qp2/src/ao_one_e_ints/pot_ao_erf_ints.irp.f

! ---

subroutine give_all_erf_kl_ao(integrals_ao,mu_in,C_center)
  implicit none
  BEGIN_DOC
  ! Subroutine that returns all integrals over $r$ of type
  ! $\frac{ \erf(\mu * | r - R_C | ) }{ | r - R_C | }$
  END_DOC
  double precision, intent(in)   :: mu_in,C_center(3)
  double precision, intent(out)  :: integrals_ao(ao_num,ao_num)
  double precision               :: NAI_pol_mult_erf_ao
  integer                        :: i,j,l,k,m
  do k = 1, ao_num
    do m = 1, ao_num
      integrals_ao(m,k) = NAI_pol_mult_erf_ao(m,k,mu_in,C_center)
    enddo
  enddo
end

! ---

double precision function NAI_pol_mult_erf_ao(i_ao, j_ao, mu_in, C_center)

  BEGIN_DOC
  !
  ! Computes the following integral :
  ! $\int_{-\infty}^{infty} dr \chi_i(r) \chi_j(r) \frac{\erf(\mu |r - R_C|)}{|r - R_C|}$.
  !
  END_DOC

  implicit none
  integer,          intent(in)   :: i_ao, j_ao
  double precision, intent(in)   :: mu_in, C_center(3)

  integer                        :: i, j, num_A, num_B, power_A(3), power_B(3), n_pt_in
  double precision               :: A_center(3), B_center(3), integral, alpha, beta

  double precision               :: NAI_pol_mult_erf

  num_A         = ao_nucl(i_ao)
  power_A(1:3)  = ao_power(i_ao,1:3)
  A_center(1:3) = nucl_coord(num_A,1:3)
  num_B         = ao_nucl(j_ao)
  power_B(1:3)  = ao_power(j_ao,1:3)
  B_center(1:3) = nucl_coord(num_B,1:3)

  n_pt_in = n_pt_max_integrals

  NAI_pol_mult_erf_ao = 0.d0
  do i = 1, ao_prim_num(i_ao)
    alpha = ao_expo_ordered_transp(i,i_ao)
    do j = 1, ao_prim_num(j_ao)
      beta = ao_expo_ordered_transp(j,j_ao)

      integral = NAI_pol_mult_erf(A_center, B_center, power_A, power_B, alpha, beta, C_center, n_pt_in,mu_in)

      NAI_pol_mult_erf_ao += integral * ao_coef_normalized_ordered_transp(j,j_ao) * ao_coef_normalized_ordered_transp(i,i_ao)
    enddo
  enddo

end function NAI_pol_mult_erf_ao

! ---

double precision function NAI_pol_mult_erf_ao_with1s(i_ao, j_ao, beta, B_center, mu_in, C_center)

  BEGIN_DOC
  !
  ! Computes the following integral :
  ! $\int_{-\infty}^{infty} dr \chi_i(r) \chi_j(r) e^{-\beta (r - B_center)^2} \frac{\erf(\mu |r - R_C|)}{|r - R_C|}$.
  !
  END_DOC

  implicit none
  integer,          intent(in)   :: i_ao, j_ao
  double precision, intent(in)   :: beta, B_center(3)
  double precision, intent(in)   :: mu_in, C_center(3)

  integer                        :: i, j, power_A1(3), power_A2(3), n_pt_in
  double precision               :: A1_center(3), A2_center(3), alpha1, alpha2, coef12, coef1, integral

  double precision, external     :: NAI_pol_mult_erf_with1s, NAI_pol_mult_erf_ao

  ASSERT(beta .ge. 0.d0)
  if(beta .lt. 1d-10) then
    NAI_pol_mult_erf_ao_with1s = NAI_pol_mult_erf_ao(i_ao, j_ao, mu_in, C_center)
    return
  endif

  power_A1(1:3) = ao_power(i_ao,1:3)
  power_A2(1:3) = ao_power(j_ao,1:3)

  A1_center(1:3) = nucl_coord(ao_nucl(i_ao),1:3)
  A2_center(1:3) = nucl_coord(ao_nucl(j_ao),1:3)

  n_pt_in = n_pt_max_integrals

  NAI_pol_mult_erf_ao_with1s = 0.d0
  do i = 1, ao_prim_num(i_ao)
    alpha1 = ao_expo_ordered_transp           (i,i_ao)
    coef1  = ao_coef_normalized_ordered_transp(i,i_ao)

    do j = 1, ao_prim_num(j_ao)
      alpha2 = ao_expo_ordered_transp(j,j_ao)
      coef12 = coef1 * ao_coef_normalized_ordered_transp(j,j_ao)
      if(dabs(coef12) .lt. 1d-14) cycle

      integral = NAI_pol_mult_erf_with1s( A1_center, A2_center, power_A1, power_A2, alpha1, alpha2 &
                                        , beta, B_center, C_center, n_pt_in, mu_in )

      NAI_pol_mult_erf_ao_with1s += integral * coef12
    enddo
  enddo

end function NAI_pol_mult_erf_ao_with1s

! ---

double precision function NAI_pol_mult_erf(A_center, B_center, power_A, power_B, alpha, beta, C_center, n_pt_in, mu_in)

  BEGIN_DOC
  !
  ! Computes the following integral :
  !
  ! .. math::
  !
  !   \int dr (x-A_x)^a (x-B_x)^b \exp(-\alpha (x-A_x)^2 - \beta (x-B_x)^2 )
  !   \frac{\erf(\mu |r - R_C |)}{| r - R_C |}$.
  !
  END_DOC

  include 'utils/constants.include.F'

  implicit none
  integer,          intent(in) :: n_pt_in
  integer,          intent(in) :: power_A(3), power_B(3)
  double precision, intent(in) :: C_center(3), A_center(3), B_center(3), alpha, beta, mu_in

  integer                      :: i, n_pt, n_pt_out
  double precision             :: P_center(3)
  double precision             :: d(0:n_pt_in), coeff, dist, const, factor
  double precision             :: const_factor, dist_integral
  double precision             :: accu, p_inv, p, rho, p_inv_2
  double precision             :: p_new

  double precision             :: rint

  p       = alpha + beta
  p_inv   = 1.d0 / p
  p_inv_2 = 0.5d0 * p_inv
  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
  if(const_factor > 80.d0) then
    NAI_pol_mult_erf = 0.d0
    return
  endif

  p_new  = mu_in / dsqrt(p + mu_in * mu_in)
  factor = dexp(-const_factor)
  coeff  = dtwo_pi * factor * p_inv * p_new

  n_pt  =  2 * ( (power_A(1) + power_B(1)) + (power_A(2) + power_B(2)) + (power_A(3) + power_B(3)) )
  const = p * dist_integral * p_new * p_new
  if(n_pt == 0) then
    NAI_pol_mult_erf = coeff * rint(0, const)
    return
  endif

  do i = 0, n_pt_in
    d(i) = 0.d0
  enddo
  ! call give_polynomial_mult_center_one_e_erf(A_center,B_center,alpha,beta,power_A,power_B,C_center,n_pt_in,d,n_pt_out,mu_in)
  p_new = p_new * p_new
  call give_polynomial_mult_center_one_e_erf_opt(A_center, B_center, power_A, power_B, C_center, n_pt_in, d, n_pt_out, p_inv_2, p_new, P_center)

  if(n_pt_out < 0) then
    NAI_pol_mult_erf = 0.d0
    return
  endif

  ! sum of integrals of type : int {t,[0,1]}  exp-(rho.(P-Q)^2 * t^2) * t^i
  accu = 0.d0
  do i = 0, n_pt_out, 2
    accu += d(i) * rint(i/2, const)
  enddo
  NAI_pol_mult_erf = accu * coeff

end function NAI_pol_mult_erf

! ---

subroutine NAI_pol_mult_erf_v(A_center, B_center, power_A, power_B, alpha, beta, C_center, LD_C, n_pt_in, mu_in, res_v, LD_resv, n_points)

  BEGIN_DOC
  !
  ! Computes the following integral :
  !
  ! .. math::
  !
  !   \int dr (x-A_x)^a (x-B_x)^b \exp(-\alpha (x-A_x)^2 - \beta (x-B_x)^2 )
  !   \frac{\erf(\mu |r - R_C |)}{| r - R_C |}$.
  !
  END_DOC

  include 'utils/constants.include.F'

  implicit none

  integer,          intent(in)  :: n_pt_in, n_points, LD_C, LD_resv
  integer,          intent(in)  :: power_A(3), power_B(3)
  double precision, intent(in)  :: A_center(3), B_center(3), alpha, beta, mu_in
  double precision, intent(in)  :: C_center(LD_C,3)
  double precision, intent(out) :: res_v(LD_resv)

  integer                       :: i, n_pt, n_pt_out, ipoint
  double precision              :: P_center(3)
  double precision              :: d(0:n_pt_in), coeff, dist, const, factor
  double precision              :: const_factor, dist_integral
  double precision              :: accu, p_inv, p, rho, p_inv_2
  double precision              :: p_new, p_new2, coef_tmp

  double precision              :: rint

  res_V(1:LD_resv) = 0.d0

  p        = alpha + beta
  p_inv    = 1.d0 / p
  p_inv_2  = 0.5d0 * p_inv
  rho      = alpha * beta * p_inv
  p_new    = mu_in / dsqrt(p + mu_in * mu_in)
  p_new2   = p_new * p_new
  coef_tmp = p * p_new2

  dist = 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))
  enddo

  const_factor = dist * rho
  if(const_factor > 80.d0) then
    return
  endif
  factor = dexp(-const_factor)
  coeff  = dtwo_pi * factor * p_inv * p_new

  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

    do ipoint = 1, n_points
      dist_integral = 0.d0
      do i = 1, 3
        dist_integral += (P_center(i) - C_center(ipoint,i)) * (P_center(i) - C_center(ipoint,i))
      enddo
      const = coef_tmp * dist_integral 

      res_v(ipoint) = coeff * rint(0, const)
    enddo

  else

    do ipoint = 1, n_points
      dist_integral = 0.d0
      do i = 1, 3
        dist_integral += (P_center(i) - C_center(ipoint,i)) * (P_center(i) - C_center(ipoint,i))
      enddo
      const = coef_tmp * dist_integral 

      do i = 0, n_pt_in
        d(i) = 0.d0
      enddo
      call give_polynomial_mult_center_one_e_erf_opt(A_center, B_center, power_A, power_B, C_center(ipoint,1:3), n_pt_in, d, n_pt_out, p_inv_2, p_new2, P_center)
     
      if(n_pt_out < 0) then
        res_v(ipoint) = 0.d0
        cycle
      endif

      ! sum of integrals of type : int {t,[0,1]}  exp-(rho.(P-Q)^2 * t^2) * t^i
      accu = 0.d0
      do i = 0, n_pt_out, 2
        accu += d(i) * rint(i/2, const)
      enddo
     
      res_v(ipoint) = accu * coeff
    enddo

  endif

end subroutine NAI_pol_mult_erf_v

! ---

double precision function NAI_pol_mult_erf_with1s( A1_center, A2_center, power_A1, power_A2, alpha1, alpha2 &
                                                 , beta, B_center, C_center, n_pt_in, mu_in )

  BEGIN_DOC
  !
  ! Computes the following integral :
  !
  ! .. math::
  !
  !   \int dx (x - A1_x)^a_1 (x - B1_x)^a_2 \exp(-\alpha_1 (x - A1_x)^2 - \alpha_2 (x - A2_x)^2)
  !   \int dy (y - A1_y)^b_1 (y - B1_y)^b_2 \exp(-\alpha_1 (y - A1_y)^2 - \alpha_2 (y - A2_y)^2)
  !   \int dz (x - A1_z)^c_1 (z - B1_z)^c_2 \exp(-\alpha_1 (z - A1_z)^2 - \alpha_2 (z - A2_z)^2)
  !   \exp(-\beta (r - B)^2)
  !   \frac{\erf(\mu |r - R_C|)}{|r - R_C|}$.
  !
  END_DOC

  include 'utils/constants.include.F'

  implicit none
  integer,          intent(in) :: n_pt_in
  integer,          intent(in) :: power_A1(3), power_A2(3)
  double precision, intent(in) :: C_center(3), A1_center(3), A2_center(3), B_center(3)
  double precision, intent(in) :: alpha1, alpha2, beta, mu_in

  integer                      :: i, n_pt, n_pt_out
  double precision             :: alpha12, alpha12_inv, alpha12_inv_2, rho12, A12_center(3), dist12, const_factor12
  double precision             :: p, p_inv, p_inv_2, rho, P_center(3), dist, const_factor
  double precision             :: dist_integral
  double precision             :: d(0:n_pt_in), coeff, const, factor
  double precision             :: accu
  double precision             :: p_new

  double precision             :: rint


  ! e^{-alpha1 (r - A1)^2} e^{-alpha2 (r - A2)^2} = e^{-K12} e^{-alpha12 (r - A12)^2}
  alpha12       = alpha1 + alpha2
  alpha12_inv   = 1.d0 / alpha12
  alpha12_inv_2 = 0.5d0 * alpha12_inv
  rho12         = alpha1 * alpha2 * alpha12_inv
  A12_center(1) = (alpha1 * A1_center(1) + alpha2 * A2_center(1)) * alpha12_inv
  A12_center(2) = (alpha1 * A1_center(2) + alpha2 * A2_center(2)) * alpha12_inv
  A12_center(3) = (alpha1 * A1_center(3) + alpha2 * A2_center(3)) * alpha12_inv
  dist12        = (A1_center(1) - A2_center(1)) * (A1_center(1) - A2_center(1)) &
                + (A1_center(2) - A2_center(2)) * (A1_center(2) - A2_center(2)) &
                + (A1_center(3) - A2_center(3)) * (A1_center(3) - A2_center(3))

  const_factor12 = dist12 * rho12
  if(const_factor12 > 80.d0) then
    NAI_pol_mult_erf_with1s = 0.d0
    return
  endif

  ! ---

  ! e^{-K12} e^{-alpha12 (r - A12)^2} e^{-beta (r - B)^2} = e^{-K} e^{-p (r - P)^2}
  p           = alpha12 + beta
  p_inv       = 1.d0 / p
  p_inv_2     = 0.5d0 * p_inv
  rho         = alpha12 * beta * p_inv
  P_center(1) = (alpha12 * A12_center(1) + beta * B_center(1)) * p_inv
  P_center(2) = (alpha12 * A12_center(2) + beta * B_center(2)) * p_inv
  P_center(3) = (alpha12 * A12_center(3) + beta * B_center(3)) * p_inv
  dist        = (A12_center(1) - B_center(1)) * (A12_center(1) - B_center(1)) &
              + (A12_center(2) - B_center(2)) * (A12_center(2) - B_center(2)) &
              + (A12_center(3) - B_center(3)) * (A12_center(3) - B_center(3))

  const_factor = const_factor12 + dist * rho
  if(const_factor > 80.d0) then
    NAI_pol_mult_erf_with1s = 0.d0
    return
  endif

  dist_integral = (P_center(1) - C_center(1)) * (P_center(1) - C_center(1)) &
                + (P_center(2) - C_center(2)) * (P_center(2) - C_center(2)) &
                + (P_center(3) - C_center(3)) * (P_center(3) - C_center(3))

  ! ---

  p_new  = mu_in / dsqrt(p + mu_in * mu_in)
  factor = dexp(-const_factor)
  coeff  = dtwo_pi * factor * p_inv * p_new

  n_pt  =  2 * ( (power_A1(1) + power_A2(1)) + (power_A1(2) + power_A2(2)) + (power_A1(3) + power_A2(3)) )
  const = p * dist_integral * p_new * p_new
  if(n_pt == 0) then
    NAI_pol_mult_erf_with1s = coeff * rint(0, const)
    return
  endif

  do i = 0, n_pt_in
    d(i) = 0.d0
  enddo
  p_new = p_new * p_new
  call give_polynomial_mult_center_one_e_erf_opt( A1_center, A2_center, power_A1, power_A2, C_center, n_pt_in, d, n_pt_out, p_inv_2, p_new, P_center)

  if(n_pt_out < 0) then
    NAI_pol_mult_erf_with1s = 0.d0
    return
  endif

  ! sum of integrals of type : int {t,[0,1]}  exp-(rho.(P-Q)^2 * t^2) * t^i
  accu = 0.d0
  do i = 0, n_pt_out, 2
    accu += d(i) * rint(i/2, const)
  enddo
  NAI_pol_mult_erf_with1s = accu * coeff

end function NAI_pol_mult_erf_with1s

! ---

subroutine NAI_pol_mult_erf_with1s_v(A1_center, A2_center, power_A1, power_A2, alpha1, alpha2, beta, B_center, LD_B, C_center, LD_C, n_pt_in, mu_in, res_v, LD_resv, n_points)

  BEGIN_DOC
  !
  ! Computes the following integral :
  !
  ! .. math                      ::
  !
  !   \int dx (x - A1_x)^a_1 (x - B1_x)^a_2 \exp(-\alpha_1 (x - A1_x)^2 - \alpha_2 (x - A2_x)^2)
  !   \int dy (y - A1_y)^b_1 (y - B1_y)^b_2 \exp(-\alpha_1 (y - A1_y)^2 - \alpha_2 (y - A2_y)^2)
  !   \int dz (x - A1_z)^c_1 (z - B1_z)^c_2 \exp(-\alpha_1 (z - A1_z)^2 - \alpha_2 (z - A2_z)^2)
  !   \exp(-\beta (r - B)^2)
  !   \frac{\erf(\mu |r - R_C|)}{|r - R_C|}$.
  !
  END_DOC

  include 'utils/constants.include.F'

  implicit none
  integer,          intent(in)  :: n_pt_in, LD_B, LD_C, LD_resv, n_points
  integer,          intent(in)  :: power_A1(3), power_A2(3)
  double precision, intent(in)  :: A1_center(3), A2_center(3)
  double precision, intent(in)  :: C_center(LD_C,3), B_center(LD_B,3)
  double precision, intent(in)  :: alpha1, alpha2, beta, mu_in
  double precision, intent(out) :: res_v(LD_resv)

  integer                       :: i, n_pt, n_pt_out, ipoint
  double precision              :: alpha12, alpha12_inv, alpha12_inv_2, rho12, A12_center(3), dist12, const_factor12
  double precision              :: p, p_inv, p_inv_2, rho, P_center(3), dist, const_factor
  double precision              :: dist_integral
  double precision              :: d(0:n_pt_in), coeff, const, factor
  double precision              :: accu
  double precision              :: p_new, p_new2, coef_tmp, cons_tmp

  double precision              :: rint


  res_V(1:LD_resv) = 0.d0

  ! e^{-alpha1 (r - A1)^2} e^{-alpha2 (r - A2)^2} = e^{-K12} e^{-alpha12 (r - A12)^2}
  alpha12       = alpha1 + alpha2
  alpha12_inv   = 1.d0 / alpha12
  alpha12_inv_2 = 0.5d0 * alpha12_inv
  rho12         = alpha1 * alpha2 * alpha12_inv
  A12_center(1) = (alpha1 * A1_center(1) + alpha2 * A2_center(1)) * alpha12_inv
  A12_center(2) = (alpha1 * A1_center(2) + alpha2 * A2_center(2)) * alpha12_inv
  A12_center(3) = (alpha1 * A1_center(3) + alpha2 * A2_center(3)) * alpha12_inv
  dist12        = (A1_center(1) - A2_center(1)) * (A1_center(1) - A2_center(1))&
                + (A1_center(2) - A2_center(2)) * (A1_center(2) - A2_center(2))&
                + (A1_center(3) - A2_center(3)) * (A1_center(3) - A2_center(3))

  const_factor12 = dist12 * rho12
  if(const_factor12 > 80.d0) then
    return
  endif

  ! e^{-K12} e^{-alpha12 (r - A12)^2} e^{-beta (r - B)^2} = e^{-K} e^{-p (r - P)^2}
  p        = alpha12 + beta
  p_inv    = 1.d0 / p
  p_inv_2  = 0.5d0 * p_inv
  rho      = alpha12 * beta * p_inv
  p_new    = mu_in / dsqrt(p + mu_in * mu_in)
  p_new2   = p_new * p_new
  coef_tmp = dtwo_pi * p_inv * p_new
  cons_tmp = p * p_new2
  n_pt     =  2 * (power_A1(1) + power_A2(1) + power_A1(2) + power_A2(2) + power_A1(3) + power_A2(3) )

  if(n_pt == 0) then

    do ipoint = 1, n_points

      dist = (A12_center(1) - B_center(ipoint,1)) * (A12_center(1) - B_center(ipoint,1))&
           + (A12_center(2) - B_center(ipoint,2)) * (A12_center(2) - B_center(ipoint,2))&
           + (A12_center(3) - B_center(ipoint,3)) * (A12_center(3) - B_center(ipoint,3))
      const_factor = const_factor12 + dist * rho
      if(const_factor > 80.d0) cycle
      coeff = coef_tmp * dexp(-const_factor)

      P_center(1) = (alpha12 * A12_center(1) + beta * B_center(ipoint,1)) * p_inv
      P_center(2) = (alpha12 * A12_center(2) + beta * B_center(ipoint,2)) * p_inv
      P_center(3) = (alpha12 * A12_center(3) + beta * B_center(ipoint,3)) * p_inv
      dist_integral = (P_center(1) - C_center(ipoint,1)) * (P_center(1) - C_center(ipoint,1))&
                    + (P_center(2) - C_center(ipoint,2)) * (P_center(2) - C_center(ipoint,2))&
                    + (P_center(3) - C_center(ipoint,3)) * (P_center(3) - C_center(ipoint,3))
      const = cons_tmp * dist_integral

      res_v(ipoint) = coeff * rint(0, const)
    enddo

  else

    do ipoint = 1, n_points
  
      dist = (A12_center(1) - B_center(ipoint,1)) * (A12_center(1) - B_center(ipoint,1))&
           + (A12_center(2) - B_center(ipoint,2)) * (A12_center(2) - B_center(ipoint,2))&
           + (A12_center(3) - B_center(ipoint,3)) * (A12_center(3) - B_center(ipoint,3)) 
      const_factor = const_factor12 + dist * rho
      if(const_factor > 80.d0) cycle
      coeff = coef_tmp * dexp(-const_factor)
  
      P_center(1) = (alpha12 * A12_center(1) + beta * B_center(ipoint,1)) * p_inv
      P_center(2) = (alpha12 * A12_center(2) + beta * B_center(ipoint,2)) * p_inv
      P_center(3) = (alpha12 * A12_center(3) + beta * B_center(ipoint,3)) * p_inv
      dist_integral = (P_center(1) - C_center(ipoint,1)) * (P_center(1) - C_center(ipoint,1))&
                    + (P_center(2) - C_center(ipoint,2)) * (P_center(2) - C_center(ipoint,2))&
                    + (P_center(3) - C_center(ipoint,3)) * (P_center(3) - C_center(ipoint,3))
      const = cons_tmp * dist_integral
  
      do i = 0, n_pt_in
        d(i) = 0.d0
      enddo
      !TODO: VECTORIZE HERE
      call give_polynomial_mult_center_one_e_erf_opt(A1_center, A2_center, power_A1, power_A2, C_center(ipoint,1:3), n_pt_in, d, n_pt_out, p_inv_2, p_new2, P_center)
  
      if(n_pt_out < 0) then
        cycle
      endif
  
      ! sum of integrals of type : int {t,[0,1]}  exp-(rho.(P-Q)^2 * t^2) * t^i
      accu = 0.d0
      do i = 0, n_pt_out, 2
        accu += d(i) * rint(i/2, const)
      enddo
  
      res_v(ipoint) = accu * coeff
    enddo

  endif

end subroutine NAI_pol_mult_erf_with1s_v

! ---

subroutine give_polynomial_mult_center_one_e_erf_opt(A_center, B_center, power_A, power_B, C_center, n_pt_in, d, n_pt_out, p_inv_2, p_new, P_center)

  BEGIN_DOC
  ! Returns the explicit polynomial in terms of the $t$ variable of the
  ! following polynomial:
  !
  ! $I_{x1}(a_x, d_x,p,q) \times I_{x1}(a_y, d_y,p,q) \times I_{x1}(a_z, d_z,p,q)$.
  END_DOC

  implicit none
  integer,          intent(in)  :: n_pt_in
  integer,          intent(in)  :: power_A(3), power_B(3)
  double precision, intent(in)  :: A_center(3), B_center(3), C_center(3), p_inv_2, p_new, P_center(3)
  integer,          intent(out) :: n_pt_out
  double precision, intent(out) :: d(0:n_pt_in)

  integer                       :: a_x, b_x, a_y, b_y, a_z, b_z
  integer                       :: n_pt1, n_pt2, n_pt3, dim, i
  integer                       :: n_pt_tmp
  double precision              :: d1(0:n_pt_in)
  double precision              :: d2(0:n_pt_in)
  double precision              :: d3(0:n_pt_in)
  double precision              :: accu
  double precision              :: R1x(0:2), B01(0:2), R1xp(0:2), R2x(0:2)

  accu = 0.d0
  ASSERT (n_pt_in > 1)

  R1x(0) = (P_center(1) - A_center(1))
  R1x(1) = 0.d0
  R1x(2) = -(P_center(1) - C_center(1))* p_new
  ! R1x = (P_x - A_x) - (P_x - C_x) ( t * mu/sqrt(p+mu^2) )^2
  R1xp(0) = (P_center(1) - B_center(1))
  R1xp(1) = 0.d0
  R1xp(2) =-(P_center(1) - C_center(1))* p_new
  !R1xp = (P_x - B_x) - (P_x - C_x) ( t * mu/sqrt(p+mu^2) )^2
  R2x(0) =  p_inv_2
  R2x(1) = 0.d0
  R2x(2) = -p_inv_2 * p_new
  !R2x  = 0.5 / p - 0.5/p ( t * mu/sqrt(p+mu^2) )^2

  do i = 0, n_pt_in
    d (i) = 0.d0
    d1(i) = 0.d0
    d2(i) = 0.d0
    d3(i) = 0.d0
  enddo

  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))* p_new
  ! R1x = (P_x - A_x) - (P_x - C_x) ( t * mu/sqrt(p+mu^2) )^2
  R1xp(0) = (P_center(2) - B_center(2))
  R1xp(1) = 0.d0
  R1xp(2) =-(P_center(2) - C_center(2))* p_new
  !R1xp = (P_x - B_x) - (P_x - C_x) ( t * mu/sqrt(p+mu^2) )^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)) * p_new
  ! R1x = (P_x - A_x) - (P_x - C_x) ( t * mu/sqrt(p+mu^2) )^2
  R1xp(0) = (P_center(3) - B_center(3))
  R1xp(1) = 0.d0
  R1xp(2) =-(P_center(3) - C_center(3)) * p_new
  !R2x  = 0.5 / p - 0.5/p ( t * mu/sqrt(p+mu^2) )^2
  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

  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 subroutine give_polynomial_mult_center_one_e_erf_opt

! ---

subroutine give_polynomial_mult_center_one_e_erf(A_center,B_center,alpha,beta,power_A,power_B,C_center,n_pt_in,d,n_pt_out,mu_in)

  BEGIN_DOC
  ! Returns the explicit polynomial in terms of the $t$ variable of the
  ! following polynomial:
  !
  ! $I_{x1}(a_x, d_x,p,q) \times I_{x1}(a_y, d_y,p,q) \times I_{x1}(a_z, d_z,p,q)$.
  END_DOC

  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,mu_in
  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))* mu_in**2 / (p+mu_in*mu_in)
  ! R1x = (P_x - A_x) - (P_x - C_x) ( t * mu/sqrt(p+mu^2) )^2
  R1xp(0)  = (P_center(1) - B_center(1))
  R1xp(1)  = 0.d0
  R1xp(2)  =-(P_center(1) - C_center(1))* mu_in**2 / (p+mu_in*mu_in)
  !R1xp = (P_x - B_x) - (P_x - C_x) ( t * mu/sqrt(p+mu^2) )^2
  R2x(0)  =  p_inv_2
  R2x(1)  = 0.d0
  R2x(2)  = -p_inv_2* mu_in**2 / (p+mu_in*mu_in)
  !R2x  = 0.5 / p - 0.5/p ( t * mu/sqrt(p+mu^2) )^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)
  ! 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))* mu_in**2 / (p+mu_in*mu_in)
  ! R1x = (P_x - A_x) - (P_x - C_x) ( t * mu/sqrt(p+mu^2) )^2
  R1xp(0)  = (P_center(2) - B_center(2))
  R1xp(1)  = 0.d0
  R1xp(2)  =-(P_center(2) - C_center(2))* mu_in**2 / (p+mu_in*mu_in)
  !R1xp = (P_x - B_x) - (P_x - C_x) ( t * mu/sqrt(p+mu^2) )^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)
  ! 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))* mu_in**2 / (p+mu_in*mu_in)
  ! R1x = (P_x - A_x) - (P_x - C_x) ( t * mu/sqrt(p+mu^2) )^2
  R1xp(0)  = (P_center(3) - B_center(3))
  R1xp(1)  = 0.d0
  R1xp(2)  =-(P_center(3) - C_center(3))* mu_in**2 / (p+mu_in*mu_in)
  !R2x  = 0.5 / p - 0.5/p ( t * mu/sqrt(p+mu^2) )^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_one_e(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