2021-02-25 10:55:08 +01:00
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subroutine UVWN3_lda_correlation_potential(nGrid,weight,nBas,AO,rho,Fc)
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! Compute unrestricted VWN3 LDA correlation potential
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implicit none
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include 'parameters.h'
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! Input variables
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integer,intent(in) :: nGrid
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double precision,intent(in) :: weight(nGrid)
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integer,intent(in) :: nBas
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double precision,intent(in) :: AO(nBas,nGrid)
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double precision,intent(in) :: rho(nGrid,nspin)
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! Local variables
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integer :: mu,nu,iG
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double precision :: ra,rb,r,rs,x,z
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double precision :: a_p,x0_p,xx0_p,b_p,c_p,x_p,q_p
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double precision :: a_f,x0_f,xx0_f,b_f,c_f,x_f,q_f
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double precision :: a_a,x0_a,xx0_a,b_a,c_a,x_a,q_a
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double precision :: dfzdz,dxdrs,dxdx_p,dxdx_f,dxdx_a,decdx_p,decdx_f,decdx_a
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double precision :: dzdra,dfzdra,drsdra,decdra_p,decdra_f,decdra_a,decdra
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double precision :: dzdrb,dfzdrb,drsdrb,decdrb_p,decdrb_f,decdrb_a,decdrb
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double precision :: ec_z,ec_p,ec_f,ec_a
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double precision :: fz,d2fz
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! Output variables
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double precision :: Fc(nBas,nBas,nspin)
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! Parameters of the functional
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a_p = +0.0621814d0/2d0
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x0_p = -0.409286d0
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b_p = +13.0720d0
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c_p = +42.7198d0
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a_f = +0.0621814d0/4d0
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x0_f = -0.743294d0
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b_f = +20.1231d0
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c_f = +101.578d0
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a_a = -1d0/(6d0*pi**2)
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x0_a = -0.0047584D0
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b_a = +1.13107d0
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c_a = +13.0045d0
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! Initialization
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Fc(:,:,:) = 0d0
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do mu=1,nBas
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do nu=1,nBas
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do iG=1,nGrid
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ra = max(0d0,rho(iG,1))
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rb = max(0d0,rho(iG,2))
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2021-03-01 14:55:29 +01:00
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r = ra + rb
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z = (ra - rb)/r
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fz = (1d0 + z)**(4d0/3d0) + (1d0 - z)**(4d0/3d0) - 2d0
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fz = fz/(2d0*(2d0**(1d0/3d0) - 1d0))
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d2fz = 4d0/(9d0*(2**(1d0/3d0) - 1d0))
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2021-02-25 10:55:08 +01:00
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! spin-up contribution
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if(ra > threshold) then
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2021-03-01 14:55:29 +01:00
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2021-02-25 10:55:08 +01:00
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rs = (4d0*pi*r/3d0)**(-1d0/3d0)
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x = sqrt(rs)
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x_p = x*x + b_p*x + c_p
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x_f = x*x + b_f*x + c_f
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x_a = x*x + b_a*x + c_a
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xx0_p = x0_p*x0_p + b_p*x0_p + c_p
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xx0_f = x0_f*x0_f + b_f*x0_f + c_f
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xx0_a = x0_a*x0_a + b_a*x0_a + c_a
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q_p = sqrt(4d0*c_p - b_p*b_p)
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q_f = sqrt(4d0*c_f - b_f*b_f)
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q_a = sqrt(4d0*c_a - b_a*b_a)
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ec_p = a_p*( log(x**2/x_p) + 2d0*b_p/q_p*atan(q_p/(2d0*x + b_p)) &
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- 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)) ) )
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ec_f = a_f*( log(x**2/x_f) + 2d0*b_f/q_f*atan(q_f/(2d0*x + b_f)) &
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- 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)) ) )
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ec_a = a_a*( log(x**2/x_a) + 2d0*b_a/q_a*atan(q_a/(2d0*x + b_a)) &
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- 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)) ) )
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ec_z = ec_p + ec_a*fz/d2fz*(1d0-z**4) + (ec_f - ec_p)*fz*z**4
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dzdra = (1d0 - z)/r
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dfzdz = (4d0/3d0)*((1d0 + z)**(1d0/3d0) - (1d0 - z)**(1d0/3d0))/(2d0*(2d0**(1d0/3d0) - 1d0))
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dfzdra = dzdra*dfzdz
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drsdra = - (36d0*pi)**(-1d0/3d0)*r**(-4d0/3d0)
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dxdrs = 0.5d0/sqrt(rs)
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dxdx_p = 2d0*x + b_p
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dxdx_f = 2d0*x + b_f
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dxdx_a = 2d0*x + b_a
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decdx_p = a_p*( 2d0/x - 4d0*b_p/( (b_p+2d0*x)**2 + q_p**2) - dxdx_p/x_p &
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- 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 ) )
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decdx_f = a_f*( 2d0/x - 4d0*b_f/( (b_f+2d0*x)**2 + q_f**2) - dxdx_f/x_f &
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- 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 ) )
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decdx_a = a_a*( 2d0/x - 4d0*b_a/( (b_a+2d0*x)**2 + q_a**2) - dxdx_a/x_a &
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- 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 ) )
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decdra_p = drsdra*dxdrs*decdx_p
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decdra_f = drsdra*dxdrs*decdx_f
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decdra_a = drsdra*dxdrs*decdx_a
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decdra = decdra_p + decdra_a*fz/d2fz*(1d0-z**4) + ec_a*dfzdra/d2fz*(1d0-z**4) - 4d0*ec_a*fz/d2fz*dzdra*z**3 &
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+ (decdra_f - decdra_p)*fz*z**4 + (ec_f - ec_p)*dfzdra*z**4 + 4d0*(ec_f - ec_p)*fz*dzdra*z**3
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Fc(mu,nu,1) = Fc(mu,nu,1) + weight(iG)*AO(mu,iG)*AO(nu,iG)*(ec_z + decdra*r)
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end if
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! spin-down contribution
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if(rb > threshold) then
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rs = (4d0*pi*r/3d0)**(-1d0/3d0)
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x = sqrt(rs)
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2021-03-01 14:55:29 +01:00
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2021-02-25 10:55:08 +01:00
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x_p = x*x + b_p*x + c_p
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x_f = x*x + b_f*x + c_f
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x_a = x*x + b_a*x + c_a
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xx0_p = x0_p*x0_p + b_p*x0_p + c_p
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xx0_f = x0_f*x0_f + b_f*x0_f + c_f
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xx0_a = x0_a*x0_a + b_a*x0_a + c_a
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q_p = sqrt(4d0*c_p - b_p*b_p)
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q_f = sqrt(4d0*c_f - b_f*b_f)
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q_a = sqrt(4d0*c_a - b_a*b_a)
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ec_p = a_p*( log(x**2/x_p) + 2d0*b_p/q_p*atan(q_p/(2d0*x + b_p)) &
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- 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)) ) )
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ec_f = a_f*( log(x**2/x_f) + 2d0*b_f/q_f*atan(q_f/(2d0*x + b_f)) &
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- 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)) ) )
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ec_a = a_a*( log(x**2/x_a) + 2d0*b_a/q_a*atan(q_a/(2d0*x + b_a)) &
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- 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)) ) )
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ec_z = ec_p + ec_a*fz/d2fz*(1d0-z**4) + (ec_f - ec_p)*fz*z**4
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dzdrb = - (1d0 + z)/r
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dfzdz = (4d0/3d0)*((1d0 + z)**(1d0/3d0) - (1d0 - z)**(1d0/3d0))/(2d0*(2d0**(1d0/3d0) - 1d0))
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dfzdrb = dzdrb*dfzdz
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drsdrb = - (36d0*pi)**(-1d0/3d0)*r**(-4d0/3d0)
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dxdrs = 0.5d0/sqrt(rs)
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dxdx_p = 2d0*x + b_p
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dxdx_f = 2d0*x + b_f
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dxdx_a = 2d0*x + b_a
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decdx_p = a_p*( 2d0/x - 4d0*b_p/( (b_p+2d0*x)**2 + q_p**2) - dxdx_p/x_p &
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- 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 ) )
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decdx_f = a_f*( 2d0/x - 4d0*b_f/( (b_f+2d0*x)**2 + q_f**2) - dxdx_f/x_f &
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- 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 ) )
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decdx_a = a_a*( 2d0/x - 4d0*b_a/( (b_a+2d0*x)**2 + q_a**2) - dxdx_a/x_a &
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- 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 ) )
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decdrb_p = drsdrb*dxdrs*decdx_p
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decdrb_f = drsdrb*dxdrs*decdx_f
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decdrb_a = drsdrb*dxdrs*decdx_a
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decdrb = decdrb_p + decdrb_a*fz/d2fz*(1d0-z**4) + ec_a*dfzdrb/d2fz*(1d0-z**4) - 4d0*ec_a*fz/d2fz*dzdrb*z**3 &
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+ (decdrb_f - decdrb_p)*fz*z**4 + (ec_f - ec_p)*dfzdrb*z**4 + 4d0*(ec_f - ec_p)*fz*dzdrb*z**3
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Fc(mu,nu,2) = Fc(mu,nu,2) + weight(iG)*AO(mu,iG)*AO(nu,iG)*(ec_z + decdrb*r)
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end if
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end do
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end do
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end do
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end subroutine UVWN3_lda_correlation_potential
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