
USE kinds, ONLY : DP
implicit none
! input
real(DP) rs, zeta
! output
real(DP) ec, vcup, vcdn, ecrs, eczeta
! local
real(DP) f, eu, ep, eurs, eprs, alfm, alfrsm, z4, fz, comm
real(DP) rs12, q0, q1, q2, q3
! parameters
real(DP) gam, fzz, thrd, thrd4
parameter(gam=0.5198421d0,fzz=1.709921d0)
parameter(thrd=0.333333333333d0,thrd4=1.333333333333d0)
!
real(DP) au, au1, bu1, bu2, bu3, bu4
parameter(au =0.0310907d0, au1=0.21370d0, bu1=7.5957d0, &
bu2=3.5876d0, bu3=1.6382d0, bu4=0.49294d0)
real(DP) ap, ap1, bp1, bp2, bp3, bp4
parameter(ap =0.01554535d0,ap1=0.20548d0, bp1=14.1189d0, &
bp2=6.1977d0, bp3=3.3662d0, bp4=0.62517d0 )
real(DP) am, am1, bm1, bm2, bm3, bm4
parameter(am =0.0168869d0, am1=0.11125d0, bm1=10.357d0, &
bm2=3.6231d0, bm3=0.88026d0, bm4=0.49671d0 )
!
rs12 = sqrt(rs)
!
q0 = -2.d0*au*(1.d0+au1*rs)
q1 = 2.d0*au*rs12*(bu1+rs12*(bu2+rs12*(bu3+bu4*rs12)))
q2 = log(1.d0+1.d0/q1)
eu = q0*q2
q3 = au*(bu1/rs12+2.d0*bu2+3.d0*bu3*rs12+2.d0*bu4*2.d0*rs)
eurs = -2.d0*au*au1*q2-q0*q3/(q1**2+q1)
!
q0 = -2.d0*ap*(1.d0+ap1*rs)
q1 = 2.d0*ap*rs12*(bp1+rs12*(bp2+rs12*(bp3+bp4*rs12)))
q2 = log(1.d0+1.d0/q1)
ep = q0*q2
q3 = ap*(bp1/rs12+2.d0*bp2+3.d0*bp3*rs12+2.d0*bp4*2.d0*rs)
eprs = -2.d0*ap*ap1*q2-q0*q3/(q1**2+q1)
!
q0 = -2.d0*am*(1.d0+am1*rs)
q1 = 2.d0*am*rs12*(bm1+rs12*(bm2+rs12*(bm3+bm4*rs12)))
q2 = log(1.d0+1.d0/q1)
! alfm is minus the spin stiffness alfc
alfm=q0*q2
q3 = am*(bm1/rs12+2.d0*bm2+3.d0*bm3*rs12+2.d0*bm4*2.d0*rs)
alfrsm=-2.d0*am*am1*q2-q0*q3/(q1**2+q1)
!
f = ((1.d0+zeta)**thrd4+(1.d0-zeta)**thrd4-2.d0)/gam
z4 = zeta**4
ec = eu*(1.d0-f*z4)+ep*f*z4-alfm*f*(1.d0-z4)/fzz
! energy done. now the potential:
ecrs = eurs*(1.d0-f*z4)+eprs*f*z4-alfrsm*f*(1.d0-z4)/fzz
fz = thrd4*((1.d0+zeta)**thrd-(1.d0-zeta)**thrd)/gam
eczeta = 4.d0*(zeta**3)*f*(ep-eu+alfm/fzz)+fz*(z4*ep-z4*eu &
& -(1.d0-z4)*alfm/fzz)
comm = ec -rs*ecrs/3.d0-zeta*eczeta
vcup = comm + eczeta
vcdn = comm - eczeta
!
return
end subroutine pwlsd
!
!______________________________________________________________________
subroutine ggapwold(nnr,nspin,gradr,rhor,exc)
! _________________________________________________________________
! perdew-wang gga
! as given in y-m juan & e kaxiras, prb 48, 14944 (1993)
! method by ja white & dm bird, prb 50, 4954 (1994)
! non-spin polarized case only
! _________________________________________________________________
! by alfredo pasquarello 22/09/1994
!
USE kinds, ONLY: DP
use constants, only: pi, fpi
!
implicit none
!
integer nspin, nnr
real(DP) gradr(nnr,3), rhor(nnr), exc
!
real(DP) bb1, bb2, bb3, bb4, bb5, alfa, beta, cc0, cc1, delt, &
c1, c2, c3, c4, c5, c6, c7, a, alfa1, bt1, bt2, bt3, bt4
parameter(bb1=0.19645d0,bb2=0.27430d0,bb3=-0.15084d0,bb4=0.004d0, &
bb5=7.7956d0,alfa=0.09d0,beta=0.0667263212d0,cc0=15.75592d0, &
cc1=0.003521d0,c1=0.001667d0,c2=0.002568d0,c3=0.023266d0,c4=7.389d-6, &
c5=8.723d0,c6=0.472d0,c7=7.389d-2,a=0.0621814d0,alfa1=0.2137d0, &
bt1=7.5957d0,bt2=3.5876d0,bt3=1.6382d0,bt4=0.49294d0,delt=1.0d-12)
real(DP) x13, x43, x76, pi2, ax, pider1, pider2, pider3, &
abder1, abder2, abder3
integer isign, ir
real(DP) &
aexp, abig, abig2, agr, aroe, byagr, ccr, ccrnum, ccrden, &
dfxd, dfxdg, dys, dfs, dh1ds, dh1dg, dh1d, dh1dt, dexcdg, &
dexcd, dh1drs, dh0da, dadec, decdrs, decd, dh0dg, dcdrs, &
dh0d, dh0dt, eclog, ecr, ecden, fx, fxnum, fxden, fxexp, &
gkf, grx, gry, grz, h0, h1, h0den, h0arg, h0num, &
roeth, roe, rs, rs12, rs2, rs3, rs32, s, sd, s2, s3, s4, &
sysl, t, td, t2, t3, t4, xchge, ys, ysl, ysr
!
!
if (nspin.ne.1) call errore('ggapw','spin not implemented',nspin)
!
x13=1.0d0/3.0d0
x43=4.0d0/3.0d0
x76=7.0d0/6.0d0
! _________________________________________________________________
! derived parameters from pi
!
pi2=pi*pi
ax=-0.75d0*(3.0d0/pi)**x13
pider1=(0.75d0/pi)**x13
pider2=(3.0d0*pi2)**x13
pider3=(3.0d0*pi2/16.0d0)**x13
! _________________________________________________________________
! derived parameters from alfa and beta
!
abder1=beta*beta/(2.0d0*alfa)
abder2=1.0d0/abder1
abder3=2.0d0*alfa/beta
! _________________________________________________________________
! main loop
!
do ir=1,nnr
roe=rhor(ir)
if(roe.eq.0.0) goto 100
aroe=abs(roe)
grx=gradr(ir,1)
gry=gradr(ir,2)
grz=gradr(ir,3)
agr=sqrt(grx*grx+gry*gry+grz*grz)
roeth=aroe**x13
rs= pider1/roeth
gkf=pider2*roeth
sd=1.0d0/(2.0d0*gkf*aroe)
s=agr*sd
s2=s*s
s3=s*s2
s4=s2*s2
! _________________________________________________________________
! exchange
!
ysr=sqrt(1.0d0+bb5*bb5*s2)
ys=bb5*s+ysr
ysl=log(ys)*bb1
sysl=s*ysl
fxexp=exp(-100.0d0*s2)
fxnum=1.0d0+sysl+(bb2+bb3*fxexp)*s2
fxden=1.0d0/(1.0d0+sysl+bb4*s4)
fx=fxnum*fxden
xchge=ax*fx*roeth
! _________________________________________________________________
! correlation ecr=ec(rho)
!
rs12=sqrt(rs)
rs32=rs12*rs
rs2=rs*rs
rs3=rs*rs2
ecden=a*(bt1*rs12+bt2*rs+bt3*rs32+bt4*rs2)
eclog=log(1.0d0+(1.0d0/ecden))
ecr=-a*(1.0d0+alfa1*rs)*eclog
! _________________________________________________________________
! correlation h0(t,ecr)
!
td=pider3*sd/rs12
t=agr*td
t2=t*t
t3=t*t2
t4=t2*t2
aexp=exp(-abder2*ecr)-1.0d0
abig=abder3/aexp
abig2=abig*abig
h0num=t2+abig*t4
h0den=1.0d0/(1.0d0+abig*t2+abig2*t4)
h0arg=1.0d0+abder3*h0num*h0den
h0=abder1*log(h0arg)
! _________________________________________________________________
! correlation h1(t,s,aroe)
!
ccrnum=c2+c3*rs+c4*rs2
ccrden=1.0d0/(1.0d0+c5*rs+c6*rs2+c7*rs3)
ccr=c1+ccrnum*ccrden
h1=cc0*(ccr-cc1)*t2*fxexp
! _________________________________________________________________
! updating of xc-energy
!
exc=exc+(xchge+ecr+h0+h1)*aroe
! _________________________________________________________________
! first part xc-potential from exchange
!
dys=bb5*(1.0d0+bb5*s/ysr)/ys
dfs=-fxnum*(ysl+bb1*s*dys+4.0d0*bb4*s3)*fxden*fxden &
& +(ysl+bb1*s*dys+2.0d0*s*(bb2+bb3*fxexp) &
& -200.0d0*s3*bb3*fxexp)*fxden
dfxd=(ax*roeth*x43)*(fx-dfs*s)
dfxdg=ax*roeth*dfs*sd
! _________________________________________________________________
! first part xc-potential from ecr
!
decdrs=-a*alfa1*eclog*rs + a*(1+alfa1*rs) &
& *a*(0.5d0*bt1*rs12+bt2*rs+1.5d0*bt3*rs32+2.0d0*bt4*rs2) &
& /(ecden*ecden+ecden)
decd=-x13*decdrs
! _________________________________________________________________
! first part xc-potential from h0
!
dh0da=abder1/h0arg*abder3*h0den* &
& (t4-h0num*h0den*(t2+2.0d0*abig*t4))
dadec=abder3*abder2*(aexp+1.0d0)/(aexp*aexp)
dh0d=dh0da*dadec*decd
dh0dt=abder1/h0arg*abder3*h0den &
& *(2.0d0*t+4.0d0*abig*t3-h0num*h0den*(2.0d0*abig*t+4.0d0*abig2*t3))
dh0d=dh0d-x76*t*dh0dt
dh0dg=dh0dt*td
! _________________________________________________________________
! first part xc-potential from h1
!
dcdrs=(c3+2.0d0*c4*rs-ccrnum*ccrden*(c5+2.0d0*c6*rs+3.0d0*c7*rs2)) &
& *ccrden
dh1drs=cc0*t2*fxexp*dcdrs
dh1d=-x13*rs*dh1drs
dh1dt=2.0d0*t*cc0*(ccr-cc1)*fxexp
dh1d=dh1d-x76*t*dh1dt
dh1ds=-200.0d0*s*cc0*(ccr-cc1)*t2*fxexp
dh1d=dh1d-x43*s*dh1ds
dh1dg=dh1dt*td+dh1ds*sd
! _________________________________________________________________
! first part of xc-potential: D(rho*exc)/D(rho)
!
dexcd=dfxd+decd+dh0d+dh1d+ecr+h0+h1
isign=sign(1.d0,agr-delt)
byagr=0.5d0*(1+isign)/(agr+(1-isign)*delt)
rhor(ir)=dexcd
!
! gradr = D(rho*exc)/D(|grad rho|) * (grad rho) / |grad rho|
!
dexcdg=(dfxdg+dh0dg+dh1dg)*aroe*byagr
gradr(ir,1)=gradr(ir,1)*dexcdg
gradr(ir,2)=gradr(ir,2)*dexcdg
gradr(ir,3)=gradr(ir,3)*dexcdg
100 continue
end do
!
return
end subroutine ggapwold
!-------------------------------------------------------------------------
subroutine expxc(nnr,nspin,rhor,exc)
!----------------------------------------------------------------------
!
! ceperley & alder's correlation energy
! after j.p. perdew & a. zunger prb 23, 5048 (1981)
!
! rhor contains rho(r) on input, vxc(r) on output
!
USE kinds, ONLY : DP
use constants, only: pi, fpi
!
implicit none
!
integer nnr, nspin
real(DP) rhor(nnr,nspin), exc
! local variables
integer ir, iflg, isup, isdw
real(DP) roe, aroe, rs, rsl, rsq, ecca, vcca, eccp, vccp, &
zeta, onemz, zp, zm, fz, dfzdz, exc1, vxc1, vxc2
! constants
real(DP) x76, x43, x13
parameter(x76=7.d0/6.d0, x43=4.d0/3.d0, x13=1.d0/3.d0)
real(DP) ax
parameter (ax = -0.916330586d0)
! Perdew and Zunger parameters
real(DP) ap, bp, cp, dp0, af, bf, cf, df, &
bp1, cp1, dp1, bf1, cf1, df1
parameter &
( ap=0.03110d0*2.0d0, bp=-0.0480d0*2.0d0, cp=0.0020d0*2.0d0, dp0=-0.0116d0*2.0d0 &
, af=0.01555d0*2.0d0, bf=-0.0269d0*2.0d0, cf=0.0007d0*2.0d0, df=-0.0048d0*2.0d0 &
, bp1=bp-ap/3.0d0, cp1=2.0d0*cp/3.0d0, dp1=(2.0d0*dp0-cp)/3.0d0 &
, bf1=bf-af/3.0d0, cf1=2.0d0*cf/3.0d0, df1=(2.0d0*df-cf)/3.0d0 )
real(DP) va(2), vb(2), vc(2), vd(2), vbt1(2), vbt2(2)
real(DP) a(2), b(2), c(2), d(2), g(2), b1(2), b2(2)
data va/ap ,af /, vb/bp1,bf1/, vc/cp1,cf1/, vd/dp1,df1/, &
vbt1/1.0529d0,1.3981d0/, vbt2/0.3334d0,0.2611d0/
data a/0.0622d0,0.0311d0/, b/-0.096d0,-0.0538d0/, c/0.0040d0,0.0014d0/, &
d/-0.0232d0,-0.0096d0/, b1/1.0529d0,1.3981d0/, b2/0.3334d0,0.2611d0/, &
g/-0.2846d0,-0.1686d0/
!
if (nspin.eq.1) then
!
! iflg=1: paramagnetic (unpolarised) results
!
iflg=1
do ir=1,nnr
roe=rhor(ir,1)
if(roe.lt.1.0d-30) goto 10
aroe=abs(roe)
rs= (3.d0/aroe/fpi)**x13
if(rs.le.1.d0) then
rsl=log(rs)
ecca= a(iflg)*rsl+ b(iflg)+ c(iflg)*rs*rsl+ d(iflg)*rs
vcca=va(iflg)*rsl+vb(iflg)+vc(iflg)*rs*rsl+vd(iflg)*rs
else
rsq=sqrt(rs)
ecca=g(iflg)/(1.d0+b1(iflg)*rsq+b2(iflg)*rs)
vcca=ecca*(1.d0+x76*vbt1(iflg)*rsq+x43*vbt2(iflg)*rs)/ &
& (1.d0+ vbt1(iflg)*rsq+ vbt2(iflg)*rs)
end if
exc1 = ( ax/rs + ecca )/2.d0
exc = exc + exc1*roe
rhor(ir,1)= ( x43*ax/rs + vcca )/2.d0
10 continue
end do
else
isup=1
isdw=2
do ir=1,nnr
roe=rhor(ir,isup)+rhor(ir,isdw)
if(roe.lt.1.0d-30) goto 20
aroe=abs(roe)
rs= (3.d0/aroe/fpi)**x13
zeta=abs(rhor(ir,isup)-rhor(ir,isdw))/aroe
zp = (1.d0+zeta)**x13
onemz=max(0.d0,1.d0-zeta)
zm = onemz**x13
fz= ((1.d0+zeta)*zp + onemz*zm - 2.d0)/ &
& (2.d0**x43 -2.d0)
dfzdz= x43*(zp - zm)/(2.d0**x43-2.d0)
!
! iflg=1: paramagnetic (unpolarised) results
! iflg=2: ferromagnetic ( polarised) results
!
if(rs.le.1.d0) then
rsl=log(rs)
ecca= a(1)*rsl+ b(1)+ c(1)*rs*rsl+ d(1)*rs
vcca=va(1)*rsl+vb(1)+vc(1)*rs*rsl+vd(1)*rs
eccp= a(2)*rsl+ b(2)+ c(2)*rs*rsl+ d(2)*rs
vccp=va(2)*rsl+vb(2)+vc(2)*rs*rsl+vd(2)*rs
else
rsq=sqrt(rs)
ecca=g(1)/(1.d0+b1(1)*rsq+b2(1)*rs)
vcca=ecca*(1.d0+x76*vbt1(1)*rsq+x43*vbt2(1)*rs)/ &
& (1.d0+ vbt1(1)*rsq+ vbt2(1)*rs)
eccp=g(2)/(1.d0+b1(2)*rsq+b2(2)*rs)
vccp=eccp*(1.d0+x76*vbt1(2)*rsq+x43*vbt2(2)*rs)/ &
& (1.d0+ vbt1(2)*rsq+ vbt2(2)*rs)
end if
! exchange part
exc1 = ax/rs*((1.d0+zeta)*zp+(1.d0-zeta)*zm)/2.d0
vxc1 = x43*ax/rs*zp
vxc2 = x43*ax/rs*zm
! correlation part
vxc1 = vxc1 + vcca + fz*(vccp-vcca) &
& + dfzdz*(eccp-ecca)*( 1.d0-zeta)
vxc2 = vxc2 + vcca + fz*(vccp-vcca) &
& + dfzdz*(eccp-ecca)*(-1.d0-zeta)
exc = exc + (exc1 + ecca+fz*(eccp-ecca))*roe/2.d0
rhor(ir,isup)=vxc1/2.d0
rhor(ir,isdw)=vxc2/2.d0
20 continue
end do
end if
return
end subroutine expxc
SUBROUTINE wrap_b88( rho, grho, sx, v1x, v2x )
USE kinds, ONLY: DP
IMPLICIT NONE
REAL(DP) :: rho, grho, sx, v1x, v2x
REAL(DP) :: b1 = 0.0042d0
REAL(DP) :: RHOA,RHOB,GRHOA,GRHOB, V1XA,V2XA,V1XB,V2XB
rhoa = 0.5d0 * rho
rhob = 0.5d0 * rho
grhoa = 0.25d0 * grho
grhob = 0.25d0 * grho
CALL LSD_B88(B1,RHOA,RHOB,GRHOA,GRHOB,sx,V1XA,V2XA,V1XB,V2XB)
v1x = V1XA
v2x = V2XA
END SUBROUTINE wrap_b88
SUBROUTINE wrap_glyp( rho, grho, sc, v1c, v2c )
USE kinds, ONLY: DP
USE corr_gga, ONLY: lsd_glyp
IMPLICIT NONE
REAL(DP) :: rho, grho, sc, v1c, v2c
REAL(DP) :: RA,RB,GRHOAA,GRHOAB,GRHOBB
REAL(DP) :: V1CA,V2CA,V1CB,V2CB,V2CAB
ra = rho * 0.5d0
rb = rho * 0.5d0
grhoaa = 0.25d0 * grho
grhobb = 0.25d0 * grho
grhoab = 0.25d0 * grho
CALL LSD_GLYP(RA,RB,GRHOAA,GRHOAB,GRHOBB,SC, &
V1CA,V2CA,V1CB,V2CB,V2CAB)
v1c = V1CA
v2c = 2.0d0*(v2ca+v2cb+v2cab*2.d0)*0.25d0
END SUBROUTINE wrap_glyp
! ==================================================================
SUBROUTINE LSD_B88(B1,RHOA,RHOB,GRHOA,GRHOB,sx,V1XA,V2XA,V1XB,V2XB)
! ==--------------------------------------------------------------==
! BECKE EXCHANGE: PRA 38, 3098 (1988)
USE kinds, ONLY: DP
IMPLICIT NONE
REAL(DP),PARAMETER :: OB3=1.D0/3.D0, SMALL=1.D-20
REAL(DP) :: xs, xs2, sa2b8, br1, br2, br4, ddd, gf, dgf, shm1, dd
REAL(DP) :: dd2, grhoa, grhob, sx, b1, rhoa, rhob, v2xb, aa, a
REAL(DP) :: v1xa, v2xa, v1xb
! ==--------------------------------------------------------------==
sx=0.0D0
V1XA=0.0D0
V2XA=0.0D0
V1XB=0.0D0
V2XB=0.0D0
IF(ABS(RHOA).GT.SMALL) THEN
AA = GRHOA
A = SQRT(AA)
BR1 = RHOA**OB3
BR2 = BR1*BR1
BR4 = BR2*BR2
XS = A/BR4
XS2 = XS*XS
SA2B8 = SQRT(1.0D0+XS2)
SHM1 = LOG(XS+SA2B8)
DD = 1.0D0 + 6.0D0*B1*XS*SHM1
DD2 = DD*DD
DDD = 6.0D0*B1*(SHM1+XS/SA2B8)
GF = -B1*XS2/DD
DGF = (-2.0D0*B1*XS*DD + B1*XS2*DDD)/DD2
sx = GF*BR4
V1XA = 4.d0/3.d0*BR1*(GF-XS*DGF)
V2XA = DGF/A
ENDIF
IF(ABS(RHOB).GT.SMALL) THEN
AA = GRHOB
A = SQRT(AA)
BR1 = RHOB**OB3
BR2 = BR1*BR1
BR4 = BR2*BR2
XS = A/BR4
XS2 = XS*XS
SA2B8 = SQRT(1.0D0+XS2)
SHM1 = LOG(XS+SA2B8)
DD = 1.0D0 + 6.0D0*B1*XS*SHM1
DD2 = DD*DD
DDD = 6.0D0*B1*(SHM1+XS/SA2B8)
GF = -B1*XS2/DD
DGF = (-2.0D0*B1*XS*DD + B1*XS2*DDD)/DD2
sx = sx+GF*BR4
V1XB = 4.d0/3.d0*BR1*(GF-XS*DGF)
V2XB = DGF/A
ENDIF
! ==--------------------------------------------------------------==
RETURN
END SUBROUTINE LSD_B88
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