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/Subroutines
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. More...

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 : indices for primitive gaussian functions

History: 10.2008 added pVp matrix elements (jens)

Author: Matthias Krack (04.10.2000)

Function/Subroutine Documentation

◆ verfc()

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
	)

Calculation of the primitive three-center nuclear potential integrals <a|Z*erfc(r)/r|b> over Cartesian Gaussian-type functions.

Parameters

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	...

Date: 04.10.2000

Author: Matthias Krack

Version: 1.0

Definition at line 138 of file ai_verfc.F.

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Functions/Subroutines

Detailed Description

Function/Subroutine Documentation

◆ verfc()