Build up the nuclear potential part of the core Hamiltonian matrix in the case of an allelectron calculation by computing the nuclear attraction integral matrix <a|-Z/r|b> and the integral matrix <a|Z*erf(r)/r|b>. The integrals <a|Z*erf(r)/r|b> can be rewritten as the three-center Coulomb integrals <ab||c> with a primitive s-type Gaussian function c multiplied by a factor N. More...

Functions
subroutine, public	verfc (la_max1, npgfa, zeta, rpgfa, la_min1, lb_max1, npgfb, zetb, rpgfb, lb_min1, zetc, rpgfc, zc, cerf, rab, rab2, rac, rac2, rbc2, vabc, verf, vnuc, f, maxder, vabc_plus, vnabc, pVp_sum, pVp, dkh_erfc)
	Calculation of the primitive three-center nuclear potential integrals <a\|Z*erfc(r)/r\|b> over Cartesian Gaussian-type functions.
Variables
CHARACTER(len=*), parameter, private	moduleN = 'ai_verfc'

Detailed Description

Build up the nuclear potential part of the core Hamiltonian matrix in the case of an allelectron calculation by computing the nuclear attraction integral matrix <a|-Z/r|b> and the integral matrix <a|Z*erf(r)/r|b>. The integrals <a|Z*erf(r)/r|b> can be rewritten as the three-center Coulomb integrals <ab||c> with a primitive s-type Gaussian function c multiplied by a factor N.

erfc(r) <a|V|b> = -Z*<a|---------|b> r

1 - erf(r) = -Z*<a|------------|b> r

1 erf(r) = -Z*(<a|---|b> - <a|--------|b>) r r

1 = -Z*(<a|---|b> - N*<ab||c>) r

1 = -Z*<a|---|b> + Z*N*<ab||c> r / / | | nuclear coulomb

Literature: S. Obara and A. Saika, J. Chem. Phys. 84, 3963 (1986)

Parameters

ax,ay,az : Angular momentum index numbers of orbital a.
bx,by,bz : Angular momentum index numbers of orbital b.
coset : Cartesian orbital set pointer.
dab : Distance between the atomic centers a and b.
dac : Distance between the atomic centers a and c.
dbc : Distance between the atomic centers b and c.
l{a,b} : Angular momentum quantum number of shell a or b.
l{a,b}_max: Maximum angular momentum quantum number of shell a or b.
l{a,b}_min: Minimum angular momentum quantum number of shell a or b.
ncoset : Number of orbitals in a Cartesian orbital set.
npgf{a,b} : Degree of contraction of shell a or b.
rab : Distance vector between the atomic centers a and b.
rab2 : Square of the distance between the atomic centers a and b.
rac : Distance vector between the atomic centers a and c.
rac2 : Square of the distance between the atomic centers a and c.
rbc2 : Square of the distance between the atomic centers b and c.
rpgf{a,b} : Radius of the primitive Gaussian-type function a or b.
zet{a,b} : Exponents of the Gaussian-type functions a or b.
zetp : Reciprocal of the sum of the exponents of orbital a and b.
zetw : Reciprocal of the sum of the exponents of orbital a, b and c. Indices for pVp matrices
ax,ay,az : Angular momentum index numbers of orbital a.
n{x1,y1,z1}{x2,y2,z2} : indices for the pVp matrix elements <p_{x1,y1,z1}a|V|p_{x2,y2,z2}>
co{a,b}{m,p}{x,y,z} : coefficients of the pVp matrix elements: co == coset(lx,ly,lz) m == li - 1; p == li +1; li= x,y,z (which l quantum number is modified); l = a,b coset ( ... -1 ...) = 1 , should be zero, lmi = 1.0 for li >=1 and lmi = 0.0 for li = 0 guarantees this as a prefactor
f{a,b}{x,y,z},ft{a,b},z : f(t)li: prefactors as a result of derivations of l with respect to i
pVpout : pVp matrix elements
pVp : local pVp matrix elements
lamin,lbmin : minimal angular quantum number used in pVp matrices _ vnucp : potential of the nuclei
vnuc : potential of the electrons _ pVpa, pVpb : pVpl=coset(lx,ly,lz)
na_pgf,nb_pgf : indice for primitive gaussian functions

History: 10.2008 added pVp matrix elements (jens)

Author:: Matthias Krack (04.10.2000)

Function Documentation

subroutine,public ai_verfc::verfc	(	INTEGER,intent(in)	la_max1,
		INTEGER,intent(in)	npgfa,
		REAL(KIND=dp),dimension(:),intent(in)	zeta,
		REAL(KIND=dp),dimension(:),intent(in)	rpgfa,
		INTEGER,intent(in)	la_min1,
		INTEGER,intent(in)	lb_max1,
		INTEGER,intent(in)	npgfb,
		REAL(KIND=dp),dimension(:),intent(in)	zetb,
		REAL(KIND=dp),dimension(:),intent(in)	rpgfb,
		INTEGER,intent(in)	lb_min1,
		REAL(KIND=dp),intent(in)	zetc,
		REAL(KIND=dp),intent(in)	rpgfc,
		REAL(KIND=dp),intent(in)	zc,
		REAL(KIND=dp),intent(in)	cerf,
		REAL(KIND=dp),dimension(3),intent(in)	rab,
		REAL(KIND=dp),intent(in)	rab2,
		REAL(KIND=dp),dimension(3),intent(in)	rac,
		REAL(KIND=dp),intent(in)	rac2,
		REAL(KIND=dp),intent(in)	rbc2,
		REAL(KIND=dp),dimension(:, :),intent(inout)	vabc,
		REAL(KIND=dp),dimension(:, :, :)	verf,
		REAL(KIND=dp),dimension(:, :, :)	vnuc,
		REAL(KIND=dp),dimension(0:)	f,
		INTEGER,intent(in),optional	maxder,
		REAL(KIND=dp),dimension(:, :),optional	vabc_plus,
		REAL(KIND=dp),dimension(:, :),optional	vnabc,
		REAL(KIND=dp),dimension(:, :),optional	pVp_sum,
		REAL(KIND=dp),dimension(:, :),optional	pVp,
		LOGICAL,optional	dkh_erfc
	)