subroutine UVWN3_lda_correlation_individual_energy(nGrid,weight,rhow,rho,doNcentered,kappa,Ec) ! Compute VWN3 LDA correlation potential implicit none include 'parameters.h' ! Input variables integer,intent(in) :: nGrid double precision,intent(in) :: weight(nGrid) double precision,intent(in) :: rhow(nGrid,nspin) double precision,intent(in) :: rho(nGrid,nspin) logical,intent(in) :: doNcentered double precision,intent(in) :: kappa ! Local variables integer :: iG double precision :: ra,rb,r,raI,rbI,rI,rs,x,z double precision :: a_p,x0_p,xx0_p,b_p,c_p,x_p,q_p double precision :: a_f,x0_f,xx0_f,b_f,c_f,x_f,q_f double precision :: a_a,x0_a,xx0_a,b_a,c_a,x_a,q_a double precision :: dfzdz,dxdrs,dxdx_p,dxdx_f,dxdx_a,decdx_p,decdx_f,decdx_a double precision :: dzdr ,dfzdr ,drsdr ,decdr_p ,decdr_f ,decdr_a, decdr double precision :: ec_z,ec_p,ec_f,ec_a double precision :: fz,d2fz double precision :: Ecrr(nsp) double precision :: EcrI(nsp) double precision :: EcrrI(nsp) ! Output variables double precision :: Ec(nsp) ! Parameters of the functional a_p = +0.0621814d0/2d0 x0_p = -0.409286d0 b_p = +13.0720d0 c_p = +42.7198d0 a_f = +0.0621814d0/4d0 x0_f = -0.743294d0 b_f = +20.1231d0 c_f = +101.578d0 a_a = -1d0/(6d0*pi**2) x0_a = -0.0047584D0 b_a = +1.13107d0 c_a = +13.0045d0 ! Initialization Ec(:) = 0d0 Ecrr(:) = 0d0 EcrI(:) = 0d0 EcrrI(:) = 0d0 do iG=1,nGrid ra = max(0d0,rhow(iG,1)) rb = max(0d0,rhow(iG,2)) raI = max(0d0,rho(iG,1)) rbI = max(0d0,rho(iG,2)) ! spin-up contribution r = ra rI = raI if(r > threshold) then rs = (4d0*pi*r/3d0)**(-1d0/3d0) x = sqrt(rs) x_f = x*x + b_f*x + c_f xx0_f = x0_f*x0_f + b_f*x0_f + c_f q_f = sqrt(4d0*c_f - b_f*b_f) ec_f = a_f*( log(x**2/x_f) + 2d0*b_f/q_f*atan(q_f/(2d0*x + b_f)) & - b_f*x0_f/xx0_f*( log((x - x0_f)**2/x_f) + 2d0*(b_f + 2d0*x0_f)/q_f*atan(q_f/(2d0*x + b_f)) ) ) drsdr = - (36d0*pi)**(-1d0/3d0)*r**(-4d0/3d0) dxdrs = 0.5d0/sqrt(rs) dxdx_f = 2d0*x + b_f decdx_f = a_f*( 2d0/x - 4d0*b_f/( (b_f+2d0*x)**2 + q_f**2) - dxdx_f/x_f & - b_f*x0_f/xx0_f*( 2/(x-x0_f) - 4d0*(b_f+2d0*x0_f)/( (b_f+2d0*x)**2 + q_f**2) - dxdx_f/x_f ) ) decdr_f = drsdr*dxdrs*decdx_f Ecrr(1) = Ecrr(1) - weight(iG)*decdr_f*r*r if(rI > threshold) then EcrI(1) = EcrI(1) + weight(iG)*ec_f*rI EcrrI(1) = EcrrI(1) + weight(iG)*decdr_f*r*rI end if end if ! up-down contribution r = ra + rb rI = raI + rbI if(r > threshold) then rs = (4d0*pi*r/3d0)**(-1d0/3d0) z = (ra - rb)/r x = sqrt(rs) fz = (1d0 + z)**(4d0/3d0) + (1d0 - z)**(4d0/3d0) - 2d0 fz = fz/(2d0*(2d0**(1d0/3d0) - 1d0)) d2fz = 4d0/(9d0*(2**(1d0/3d0) - 1d0)) x_p = x*x + b_p*x + c_p x_f = x*x + b_f*x + c_f x_a = x*x + b_a*x + c_a xx0_p = x0_p*x0_p + b_p*x0_p + c_p xx0_f = x0_f*x0_f + b_f*x0_f + c_f xx0_a = x0_a*x0_a + b_a*x0_a + c_a q_p = sqrt(4d0*c_p - b_p*b_p) q_f = sqrt(4d0*c_f - b_f*b_f) q_a = sqrt(4d0*c_a - b_a*b_a) ec_p = a_p*( log(x**2/x_p) + 2d0*b_p/q_p*atan(q_p/(2d0*x + b_p)) & - b_p*x0_p/xx0_p*( log((x - x0_p)**2/x_p) + 2d0*(b_p + 2d0*x0_p)/q_p*atan(q_p/(2d0*x + b_p)) ) ) ec_f = a_f*( log(x**2/x_f) + 2d0*b_f/q_f*atan(q_f/(2d0*x + b_f)) & - b_f*x0_f/xx0_f*( log((x - x0_f)**2/x_f) + 2d0*(b_f + 2d0*x0_f)/q_f*atan(q_f/(2d0*x + b_f)) ) ) ec_a = a_a*( log(x**2/x_a) + 2d0*b_a/q_a*atan(q_a/(2d0*x + b_a)) & - b_a*x0_a/xx0_a*( log((x - x0_a)**2/x_a) + 2d0*(b_a + 2d0*x0_a)/q_a*atan(q_a/(2d0*x + b_a)) ) ) ec_z = ec_p + ec_a*fz/d2fz*(1d0-z**4) + (ec_f - ec_p)*fz*z**4 dzdr = (1d0 - z)/r dfzdz = (4d0/3d0)*((1d0 + z)**(1d0/3d0) - (1d0 - z)**(1d0/3d0))/(2d0*(2d0**(1d0/3d0) - 1d0)) dfzdr = dzdr*dfzdz drsdr = - (36d0*pi)**(-1d0/3d0)*r**(-4d0/3d0) dxdrs = 0.5d0/sqrt(rs) dxdx_p = 2d0*x + b_p dxdx_f = 2d0*x + b_f dxdx_a = 2d0*x + b_a decdx_p = a_p*( 2d0/x - 4d0*b_p/( (b_p+2d0*x)**2 + q_p**2) - dxdx_p/x_p & - b_p*x0_p/xx0_p*( 2/(x-x0_p) - 4d0*(b_p+2d0*x0_p)/( (b_p+2d0*x)**2 + q_p**2) - dxdx_p/x_p ) ) decdx_f = a_f*( 2d0/x - 4d0*b_f/( (b_f+2d0*x)**2 + q_f**2) - dxdx_f/x_f & - b_f*x0_f/xx0_f*( 2/(x-x0_f) - 4d0*(b_f+2d0*x0_f)/( (b_f+2d0*x)**2 + q_f**2) - dxdx_f/x_f ) ) decdx_a = a_a*( 2d0/x - 4d0*b_a/( (b_a+2d0*x)**2 + q_a**2) - dxdx_a/x_a & - b_a*x0_a/xx0_a*( 2/(x-x0_a) - 4d0*(b_a+2d0*x0_a)/( (b_a+2d0*x)**2 + q_a**2) - dxdx_a/x_a ) ) decdr_p = drsdr*dxdrs*decdx_p decdr_f = drsdr*dxdrs*decdx_f decdr_a = drsdr*dxdrs*decdx_a decdr = decdr_p + decdr_a*fz/d2fz*(1d0-z**4) + ec_a*dfzdr/d2fz*(1d0-z**4) - 4d0*ec_a*fz/d2fz*dzdr*z**3 & + (decdr_f - decdr_p)*fz*z**4 + (ec_f - ec_p)*dfzdr*z**4 + 4d0*(ec_f - ec_p)*fz*dzdr*z**3 Ecrr(2) = Ecrr(2) - weight(iG)*decdr*r*r if(rI > threshold) then EcrI(2) = EcrI(2) + weight(iG)*ec_z*rI EcrrI(2) = EcrrI(2) + weight(iG)*decdr*r*rI end if end if ! spin-down contribution r = rb rI = rbI if(r > threshold) then rs = (4d0*pi*r/3d0)**(-1d0/3d0) x = sqrt(rs) x_f = x*x + b_f*x + c_f xx0_f = x0_f*x0_f + b_f*x0_f + c_f q_f = sqrt(4d0*c_f - b_f*b_f) ec_f = a_f*( log(x**2/x_f) + 2d0*b_f/q_f*atan(q_f/(2d0*x + b_f)) & - b_f*x0_f/xx0_f*( log((x - x0_f)**2/x_f) + 2d0*(b_f + 2d0*x0_f)/q_f*atan(q_f/(2d0*x + b_f)) ) ) drsdr = - (36d0*pi)**(-1d0/3d0)*r**(-4d0/3d0) dxdrs = 0.5d0/sqrt(rs) dxdx_f = 2d0*x + b_f decdx_f = a_f*( 2d0/x - 4d0*b_f/( (b_f+2d0*x)**2 + q_f**2) - dxdx_f/x_f & - b_f*x0_f/xx0_f*( 2/(x-x0_f) - 4d0*(b_f+2d0*x0_f)/( (b_f+2d0*x)**2 + q_f**2) - dxdx_f/x_f ) ) decdr_f = drsdr*dxdrs*decdx_f Ecrr(3) = Ecrr(3) - weight(iG)*decdr_f*r*r if(rI > threshold) then EcrI(3) = EcrI(3) + weight(iG)*ec_f*rI EcrrI(3) = EcrrI(3) + weight(iG)*decdr_f*r*rI end if end if end do Ecrr(2) = Ecrr(2) - Ecrr(1) - Ecrr(3) EcrI(2) = EcrI(2) - EcrI(1) - EcrI(3) EcrrI(2) = EcrrI(2) - EcrrI(1) - EcrrI(3) ! De-scaling for N-centered ensemble if(doNcentered) then Ecrr(:) = kappa*Ecrr(:) EcrI(:) = kappa*EcrI(:) endif Ec(:) = Ecrr(:) + EcrI(:) + EcrrI(:) end subroutine UVWN3_lda_correlation_individual_energy