xc_vbh.f90

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! Copyright (C) 2002-2005 F. Cricchio, J. K. Dewhurst and S. Sharma.
! This file is distributed under the terms of the GNU Lesser General Public
! License. See the file COPYING for license details.
 
!BOP
! !ROUTINE: xc_vbh
! !INTERFACE:
subroutine xc_vbh(n,rhoup,rhodn,ex,ec,vxup,vxdn,vcup,vcdn)
! !INPUT/OUTPUT PARAMETERS:
!   n     : number of density points (in,integer)
!   rhoup : spin-up charge density (in,real(n))
!   rhodn : spin-down charge density (in,real(n))
!   ex    : exchange energy density (out,real(n))
!   ec    : correlation energy density (out,real(n))
!   vxup  : spin-up exchange potential (out,real(n))
!   vxdn  : spin-down exchange potential (out,real(n))
!   vcup  : spin-up correlation potential (out,real(n))
!   vcdn  : spin-down correlation potential (out,real(n))
! !DESCRIPTION:
!   Spin-polarised exchange-correlation potential and energy functional of
!   von Barth and Hedin: {\it J. Phys. C} {\bf 5}, 1629 (1972). Note that the
!   implementation is in Rydbergs in order to follow the paper step by step, at
!   the end the potential and energy are converted to Hartree.
!
! !REVISION HISTORY:
!   Created September 2007 (F. Cricchio)
!EOP
!BOC
implicit none
! arguments
integer, intent(in) :: n
real(8), intent(in) :: rhoup(n),rhodn(n)
real(8), intent(out) :: ex(n),ec(n)
real(8), intent(out) :: vxup(n),vxdn(n)
real(8), intent(out) :: vcup(n),vcdn(n)
! local variables
integer i
real(8), parameter :: pi=3.1415926535897932385d0
real(8), parameter :: cp=0.0504d0
real(8), parameter :: cf=0.0254d0
real(8), parameter :: rp=30.d0
real(8), parameter :: rf=75.d0
real(8) alpha0,eps0_x,a,gamma
real(8) rup,rdn,r,rs,x,zf,zp
real(8) fx,fp,ff,epsp_x,mup_x
real(8) epsp_c,epsf_c,mup_c,muf_c,vc,tau_c
alpha0=(4.d0/(9.d0*pi))**(1.d0/3.d0)
eps0_x=(3.d0/2.d0)/(pi*alpha0)
a=2.d0**(-1.d0/3.d0)
gamma=(4.d0/3.d0)*a/(1.d0-a)
do i=1,n
  rup=rhoup(i); rdn=rhodn(i)
! total density
  r=rup+rdn
  if ((rup >= 0.d0).and.(rdn >= 0.d0).and.(r > 1.d-12)) then
! Wigner-Seitz radius in atomic units (a0=1)
    rs=(3.d0/(4.d0*pi*r))**(1.d0/3.d0)
    x=rup/r
    fx=(1.d0/(1.d0-a))*(x**(4.d0/3.d0)+(1.d0-x)**(4.d0/3.d0)-a)
    epsp_x=-eps0_x/rs
    mup_x=(4.d0/3.d0)*epsp_x
! exchange energy
    ex(i)=epsp_x+(1.d0/gamma)*mup_x*fx
    zp=rs/rp
    fp=(1.d0+zp**3)*log(1.d0+1.d0/zp)+0.5d0*zp-zp**2-1.d0/3.d0
    zf=rs/rf
    ff=(1.d0+zf**3)*log(1.d0+1.d0/zf)+0.5d0*zf-zf**2-1.d0/3.d0
    epsp_c=-cp*fp
    epsf_c=-cf*ff
    vc=gamma*(epsf_c-epsp_c)
! correlation energy
    ec(i)=epsp_c+(1.d0/gamma)*vc*fx
    mup_c=-cp*log(1.d0+rp/rs)
    muf_c=-cf*log(1.d0+rf/rs)
    tau_c=muf_c-mup_c-(4.d0/3.d0)*(epsf_c-epsp_c)
! exchange potential
    vxup(i)=mup_x*(2.d0*x)**(1.d0/3.d0)
    vxdn(i)=mup_x*(2.d0*(1.d0-x))**(1.d0/3.d0)
! correlation potential
    vcup(i)=vc*(2.d0*x)**(1.d0/3.d0)+mup_c-vc+tau_c*fx
    vcdn(i)=vc*(2.d0*(1.d0-x))**(1.d0/3.d0)+mup_c-vc+tau_c*fx
! convert from Rybergs to Hartree
    ex(i)=0.5d0*ex(i)
    ec(i)=0.5d0*ec(i)
    vxup(i)=0.5d0*vxup(i)
    vxdn(i)=0.5d0*vxdn(i)
    vcup(i)=0.5d0*vcup(i)
    vcdn(i)=0.5d0*vcdn(i)
  else
    ex(i)=0.d0
    ec(i)=0.d0
    vxup(i)=0.d0
    vxdn(i)=0.d0
    vcup(i)=0.d0
    vcdn(i)=0.d0
  end if
end do
end subroutine
!EOC