quantum_package/src/Integrals_Monoelec/pot_ao_ints.irp.f

 BEGIN_PROVIDER [ double precision, ao_nucl_elec_integral, (ao_num_align,ao_num)]
 BEGIN_DOC
! interaction nuclear electron
 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

  ao_nucl_elec_integral = 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_nucl_elec_integral,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_nucl_elec_integral(i,j) = ao_nucl_elec_integral(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

 END_PROVIDER

 BEGIN_PROVIDER [ double precision, ao_nucl_elec_integral_per_atom, (ao_num_align,ao_num,nucl_num)]
 BEGIN_DOC
! ao_nucl_elec_integral_per_atom(i,j,k) = -<AO(i)|1/|r-Rk|AO(j)> 
! where Rk is the geometry of the kth atom
 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_nucl_elec_integral_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_nucl_elec_integral_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_nucl_elec_integral_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)
! function that calculate the folowing integral :
!       int{dr} of (x-A_x)^ax (x-B_X)^bx exp(-alpha (x-A_x)^2 - beta (x-B_x)^2 ) 1/(r-R_c)

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
!  print*, "a"
  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_polynom_mult_center_mono_elec(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_polynom_mult_center_mono_elec(A_center,B_center,alpha,beta,power_A,power_B,C_center,n_pt_in,d,n_pt_out)
!!!! subroutine that returns the explicit polynom in term of the "t" variable of the following polynomw ::
!!!!         I_x1(a_x, d_x,p,q) * I_x1(a_y, d_y,p,q) * I_x1(a_z, d_z,p,q)
!!!! it is for the nuclear electron atraction
implicit none
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
!print*,'n_pt_in = ',n_pt_in
 accu = 0.d0
!COMPTEUR irp_rdtsc1 = irp_rdtsc()
 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
! print*,'passed the P_center'
 
 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))
 ! R1x = (P_x - A_x) - (P_x - C_x) t^2
 R1xp(0)  = (P_center(1) - B_center(1))
 R1xp(1)  = 0.d0
 R1xp(2)  =-(P_center(1) - C_center(1))
 !R1xp = (P_x - B_x) - (P_x - C_x) t^2
 R2x(0)  =  p_inv_2
 R2x(1)  = 0.d0
 R2x(2)  = -p_inv_2
 !R2x  = 0.5 / p - 0.5/p t^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_mono_elec(a_x,b_x,R1x,R1xp,R2x,d1,n_pt1,n_pt_in) 
! print*,'passed the first I_x1'
  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))
 ! R1x = (P_x - A_x) - (P_x - C_x) t^2
 R1xp(0)  = (P_center(2) - B_center(2))
 R1xp(1)  = 0.d0
 R1xp(2)  =-(P_center(2) - C_center(2))
 !R1xp = (P_x - B_x) - (P_x - C_x) t^2
 a_y = power_A(2)
 b_y = power_B(2)
  call I_x1_pol_mult_mono_elec(a_y,b_y,R1x,R1xp,R2x,d2,n_pt2,n_pt_in) 
! print*,'passed the second I_x1'
  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))
 ! R1x = (P_x - A_x) - (P_x - C_x) t^2
 R1xp(0)  = (P_center(3) - B_center(3))
 R1xp(1)  = 0.d0
 R1xp(2)  =-(P_center(3) - C_center(3))
 !R2x  = 0.5 / p - 0.5/p t^2
 a_z = power_A(3)
 b_z = power_B(3)

! print*,'a_z = ',a_z
! print*,'b_z = ',b_z
  call I_x1_pol_mult_mono_elec(a_z,b_z,R1x,R1xp,R2x,d3,n_pt3,n_pt_in) 
! print*,'passed the third I_x1'
  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_mono_elec(a,c,R1x,R1xp,R2x,d,nd,n_pt_in) 
!!!!  recursive function involved in the electron nucleus potential
 implicit none
 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_mono_elec(c,R1x,R1xp,R2x,d,nd,n_pt_in)
!    print*,'nd 0,c',nd
  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_mono_elec(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_mono_elec(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_mono_elec(a-2,c,R1x,R1xp,R2x,X,nx,n_pt_in)
!    print*,'nx a-2,c= ',nx
     do ix=0,nx
       X(ix) *= dble(a-1)
     enddo
     call multiply_poly(X,nx,R2x,2,d,nd)
!    print*,'nd out = ',nd

     nx = nd
     do ix=0,n_pt_in
       X(ix) = 0.d0
     enddo
     call I_x1_pol_mult_mono_elec(a-1,c-1,R1x,R1xp,R2x,X,nx,n_pt_in)
!      print*,'nx a-1,c-1 = ',nx
       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_mono_elec(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_mono_elec(c,R1x,R1xp,R2x,d,nd,dim) 
 implicit none
 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
!print*,'X2,c = ',c
!print*,'nd_in = ',nd

 if(c==0) then
   nd = 0
   d(0) = 1.d0
!  print*,'nd  IX2 = ',nd
   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_mono_elec(0,c-2,R1x,R1xp,R2x,X,nx,dim)
!      print*,'nx 0,c-2 = ',nx
       do ix=0,nx
         X(ix) *= dble(c-1)
       enddo
       call multiply_poly(X,nx,R2x,2,d,nd)
!      print*,'nd = ',nd
       ny = 0
       do ix=0,dim
        Y(ix) = 0.d0
       enddo
       
       call I_x1_pol_mult_mono_elec(0,c-1,R1x,R1xp,R2x,Y,ny,dim)
!      print*,'ny = ',ny
!      do ix=0,ny
!        print*,'Y(ix) = ',Y(ix)
!      enddo
       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
!!! primitve nuclear attraction between the two primitves centered on the same atom ::
!!!!         primitive_1 = x**(a_x) y**(a_y) z**(a_z) exp(-alpha * r**2)
!!!!         primitive_2 = x**(b_x) y**(b_y) z**(b_z) exp(- beta * r**2)
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)
!!!! calculate the integral of 
!!     integral on "x" with boundaries (- infinity; + infinity) of [ x**n  exp(-alpha * x**2) ]
implicit none
double precision :: alpha
integer :: n
double precision :: dble_fact
include 'Utils/constants.include.F'

!if(iand(n,1).eq.1)then
!  int_gaus_pol= 0.d0
!else
!  int_gaus_pol = dsqrt(pi/alpha) * dble_fact(n -1)/(alpha+alpha)**(n/2)
!endif

 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)
!!!! calculate the radial part of the nuclear attraction integral which is the following integral :
!!     integral on "r" with boundaries ( 0 ; + infinity) of [ r**n  exp(-alpha * r**2) ]
!!! CAUTION  :: this function requires the constant sqpi = dsqrt(pi)
implicit none
double precision ::  alpha, fact
integer :: n
include 'Utils/constants.include.F'
if(iand(n,1).eq.1)then
 V_r = 0.5d0 * fact(ishft(n,-1)) / (alpha ** (ishft(n,-1) + 1))
else
 V_r = sqpi * fact(n) / fact(ishft(n,-1)) * (0.5d0/sqrt(alpha)) ** (n+1)
endif
end


double precision function V_phi(n,m)
implicit none
!!!! calculate the angular "phi" part of the nuclear attraction integral wich is the following integral :
!!     integral on "phi" with boundaries ( 0 ; 2 pi) of [ cos(phi) **n  sin(phi) **m ]
integer :: n,m, i
double precision ::  prod, Wallis
  prod = 1.d0
  do i = 0,ishft(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
!!!! calculate the angular "theta" part of the nuclear attraction integral wich is the following integral :
!!     integral on "theta" with boundaries ( 0 ; pi) of [ cos(theta) **n  sin(theta) **m ]
integer :: n,m,i
double precision ::  Wallis, prod
include 'Utils/constants.include.F'
  V_theta = 0.d0
  prod = 1.d0
  do i = 0,ishft(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)
!!!! calculate the Wallis integral :
!!     integral on "theta" with boundaries ( 0 ; pi/2) of [ cos(theta) **n ]
implicit none
double precision :: fact
integer :: n,p
include 'Utils/constants.include.F'
if(iand(n,1).eq.0)then
  Wallis = fact(ishft(n,-1))
  Wallis = pi * fact(n) / (dble(ibset(0_8,n)) * (Wallis+Wallis)*Wallis) 
else
 p = ishft(n,-1)
 Wallis = fact(p)
 Wallis = dble(ibset(0_8,p+p)) * Wallis*Wallis / fact(p+p+1)
endif

end