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qp2/src/utils/integration.irp.f

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subroutine give_explicit_poly_and_gaussian_x(P_new,P_center,p,fact_k,iorder,alpha,beta,a,b,A_center,B_center,dim)
BEGIN_DOC
! Transform the product of
! (x-x_A)^a(1) (x-x_B)^b(1) (x-x_A)^a(2) (y-y_B)^b(2) (z-z_A)^a(3) (z-z_B)^b(3) exp(-(r-A)^2 alpha) exp(-(r-B)^2 beta)
! into
! fact_k (x-x_P)^iorder(1) (y-y_P)^iorder(2) (z-z_P)^iorder(3) exp(-p(r-P)^2)
END_DOC
implicit none
include 'constants.include.F'
integer, intent(in) :: dim
integer, intent(in) :: a,b ! powers : (x-xa)**a_x = (x-A(1))**a(1)
double precision, intent(in) :: alpha, beta ! exponents
double precision, intent(in) :: A_center ! A center
double precision, intent(in) :: B_center ! B center
double precision, intent(out) :: P_center ! new center
double precision, intent(out) :: p ! new exponent
double precision, intent(out) :: fact_k ! constant factor
double precision, intent(out) :: P_new(0:max_dim) ! polynomial
integer, intent(out) :: iorder ! order of the polynomials
double precision :: P_a(0:max_dim), P_b(0:max_dim)
integer :: n_new,i,j
double precision :: p_inv,ab,d_AB
!DIR$ ATTRIBUTES ALIGN : $IRP_ALIGN :: P_a, P_b
! Do the gaussian product to get the new center and the new exponent
P_new = 0.d0
p = alpha+beta
p_inv = 1.d0/p
ab = alpha * beta
d_AB = (A_center - B_center) * (A_center - B_center)
P_center = (alpha * A_center + beta * B_center) * p_inv
fact_k = exp(-ab*p_inv * d_AB)
! Recenter the polynomials P_a and P_b on x
!DIR$ FORCEINLINE
call recentered_poly2(P_a(0),A_center,P_center,a,P_b(0),B_center,P_center,b)
n_new = 0
!DIR$ FORCEINLINE
call multiply_poly(P_a(0),a,P_b(0),b,P_new(0),n_new)
iorder = a + b
end
subroutine give_explicit_poly_and_gaussian(P_new,P_center,p,fact_k,iorder,alpha,beta,a,b,A_center,B_center,dim)
BEGIN_DOC
! Transforms the product of
! (x-x_A)^a(1) (x-x_B)^b(1) (x-x_A)^a(2) (y-y_B)^b(2) (z-z_A)^a(3) (z-z_B)^b(3) exp(-(r-A)^2 alpha) exp(-(r-B)^2 beta)
! into
! fact_k * [ sum (l_x = 0,i_order(1)) P_new(l_x,1) * (x-P_center(1))^l_x ] exp (- p (x-P_center(1))^2 )
! * [ sum (l_y = 0,i_order(2)) P_new(l_y,2) * (y-P_center(2))^l_y ] exp (- p (y-P_center(2))^2 )
! * [ sum (l_z = 0,i_order(3)) P_new(l_z,3) * (z-P_center(3))^l_z ] exp (- p (z-P_center(3))^2 )
END_DOC
implicit none
include 'constants.include.F'
integer, intent(in) :: dim
integer, intent(in) :: a(3),b(3) ! powers : (x-xa)**a_x = (x-A(1))**a(1)
double precision, intent(in) :: alpha, beta ! exponents
double precision, intent(in) :: A_center(3) ! A center
double precision, intent(in) :: B_center (3) ! B center
double precision, intent(out) :: P_center(3) ! new center
double precision, intent(out) :: p ! new exponent
double precision, intent(out) :: fact_k ! constant factor
double precision, intent(out) :: P_new(0:max_dim,3)! polynomial
integer, intent(out) :: iorder(3) ! i_order(i) = order of the polynomials
double precision :: P_a(0:max_dim,3), P_b(0:max_dim,3)
integer :: n_new,i,j
!DIR$ ATTRIBUTES ALIGN : $IRP_ALIGN :: P_a, P_b
iorder(1) = 0
iorder(2) = 0
iorder(3) = 0
P_new(0,1) = 0.d0
P_new(0,2) = 0.d0
P_new(0,3) = 0.d0
!DIR$ FORCEINLINE
call gaussian_product(alpha,A_center,beta,B_center,fact_k,p,P_center)
if (fact_k < thresh) then
fact_k = 0.d0
return
endif
!DIR$ FORCEINLINE
call recentered_poly2(P_a(0,1),A_center(1),P_center(1),a(1),P_b(0,1),B_center(1),P_center(1),b(1))
iorder(1) = a(1) + b(1)
do i=0,iorder(1)
P_new(i,1) = 0.d0
enddo
n_new=0
!DIR$ FORCEINLINE
call multiply_poly(P_a(0,1),a(1),P_b(0,1),b(1),P_new(0,1),n_new)
!DIR$ FORCEINLINE
call recentered_poly2(P_a(0,2),A_center(2),P_center(2),a(2),P_b(0,2),B_center(2),P_center(2),b(2))
iorder(2) = a(2) + b(2)
do i=0,iorder(2)
P_new(i,2) = 0.d0
enddo
n_new=0
!DIR$ FORCEINLINE
call multiply_poly(P_a(0,2),a(2),P_b(0,2),b(2),P_new(0,2),n_new)
!DIR$ FORCEINLINE
call recentered_poly2(P_a(0,3),A_center(3),P_center(3),a(3),P_b(0,3),B_center(3),P_center(3),b(3))
iorder(3) = a(3) + b(3)
do i=0,iorder(3)
P_new(i,3) = 0.d0
enddo
n_new=0
!DIR$ FORCEINLINE
call multiply_poly(P_a(0,3),a(3),P_b(0,3),b(3),P_new(0,3),n_new)
end
subroutine give_explicit_poly_and_gaussian_double(P_new,P_center,p,fact_k,iorder,alpha,beta,gama,a,b,A_center,B_center,Nucl_center,dim)
BEGIN_DOC
! Transforms the product of
! (x-x_A)^a(1) (x-x_B)^b(1) (x-x_A)^a(2) (y-y_B)^b(2) (z-z_A)^a(3) (z-z_B)^b(3)
! exp(-(r-A)^2 alpha) exp(-(r-B)^2 beta) exp(-(r-Nucl_center)^2 gama
!
! into
! fact_k * [ sum (l_x = 0,i_order(1)) P_new(l_x,1) * (x-P_center(1))^l_x ] exp (- p (x-P_center(1))^2 )
! * [ sum (l_y = 0,i_order(2)) P_new(l_y,2) * (y-P_center(2))^l_y ] exp (- p (y-P_center(2))^2 )
! * [ sum (l_z = 0,i_order(3)) P_new(l_z,3) * (z-P_center(3))^l_z ] exp (- p (z-P_center(3))^2 )
END_DOC
implicit none
include 'constants.include.F'
integer, intent(in) :: dim
integer, intent(in) :: a(3),b(3) ! powers : (x-xa)**a_x = (x-A(1))**a(1)
double precision, intent(in) :: alpha, beta, gama ! exponents
double precision, intent(in) :: A_center(3) ! A center
double precision, intent(in) :: B_center (3) ! B center
double precision, intent(in) :: Nucl_center(3) ! B center
double precision, intent(out) :: P_center(3) ! new center
double precision, intent(out) :: p ! new exponent
double precision, intent(out) :: fact_k ! constant factor
double precision, intent(out) :: P_new(0:max_dim,3)! polynomial
integer , intent(out) :: iorder(3) ! i_order(i) = order of the polynomials
double precision :: P_center_tmp(3) ! new center
double precision :: p_tmp ! new exponent
double precision :: fact_k_tmp,fact_k_bis ! constant factor
double precision :: P_new_tmp(0:max_dim,3)! polynomial
integer :: i,j
double precision :: binom_func
! First you transform the two primitives into a sum of primitive with the same center P_center_tmp and gaussian exponent p_tmp
call give_explicit_poly_and_gaussian(P_new_tmp,P_center_tmp,p_tmp,fact_k_tmp,iorder,alpha,beta,a,b,A_center,B_center,dim)
! Then you create the new gaussian from the product of the new one per the Nuclei one
call gaussian_product(p_tmp,P_center_tmp,gama,Nucl_center,fact_k_bis,p,P_center)
fact_k = fact_k_bis * fact_k_tmp
! Then you build the coefficient of the new polynom
do i = 0, iorder(1)
P_new(i,1) = 0.d0
do j = i,iorder(1)
P_new(i,1) = P_new(i,1) + P_new_tmp(j,1) * binom_func(j,j-i) * (P_center(1) - P_center_tmp(1))**(j-i)
enddo
enddo
do i = 0, iorder(2)
P_new(i,2) = 0.d0
do j = i,iorder(2)
P_new(i,2) = P_new(i,2) + P_new_tmp(j,2) * binom_func(j,j-i) * (P_center(2) - P_center_tmp(2))**(j-i)
enddo
enddo
do i = 0, iorder(3)
P_new(i,3) = 0.d0
do j = i,iorder(3)
P_new(i,3) = P_new(i,3) + P_new_tmp(j,3) * binom_func(j,j-i) * (P_center(3) - P_center_tmp(3))**(j-i)
enddo
enddo
end
subroutine gaussian_product(a,xa,b,xb,k,p,xp)
implicit none
BEGIN_DOC
! Gaussian product in 1D.
! e^{-a (x-x_A)^2} e^{-b (x-x_B)^2} = K_{ab}^x e^{-p (x-x_P)^2}
END_DOC
double precision, intent(in) :: a,b ! Exponents
double precision, intent(in) :: xa(3),xb(3) ! Centers
double precision, intent(out) :: p ! New exponent
double precision, intent(out) :: xp(3) ! New center
double precision, intent(out) :: k ! Constant
double precision :: p_inv
ASSERT (a>0.)
ASSERT (b>0.)
double precision :: xab(3), ab
!DIR$ ATTRIBUTES ALIGN : $IRP_ALIGN :: xab
p = a+b
p_inv = 1.d0/(a+b)
ab = a*b
xab(1) = xa(1)-xb(1)
xab(2) = xa(2)-xb(2)
xab(3) = xa(3)-xb(3)
ab = ab*p_inv
k = ab*(xab(1)*xab(1)+xab(2)*xab(2)+xab(3)*xab(3))
if (k > 40.d0) then
k=0.d0
return
endif
k = dexp(-k)
xp(1) = (a*xa(1)+b*xb(1))*p_inv
xp(2) = (a*xa(2)+b*xb(2))*p_inv
xp(3) = (a*xa(3)+b*xb(3))*p_inv
end subroutine
subroutine gaussian_product_x(a,xa,b,xb,k,p,xp)
implicit none
BEGIN_DOC
! Gaussian product in 1D.
! e^{-a (x-x_A)^2} e^{-b (x-x_B)^2} = K_{ab}^x e^{-p (x-x_P)^2}
END_DOC
double precision , intent(in) :: a,b ! Exponents
double precision , intent(in) :: xa,xb ! Centers
double precision , intent(out) :: p ! New exponent
double precision , intent(out) :: xp ! New center
double precision , intent(out) :: k ! Constant
double precision :: p_inv
ASSERT (a>0.)
ASSERT (b>0.)
double precision :: xab, ab
p = a+b
p_inv = 1.d0/(a+b)
ab = a*b
xab = xa-xb
ab = ab*p_inv
k = ab*xab*xab
if (k > 40.d0) then
k=0.d0
return
endif
k = exp(-k)
xp = (a*xa+b*xb)*p_inv
end subroutine
subroutine multiply_poly(b,nb,c,nc,d,nd)
implicit none
BEGIN_DOC
! Multiply two polynomials
! D(t) += B(t)*C(t)
END_DOC
integer, intent(in) :: nb, nc
integer, intent(out) :: nd
double precision, intent(in) :: b(0:nb), c(0:nc)
double precision, intent(inout) :: d(0:nb+nc)
integer :: ndtmp
integer :: ib, ic, id, k
if(ior(nc,nb) >= 0) then ! True if nc>=0 and nb>=0
continue
else
return
endif
ndtmp = nb+nc
do ic = 0,nc
d(ic) = d(ic) + c(ic) * b(0)
enddo
do ib=1,nb
d(ib) = d(ib) + c(0) * b(ib)
do ic = 1,nc
d(ib+ic) = d(ib+ic) + c(ic) * b(ib)
enddo
enddo
do nd = ndtmp,0,-1
if (d(nd) == 0.d0) then
cycle
endif
exit
enddo
end
subroutine add_poly(b,nb,c,nc,d,nd)
implicit none
BEGIN_DOC
! Add two polynomials
! D(t) += B(t)+C(t)
END_DOC
integer, intent(inout) :: nb, nc
integer, intent(out) :: nd
double precision, intent(in) :: b(0:nb), c(0:nc)
double precision, intent(out) :: d(0:nb+nc)
nd = nb+nc
integer :: ib, ic, id
do ib=0,max(nb,nc)
d(ib) = d(ib) + c(ib) + b(ib)
enddo
do while ( (d(nd) == 0.d0).and.(nd>=0) )
nd -= 1
if (nd < 0) then
exit
endif
enddo
end
subroutine add_poly_multiply(b,nb,cst,d,nd)
implicit none
BEGIN_DOC
! Add a polynomial multiplied by a constant
! D(t) += cst * B(t)
END_DOC
integer, intent(in) :: nb
integer, intent(inout) :: nd
double precision, intent(in) :: b(0:nb),cst
double precision, intent(inout) :: d(0:max(nb,nd))
nd = max(nd,nb)
if (nd /= -1) then
integer :: ib, ic, id
do ib=0,nb
d(ib) = d(ib) + cst*b(ib)
enddo
do while ( d(nd) == 0.d0 )
nd -= 1
if (nd < 0) then
exit
endif
enddo
endif
end
subroutine recentered_poly2(P_new,x_A,x_P,a,P_new2,x_B,x_Q,b)
implicit none
BEGIN_DOC
! Recenter two polynomials
END_DOC
integer, intent(in) :: a,b
double precision, intent(in) :: x_A,x_P,x_B,x_Q
double precision, intent(out) :: P_new(0:a),P_new2(0:b)
double precision :: pows_a(-2:a+b+4), pows_b(-2:a+b+4)
double precision :: binom_func
integer :: i,j,k,l, minab, maxab
if ((a<0).or.(b<0) ) return
maxab = max(a,b)
minab = max(min(a,b),0)
pows_a(0) = 1.d0
pows_a(1) = (x_P - x_A)
pows_b(0) = 1.d0
pows_b(1) = (x_Q - x_B)
do i = 2,maxab
pows_a(i) = pows_a(i-1)*pows_a(1)
pows_b(i) = pows_b(i-1)*pows_b(1)
enddo
P_new (0) = pows_a(a)
P_new2(0) = pows_b(b)
do i = 1,min(minab,20)
P_new (i) = binom_transp(a-i,a) * pows_a(a-i)
P_new2(i) = binom_transp(b-i,b) * pows_b(b-i)
enddo
do i = minab+1,min(a,20)
P_new (i) = binom_transp(a-i,a) * pows_a(a-i)
enddo
do i = minab+1,min(b,20)
P_new2(i) = binom_transp(b-i,b) * pows_b(b-i)
enddo
do i = 101,a
P_new(i) = binom_func(a,a-i) * pows_a(a-i)
enddo
do i = 101,b
P_new2(i) = binom_func(b,b-i) * pows_b(b-i)
enddo
end
double precision function F_integral(n,p)
BEGIN_DOC
! function that calculates the following integral
! \int_{\-infty}^{+\infty} x^n \exp(-p x^2) dx
END_DOC
implicit none
integer :: n
double precision :: p
integer :: i,j
double precision :: accu,sqrt_p,fact_ratio,tmp,fact
include 'constants.include.F'
if(n < 0)then
F_integral = 0.d0
endif
if(iand(n,1).ne.0)then
F_integral = 0.d0
return
endif
sqrt_p = 1.d0/dsqrt(p)
if(n==0)then
F_integral = sqpi * sqrt_p
return
endif
F_integral = sqpi * 0.5d0**n * sqrt_p**(n+1) * fact(n)/fact(shiftr(n,1))
end
double precision function rint(n,rho)
implicit none
BEGIN_DOC
!.. math::
!
! \int_0^1 dx \exp(-p x^2) x^n
!
END_DOC
include 'constants.include.F'
double precision :: rho,u,rint1,v,val0,rint_large_n,u_inv
integer :: n,k
double precision :: two_rho_inv
if(n.eq.0)then
if(rho == 0.d0)then
rint=1.d0
else
u_inv=1.d0/dsqrt(rho)
u=rho*u_inv
rint=0.5d0*u_inv*sqpi*erf(u)
endif
return
endif
if(rho.lt.1.d0)then
rint=rint1(n,rho)
else
if(n.le.20)then
u_inv=1.d0/dsqrt(rho)
if(rho.gt.80.d0)then
v=0.d0
else
v=dexp(-rho)
endif
u=rho*u_inv
two_rho_inv = 0.5d0*u_inv*u_inv
val0=0.5d0*u_inv*sqpi*erf(u)
rint=(val0-v)*two_rho_inv
do k=2,n
rint=(rint*dfloat(k+k-1)-v)*two_rho_inv
enddo
else
rint=rint_large_n(n,rho)
endif
endif
end
double precision function rint_sum(n_pt_out,rho,d1)
implicit none
BEGIN_DOC
! Needed for the calculation of two-electron integrals.
END_DOC
include 'constants.include.F'
integer, intent(in) :: n_pt_out
double precision, intent(in) :: rho,d1(0:n_pt_out)
double precision :: u,rint1,v,val0,rint_large_n,u_inv
integer :: n,k,i
double precision :: two_rho_inv, rint_tmp, di
if(rho < 1.d0)then
if(rho == 0.d0)then
rint_sum=d1(0)
else
u_inv=1.d0/dsqrt(rho)
u=rho*u_inv
rint_sum=0.5d0*u_inv*sqpi*erf(u) *d1(0)
endif
do i=2,n_pt_out,2
n = shiftr(i,1)
rint_sum = rint_sum + d1(i)*rint1(n,rho)
enddo
else
if(rho.gt.80.d0)then
v=0.d0
else
v=dexp(-rho)
endif
u_inv=1.d0/dsqrt(rho)
u=rho*u_inv
two_rho_inv = 0.5d0*u_inv*u_inv
val0=0.5d0*u_inv*sqpi*erf(u)
rint_sum=val0*d1(0)
rint_tmp=(val0-v)*two_rho_inv
di = 3.d0
do i=2,min(n_pt_out,40),2
rint_sum = rint_sum + d1(i)*rint_tmp
rint_tmp = (rint_tmp*di-v)*two_rho_inv
di = di+2.d0
enddo
do i=42,n_pt_out,2
n = shiftr(i,1)
rint_sum = rint_sum + d1(i)*rint_large_n(n,rho)
enddo
endif
end
double precision function hermite(n,x)
implicit none
BEGIN_DOC
! Hermite polynomial
END_DOC
integer :: n,k
double precision :: h0,x,h1,h2
h0=1.d0
if(n.eq.0)then
hermite=h0
return
endif
h1=x+x
if(n.eq.1)then
hermite=h1
return
endif
do k=1,n-1
h2=(x+x)*h1-dfloat(k+k)*h0
h0=h1
h1=h2
enddo
hermite=h2
end
double precision function rint_large_n(n,rho)
implicit none
BEGIN_DOC
! Version of rint for large values of n
END_DOC
integer :: n,k,l
double precision :: rho,u,accu,eps,t1,t2,fact,alpha_k,rajout,hermite
u=dsqrt(rho)
accu=0.d0
k=0
eps=1.d0
do while (eps.gt.1.d-15)
t1=1.d0
do l=0,k
t1=t1*(n+n+l+1.d0)
enddo
t2=0.d0
do l=0,k
t2=t2+(-1.d0)**l/(fact(l+1)*fact(k-l))
enddo
alpha_k=t2*fact(k+1)*fact(k)*(-1.d0)**k
alpha_k= alpha_k/t1
rajout=(-1.d0)**k*u**k*hermite(k,u)*alpha_k/fact(k)
accu=accu+rajout
eps=dabs(rajout)/accu
k=k+1
enddo
rint_large_n=dexp(-rho)*accu
end
double precision function rint1(n,rho)
implicit none
BEGIN_DOC
! Standard version of rint
END_DOC
integer, intent(in) :: n
double precision, intent(in) :: rho
double precision, parameter :: eps=1.d-15
double precision :: rho_tmp, diff
integer :: k
rint1=inv_int(n+n+1)
rho_tmp = 1.d0
do k=1,20
rho_tmp = -rho_tmp*rho
diff=rho_tmp*fact_inv(k)*inv_int(shiftl(k+n,1)+1)
rint1=rint1+diff
if (dabs(diff) > eps) then
cycle
endif
return
enddo
write(*,*)'pb in rint1 k too large!'
stop 1
end
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