BEGIN_PROVIDER [ double precision, ao_integrals_n_e, (ao_num,ao_num)] BEGIN_DOC ! Nucleus-electron interaction, in the |AO| basis set. ! ! :math:`\langle \chi_i | -\sum_A \frac{1}{|r-R_A|} | \chi_j \rangle` END_DOC implicit none double precision :: alpha, beta, gama, delta integer :: num_A,num_B double precision :: A_center(3),B_center(3),C_center(3) integer :: power_A(3),power_B(3) integer :: i,j,k,l,n_pt_in,m double precision :: overlap_x,overlap_y,overlap_z,overlap,dx,NAI_pol_mult if (read_ao_integrals_e_n) then call ezfio_get_ao_one_e_ints_ao_integrals_e_n(ao_integrals_n_e) print *, 'AO N-e integrals read from disk' else ao_integrals_n_e = 0.d0 ! _ ! /| / |_) ! | / | \ ! !$OMP PARALLEL & !$OMP DEFAULT (NONE) & !$OMP PRIVATE (i,j,k,l,m,alpha,beta,A_center,B_center,C_center,power_A,power_B,& !$OMP num_A,num_B,Z,c,n_pt_in) & !$OMP SHARED (ao_num,ao_prim_num,ao_expo_ordered_transp,ao_power,ao_nucl,nucl_coord,ao_coef_normalized_ordered_transp,& !$OMP n_pt_max_integrals,ao_integrals_n_e,nucl_num,nucl_charge) n_pt_in = n_pt_max_integrals !$OMP DO SCHEDULE (dynamic) do j = 1, ao_num num_A = ao_nucl(j) power_A(1:3)= ao_power(j,1:3) A_center(1:3) = nucl_coord(num_A,1:3) do i = 1, ao_num num_B = ao_nucl(i) power_B(1:3)= ao_power(i,1:3) B_center(1:3) = nucl_coord(num_B,1:3) do l=1,ao_prim_num(j) alpha = ao_expo_ordered_transp(l,j) do m=1,ao_prim_num(i) beta = ao_expo_ordered_transp(m,i) double precision :: c c = 0.d0 do k = 1, nucl_num double precision :: Z Z = nucl_charge(k) C_center(1:3) = nucl_coord(k,1:3) c = c - Z * NAI_pol_mult(A_center,B_center, & power_A,power_B,alpha,beta,C_center,n_pt_in) enddo ao_integrals_n_e(i,j) = ao_integrals_n_e(i,j) & + ao_coef_normalized_ordered_transp(l,j) & * ao_coef_normalized_ordered_transp(m,i) * c enddo enddo enddo enddo !$OMP END DO !$OMP END PARALLEL endif if (write_ao_integrals_e_n) then call ezfio_set_ao_one_e_ints_ao_integrals_e_n(ao_integrals_n_e) print *, 'AO N-e integrals written to disk' endif END_PROVIDER BEGIN_PROVIDER [ double precision, ao_integrals_n_e_per_atom, (ao_num,ao_num,nucl_num)] BEGIN_DOC ! Nucleus-electron interaction in the |AO| basis set, per atom A. ! ! :math:`\langle \chi_i | -\frac{1}{|r-R_A|} | \chi_j \rangle` END_DOC implicit none double precision :: alpha, beta, gama, delta integer :: i_c,num_A,num_B double precision :: A_center(3),B_center(3),C_center(3) integer :: power_A(3),power_B(3) integer :: i,j,k,l,n_pt_in,m double precision :: overlap_x,overlap_y,overlap_z,overlap,dx,NAI_pol_mult ao_integrals_n_e_per_atom = 0.d0 !$OMP PARALLEL & !$OMP DEFAULT (NONE) & !$OMP PRIVATE (i,j,k,l,m,alpha,beta,A_center,B_center,power_A,power_B,& !$OMP num_A,num_B,c,n_pt_in,C_center) & !$OMP SHARED (ao_num,ao_prim_num,ao_expo_ordered_transp,ao_power,ao_nucl,nucl_coord,ao_coef_normalized_ordered_transp,& !$OMP n_pt_max_integrals,ao_integrals_n_e_per_atom,nucl_num) n_pt_in = n_pt_max_integrals !$OMP DO SCHEDULE (dynamic) double precision :: c do j = 1, ao_num power_A(1)= ao_power(j,1) power_A(2)= ao_power(j,2) power_A(3)= ao_power(j,3) num_A = ao_nucl(j) A_center(1) = nucl_coord(num_A,1) A_center(2) = nucl_coord(num_A,2) A_center(3) = nucl_coord(num_A,3) do k = 1, nucl_num C_center(1) = nucl_coord(k,1) C_center(2) = nucl_coord(k,2) C_center(3) = nucl_coord(k,3) do i = 1, ao_num power_B(1)= ao_power(i,1) power_B(2)= ao_power(i,2) power_B(3)= ao_power(i,3) num_B = ao_nucl(i) B_center(1) = nucl_coord(num_B,1) B_center(2) = nucl_coord(num_B,2) B_center(3) = nucl_coord(num_B,3) c = 0.d0 do l=1,ao_prim_num(j) alpha = ao_expo_ordered_transp(l,j) do m=1,ao_prim_num(i) beta = ao_expo_ordered_transp(m,i) c = c + NAI_pol_mult(A_center,B_center,power_A,power_B, & alpha,beta,C_center,n_pt_in) & * ao_coef_normalized_ordered_transp(l,j) & * ao_coef_normalized_ordered_transp(m,i) enddo enddo ao_integrals_n_e_per_atom(i,j,k) = -c enddo enddo enddo !$OMP END DO !$OMP END PARALLEL END_PROVIDER double precision function NAI_pol_mult(A_center,B_center,power_A,power_B,alpha,beta,C_center,n_pt_in) BEGIN_DOC ! Computes the electron-nucleus attraction with two primitves. ! ! :math:`\langle g_i | \frac{1}{|r-R_c|} | g_j \rangle` END_DOC implicit none integer, intent(in) :: n_pt_in double precision,intent(in) :: C_center(3),A_center(3),B_center(3),alpha,beta integer :: power_A(3),power_B(3) integer :: i,j,k,l,n_pt double precision :: P_center(3) double precision :: d(0:n_pt_in),pouet,coeff,rho,dist,const,pouet_2,p,p_inv,factor double precision :: I_n_special_exact,integrate_bourrin,I_n_bibi double precision :: V_e_n,const_factor,dist_integral,tmp double precision :: accu,epsilo,rint integer :: n_pt_out,lmax include 'utils/constants.include.F' if ( (A_center(1)/=B_center(1)).or. & (A_center(2)/=B_center(2)).or. & (A_center(3)/=B_center(3)).or. & (A_center(1)/=C_center(1)).or. & (A_center(2)/=C_center(2)).or. & (A_center(3)/=C_center(3))) then continue else NAI_pol_mult = V_e_n(power_A(1),power_A(2),power_A(3), & power_B(1),power_B(2),power_B(3),alpha,beta) return endif p = alpha + beta p_inv = 1.d0/p rho = alpha * beta * p_inv dist = 0.d0 dist_integral = 0.d0 do i = 1, 3 P_center(i) = (alpha * A_center(i) + beta * B_center(i)) * p_inv dist += (A_center(i) - B_center(i))*(A_center(i) - B_center(i)) dist_integral += (P_center(i) - C_center(i))*(P_center(i) - C_center(i)) enddo const_factor = dist*rho const = p * dist_integral if(const_factor > 80.d0)then NAI_pol_mult = 0.d0 return endif factor = dexp(-const_factor) coeff = dtwo_pi * factor * p_inv lmax = 20 ! print*, "b" do i = 0, n_pt_in d(i) = 0.d0 enddo n_pt = 2 * ( (power_A(1) + power_B(1)) +(power_A(2) + power_B(2)) +(power_A(3) + power_B(3)) ) if (n_pt == 0) then epsilo = 1.d0 pouet = rint(0,const) NAI_pol_mult = coeff * pouet return endif call give_polynomial_mult_center_one_e(A_center,B_center,alpha,beta,power_A,power_B,C_center,n_pt_in,d,n_pt_out) if(n_pt_out<0)then NAI_pol_mult = 0.d0 return endif accu = 0.d0 ! 1/r1 standard attraction integral epsilo = 1.d0 ! sum of integrals of type : int {t,[0,1]} exp-(rho.(P-Q)^2 * t^2) * t^i do i =0 ,n_pt_out,2 accu += d(i) * rint(i/2,const) enddo NAI_pol_mult = accu * coeff end subroutine give_polynomial_mult_center_one_e(A_center,B_center,alpha,beta,power_A,power_B,C_center,n_pt_in,d,n_pt_out) implicit none BEGIN_DOC ! Returns the explicit polynomial in terms of the "t" variable of the following ! ! $I_{x1}(a_x, d_x,p,q) \times I_{x1}(a_y, d_y,p,q) \times I_{x1}(a_z, d_z,p,q)$. END_DOC integer, intent(in) :: n_pt_in integer,intent(out) :: n_pt_out double precision, intent(in) :: A_center(3), B_center(3),C_center(3) double precision, intent(in) :: alpha,beta integer, intent(in) :: power_A(3), power_B(3) integer :: a_x,b_x,a_y,b_y,a_z,b_z double precision :: d(0:n_pt_in) double precision :: d1(0:n_pt_in) double precision :: d2(0:n_pt_in) double precision :: d3(0:n_pt_in) double precision :: accu, pq_inv, p10_1, p10_2, p01_1, p01_2 double precision :: p,P_center(3),rho,p_inv,p_inv_2 accu = 0.d0 ASSERT (n_pt_in > 1) p = alpha+beta p_inv = 1.d0/p p_inv_2 = 0.5d0/p do i =1, 3 P_center(i) = (alpha * A_center(i) + beta * B_center(i)) * p_inv enddo double precision :: R1x(0:2), B01(0:2), R1xp(0:2),R2x(0:2) R1x(0) = (P_center(1) - A_center(1)) R1x(1) = 0.d0 R1x(2) = -(P_center(1) - C_center(1)) R1xp(0) = (P_center(1) - B_center(1)) R1xp(1) = 0.d0 R1xp(2) =-(P_center(1) - C_center(1)) R2x(0) = p_inv_2 R2x(1) = 0.d0 R2x(2) = -p_inv_2 do i = 0,n_pt_in d(i) = 0.d0 enddo do i = 0,n_pt_in d1(i) = 0.d0 enddo do i = 0,n_pt_in d2(i) = 0.d0 enddo do i = 0,n_pt_in d3(i) = 0.d0 enddo integer :: n_pt1,n_pt2,n_pt3,dim,i n_pt1 = n_pt_in n_pt2 = n_pt_in n_pt3 = n_pt_in a_x = power_A(1) b_x = power_B(1) call I_x1_pol_mult_one_e(a_x,b_x,R1x,R1xp,R2x,d1,n_pt1,n_pt_in) if(n_pt1<0)then n_pt_out = -1 do i = 0,n_pt_in d(i) = 0.d0 enddo return endif R1x(0) = (P_center(2) - A_center(2)) R1x(1) = 0.d0 R1x(2) = -(P_center(2) - C_center(2)) R1xp(0) = (P_center(2) - B_center(2)) R1xp(1) = 0.d0 R1xp(2) =-(P_center(2) - C_center(2)) a_y = power_A(2) b_y = power_B(2) call I_x1_pol_mult_one_e(a_y,b_y,R1x,R1xp,R2x,d2,n_pt2,n_pt_in) if(n_pt2<0)then n_pt_out = -1 do i = 0,n_pt_in d(i) = 0.d0 enddo return endif R1x(0) = (P_center(3) - A_center(3)) R1x(1) = 0.d0 R1x(2) = -(P_center(3) - C_center(3)) R1xp(0) = (P_center(3) - B_center(3)) R1xp(1) = 0.d0 R1xp(2) =-(P_center(3) - C_center(3)) a_z = power_A(3) b_z = power_B(3) call I_x1_pol_mult_one_e(a_z,b_z,R1x,R1xp,R2x,d3,n_pt3,n_pt_in) if(n_pt3<0)then n_pt_out = -1 do i = 0,n_pt_in d(i) = 0.d0 enddo return endif integer :: n_pt_tmp n_pt_tmp = 0 call multiply_poly(d1,n_pt1,d2,n_pt2,d,n_pt_tmp) do i = 0,n_pt_tmp d1(i) = 0.d0 enddo n_pt_out = 0 call multiply_poly(d ,n_pt_tmp ,d3,n_pt3,d1,n_pt_out) do i = 0, n_pt_out d(i) = d1(i) enddo end recursive subroutine I_x1_pol_mult_one_e(a,c,R1x,R1xp,R2x,d,nd,n_pt_in) implicit none BEGIN_DOC ! Recursive routine involved in the electron-nucleus potential END_DOC integer , intent(in) :: n_pt_in double precision,intent(inout) :: d(0:n_pt_in) integer,intent(inout) :: nd integer, intent(in) :: a,c double precision, intent(in) :: R1x(0:2),R1xp(0:2),R2x(0:2) include 'utils/constants.include.F' double precision :: X(0:max_dim) double precision :: Y(0:max_dim) !DIR$ ATTRIBUTES ALIGN : $IRP_ALIGN :: X, Y integer :: nx, ix,dim,iy,ny dim = n_pt_in ! print*,'a,c = ',a,c ! print*,'nd_in = ',nd if( (a==0) .and. (c==0))then nd = 0 d(0) = 1.d0 return elseif( (c<0).or.(nd<0) )then nd = -1 return else if ((a==0).and.(c.ne.0)) then call I_x2_pol_mult_one_e(c,R1x,R1xp,R2x,d,nd,n_pt_in) else if (a==1) then nx = nd do ix=0,n_pt_in X(ix) = 0.d0 Y(ix) = 0.d0 enddo call I_x2_pol_mult_one_e(c-1,R1x,R1xp,R2x,X,nx,n_pt_in) do ix=0,nx X(ix) *= dble(c) enddo call multiply_poly(X,nx,R2x,2,d,nd) ny=0 call I_x2_pol_mult_one_e(c,R1x,R1xp,R2x,Y,ny,n_pt_in) call multiply_poly(Y,ny,R1x,2,d,nd) else do ix=0,n_pt_in X(ix) = 0.d0 Y(ix) = 0.d0 enddo nx = 0 call I_x1_pol_mult_one_e(a-2,c,R1x,R1xp,R2x,X,nx,n_pt_in) do ix=0,nx X(ix) *= dble(a-1) enddo call multiply_poly(X,nx,R2x,2,d,nd) nx = nd do ix=0,n_pt_in X(ix) = 0.d0 enddo call I_x1_pol_mult_one_e(a-1,c-1,R1x,R1xp,R2x,X,nx,n_pt_in) do ix=0,nx X(ix) *= dble(c) enddo call multiply_poly(X,nx,R2x,2,d,nd) ny=0 call I_x1_pol_mult_one_e(a-1,c,R1x,R1xp,R2x,Y,ny,n_pt_in) call multiply_poly(Y,ny,R1x,2,d,nd) endif end recursive subroutine I_x2_pol_mult_one_e(c,R1x,R1xp,R2x,d,nd,dim) implicit none BEGIN_DOC ! Recursive routine involved in the electron-nucleus potential END_DOC integer , intent(in) :: dim include 'utils/constants.include.F' double precision :: d(0:max_dim) integer,intent(inout) :: nd integer, intent(in) :: c double precision, intent(in) :: R1x(0:2),R1xp(0:2),R2x(0:2) integer :: i if(c==0) then nd = 0 d(0) = 1.d0 return elseif ((nd<0).or.(c<0))then nd = -1 return else integer :: nx, ix,ny double precision :: X(0:max_dim),Y(0:max_dim) !DIR$ ATTRIBUTES ALIGN : $IRP_ALIGN :: X, Y do ix=0,dim X(ix) = 0.d0 Y(ix) = 0.d0 enddo nx = 0 call I_x1_pol_mult_one_e(0,c-2,R1x,R1xp,R2x,X,nx,dim) do ix=0,nx X(ix) *= dble(c-1) enddo call multiply_poly(X,nx,R2x,2,d,nd) ny = 0 do ix=0,dim Y(ix) = 0.d0 enddo call I_x1_pol_mult_one_e(0,c-1,R1x,R1xp,R2x,Y,ny,dim) if(ny.ge.0)then call multiply_poly(Y,ny,R1xp,2,d,nd) endif endif end double precision function V_e_n(a_x,a_y,a_z,b_x,b_y,b_z,alpha,beta) implicit none BEGIN_DOC ! Primitve nuclear attraction between the two primitves centered on the same atom. ! ! $p_1 = x^{a_x} y^{a_y} z^{a_z} \exp(-\alpha r^2)$ ! ! $p_2 = x^{b_x} y^{b_y} z^{b_z} \exp(-\beta r^2)$ END_DOC integer :: a_x,a_y,a_z,b_x,b_y,b_z double precision :: alpha,beta double precision :: V_r, V_phi, V_theta if(iand((a_x+b_x),1)==1.or.iand(a_y+b_y,1)==1.or.iand((a_z+b_z),1)==1)then V_e_n = 0.d0 else V_e_n = V_r(a_x+b_x+a_y+b_y+a_z+b_z+1,alpha+beta) & * V_phi(a_x+b_x,a_y+b_y) & * V_theta(a_z+b_z,a_x+b_x+a_y+b_y+1) endif end double precision function int_gaus_pol(alpha,n) implicit none BEGIN_DOC ! Computes the integral: ! ! $\int_{-\infty}^{\infty} x^n \exp(-\alpha x^2) dx$. END_DOC double precision :: alpha integer :: n double precision :: dble_fact include 'utils/constants.include.F' int_gaus_pol = 0.d0 if(iand(n,1).eq.0)then int_gaus_pol = dsqrt(alpha/pi) double precision :: two_alpha two_alpha = alpha+alpha integer :: i do i=1,n,2 int_gaus_pol = int_gaus_pol * two_alpha enddo int_gaus_pol = dble_fact(n -1) / int_gaus_pol endif end double precision function V_r(n,alpha) implicit none BEGIN_DOC ! Computes the radial part of the nuclear attraction integral: ! ! $\int_{0}^{\infty} r^n \exp(-\alpha r^2) dr$ ! END_DOC double precision :: alpha, fact integer :: n include 'utils/constants.include.F' if(iand(n,1).eq.1)then V_r = 0.5d0 * fact(shiftr(n,1)) / (alpha ** (shiftr(n,1) + 1)) else V_r = sqpi * fact(n) / fact(shiftr(n,1)) * (0.5d0/sqrt(alpha)) ** (n+1) endif end double precision function V_phi(n,m) implicit none BEGIN_DOC ! Computes the angular $\phi$ part of the nuclear attraction integral: ! ! $\int_{0}^{2 \pi} \cos(\phi)^n \sin(\phi)^m d\phi$. END_DOC integer :: n,m, i double precision :: prod, Wallis prod = 1.d0 do i = 0,shiftr(n,1)-1 prod = prod/ (1.d0 + dfloat(m+1)/dfloat(n-i-i-1)) enddo V_phi = 4.d0 * prod * Wallis(m) end double precision function V_theta(n,m) implicit none BEGIN_DOC ! Computes the angular $\theta$ part of the nuclear attraction integral: ! ! $\int_{0}^{\pi} \cos(\theta)^n \sin(\theta)^m d\theta$ END_DOC integer :: n,m,i double precision :: Wallis, prod include 'utils/constants.include.F' V_theta = 0.d0 prod = 1.d0 do i = 0,shiftr(n,1)-1 prod = prod / (1.d0 + dfloat(m+1)/dfloat(n-i-i-1)) enddo V_theta = (prod+prod) * Wallis(m) end double precision function Wallis(n) implicit none BEGIN_DOC ! Wallis integral: ! ! $\int_{0}^{\pi} \cos(\theta)^n d\theta$. END_DOC double precision :: fact integer :: n,p include 'utils/constants.include.F' if(iand(n,1).eq.0)then Wallis = fact(shiftr(n,1)) Wallis = pi * fact(n) / (dble(ibset(0_8,n)) * (Wallis+Wallis)*Wallis) else p = shiftr(n,1) Wallis = fact(p) Wallis = dble(ibset(0_8,p+p)) * Wallis*Wallis / fact(p+p+1) endif end