Analytical two-center integrals over Slater geminal functions

TL;DR

This paper derives analytical formulas for two-center two-electron integrals over Slater geminal functions, introducing new special functions and recursion relations, validated through numerical examples and comparisons.

Contribution

It provides the first analytical solutions for these integrals, including new special functions and recursion methods, advancing computational quantum chemistry techniques.

Findings

Derived inhomogeneous differential equation for the master integral

Introduced new special functions for solving the differential equation

Validated formulas with numerical examples and comparisons

Abstract

We present analytical formulas for the calculation of the two-center two-electron integrals in the basis of Slater geminals and products of Slater orbitals. Our derivation starts with establishing a inhomogeneous fourth-order ordinary differential equation that is obeyed by the master integral, the simplest integral with inverse powers of all interparticle distances. To solve this equation it was necessary to introduce a new family of special functions which are defined through their series expansions around regular singular points of the differential equation. To increase the power of the interparticle distances under the sign of the integral we developed a family of open-ended recursion relations. A handful of special cases of the integrals is also analysed with some remarks on simplifications that occur. Additionally, we present some numerical examples of the master integral that validate the usefulness and correctness of the key equations derived in this paper. In particular, we compare our results with the calculations based on the series expansion of the exp(-\gamma r12) term in the master integral.