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QuantumPackage/src/ao_one_e_ints/pseudopot.f90
Anthony Scemama 24d9745056 Initial commit
2019-01-25 11:39:31 +01:00

2173 lines
53 KiB
Fortran
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!! INFO : You can display equations using : http://www.codecogs.com/latex/eqneditor.php
!!
!! {\tt Vps}(C) = \langle \Phi_A|{\tt Vloc}(C)+{\tt Vpp}(C)| \Phi_B \rangle
!!
!! with :
!!
!! {\tt Vloc}(C)=\sum_{k=1}^{\tt klocmax} v_k r_C^{n_k} \exp(-dz_k r_C^2) \\
!!
!! {\tt Vpp}(C)=\sum_{l=0}^{\tt lmax}\left( \sum_{k=1}^{\tt kmax} v_{kl}
!! r_C^{n_{kl}} \exp(-dz_{kl} r_C)^2 \right) |l\rangle \langle l|
!!
double precision function Vps &
(a,n_a,g_a,b,n_b,g_b,c,klocmax,v_k,n_k,dz_k,lmax,kmax,v_kl,n_kl,dz_kl)
implicit none
integer n_a(3),n_b(3)
double precision g_a,g_b,a(3),b(3),c(3)
integer lmax,kmax,n_kl(kmax,0:lmax)
double precision v_kl(kmax,0:lmax),dz_kl(kmax,0:lmax)
integer klocmax,n_k(klocmax)
double precision v_k(klocmax),dz_k(klocmax)
double precision Vloc,Vpseudo
Vps=Vloc(klocmax,v_k,n_k,dz_k,a,n_a,g_a,b,n_b,g_b,c) &
+Vpseudo(lmax,kmax,v_kl,n_kl,dz_kl,a,n_a,g_a,b,n_b,g_b,c)
end
!!
!! Vps_num: brute force numerical evaluation of the same matrix element Vps
!!
double precision function Vps_num &
(npts,rmax,a,n_a,g_a,b,n_b,g_b,c,klocmax,v_k,n_k,dz_k,lmax,kmax,v_kl,n_kl,dz_kl)
implicit none
integer n_a(3),n_b(3)
double precision g_a,g_b,a(3),b(3),c(3),rmax
integer lmax,kmax,n_kl(kmax,0:lmax)
double precision v_kl(kmax,0:lmax),dz_kl(kmax,0:lmax)
integer klocmax,n_k(klocmax)
double precision v_k(klocmax),dz_k(klocmax)
double precision Vloc_num,Vpseudo_num,v1,v2
integer npts,nptsgrid
nptsgrid=50
call initpseudos(nptsgrid)
v1=Vloc_num(npts,rmax,klocmax,v_k,n_k,dz_k,a,n_a,g_a,b,n_b,g_b,c)
v2=Vpseudo_num(nptsgrid,rmax,lmax,kmax,v_kl,n_kl,dz_kl,a,n_a,g_a,b,n_b,g_b,c)
Vps_num=v1+v2
end
double precision function Vloc_num(npts_over,xmax,klocmax,v_k,n_k,dz_k,a,n_a,g_a,b,n_b,g_b,c)
implicit none
integer klocmax
double precision v_k(klocmax),dz_k(klocmax)
integer n_k(klocmax)
integer npts_over,ix,iy,iz
double precision xmax,dx,x,y,z
double precision a(3),b(3),c(3),term,r,orb_phi,g_a,g_b,ac(3),bc(3)
integer n_a(3),n_b(3),k,l
do l=1,3
ac(l)=a(l)-c(l)
bc(l)=b(l)-c(l)
enddo
dx=2.d0*xmax/npts_over
Vloc_num=0.d0
do ix=1,npts_over
do iy=1,npts_over
do iz=1,npts_over
x=-xmax+dx*ix+dx/2.d0
y=-xmax+dx*iy+dx/2.d0
z=-xmax+dx*iz+dx/2.d0
term=orb_phi(x,y,z,n_a,ac,g_a)*orb_phi(x,y,z,n_b,bc,g_b)
r=dsqrt(x**2+y**2+z**2)
do k=1,klocmax
Vloc_num=Vloc_num+dx**3*v_k(k)*r**n_k(k)*dexp(-dz_k(k)*r**2)*term
enddo
enddo
enddo
enddo
end
double precision function orb_phi(x,y,z,npower,center,gamma)
implicit none
integer npower(3)
double precision x,y,z,r2,gamma,center(3)
r2=(x-center(1))**2+(y-center(2))**2+(z-center(3))**2
orb_phi=(x-center(1))**npower(1)*(y-center(2))**npower(2)*(z-center(3))**npower(3)
orb_phi=orb_phi*dexp(-gamma*r2)
end
!! Real spherical harmonics Ylm
! factor = ([(2l+1)*(l-|m|)!]/[4pi*(l+|m|)!])^1/2
! Y_lm(theta,phi) =
! m > 0 factor* P_l^|m|(cos(theta)) cos (|m| phi)
! m = 0 1/sqrt(2) *factor* P_l^0(cos(theta))
! m < 0 factor* P_l^|m|(cos(theta)) sin (|m| phi)
!
! x=cos(theta)
double precision function ylm_real(l,m,x,phi)
implicit double precision (a-h,o-z)
DIMENSION PM(0:100,0:100)
MM=100
pi=dacos(-1.d0)
fourpi=4.d0*pi
iabs_m=iabs(m)
if(iabs_m.gt.l)stop 'm must be between -l and l'
factor= dsqrt( ((l+l+1)*fact(l-iabs_m))/(fourpi*fact(l+iabs_m)) )
if(dabs(x).gt.1.d0)then
print*,'pb. in ylm_no'
print*,'x=',x
stop
endif
call LPMN(MM,l,l,X,PM)
plm=PM(iabs_m,l)
coef=factor*plm
if(m.gt.0)ylm_real=dsqrt(2.d0)*coef*dcos(iabs_m*phi)
if(m.eq.0)ylm_real=coef
if(m.lt.0)ylm_real=dsqrt(2.d0)*coef*dsin(iabs_m*phi)
if(l.eq.0)ylm_real=dsqrt(1.d0/fourpi)
xchap=dsqrt(1.d0-x**2)*dcos(phi)
ychap=dsqrt(1.d0-x**2)*dsin(phi)
zchap=x
if(l.eq.1.and.m.eq.1)ylm_real=dsqrt(3.d0/fourpi)*xchap
if(l.eq.1.and.m.eq.0)ylm_real=dsqrt(3.d0/fourpi)*zchap
if(l.eq.1.and.m.eq.-1)ylm_real=dsqrt(3.d0/fourpi)*ychap
if(l.eq.2.and.m.eq.2)ylm_real=dsqrt(15.d0/16.d0/pi)*(xchap*xchap-ychap*ychap)
if(l.eq.2.and.m.eq.1)ylm_real=dsqrt(15.d0/fourpi)*xchap*zchap
if(l.eq.2.and.m.eq.0)ylm_real=dsqrt(5.d0/16.d0/pi)*(2.d0*zchap*zchap-xchap*xchap-ychap*ychap)
if(l.eq.2.and.m.eq.-1)ylm_real=dsqrt(15.d0/fourpi)*ychap*zchap
if(l.eq.2.and.m.eq.-2)ylm_real=dsqrt(15.d0/fourpi)*xchap*ychap
if(l.gt.2)stop 'l > 2 not coded!'
end
! _
! | |
! __ __ _ __ ___ ___ _ _ __| | ___
! \ \ / / | '_ \/ __|/ _ \ | | |/ _` |/ _ \
! \ V / | |_) \__ \ __/ |_| | (_| | (_) |
! \_/ | .__/|___/\___|\__,_|\____|\___/
! | |
! |_|
!! Routine Vpseudo is based on formumla (66)
!! of Kahn Baybutt TRuhlar J.Chem.Phys. vol.65 3826 (1976):
!!
!! Vpseudo= (4pi)**2* \sum_{l=0}^lmax \sum_{m=-l}^{l}
!! \sum{lambda=0}^{l+nA} \sum_{mu=-lambda}^{lambda}
!! \sum{k1=0}^{nAx} \sum{k2=0}^{nAy} \sum{k3=0}^{nAz}
!! binom(nAx,k1)*binom(nAy,k2)*binom(nAz,k3)* Y_{lambda mu}(AC_unit)
!! *CAx**(nAx-k1)*CAy**(nAy-k2)*CAz**(nAz-k3)*
!! bigI(lambda,mu,l,m,k1,k2,k3)
!! \sum{lambdap=0}^{l+nB} \sum_{mup=-lambdap}^{lambdap}
!! \sum{k1p=0}^{nBx} \sum{k2p=0}^{nBy} \sum{k3p=0}^{nBz}
!! binom(nBx,k1p)*binom(nBy,k2p)*binom(nBz,k3p)* Y_{lambdap mup}(BC_unit)
!! *CBx**(nBx-k1p)*CBy**(nBy-k2p)*CBz**(nBz-k3p)*
!! bigI(lambdap,mup,l,m,k1p,k2p,k3p)*
!! \sum_{k=1}^{kmax} v_kl(k,l)*
!! bigR(lambda,lambdap,k1+k2+k3+k1p+k2p+k3p+n_kl(k,l),g_a,g_b,AC,BC,dz_kl(k,l))
!!
!! nA=nAx+nAy+nAz
!! nB=nBx+nBy+nBz
!! AC=|A-C|
!! AC_x= A_x-C_x, etc.
!! BC=|B-C|
!! AC_unit= vect(AC)/AC
!! BC_unit= vect(BC)/BC
!! bigI(lambda,mu,l,m,k1,k2,k3)=
!! \int dOmega Y_{lambda mu}(Omega) xchap^k1 ychap^k2 zchap^k3 Y_{l m}(Omega)
!!
!! bigR(lambda,lambdap,N,g_a,g_b,gamm_k,AC,BC)
!! = exp(-g_a* AC**2 -g_b* BC**2) * int_prod_bessel_loc(ktot+2,g_a+g_b+dz_k(k),l,dreal)
!! /int dx x^{ktot} exp(-g_k)*x**2) M_lambda(2 g_k D x)
double precision function Vpseudo &
(lmax,kmax,v_kl,n_kl,dz_kl,a,n_a,g_a,b,n_b,g_b,c)
implicit none
! ___
! | ._ ._ _|_
! _|_ | | |_) |_| |_
! |
double precision, intent(in) :: a(3),g_a,b(3),g_b,c(3)
integer, intent(in) :: lmax,kmax,n_kl(kmax,0:lmax)
integer, intent(in) :: n_a(3),n_b(3)
double precision, intent(in) :: v_kl(kmax,0:lmax),dz_kl(kmax,0:lmax)
!
! | _ _ _. |
! |_ (_) (_ (_| |
!
double precision :: fourpi,f,prod,prodp,binom_func,accu,bigR,bigI,ylm
double precision :: theta_AC0,phi_AC0,theta_BC0,phi_BC0,ac,bc,big
double precision :: areal,freal,breal,t1,t2,int_prod_bessel
double precision :: arg
integer :: ntot,ntotA,m,mu,mup,k1,k2,k3,ntotB,k1p,k2p,k3p,lambda,lambdap,ktot
integer :: l,k, nkl_max
! _
! |_) o _ _. ._ ._ _.
! |_) | (_| (_| | | (_| \/
! _| /
double precision, allocatable :: array_coefs_A(:,:)
double precision, allocatable :: array_coefs_B(:,:)
double precision, allocatable :: array_R(:,:,:,:,:)
double precision, allocatable :: array_I_A(:,:,:,:,:)
double precision, allocatable :: array_I_B(:,:,:,:,:)
double precision :: f1, f2, f3
if (kmax.eq.1.and.lmax.eq.0.and.v_kl(1,0).eq.0.d0) then
Vpseudo=0.d0
return
end if
fourpi=4.d0*dacos(-1.d0)
ac=dsqrt((a(1)-c(1))**2+(a(2)-c(2))**2+(a(3)-c(3))**2)
bc=dsqrt((b(1)-c(1))**2+(b(2)-c(2))**2+(b(3)-c(3))**2)
arg= g_a*ac*ac + g_b*bc*bc
if(arg.gt.-dlog(1.d-20))then
Vpseudo=0.d0
return
endif
freal=dexp(-arg)
areal=2.d0*g_a*ac
breal=2.d0*g_b*bc
ntotA=n_a(1)+n_a(2)+n_a(3)
ntotB=n_b(1)+n_b(2)+n_b(3)
ntot=ntotA+ntotB
nkl_max=4
!=!=!=!=!=!=!=!=!=!
! A l l o c a t e !
!=!=!=!=!=!=!=!=!=!
allocate (array_coefs_A(0:ntot,3))
allocate (array_coefs_B(0:ntot,3))
allocate (array_R(kmax,0:ntot+nkl_max,0:lmax,0:lmax+ntot,0:lmax+ntot))
allocate (array_I_A(-(lmax+ntot):lmax+ntot,0:lmax+ntot,0:ntot,0:ntot,0:ntot))
allocate (array_I_B(-(lmax+ntot):lmax+ntot,0:lmax+ntot,0:ntot,0:ntot,0:ntot))
if(ac.eq.0.d0.and.bc.eq.0.d0)then
!=!=!=!=!=!
! I n i t !
!=!=!=!=!=!
accu=0.d0
!=!=!=!=!=!=!=!
! c a l c u l !
!=!=!=!=!=!=!=!
do k=1,kmax
do l=0,lmax
ktot=ntot+n_kl(k,l)
if (v_kl(k,l) == 0.d0) cycle
do m=-l,l
prod=bigI(0,0,l,m,n_a(1),n_a(2),n_a(3))
if (prod == 0.d0) cycle
prodp=bigI(0,0,l,m,n_b(1),n_b(2),n_b(3))
if (prodp == 0.d0) cycle
accu=accu+prod*prodp*v_kl(k,l)*int_prod_bessel(ktot+2,g_a+g_b+dz_kl(k,l),0,0,areal,breal,arg)
enddo
enddo
enddo
!=!=!=!=!
! E n d !
!=!=!=!=!
Vpseudo=accu*fourpi
else if(ac.ne.0.d0.and.bc.ne.0.d0)then
!=!=!=!=!=!
! I n i t !
!=!=!=!=!=!
f=fourpi*fourpi
theta_AC0=dacos( (a(3)-c(3))/ac )
phi_AC0=datan2((a(2)-c(2))/ac,(a(1)-c(1))/ac)
theta_BC0=dacos( (b(3)-c(3))/bc )
phi_BC0=datan2((b(2)-c(2))/bc,(b(1)-c(1))/bc)
do lambdap=0,lmax+ntotB
do lambda=0,lmax+ntotA
do l=0,lmax
do ktot=0,ntotA+ntotB+nkl_max
do k=1,kmax
array_R(k,ktot,l,lambda,lambdap)= int_prod_bessel(ktot+2,g_a+g_b+dz_kl(k,l),lambda,lambdap,areal,breal,arg)
enddo
enddo
enddo
enddo
enddo
do k1=0,n_a(1)
array_coefs_A(k1,1) = binom_func(n_a(1),k1)*(c(1)-a(1))**(n_a(1)-k1)
enddo
do k2=0,n_a(2)
array_coefs_A(k2,2) = binom_func(n_a(2),k2)*(c(2)-a(2))**(n_a(2)-k2)
enddo
do k3=0,n_a(3)
array_coefs_A(k3,3) = binom_func(n_a(3),k3)*(c(3)-a(3))**(n_a(3)-k3)
enddo
do k1p=0,n_b(1)
array_coefs_B(k1p,1) = binom_func(n_b(1),k1p)*(c(1)-b(1))**(n_b(1)-k1p)
enddo
do k2p=0,n_b(2)
array_coefs_B(k2p,2) = binom_func(n_b(2),k2p)*(c(2)-b(2))**(n_b(2)-k2p)
enddo
do k3p=0,n_b(3)
array_coefs_B(k3p,3) = binom_func(n_b(3),k3p)*(c(3)-b(3))**(n_b(3)-k3p)
enddo
!=!=!=!=!=!=!=!
! c a l c u l !
!=!=!=!=!=!=!=!
accu=0.d0
do l=0,lmax
do m=-l,l
do k3=0,n_a(3)
do k2=0,n_a(2)
do k1=0,n_a(1)
do lambda=0,l+ntotA
do mu=-lambda,lambda
array_I_A(mu,lambda,k1,k2,k3)=bigI(lambda,mu,l,m,k1,k2,k3)
enddo
enddo
enddo
enddo
enddo
do k3p=0,n_b(3)
do k2p=0,n_b(2)
do k1p=0,n_b(1)
do lambdap=0,l+ntotB
do mup=-lambdap,lambdap
array_I_B(mup,lambdap,k1p,k2p,k3p)=bigI(lambdap,mup,l,m,k1p,k2p,k3p)
enddo
enddo
enddo
enddo
enddo
do k3=0,n_a(3)
if (array_coefs_A(k3,3) == 0.d0) cycle
do k2=0,n_a(2)
if (array_coefs_A(k2,2) == 0.d0) cycle
do k1=0,n_a(1)
if (array_coefs_A(k1,1) == 0.d0) cycle
do lambda=0,l+ntotA
do mu=-lambda,lambda
prod=ylm(lambda,mu,theta_AC0,phi_AC0)*array_coefs_A(k1,1)*array_coefs_A(k2,2)*array_coefs_A(k3,3)*array_I_A(mu,lambda,k1,k2,k3)
if (prod == 0.d0) cycle
do k3p=0,n_b(3)
do k2p=0,n_b(2)
do k1p=0,n_b(1)
do lambdap=0,l+ntotB
do mup=-lambdap,lambdap
prodp=prod*ylm(lambdap,mup,theta_BC0,phi_BC0)* &
array_coefs_B(k1p,1)*array_coefs_B(k2p,2)*array_coefs_B(k3p,3)* &
array_I_B(mup,lambdap,k1p,k2p,k3p)
if (prodp == 0.d0) cycle
do k=1,kmax
ktot=k1+k2+k3+k1p+k2p+k3p+n_kl(k,l)
accu=accu+prodp*v_kl(k,l)*array_R(k,ktot,l,lambda,lambdap)
enddo
enddo
enddo
enddo
enddo
enddo
enddo
enddo
enddo
enddo
enddo
enddo
enddo
!=!=!=!=!
! E n d !
!=!=!=!=!
Vpseudo=f*accu
else if(ac.eq.0.d0.and.bc.ne.0.d0)then
!=!=!=!=!=!
! I n i t !
!=!=!=!=!=!
f=fourpi**1.5d0
theta_BC0=dacos( (b(3)-c(3))/bc )
phi_BC0=datan2((b(2)-c(2))/bc,(b(1)-c(1))/bc)
areal=2.d0*g_a*ac
breal=2.d0*g_b*bc
freal=dexp(-g_a*ac**2-g_b*bc**2)
do lambdap=0,lmax+ntotB
do l=0,lmax
do ktot=0,ntotA+ntotB+nkl_max
do k=1,kmax
array_R(k,ktot,l,0,lambdap)= int_prod_bessel(ktot+2,g_a+g_b+dz_kl(k,l),0,lambdap,areal,breal,arg)
enddo
enddo
enddo
enddo
do k1p=0,n_b(1)
array_coefs_B(k1p,1) = binom_func(n_b(1),k1p)*(c(1)-b(1))**(n_b(1)-k1p)
enddo
do k2p=0,n_b(2)
array_coefs_B(k2p,2) = binom_func(n_b(2),k2p)*(c(2)-b(2))**(n_b(2)-k2p)
enddo
do k3p=0,n_b(3)
array_coefs_B(k3p,3) = binom_func(n_b(3),k3p)*(c(3)-b(3))**(n_b(3)-k3p)
enddo
!=!=!=!=!=!=!=!
! c a l c u l !
!=!=!=!=!=!=!=!
accu=0.d0
do l=0,lmax
do m=-l,l
do k3p=0,n_b(3)
do k2p=0,n_b(2)
do k1p=0,n_b(1)
do lambdap=0,l+ntotB
do mup=-lambdap,lambdap
array_I_B(mup,lambdap,k1p,k2p,k3p)=bigI(lambdap,mup,l,m,k1p,k2p,k3p)
enddo
enddo
enddo
enddo
enddo
prod=bigI(0,0,l,m,n_a(1),n_a(2),n_a(3))
do k3p=0,n_b(3)
if (array_coefs_B(k3p,3) == 0.d0) cycle
do k2p=0,n_b(2)
if (array_coefs_B(k2p,2) == 0.d0) cycle
do k1p=0,n_b(1)
if (array_coefs_B(k1p,1) == 0.d0) cycle
do lambdap=0,l+ntotB
do mup=-lambdap,lambdap
prodp=prod*array_coefs_B(k1p,1)*array_coefs_B(k2p,2)*array_coefs_B(k3p,3)*ylm(lambdap,mup,theta_BC0,phi_BC0)*array_I_B(mup,lambdap,k1p,k2p,k3p)
if (prodp == 0.d0) cycle
do k=1,kmax
ktot=ntotA+k1p+k2p+k3p+n_kl(k,l)
accu=accu+prodp*v_kl(k,l)*array_R(k,ktot,l,0,lambdap)
enddo
enddo
enddo
enddo
enddo
enddo
enddo
enddo
!=!=!=!=!
! E n d !
!=!=!=!=!
Vpseudo=f*accu
else if(ac.ne.0.d0.and.bc.eq.0.d0)then
!=!=!=!=!=!
! I n i t !
!=!=!=!=!=!
f=fourpi**1.5d0
theta_AC0=dacos( (a(3)-c(3))/ac )
phi_AC0=datan2((a(2)-c(2))/ac,(a(1)-c(1))/ac)
areal=2.d0*g_a*ac
breal=2.d0*g_b*bc
freal=dexp(-g_a*ac**2-g_b*bc**2)
do lambda=0,lmax+ntotA
do l=0,lmax
do ktot=0,ntotA+ntotB+nkl_max
do k=1,kmax
array_R(k,ktot,l,lambda,0)= int_prod_bessel(ktot+2,g_a+g_b+dz_kl(k,l),lambda,0,areal,breal,arg)
enddo
enddo
enddo
enddo
do k1=0,n_a(1)
array_coefs_A(k1,1) = binom_func(n_a(1),k1)*(c(1)-a(1))**(n_a(1)-k1)
enddo
do k2=0,n_a(2)
array_coefs_A(k2,2) = binom_func(n_a(2),k2)*(c(2)-a(2))**(n_a(2)-k2)
enddo
do k3=0,n_a(3)
array_coefs_A(k3,3) = binom_func(n_a(3),k3)*(c(3)-a(3))**(n_a(3)-k3)
enddo
!=!=!=!=!=!=!=!
! c a l c u l !
!=!=!=!=!=!=!=!
accu=0.d0
do l=0,lmax
do m=-l,l
do k3=0,n_a(3)
do k2=0,n_a(2)
do k1=0,n_a(1)
do lambda=0,l+ntotA
do mu=-lambda,lambda
array_I_A(mu,lambda,k1,k2,k3)=bigI(lambda,mu,l,m,k1,k2,k3)
enddo
enddo
enddo
enddo
enddo
do k3=0,n_a(3)
if (array_coefs_A(k3,3) == 0.d0) cycle
do k2=0,n_a(2)
if (array_coefs_A(k2,2) == 0.d0) cycle
do k1=0,n_a(1)
if (array_coefs_A(k1,1) == 0.d0) cycle
do lambda=0,l+ntotA
do mu=-lambda,lambda
prod=array_coefs_A(k1,1)*array_coefs_A(k2,2)*array_coefs_A(k3,3)*ylm(lambda,mu,theta_AC0,phi_AC0)*array_I_A(mu,lambda,k1,k2,k3)
if (prod == 0.d0) cycle
prodp=prod*bigI(0,0,l,m,n_b(1),n_b(2),n_b(3))
if (prodp == 0.d0) cycle
do k=1,kmax
ktot=k1+k2+k3+ntotB+n_kl(k,l)
accu=accu+prodp*v_kl(k,l)*array_R(k,ktot,l,lambda,0)
enddo
enddo
enddo
enddo
enddo
enddo
enddo
enddo
!=!=!=!=!
! E n d !
!=!=!=!=!
Vpseudo=f*accu
endif
! _
! |_ o ._ _. | o _ _
! | | | | (_| | | _> (/_
!
deallocate (array_R, array_I_A, array_I_B)
deallocate (array_coefs_A, array_coefs_B)
return
end
! _
! | |
!__ __ _ __ ___ ___ _ _ __| | ___ _ __ _ _ _ __ ___
!\ \ / / | '_ \/ __|/ _ \ | | |/ _` |/ _ \ | '_ \| | | | '_ ` _ \
! \ V / | |_) \__ \ __/ |_| | (_| | (_) | | | | | |_| | | | | | |
! \_/ | .__/|___/\___|\__,_|\__,_|\___/ |_| |_|\__,_|_| |_| |_|
! | |
! |_|
double precision function Vpseudo_num(npts,rmax,lmax,kmax,v_kl,n_kl,dz_kl,a,n_a,g_a,b,n_b,g_b,c)
implicit none
! ___
! | ._ ._ _|_
! _|_ | | |_) |_| |_
! |
double precision, intent(in) :: a(3),g_a,b(3),g_b,c(3)
integer, intent(in) :: lmax,kmax,npts
integer, intent(in) :: n_a(3),n_b(3), n_kl(kmax,0:lmax)
double precision, intent(in) :: v_kl(kmax,0:lmax),dz_kl(kmax,0:lmax)
double precision, intent(in) :: rmax
!
! | _ _ _. |
! |_ (_) (_ (_| |
!
integer :: l,m,k,kk
double precision ac(3),bc(3)
double precision dr,sum,rC
double precision overlap_orb_ylm_brute
! _
! / _. | _ |
! \_ (_| | (_ |_| |
!
do l=1,3
ac(l)=a(l)-c(l)
bc(l)=b(l)-c(l)
enddo
dr=rmax/npts
sum=0.d0
do l=0,lmax
do m=-l,l
do k=1,npts
rC=(k-1)*dr+dr/2.d0
do kk=1,kmax
sum=sum+dr*v_kl(kk,l)*rC**(n_kl(kk,l)+2)*dexp(-dz_kl(kk,l)*rC**2) &
*overlap_orb_ylm_brute(npts,rC,n_a,ac,g_a,l,m) &
*overlap_orb_ylm_brute(npts,rC,n_b,bc,g_b,l,m)
enddo
enddo
enddo
enddo
Vpseudo_num=sum
return
end
!! Routine Vloc is a variation of formumla (66)
!! of Kahn Baybutt TRuhlar J.Chem.Phys. vol.65 3826 (1976)
!! without the projection operator
!!
!! Vloc= (4pi)**3/2* \sum_{k=1}^{klocmax} \sum_{l=0}^lmax \sum_{m=-l}^{l}
!!\sum{k1=0}^{nAx} \sum{k2=0}^{nAy} \sum{k3=0}^{nAz}
!! binom(nAx,k1)*binom(nAy,k2)*binom(nAz,k3)
!! *CAx**(nAx-k1)*CAy**(nAy-k2)*CAz**(nAz-k3)*
!! \sum{k1p=0}^{nBx} \sum{k2p=0}^{nBy} \sum{k3p=0}^{nBz}
!! binom(nBx,k1p)*binom(nBy,k2p)*binom(nBz,k3p)
!! *CBx**(nBx-k1p)*CBy**(nBy-k2p)*CBz**(nBz-k3p)*
!!\sum_{l=0}^lmax \sum_{m=-l}^{l}
!! bigI(0,0,l,m,k1+k1p,k2+k2p,k3+k3p)*Y_{l m}(D_unit)
!! *v_k(k)* bigR(lambda,k1+k2+k3+k1p+k2p+k3p+n_k(k),g_a,g_b,AC,BC,dz_k(k))
!!
!! nA=nAx+nAy+nAz
!! nB=nBx+nBy+nBz
!! D=(g_a AC+g_b BC)
!! D_unit= vect(D)/D
!! AC_x= A_x-C_x, etc.
!! BC=|B-C|
!! AC_unit= vect(AC)/AC
!! BC_unit= vect(BC)/BCA
!!
!! bigR(lambda,g_a,g_b,g_k,AC,BC)
!! = exp(-g_a* AC**2 -g_b* BC**2)*
!! I_loc= \int dx x**l *exp(-gam*x**2) M_n(ax) l=ktot+2 gam=g_a+g_b+dz_k(k) a=dreal n=l
!! M_n(x) modified spherical bessel function
double precision function Vloc(klocmax,v_k,n_k,dz_k,a,n_a,g_a,b,n_b,g_b,c)
implicit none
integer klocmax
double precision v_k(klocmax),dz_k(klocmax),crochet,bigA
integer n_k(klocmax)
double precision a(3),g_a,b(3),g_b,c(3),d(3)
integer n_a(3),n_b(3),ntotA,ntotB,ntot,m
integer i,l,k,ktot,k1,k2,k3,k1p,k2p,k3p
double precision f,fourpi,ac,bc,freal,d2,dreal,theta_DC0,phi_DC0,coef
double precision,allocatable :: array_R_loc(:,:,:)
double precision,allocatable :: array_coefs(:,:,:,:,:,:)
double precision int_prod_bessel_loc,binom_func,accu,prod,ylm,bigI,arg
fourpi=4.d0*dacos(-1.d0)
f=fourpi**1.5d0
ac=dsqrt((a(1)-c(1))**2+(a(2)-c(2))**2+(a(3)-c(3))**2)
bc=dsqrt((b(1)-c(1))**2+(b(2)-c(2))**2+(b(3)-c(3))**2)
arg=g_a*ac**2+g_b*bc**2
if(arg.gt.-dlog(10.d-20))then
Vloc=0.d0
return
endif
ntotA=n_a(1)+n_a(2)+n_a(3)
ntotB=n_b(1)+n_b(2)+n_b(3)
ntot=ntotA+ntotB
if(ac.eq.0.d0.and.bc.eq.0.d0)then
accu=0.d0
do k=1,klocmax
accu=accu+v_k(k)*crochet(n_k(k)+2+ntot,g_a+g_b+dz_k(k))
enddo
Vloc=accu*fourpi*bigI(0,0,0,0,n_a(1)+n_b(1),n_a(2)+n_b(2),n_a(3)+n_b(3))
!bigI frequently is null
return
endif
freal=dexp(-g_a*ac**2-g_b*bc**2)
d2 = 0.d0
do i=1,3
d(i)=g_a*(a(i)-c(i))+g_b*(b(i)-c(i))
d2=d2+d(i)*d(i)
enddo
d2=dsqrt(d2)
dreal=2.d0*d2
allocate (array_R_loc(-2:ntot+klocmax,klocmax,0:ntot))
allocate (array_coefs(0:ntot,0:ntot,0:ntot,0:ntot,0:ntot,0:ntot))
do ktot=-2,ntotA+ntotB+klocmax
do l=0,ntot
do k=1,klocmax
array_R_loc(ktot,k,l)=freal*int_prod_bessel_loc(ktot+2,g_a+g_b+dz_k(k),l,dreal)
enddo
enddo
enddo
do k1=0,n_a(1)
do k2=0,n_a(2)
do k3=0,n_a(3)
do k1p=0,n_b(1)
do k2p=0,n_b(2)
do k3p=0,n_b(3)
array_coefs(k1,k2,k3,k1p,k2p,k3p)=binom_func(n_a(1),k1)*binom_func(n_a(2),k2)*binom_func(n_a(3),k3)&
*(c(1)-a(1))**(n_a(1)-k1)*(c(2)-a(2))**(n_a(2)-k2)*(c(3)-a(3))**(n_a(3)-k3)&
*binom_func(n_b(1),k1p)*binom_func(n_b(2),k2p)*binom_func(n_b(3),k3p)&
*(c(1)-b(1))**(n_b(1)-k1p)*(c(2)-b(2))**(n_b(2)-k2p)*(c(3)-b(3))**(n_b(3)-k3p)
enddo
enddo
enddo
enddo
enddo
enddo
accu=0.d0
if(d2 == 0.d0)then
l=0
m=0
coef=1.d0/dsqrt(4.d0*dacos(-1.d0))
do k=1,klocmax
do k1=0,n_a(1)
do k2=0,n_a(2)
do k3=0,n_a(3)
do k1p=0,n_b(1)
do k2p=0,n_b(2)
do k3p=0,n_b(3)
prod=coef*array_coefs(k1,k2,k3,k1p,k2p,k3p) &
*bigI(l,m,0,0,k1+k1p,k2+k2p,k3+k3p)
ktot=k1+k2+k3+k1p+k2p+k3p+n_k(k)
accu=accu+prod*v_k(k)*array_R_loc(ktot,k,l)
enddo
enddo
enddo
enddo
enddo
enddo
enddo
else
theta_DC0=dacos(d(3)/d2)
phi_DC0=datan2(d(2)/d2,d(1)/d2)
do k=1,klocmax
if (v_k(k) == 0.d0) cycle
do k1=0,n_a(1)
do k2=0,n_a(2)
do k3=0,n_a(3)
do k1p=0,n_b(1)
do k2p=0,n_b(2)
do k3p=0,n_b(3)
if (array_coefs(k1,k2,k3,k1p,k2p,k3p) == 0.d0) cycle
do l=0,ntot
do m=-l,l
coef=ylm(l,m,theta_DC0,phi_DC0)
if (coef == 0.d0) cycle
ktot=k1+k2+k3+k1p+k2p+k3p+n_k(k)
if (array_R_loc(ktot,k,l) == 0.d0) cycle
prod=coef*array_coefs(k1,k2,k3,k1p,k2p,k3p) &
*bigI(l,m,0,0,k1+k1p,k2+k2p,k3+k3p)
accu=accu+prod*v_k(k)*array_R_loc(ktot,k,l)
enddo
enddo
enddo
enddo
enddo
enddo
enddo
enddo
enddo
endif
Vloc=f*accu
deallocate (array_R_loc)
deallocate (array_coefs)
end
double precision function bigA(i,j,k)
implicit none
integer i,j,k
double precision fourpi,dble_fact
fourpi=4.d0*dacos(-1.d0)
bigA=0.d0
if(mod(i,2).eq.1)return
if(mod(j,2).eq.1)return
if(mod(k,2).eq.1)return
bigA=fourpi*dble_fact(i-1)*dble_fact(j-1)*dble_fact(k-1)/dble_fact(i+j+k+1)
end
!!
!! I_{lambda,mu,l,m}^{k1,k2,k3} = /int dOmega Y_{lambda mu} xchap^k1 ychap^k2 zchap^k3 Y_{lm}
!!
double precision function bigI(lambda,mu,l,m,k1,k2,k3)
implicit none
integer lambda,mu,l,m,k1,k2,k3
integer k,i,kp,ip
double precision pi,sum,factor1,factor2,cylm,cylmp,bigA,binom_func,fact,coef_pm
double precision sgn, sgnp
pi=dacos(-1.d0)
bigI=0.d0
if(mu.gt.0.and.m.gt.0)then
sum=0.d0
factor1=dsqrt((2*lambda+1)*fact(lambda-iabs(mu))/(2.d0*pi*fact(lambda+iabs(mu))))
if (factor1== 0.d0) return
factor2=dsqrt((2*l+1)*fact(l-iabs(m))/(2.d0*pi*fact(l+iabs(m))))
if (factor2== 0.d0) return
sgn = 1.d0
do k=0,mu/2
do i=0,lambda-mu
if (coef_pm(lambda,i+mu) == 0.d0) cycle
sgnp = 1.d0
do kp=0,m/2
do ip=0,l-m
cylm=sgn*factor1*binom_func(mu,2*k)*fact(mu+i)/fact(i)*coef_pm(lambda,i+mu)
if (cylm == 0.d0) cycle
cylmp=sgnp*factor2*binom_func(m,2*kp)*fact(m+ip)/fact(ip)*coef_pm(l,ip+m)
if (cylmp == 0.d0) cycle
sum=sum+cylm*cylmp*bigA(mu-2*k+m-2*kp+k1,2*k+2*kp+k2,i+ip+k3)
enddo
sgnp = -sgnp
enddo
enddo
sgn = -sgn
enddo
bigI=sum
return
endif
if(mu.eq.0.and.m.eq.0)then
factor1=dsqrt((2*lambda+1)/(4.d0*pi))
if (factor1== 0.d0) return
factor2=dsqrt((2*l+1)/(4.d0*pi))
if (factor2== 0.d0) return
sum=0.d0
do i=0,lambda
do ip=0,l
cylm=factor1*coef_pm(lambda,i)
if (cylm == 0.d0) cycle
cylmp=factor2*coef_pm(l,ip)
if (cylmp == 0.d0) cycle
sum=sum+cylm*cylmp*bigA(k1,k2,i+ip+k3)
enddo
enddo
bigI=sum
return
endif
if(mu.eq.0.and.m.gt.0)then
factor1=dsqrt((2*lambda+1)/(4.d0*pi))
if (factor1== 0.d0) return
factor2=dsqrt((2*l+1)*fact(l-iabs(m))/(2.d0*pi*fact(l+iabs(m))))
if (factor2== 0.d0) return
sum=0.d0
do i=0,lambda
sgnp = 1.d0
do kp=0,m/2
do ip=0,l-m
cylm=factor1*coef_pm(lambda,i)
if (cylm == 0.d0) cycle
cylmp=sgnp*factor2*binom_func(m,2*kp)*fact(m+ip)/fact(ip)*coef_pm(l,ip+m)
if (cylmp == 0.d0) cycle
sum=sum+cylm*cylmp*bigA(m-2*kp+k1,2*kp+k2,i+ip+k3)
enddo
sgnp = -sgnp
enddo
enddo
bigI=sum
return
endif
if(mu.gt.0.and.m.eq.0)then
sum=0.d0
factor1=dsqrt((2*lambda+1)*fact(lambda-iabs(mu))/(2.d0*pi*fact(lambda+iabs(mu))))
if (factor1== 0.d0) return
factor2=dsqrt((2*l+1)/(4.d0*pi))
if (factor2== 0.d0) return
sgn = 1.d0
do k=0,mu/2
do i=0,lambda-mu
if (coef_pm(lambda,i+mu) == 0.d0) cycle
do ip=0,l
cylm=sgn*factor1*binom_func(mu,2*k)*fact(mu+i)/fact(i)*coef_pm(lambda,i+mu)
if (cylm == 0.d0) cycle
cylmp=factor2*coef_pm(l,ip)
if (cylmp == 0.d0) cycle
sum=sum+cylm*cylmp*bigA(mu-2*k +k1,2*k +k2,i+ip +k3)
enddo
enddo
sgn = -sgn
enddo
bigI=sum
return
endif
if(mu.lt.0.and.m.lt.0)then
mu=-mu
m=-m
factor1=dsqrt((2*lambda+1)*fact(lambda-iabs(mu))/(2.d0*pi*fact(lambda+iabs(mu))))
if (factor1== 0.d0) return
factor2=dsqrt((2*l+1)*fact(l-iabs(m))/(2.d0*pi*fact(l+iabs(m))))
if (factor2== 0.d0) return
sum=0.d0
sgn = 1.d0
do k=0,(mu-1)/2
do i=0,lambda-mu
if (coef_pm(lambda,i+mu) == 0.d0) cycle
sgnp = 1.d0
do kp=0,(m-1)/2
do ip=0,l-m
if (coef_pm(l,ip+m) == 0.d0) cycle
cylm=sgn*factor1*binom_func(mu,2*k+1)*fact(mu+i)/fact(i)*coef_pm(lambda,i+mu)
if (cylm == 0.d0) cycle
cylmp=sgnp*factor2*binom_func(m,2*kp+1)*fact(m+ip)/fact(ip)*coef_pm(l,ip+m)
if (cylmp == 0.d0) cycle
sum=sum+cylm*cylmp*bigA(mu-(2*k+1)+m-(2*kp+1)+k1,(2*k+1)+(2*kp+1)+k2,i+ip+k3)
enddo
sgnp = -sgnp
enddo
enddo
sgn = -sgn
enddo
mu=-mu
m=-m
bigI=sum
return
endif
if(mu.eq.0.and.m.lt.0)then
m=-m
factor1=dsqrt((2*lambda+1)/(4.d0*pi))
if (factor1 == 0.d0) return
factor2=dsqrt((2*l+1)*fact(l-iabs(m))/(2.d0*pi*fact(l+iabs(m))))
if (factor2 == 0.d0) return
sum=0.d0
do i=0,lambda
sgnp = 1.d0
do kp=0,(m-1)/2
do ip=0,l-m
cylm=factor1*coef_pm(lambda,i)
if (cylm == 0.d0) cycle
cylmp=sgnp*factor2*binom_func(m,2*kp+1)*fact(m+ip)/fact(ip)*coef_pm(l,ip+m)
if (cylmp == 0.d0) cycle
sum=sum+cylm*cylmp*bigA(m-(2*kp+1)+k1,2*kp+1+k2,i+ip+k3)
enddo
sgnp = -sgnp
enddo
enddo
m=-m
bigI=sum
return
endif
if(mu.lt.0.and.m.eq.0)then
sum=0.d0
mu=-mu
factor1=dsqrt((2*lambda+1)*fact(lambda-iabs(mu))/(2.d0*pi*fact(lambda+iabs(mu))))
if (factor1== 0.d0) return
factor2=dsqrt((2*l+1)/(4.d0*pi))
if (factor2== 0.d0) return
sgn = 1.d0
do k=0,(mu-1)/2
do i=0,lambda-mu
do ip=0,l
cylm=sgn*factor1*binom_func(mu,2*k+1)*fact(mu+i)/fact(i)*coef_pm(lambda,i+mu)
if (cylm == 0.d0) cycle
cylmp=factor2*coef_pm(l,ip)
if (cylmp == 0.d0) cycle
sum=sum+cylm*cylmp*bigA(mu-(2*k+1)+k1,2*k+1+k2,i+ip+k3)
enddo
enddo
sgn = -sgn
enddo
mu=-mu
bigI=sum
return
endif
if(mu.gt.0.and.m.lt.0)then
sum=0.d0
factor1=dsqrt((2*lambda+1)*fact(lambda-iabs(mu))/(2.d0*pi*fact(lambda+iabs(mu))))
if (factor1== 0.d0) return
factor2=dsqrt((2*l+1)*fact(l-iabs(m))/(2.d0*pi*fact(l+iabs(m))))
if (factor2== 0.d0) return
m=-m
sgn=1.d0
do k=0,mu/2
do i=0,lambda-mu
if (coef_pm(lambda,i+mu) == 0.d0) cycle
sgnp=1.d0
do kp=0,(m-1)/2
do ip=0,l-m
if (coef_pm(l,ip+m) == 0.d0) cycle
cylm =sgn *factor1*binom_func(mu,2*k)*fact(mu+i)/fact(i)*coef_pm(lambda,i+mu)
if (cylm == 0.d0) cycle
cylmp=sgnp*factor2*binom_func(m,2*kp+1)*fact(m+ip)/fact(ip)*coef_pm(l,ip+m)
if (cylmp == 0.d0) cycle
sum=sum+cylm*cylmp*bigA(mu-2*k+m-(2*kp+1)+k1,2*k+2*kp+1+k2,i+ip+k3)
enddo
sgnp = -sgnp
enddo
enddo
sgn = -sgn
enddo
m=-m
bigI=sum
return
endif
if(mu.lt.0.and.m.gt.0)then
mu=-mu
factor1=dsqrt((2*lambda+1)*fact(lambda-iabs(mu))/(2.d0*pi*fact(lambda+iabs(mu))))
if (factor1== 0.d0) return
factor2=dsqrt((2*l+1)*fact(l-iabs(m))/(2.d0*pi*fact(l+iabs(m))))
if (factor2== 0.d0) return
sum=0.d0
sgn = 1.d0
do k=0,(mu-1)/2
do i=0,lambda-mu
if (coef_pm(lambda,i+mu) == 0.d0) cycle
sgnp = 1.d0
do kp=0,m/2
do ip=0,l-m
if (coef_pm(l,ip+m) == 0.d0) cycle
cylm=sgn*factor1 *binom_func(mu,2*k+1)*fact(mu+i)/fact(i)*coef_pm(lambda,i+mu)
if (cylm == 0.d0) cycle
cylmp=sgnp*factor2*binom_func(m,2*kp)*fact(m+ip)/fact(ip)*coef_pm(l,ip+m)
if (cylmp == 0.d0) cycle
sum=sum+cylm*cylmp*bigA(mu-(2*k+1)+m-2*kp+k1,2*k+1+2*kp+k2,i+ip+k3)
enddo
sgnp = -sgnp
enddo
enddo
sgn = -sgn
enddo
bigI=sum
mu=-mu
return
endif
stop 'pb in bigI!'
end
double precision function crochet(n,g)
implicit none
integer n
double precision g,dble_fact,expo
double precision, parameter :: sq_pi_ov_2=dsqrt(dacos(-1.d0)*0.5d0)
expo=0.5d0*dfloat(n+1)
crochet=dble_fact(n-1)/(g+g)**expo
if(mod(n,2).eq.0)crochet=crochet*sq_pi_ov_2
end
!!
!! overlap= <phi|Ylm>= /int dOmega Ylm (x-center_x)**nx*(y-center_y)**nx*(z-center)**nx
!! *exp(-g*(r-center)**2)
!!
double precision function overlap_orb_ylm_brute(npts,r,npower_orb,center_orb,g_orb,l,m)
implicit none
integer npower_orb(3),l,m,i,j,npts
double precision u,g_orb,du,dphi,term,orb_phi,ylm_real,sintheta,r_orb,phi,center_orb(3)
double precision x_orb,y_orb,z_orb,twopi,r
twopi=2.d0*dacos(-1.d0)
du=2.d0/npts
dphi=twopi/npts
overlap_orb_ylm_brute=0.d0
do i=1,npts
u=-1.d0+du*(i-1)+du/2.d0
sintheta=dsqrt(1.d0-u**2)
do j=1,npts
phi=dphi*(j-1)+dphi/2.d0
x_orb=r*dcos(phi)*sintheta
y_orb=r*dsin(phi)*sintheta
z_orb=r*u
term=orb_phi(x_orb,y_orb,z_orb,npower_orb,center_orb,g_orb)*ylm_real(l,m,u,phi)
overlap_orb_ylm_brute= overlap_orb_ylm_brute+term*du*dphi
enddo
enddo
end
double precision function overlap_orb_ylm_grid(nptsgrid,r_orb,npower_orb,center_orb,g_orb,l,m)
implicit none
!! PSEUDOS
integer nptsgridmax,nptsgrid
double precision coefs_pseudo,ptsgrid
parameter(nptsgridmax=50)
common/pseudos/coefs_pseudo(nptsgridmax),ptsgrid(nptsgridmax,3)
!!!!!
integer npower_orb(3),l,m,i
double precision x,g_orb,two_pi,dx,dphi,term,orb_phi,ylm_real,sintheta,r_orb,phi,center_orb(3)
double precision x_orb,y_orb,z_orb,twopi,pi,cosphi,sinphi,xbid
pi=dacos(-1.d0)
twopi=2.d0*pi
overlap_orb_ylm_grid=0.d0
do i=1,nptsgrid
x_orb=r_orb*ptsgrid(i,1)
y_orb=r_orb*ptsgrid(i,2)
z_orb=r_orb*ptsgrid(i,3)
x=ptsgrid(i,3)
phi=datan2(ptsgrid(i,2),ptsgrid(i,1))
term=orb_phi(x_orb,y_orb,z_orb,npower_orb,center_orb,g_orb)*ylm_real(l,m,x,phi)
overlap_orb_ylm_grid= overlap_orb_ylm_grid+coefs_pseudo(i)*term
enddo
overlap_orb_ylm_grid=2.d0*twopi*overlap_orb_ylm_grid
end
! Y_l^m(theta,phi) = i^(m+|m|) ([(2l+1)*(l-|m|)!]/[4pi*(l+|m|)!])^1/2 P_l^|m|(cos(theta)) exp(i m phi)
! l=0,1,2,....
! m=0,1,...,l
! Here:
! n=l (n=0,1,...)
! m=0,1,...,n
! x=cos(theta) 0 < x < 1
!
!
! This routine computes: PM(m,n) for n=0,...,N (number N in input) and m=0,..,n
! Exemples (see 'Associated Legendre Polynomilas wikipedia')
! P_{0}^{0}(x)=1
! P_{1}^{-1}(x)=-1/2 P_{1}^{1}(x)
! P_{1}^{0}(x)=x
! P_{1}^{1}(x)=-(1-x^2)^{1/2}
! P_{2}^{-2}(x)=1/24 P_{2}^{2}(x)
! P_{2}^{-1}(x)=-1/6 P_{2}^{1}(x)
! P_{2}^{0}(x)=1/2 (3x^{2}-1)
! P_{2}^{1}(x)=-3x(1-x^2)^{1/2}
! P_{2}^{2}(x)=3(1-x^2)
SUBROUTINE LPMN(MM,M,N,X,PM)
!
! Here N = LMAX
! Here M= MMAX (we take M=LMAX in input)
!
! =====================================================
! Purpose: Compute the associated Legendre functions Pmn(x)
! Input : x --- Argument of Pmn(x)
! m --- Order of Pmn(x), m = 0,1,2,...,n
! n --- Degree of Pmn(x), n = 0,1,2,...,N
! mm --- Physical dimension of PM
! Output: PM(m,n) --- Pmn(x)
! =====================================================
!
IMPLICIT DOUBLE PRECISION (P,X)
DIMENSION PM(0:MM,0:(N+1))
DOUBLE PRECISION, SAVE :: INVERSE(100) = 0.D0
DOUBLE PRECISION :: LS, II, JJ
IF (INVERSE(1) == 0.d0) THEN
DO I=1,100
INVERSE(I) = 1.D0/DBLE(I)
ENDDO
ENDIF
DO I=0,N
DO J=0,M
PM(J,I)=0.0D0
ENDDO
ENDDO
PM(0,0)=1.0D0
IF (DABS(X).EQ.1.0D0) THEN
DO I=1,N
PM(0,I)=X**I
ENDDO
RETURN
ENDIF
LS=1.D0
IF (DABS(X).GT.1.0D0) LS=-1.D0
XQ=DSQRT(LS*(1.0D0-X*X))
XS=LS*(1.0D0-X*X)
II = 1.D0
DO I=1,M
PM(I,I)=-LS*II*XQ*PM(I-1,I-1)
II = II+2.D0
ENDDO
II = 1.D0
DO I=0,M
PM(I,I+1)=II*X*PM(I,I)
II = II+2.D0
ENDDO
II = 0.D0
DO I=0,M
JJ = II+2.D0
DO J=I+2,N
PM(I,J)=((2.0D0*JJ-1.0D0)*X*PM(I,J-1)- (II+JJ-1.0D0)*PM(I,J-2))*INVERSE(J-I)
JJ = JJ+1.D0
ENDDO
II = II+1.D0
ENDDO
END
! Y_l^m(theta,phi) = i^(m+|m|) ([(2l+1)*(l-|m|)!]/[4pi*(l+|m|)!])^1/2
! P_l^|m|(cos(theta)) exp(i m phi)
subroutine erreur(x,n,rmoy,error)
implicit double precision(a-h,o-z)
dimension x(n)
! calcul de la moyenne
rmoy=0.d0
do i=1,n
rmoy=rmoy+x(i)
enddo
rmoy=rmoy/dfloat(n)
! calcul de l'erreur
error=0.d0
do i=1,n
error=error+(x(i)-rmoy)**2
enddo
if(n.gt.1)then
rn=dfloat(n)
rn1=dfloat(n-1)
error=dsqrt(error)/dsqrt(rn*rn1)
else
write(2,*)'Seulement un block Erreur nondefinie'
error=0.d0
endif
end
subroutine initpseudos(nptsgrid)
implicit none
integer nptsgridmax,nptsgrid,ik
double precision coefs_pseudo,ptsgrid
double precision p,q,r,s
parameter(nptsgridmax=50)
common/pseudos/coefs_pseudo(nptsgridmax),ptsgrid(nptsgridmax,3)
p=1.d0/dsqrt(2.d0)
q=1.d0/dsqrt(3.d0)
r=1.d0/dsqrt(11.d0)
s=3.d0/dsqrt(11.d0)
if(nptsgrid.eq.4)then
ptsgrid(1,1)=q
ptsgrid(1,2)=q
ptsgrid(1,3)=q
ptsgrid(2,1)=q
ptsgrid(2,2)=-q
ptsgrid(2,3)=-q
ptsgrid(3,1)=-q
ptsgrid(3,2)=q
ptsgrid(3,3)=-q
ptsgrid(4,1)=-q
ptsgrid(4,2)=-q
ptsgrid(4,3)=q
do ik=1,4
coefs_pseudo(ik)=1.d0/4.d0
enddo
return
endif
ptsgrid(1,1)=1.d0
ptsgrid(1,2)=0.d0
ptsgrid(1,3)=0.d0
ptsgrid(2,1)=-1.d0
ptsgrid(2,2)=0.d0
ptsgrid(2,3)=0.d0
ptsgrid(3,1)=0.d0
ptsgrid(3,2)=1.d0
ptsgrid(3,3)=0.d0
ptsgrid(4,1)=0.d0
ptsgrid(4,2)=-1.d0
ptsgrid(4,3)=0.d0
ptsgrid(5,1)=0.d0
ptsgrid(5,2)=0.d0
ptsgrid(5,3)=1.d0
ptsgrid(6,1)=0.d0
ptsgrid(6,2)=0.d0
ptsgrid(6,3)=-1.d0
do ik=1,6
coefs_pseudo(ik)=1.d0/6.d0
enddo
if(nptsgrid.eq.6)return
ptsgrid(7,1)=p
ptsgrid(7,2)=p
ptsgrid(7,3)=0.d0
ptsgrid(8,1)=p
ptsgrid(8,2)=-p
ptsgrid(8,3)=0.d0
ptsgrid(9,1)=-p
ptsgrid(9,2)=p
ptsgrid(9,3)=0.d0
ptsgrid(10,1)=-p
ptsgrid(10,2)=-p
ptsgrid(10,3)=0.d0
ptsgrid(11,1)=p
ptsgrid(11,2)=0.d0
ptsgrid(11,3)=p
ptsgrid(12,1)=p
ptsgrid(12,2)=0.d0
ptsgrid(12,3)=-p
ptsgrid(13,1)=-p
ptsgrid(13,2)=0.d0
ptsgrid(13,3)=p
ptsgrid(14,1)=-p
ptsgrid(14,2)=0.d0
ptsgrid(14,3)=-p
ptsgrid(15,1)=0.d0
ptsgrid(15,2)=p
ptsgrid(15,3)=p
ptsgrid(16,1)=0.d0
ptsgrid(16,2)=p
ptsgrid(16,3)=-p
ptsgrid(17,1)=0.d0
ptsgrid(17,2)=-p
ptsgrid(17,3)=p
ptsgrid(18,1)=0.d0
ptsgrid(18,2)=-p
ptsgrid(18,3)=-p
do ik=1,6
coefs_pseudo(ik)=1.d0/30.d0
enddo
do ik=7,18
coefs_pseudo(ik)=1.d0/15.d0
enddo
if(nptsgrid.eq.18)return
ptsgrid(19,1)=q
ptsgrid(19,2)=q
ptsgrid(19,3)=q
ptsgrid(20,1)=-q
ptsgrid(20,2)=q
ptsgrid(20,3)=q
ptsgrid(21,1)=q
ptsgrid(21,2)=-q
ptsgrid(21,3)=q
ptsgrid(22,1)=q
ptsgrid(22,2)=q
ptsgrid(22,3)=-q
ptsgrid(23,1)=-q
ptsgrid(23,2)=-q
ptsgrid(23,3)=q
ptsgrid(24,1)=-q
ptsgrid(24,2)=q
ptsgrid(24,3)=-q
ptsgrid(25,1)=q
ptsgrid(25,2)=-q
ptsgrid(25,3)=-q
ptsgrid(26,1)=-q
ptsgrid(26,2)=-q
ptsgrid(26,3)=-q
do ik=1,6
coefs_pseudo(ik)=1.d0/21.d0
enddo
do ik=7,18
coefs_pseudo(ik)=4.d0/105.d0
enddo
do ik=19,26
coefs_pseudo(ik)=27.d0/840.d0
enddo
if(nptsgrid.eq.26)return
ptsgrid(27,1)=r
ptsgrid(27,2)=r
ptsgrid(27,3)=s
ptsgrid(28,1)=r
ptsgrid(28,2)=-r
ptsgrid(28,3)=s
ptsgrid(29,1)=-r
ptsgrid(29,2)=r
ptsgrid(29,3)=s
ptsgrid(30,1)=-r
ptsgrid(30,2)=-r
ptsgrid(30,3)=s
ptsgrid(31,1)=r
ptsgrid(31,2)=r
ptsgrid(31,3)=-s
ptsgrid(32,1)=r
ptsgrid(32,2)=-r
ptsgrid(32,3)=-s
ptsgrid(33,1)=-r
ptsgrid(33,2)=r
ptsgrid(33,3)=-s
ptsgrid(34,1)=-r
ptsgrid(34,2)=-r
ptsgrid(34,3)=-s
ptsgrid(35,1)=r
ptsgrid(35,2)=s
ptsgrid(35,3)=r
ptsgrid(36,1)=-r
ptsgrid(36,2)=s
ptsgrid(36,3)=r
ptsgrid(37,1)=r
ptsgrid(37,2)=s
ptsgrid(37,3)=-r
ptsgrid(38,1)=-r
ptsgrid(38,2)=s
ptsgrid(38,3)=-r
ptsgrid(39,1)=r
ptsgrid(39,2)=-s
ptsgrid(39,3)=r
ptsgrid(40,1)=r
ptsgrid(40,2)=-s
ptsgrid(40,3)=-r
ptsgrid(41,1)=-r
ptsgrid(41,2)=-s
ptsgrid(41,3)=r
ptsgrid(42,1)=-r
ptsgrid(42,2)=-s
ptsgrid(42,3)=-r
ptsgrid(43,1)=s
ptsgrid(43,2)=r
ptsgrid(43,3)=r
ptsgrid(44,1)=s
ptsgrid(44,2)=r
ptsgrid(44,3)=-r
ptsgrid(45,1)=s
ptsgrid(45,2)=-r
ptsgrid(45,3)=r
ptsgrid(46,1)=s
ptsgrid(46,2)=-r
ptsgrid(46,3)=-r
ptsgrid(47,1)=-s
ptsgrid(47,2)=r
ptsgrid(47,3)=r
ptsgrid(48,1)=-s
ptsgrid(48,2)=r
ptsgrid(48,3)=-r
ptsgrid(49,1)=-s
ptsgrid(49,2)=-r
ptsgrid(49,3)=r
ptsgrid(50,1)=-s
ptsgrid(50,2)=-r
ptsgrid(50,3)=-r
do ik=1,6
coefs_pseudo(ik)=4.d0/315.d0
enddo
do ik=7,18
coefs_pseudo(ik)=64.d0/2835.d0
enddo
do ik=19,26
coefs_pseudo(ik)=27.d0/1280.d0
enddo
do ik=27,50
coefs_pseudo(ik)=14641.d0/725760.d0
enddo
if(nptsgrid.eq.50)return
write(*,*)'Grid for pseudos not available!'
write(*,*)'N=4-6-18-26-50 only!'
stop
end
!!
!! R_{lambda,lamba',N}= exp(-ga_a AC**2 -g_b BC**2) /int_{0}{+infty} r**(2+n) exp(-(g_a+g_b+g_k)r**2)
!! * M_{lambda}( 2g_a ac r) M_{lambda'}(2g_b bc r)
!!
double precision function bigR(lambda,lambdap,n,g_a,g_b,ac,bc,g_k)
implicit none
integer lambda,lambdap,n,npts,i
double precision g_a,g_b,ac,bc,g_k,arg,factor,delta1,delta2,cc,rmax,dr,sum,x1,x2,r
double precision bessel_mod
arg=g_a*ac**2+g_b*bc**2
factor=dexp(-arg)
delta1=2.d0*g_a*ac
delta2=2.d0*g_b*bc
cc=g_a+g_b+g_k
if(cc.eq.0.d0)stop 'pb. in bigR'
rmax=dsqrt(-dlog(10.d-20)/cc)
npts=500
dr=rmax/npts
sum=0.d0
do i=1,npts
r=(i-1)*dr
x1=delta1*r
x2=delta2*r
sum=sum+dr*r**(n+2)*dexp(-cc*r*r)*bessel_mod(x1,lambda)*bessel_mod(x2,lambdap)
enddo
bigR=sum*factor
end
double precision function bessel_mod(x,n)
implicit none
integer n
double precision x,bessel_mod_exp,bessel_mod_recur
if(x.le.0.8d0)then
bessel_mod=bessel_mod_exp(n,x)
else
bessel_mod=bessel_mod_recur(n,x)
endif
end
recursive function bessel_mod_recur(n,x) result(a)
implicit none
integer n
double precision x,a,bessel_mod_exp
if(x.le.0.8d0)then
a=bessel_mod_exp(n,x)
return
endif
if(n.eq.0)a=dsinh(x)/x
if(n.eq.1)a=(x*dcosh(x)-dsinh(x))/(x*x)
if(n.ge.2)a=bessel_mod_recur(n-2,x)-(n+n-1)/x*bessel_mod_recur(n-1,x)
end
double precision function bessel_mod_exp(n,x)
implicit none
integer n,k
double precision x,coef,accu,fact,dble_fact
accu=0.d0
do k=0,10
coef=1.d0/(fact(k)*dble_fact(2*(n+k)+1))
accu=accu+(0.5d0*x*x)**k*coef
enddo
bessel_mod_exp=x**n*accu
end
!c Computation of real spherical harmonics Ylm(theta,phi)
!c
!c l=0,1,....
!c m=-l,l
!c
!c m>0: Y_lm = sqrt(2) ([(2l+1)*(l-|m|)!]/[4pi*(l+|m|)!])^1/2 P_l^|m|(cos(theta)) cos(m phi)
!c m=0: Y_l0 = ([(2l+1)*(l-|m|)!]/[4pi*(l+|m|)!])^1/2 P_l^0 (cos(theta))
!c m<0: Y_lm = sqrt(2) ([(2l+1)*(l-|m|)!]/[4pi*(l+|m|)!])^1/2 P_l^|m|(cos(theta)) sin(|m|phi)
!Examples(wikipedia http://en.wikipedia.org/wiki/Table_of_spherical_harmonics#Real_spherical_harmonics)
! l = 0
! Y_00 = \sqrt{1/(4pi)}
! l = 1
! Y_1-1= \sqrt{3/(4pi)} y/r
! Y_10 = \sqrt{3/(4pi)} z/r
! Y_11 = \sqrt{3/(4pi)} x/r
!
! l = 2
!
! Y_2,-2= 1/2 \sqrt{15/pi} xy/r^2
! Y_2,-1= 1/2 \sqrt{15/pi} yz/r^2
! Y_20 = 1/4 \sqrt{15/pi} (-x^2-y^2 +2z^2)/r^2
! Y_21 = 1/2 \sqrt{15/pi} zx/r^2
! Y_22 = 1/4 \sqrt{15/pi} (x^2-y^2)/r^2
!
!c
double precision function ylm(l,m,theta,phi)
implicit none
integer l,m,i
double precision theta,phi,pm,factor,twopi,x,fact,sign
DIMENSION PM(0:100,0:100)
twopi=2.d0*dacos(-1.d0)
x=dcos(theta)
if (iand(m,1) == 1) then
sign=-1.d0
else
sign=1.d0
endif
CALL LPMN(100,l,l,X,PM)
if (m > 0) then
factor=dsqrt((l+l+1)*fact(l-m) /(twopi*fact(l+m)) )
! factor = dble(l+m)
! do i=-m,m-1
! factor = factor * (factor - 1.d0)
! enddo
! factor=dsqrt(dble(l+l+1)/(twopi*factor) )
ylm=sign*factor*pm(m,l)*dcos(dfloat(m)*phi)
else if (m == 0) then
factor=dsqrt( 0.5d0*(l+l+1) /twopi )
ylm=factor*pm(m,l)
else if (m < 0) then
factor=dsqrt( (l+l+1)*fact(l+m) /(twopi*fact(l-m)) )
! factor = dble(l-m)
! do i=m,-m-1
! factor = factor * (factor - 1.d0)
! enddo
! factor=dsqrt(dble(l+l+1)/(twopi*factor) )
ylm=sign*factor*pm(-m,l)*dsin(dfloat(-m)*phi)
endif
end
!c Explicit representation of Legendre polynomials P_n(x)
!!
!! P_n0(x) = P_n(x)= \sum_{k=0}^n a_k x^k
!!
!! with a_k= 2^n binom(n,k) binom( (n+k-1)/2, n )
!! coef_pm(n,k) is the k_th coefficient of P_n(x)
double precision function coef_pm(n,k)
implicit none
integer n,k
double precision arg,binom_func,binom_gen
if(n.eq.0.and.k.ne.0)stop 'coef_pm not defined'
if(n.eq.0.and.k.eq.0)then
coef_pm=1.d0
return
endif
arg=0.5d0*dfloat(n+k-1)
coef_pm=2.d0**n*binom_func(n,k)*binom_gen(arg,n)
end
!! Ylm_bis uses the series expansion of Ylm in xchap^i ychap^j zchap^k
!! xchap=x/r etc.
!c m>0: Y_lm = sqrt(2)*factor* P_l^|m|(cos(theta)) cos(m phi)
!c m=0: Y_l0 = factor* P_l^0 (cos(theta))
!c m<0: Y_lm = sqrt(2) factor* P_l^|m|(cos(theta)) sin(|m|phi)
!c factor= ([(2l+1)*(l-|m|)!]/[4pi*(l+|m|)!])^1/2
!! P_l^m (x) = (-1)**m (1-x**2)^m/2 d^m/dx^m P_l(x) m >0 or 0
!! the series expansion of P_m (x) is known
!!
!! sin(theta)**m cos(mphi) = \sum_0^[m/2] binom(m,2k) x^(m-2k) y^2k (-1)**k (easy to proove with
!! Moivre formula)
!! (here x = xchap...)
!!
!! Ylm m> 0 = \sum_{k=0}^[m/2] \sum_{i=0}^(l-m) c_ki x^(m-2k) y^2k z^i
!!
!! c_ki= (-1)^k sqrt(2)*factor*binom(m,2k)*(m+i)!/i!*coef_pm(l,i+m)
!!
!! Ylm m< 0 = \sum_{k=0}^[(m-1)/2] \sum_{i=0}^(l-m) c_ki x^(m-(2k+1)) y^(2k+1) z^i
!!
!! c_ki= (-1)^k sqrt(2)*factor*binom(m,2k+1)*(m+i)!/i!*coef_pm(l,i+m)
double precision function ylm_bis(l,m,theta,phi)
implicit none
integer l,m,k,i
double precision x,y,z,theta,phi,sum,factor,pi,binom_func,fact,coef_pm,cylm
pi=dacos(-1.d0)
x=dsin(theta)*dcos(phi)
y=dsin(theta)*dsin(phi)
z=dcos(theta)
factor=dsqrt((2*l+1)*fact(l-iabs(m))/(4.d0*pi*fact(l+iabs(m))))
if(m.gt.0)then
sum=0.d0
do k=0,m/2
do i=0,l-m
cylm=(-1.d0)**k*factor*dsqrt(2.d0)*binom_func(m,2*k)*fact(m+i)/fact(i)*coef_pm(l,i+m)
sum=sum+cylm*x**(m-2*k)*y**(2*k)*z**i
enddo
enddo
ylm_bis=sum
return
endif
if(m.eq.0)then
sum=0.d0
do i=0,l
sum=sum+factor*coef_pm(l,i)*z**i
enddo
ylm_bis=sum
return
endif
if(m.lt.0)then
m=-m
sum=0.d0
do k=0,(m-1)/2
do i=0,l-m
cylm=(-1.d0)**k*factor*dsqrt(2.d0)*binom_func(m,2*k+1)*fact(m+i)/fact(i)*coef_pm(l,i+m)
sum=sum+cylm*x**(m-(2*k+1))*y**(2*k+1)*z**i
enddo
enddo
ylm_bis=sum
m=-m
return
endif
end
!c
!c Computation of associated Legendre Polynomials PM(m,n) for n=0,...,N
!c Here:
!c n=l (n=0,1,...)
!c m=0,1,...,n
!c x=cos(theta) 0 < x < 1
!c
!c This routine computes: PM(m,n) for n=0,...,N (number N in input) and m=0,..,n
!c Exemples (see 'Associated Legendre Polynomilas wikipedia')
!c P_{0}^{0}(x)=1
!c P_{1}^{-1}(x)=-1/2 P_{1}^{1}(x)
!c P_{1}^{0}(x)=x
!c P_{1}^{1}(x)=-(1-x^2)^{1/2}
!c P_{2}^{-2}(x)=1/24 P_{2}^{2}(x)
!c P_{2}^{-1}(x)=-1/6 P_{2}^{1}(x)
!c P_{2}^{0}(x)=1/2 (3x^{2}-1)
!c P_{2}^{1}(x)=-3x(1-x^2)^{1/2}
!c P_{2}^{2}(x)=3(1-x^2)
!c
!c Explicit representation:
!!
!! P_n0(x) = P_n(x)= \sum_{k=0}^n a_k x^k
!!
!! with a_k= 2^n binom(n,k) binom( (n+k-1)/2, n )
double precision function binom_gen(alpha,n)
implicit none
integer :: n,k
double precision :: fact,alpha,prod, factn_inv
prod=1.d0
factn_inv = 1.d0/fact(n)
do k=0,n-1
prod=prod*(alpha-k)
binom_gen = prod*factn_inv
enddo
end
double precision function coef_nk(n,k)
implicit none
integer n,k
double precision gam,dble_fact,fact
if (k<0) stop 'pseudopot.f90 : coef_nk'
if (k>63) stop 'pseudopot.f90 : coef_nk'
gam=dble_fact(n+n+k+k+1)
! coef_nk=1.d0/(2.d0**k*fact(k)*gam)
coef_nk=1.d0/(dble(ibset(0_8,k))*fact(k)*gam)
return
end
!! Calculation of
!!
!! I= \int dx x**l *exp(-gam*x**2) M_n(ax) M_m(bx)
!!
!! M_n(x) modified spherical bessel function
!!
double precision function int_prod_bessel(l,gam,n,m,a,b,arg)
implicit none
integer n,k,m,q,l,kcp
double precision gam,dble_fact,fact,pi,a,b
double precision int,intold,sum,coef_nk,crochet,u,int_prod_bessel_large,freal,arg
integer :: n_1,m_1,nlm
double precision :: term_A, term_B, term_rap, expo
double precision :: s_q_0, s_q_k, s_0_0, a_over_b_square
double precision :: int_prod_bessel_loc
double precision :: inverses(0:300)
double precision :: two_qkmp1, qk, mk, nk
logical done
u=(a+b)*0.5d0/dsqrt(gam)
freal=dexp(-arg)
if(a.eq.0.d0.and.b.eq.0.d0)then
if(n.ne.0.or.m.ne.0)then
int_prod_bessel=0.d0
return
endif
int_prod_bessel=crochet(l,gam)*freal
return
endif
if(u.gt.6.d0)then
int_prod_bessel=int_prod_bessel_large(l,gam,n,m,a,b,arg)
return
endif
if(a.ne.0.d0.and.b.ne.0.d0)then
q=0
intold=-1.d0
int=0.d0
done=.false.
n_1 = n+n+1
m_1 = m+m+1
nlm = n+m+l
pi=dacos(-1.d0)
a_over_b_square = (a/b)**2
! Calcul first term of the sequence
term_a =dble_fact(nlm-1) / (dble_fact(n_1)*dble_fact(m_1))
expo=0.5d0*dfloat(nlm+1)
term_rap = term_a / (2.d0*gam)**expo
s_0_0=term_rap*a**(n)*b**(m)
if(mod(nlm,2).eq.0)s_0_0=s_0_0*dsqrt(pi*.5d0)
! Initialise the first recurence terme for the q loop
s_q_0 = s_0_0
mk = dble(m)
! Loop over q for the convergence of the sequence
do while (.not.done)
! Init
s_q_k=s_q_0
sum=s_q_0
if (q>300) then
stop 'pseudopot.f90 : q > 300'
endif
qk = dble(q)
two_qkmp1 = 2.d0*(qk+mk)+1.d0
do k=0,q-1
s_q_k = two_qkmp1*qk*inverses(k)*s_q_k
sum=sum+s_q_k
two_qkmp1 = two_qkmp1-2.d0
qk = qk-1.d0
enddo
inverses(q) = a_over_b_square/(dble(q+n+q+n+3) * dble(q+1))
! do k=0,q
! sum=sum+s_q_k
! s_q_k = a_over_b_square * ( dble(2*(q-k+m)+1)*dble(q-k)/(dble(2*(k+n)+3) * dble(k+1)) ) * s_q_k
! enddo
int=int+sum
if(dabs(int-intold).lt.1d-15)then
done=.true.
else
!Compute the s_q+1_0
! s_q_0=s_q_0*(2.d0*q+nlm+1)*b**2/((2.d0*(m+q)+3)*4.d0*(q+1)*gam)
s_q_0=s_q_0*(q+q+nlm+1)*b*b/(dble(8*(m+q)+12)*(q+1)*gam)
if(mod(n+m+l,2).eq.1)s_q_0=s_q_0*dsqrt(pi*.5d0)
! Increment q
q=q+1
intold=int
endif
enddo
int_prod_bessel=int*freal
return
endif
if(a.eq.0.d0.and.b.ne.0.d0)then
int = int_prod_bessel_loc(l,gam,m,b)
int_prod_bessel=int*freal
return
endif
if(a.ne.0.d0.and.b.eq.0.d0)then
int = int_prod_bessel_loc(l,gam,n,a)
int_prod_bessel=int*freal
return
endif
stop 'pb in int_prod_bessel!!'
end
double precision function int_prod_bessel_large(l,gam,n,m,a,b,arg)
implicit none
integer n,m,i,npts,l
double precision gam,a,b
double precision sum,x,bessel_mod,u,factor,arg
double precision xq(100),wq(100)
u=(a+b)/(2.d0*dsqrt(gam))
factor=dexp(u*u-arg)/dsqrt(gam)
xq(1)= 5.38748089001123
xq(2)= 4.60368244955074
xq(3)= 3.94476404011563
xq(4)= 3.34785456738322
xq(5)= 2.78880605842813
xq(6)= 2.25497400208928
xq(7)= 1.73853771211659
xq(8)= 1.23407621539532
xq(9)= 0.737473728545394
xq(10)= 0.245340708300901
xq(11)=-0.245340708300901
xq(12)=-0.737473728545394
xq(13)=-1.23407621539532
xq(14)=-1.73853771211659
xq(15)=-2.25497400208928
xq(16)=-2.78880605842813
xq(17)=-3.34785456738322
xq(18)=-3.94476404011563
xq(19)=-4.60368244955074
xq(20)=-5.38748089001123
wq(1)= 2.229393645534151E-013
wq(2)= 4.399340992273176E-010
wq(3)= 1.086069370769280E-007
wq(4)= 7.802556478532063E-006
wq(5)= 2.283386360163528E-004
wq(6)= 3.243773342237853E-003
wq(7)= 2.481052088746362E-002
wq(8)= 0.109017206020022
wq(9)= 0.286675505362834
wq(10)= 0.462243669600610
wq(11)= 0.462243669600610
wq(12)= 0.286675505362834
wq(13)= 0.109017206020022
wq(14)= 2.481052088746362E-002
wq(15)= 3.243773342237853E-003
wq(16)= 2.283386360163528E-004
wq(17)= 7.802556478532063E-006
wq(18)= 1.086069370769280E-007
wq(19)= 4.399340992273176E-010
wq(20)= 2.229393645534151E-013
npts=20
! call gauher(xq,wq,npts)
sum=0.d0
do i=1,npts
x=(xq(i)+u)/dsqrt(gam)
sum=sum+wq(i)*x**l*bessel_mod(a*x,n)*bessel_mod(b*x,m)*dexp(-(a+b)*x)
enddo
int_prod_bessel_large=sum*factor
end
!! Calculation of
!!
!! I= \int dx x**l *exp(-gam*x**2) M_n(ax)
!!
!! M_n(x) modified spherical bessel function
!!
double precision function int_prod_bessel_loc(l,gam,n,a)
implicit none
integer n,k,l,kcp
double precision gam,a
double precision int,intold,coef_nk,crochet,dble_fact, fact, pi, expo
double precision :: f_0, f_k
logical done
pi=dacos(-1.d0)
intold=-1.d0
done=.false.
int=0
! Int f_0
coef_nk=1.d0/dble_fact( n+n+1 )
expo=0.5d0*dfloat(n+l+1)
crochet=dble_fact(n+l-1)/(gam+gam)**expo
if(mod(n+l,2).eq.0)crochet=crochet*dsqrt(0.5d0*pi)
f_0 = coef_nk*a**n*crochet
k=0
f_k=f_0
do while (.not.done)
int=int+f_k
! f_k = f_k*(a**2*(2*(k+1)+n+l-1)) / (2*(k+1)*(2*(n+k+1)+1)*2*gam)
f_k = f_k*(a*a*dble(k+k+1+n+l)) / (dble((k+k+2)*(4*(n+k+1)+2))*gam)
if(dabs(int-intold).lt.1d-15)then
done=.true.
else
k=k+1
intold=int
endif
enddo
int_prod_bessel_loc=int
end
double precision function int_prod_bessel_num(l,gam,n,m,a,b)
implicit none
integer n,m,l,i,npoints
double precision gam,a,b
double precision sum,dx,x,bessel_mod
sum=0.d0
npoints=20000
dx=30.d0/npoints
do i=1,npoints
x=(i-1)*dx+0.5d0*dx
sum=sum+dx*x**l*dexp(-gam*x**2)*bessel_mod(a*x,n)*bessel_mod(b*x,m)
enddo
int_prod_bessel_num=sum
end
! l,m : Y(l,m) parameters
! c(3) : pseudopotential center
! a(3) : Atomic Orbital center
! n_a(3) : Powers of x,y,z in the Atomic Orbital
! g_a : Atomic Orbital exponent
! r : Distance between the Atomic Orbital center and the considered point
double precision function ylm_orb(l,m,c,a,n_a,g_a,r)
implicit none
integer lmax_max
integer l,m
double precision a(3),g_a,c(3)
double precision prod,binom_func,accu,bigI,ylm,bessel_mod
double precision theta_AC0,phi_AC0,ac,ac2,factor,fourpi,arg,r,areal
integer ntotA,mu,k1,k2,k3,lambda
integer n_a(3)
double precision y, f1, f2
double precision, allocatable :: array_coefs_A(:,:)
ac2=(a(1)-c(1))**2+(a(2)-c(2))**2+(a(3)-c(3))**2
ac=dsqrt(ac2)
arg=g_a*(ac2+r*r)
fourpi=4.d0*dacos(-1.d0)
factor=fourpi*dexp(-arg)
areal=2.d0*g_a*ac
ntotA=n_a(1)+n_a(2)+n_a(3)
if(ac.eq.0.d0)then
ylm_orb=dsqrt(fourpi)*r**ntotA*dexp(-g_a*r**2)*bigI(0,0,l,m,n_a(1),n_a(2),n_a(3))
return
else
theta_AC0=dacos( (a(3)-c(3))/ac )
phi_AC0=datan2((a(2)-c(2))/ac,(a(1)-c(1))/ac)
allocate (array_coefs_A(0:ntotA,3))
do k1=0,n_a(1)
array_coefs_A(k1,1) = binom_func(n_a(1),k1)*(c(1)-a(1))**(n_a(1)-k1)*r**(k1)
enddo
do k2=0,n_a(2)
array_coefs_A(k2,2) = binom_func(n_a(2),k2)*(c(2)-a(2))**(n_a(2)-k2)*r**(k2)
enddo
do k3=0,n_a(3)
array_coefs_A(k3,3) = binom_func(n_a(3),k3)*(c(3)-a(3))**(n_a(3)-k3)*r**(k3)
enddo
accu=0.d0
do lambda=0,l+ntotA
do mu=-lambda,lambda
y = ylm(lambda,mu,theta_AC0,phi_AC0)
if (y == 0.d0) then
cycle
endif
do k3=0,n_a(3)
f1 = y*array_coefs_A(k3,3)
if (f1 == 0.d0) cycle
do k2=0,n_a(2)
f2 = f1*array_coefs_A(k2,2)
if (f2 == 0.d0) cycle
do k1=0,n_a(1)
prod=f2*array_coefs_A(k1,1)*bigI(lambda,mu,l,m,k1,k2,k3)
if (prod == 0.d0) then
cycle
endif
if (areal*r < 100.d0) then ! overflow!
accu=accu+prod*bessel_mod(areal*r,lambda)
endif
enddo
enddo
enddo
enddo
enddo
ylm_orb=factor*accu
deallocate (array_coefs_A)
return
endif
end
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