PAOFLOW.utils.pyints#

pyints — Two-electron Coulomb repulsion integrals over Gaussian-type orbitals.

This module provides a pure-Python implementation of the analytical two-electron electron-repulsion integrals (ERIs) needed by the ACBN0 and eACBN0 Hartree exchange-correlation routines in PAOFLOW. The algorithms follow the Obara–Saika / THO (Taketa–Huzinaga–O-Ohata) recurrence-relation formalism.

Overview#

The central quantity computed is the four-centre two-electron repulsion integral over contracted Gaussian-type orbitals (cGTOs):

\[J_{\mu\nu\lambda\sigma} = \int\!\int \phi_\mu(\mathbf{r}_1)\,\phi_\nu(\mathbf{r}_1)\, \frac{1}{r_{12}}\, \phi_\lambda(\mathbf{r}_2)\,\phi_\sigma(\mathbf{r}_2)\, d\mathbf{r}_1\,d\mathbf{r}_2\]

The evaluation chain is:

Primitive ERI — coulomb_repulsion() computes the integral for a single Gaussian primitive on each of the four centres, using the Boys function Fgamma() and the Cartesian B-array expansion (B_array(), fB(), B0()) from THO eq. 2.22.
Contraction — contr_coulomb() sums the primitive integrals over all contraction coefficients and normalisation factors, yielding the contracted ERI used by ACBN0.

Special functions#

gammln(): Natural logarithm of the Gamma function via the Lanczos approximation (Numerical Recipes sect. 6.1).
gammp(): Incomplete Gamma function \(P(a; x)\) via series expansion (_gser()) for \(x < a+1\) or continued-fraction representation (_gcf()) otherwise (Numerical Recipes sect. 6.2).
gamm_inc(): Unnormalised incomplete Gamma function \(\gamma(a, x)\).
Fgamma(): Boys function \(F_m(x) = \int_0^1 t^{2m} e^{-xt^2}\,dt\), evaluated via the relation \(F_m(x) = \gamma(m+\tfrac{1}{2}, x)/(2x^{m+\tfrac{1}{2}})\).

Combinatorial utilities#

fact(), fact_ratio(), binomial(), binomial_prefactor(): Factorial, double-factorial ratio, and binomial-coefficient helpers used in the Cartesian angular-momentum reduction (Augspurger & Dykstra).

Geometry helpers#

dist(): Squared Euclidean distance \(|A - B|^2\) between two 3D points.
gaussian_product_center(): Gaussian product theorem centre \(P = (\alpha_1 A + \alpha_2 B)/(\alpha_1+\alpha_2)\).

Data structure#

CGBF: Lightweight container for a contracted Gaussian basis function, storing the centre origin, angular-momentum powers, primitive exponents, contraction coefficients, and normalisation factors.

Classes#

CGBF

Class for a contracted Gaussian basis function

Functions#

`gammln`(x)	Numerical recipes, section 6.1
`dist`(A, B)
`gaussian_product_center`(alpha1, A, alpha2, B)
`Fgamma`(m, x)	Incomplete gamma function
`gammp`(a, x)	Returns the incomplete gamma function P(a;x). NumRec sect 6.2.
`gamm_inc`(a, x)	Incomple gamma function; computed from NumRec routine gammp.
`fact`(i)	Normal factorial
`fact_ratio`(a, b)
`binomial`(a, b)	Binomial coefficient
`binomial_prefactor`(s, ia, ib, xpa, xpb)	From Augspurger and Dykstra
`B0`(i, r, g)
`fB`(i, l1, l2, P, A, B, r, g)
`B_array`(l1, l2, l3, l4, p, a, b, q, c, d, g1, g2, delta)
`coulomb_repulsion`(axyz, anorm, almn, aalpha, bxyz, ...)
`contr_coulomb`(aexps, acoefs, anorms, xyza, powa, ...)

Module Contents#

class PAOFLOW.utils.pyints.CGBF(origin, atid=0)[source]#

Class for a contracted Gaussian basis function

origin[source]#

powers = [][source]#

pnorms = [][source]#

prims = [][source]#

pexps = [][source]#

pcoefs = [][source]#

atid = 0[source]#

PAOFLOW.utils.pyints.gammln(x)[source]#: Numerical recipes, section 6.1

PAOFLOW.utils.pyints.dist(A, B)[source]#

PAOFLOW.utils.pyints.gaussian_product_center(alpha1, A, alpha2, B)[source]#

PAOFLOW.utils.pyints.Fgamma(m, x)[source]#: Incomplete gamma function

PAOFLOW.utils.pyints.gammp(a, x)[source]#: Returns the incomplete gamma function P(a;x). NumRec sect 6.2.

PAOFLOW.utils.pyints.gamm_inc(a, x)[source]#: Incomple gamma function; computed from NumRec routine gammp.

PAOFLOW.utils.pyints.fact(i)[source]#: Normal factorial

PAOFLOW.utils.pyints.fact_ratio(a, b)[source]#

PAOFLOW.utils.pyints.binomial(a, b)[source]#: Binomial coefficient

PAOFLOW.utils.pyints.binomial_prefactor(s, ia, ib, xpa, xpb)[source]#: From Augspurger and Dykstra

PAOFLOW.utils.pyints.B0(i, r, g)[source]#

PAOFLOW.utils.pyints.fB(i, l1, l2, P, A, B, r, g)[source]#

PAOFLOW.utils.pyints.B_array(l1, l2, l3, l4, p, a, b, q, c, d, g1, g2, delta)[source]#

PAOFLOW.utils.pyints.coulomb_repulsion(axyz, anorm, almn, aalpha, bxyz, bnorm, blmn, balpha, cxyz, cnorm, clmn, calpha, dxyz, dnorm, dlmn, dalpha)[source]#

PAOFLOW.utils.pyints.contr_coulomb(aexps, acoefs, anorms, xyza, powa, bexps, bcoefs, bnorms, xyzb, powb, cexps, ccoefs, cnorms, xyzc, powc, dexps, dcoefs, dnorms, xyzd, powd)[source]#