On the Evaluation of the electron repulsion integrals

TL;DR

This paper develops new methods to evaluate electron repulsion integrals over non-integer Slater-type orbitals, simplifying their computation by removing hyper-geometric functions and expressing integrals as finite power series.

Contribution

It introduces relationships that eliminate hyper-geometric functions from integral evaluations, enabling more efficient computation of electron repulsion integrals.

Findings

Derived hyper-geometric free relationships for Coulomb expectation values

Expressed integrals as finite power series

Removed convergence and numerical issues in integral calculations

Abstract

The electron repulsion integrals over the Slater-type orbitals with non-integer principal quantum numbers are considered. These integrals are useful in both non-relativistic and relativistic calculations of many-electron systems. They involve hyper-geometric functions. Due to the non-trivial structure of infinite series that are used to define them the hyper-geometric functions are practically difficult to compute. Convergence of their series are strictly depends on the values of parameters. Computational issues such as cancellation or round-off error emerge. Relationships free from hyper$-$geometric functions for expectation values of Coulomb potential $\left(r_{21}^{-1}\right)$ are derived. These relationships are new and show that the complication coming from two-range nature of Laplace expansion for the Coulomb potential is removed. These integrals also form an initial condition for expectation values of a potential with arbitrary power. The electron repulsion integrals are expressed by finite series of power functions. The methodology given here for evaluation of electron repulsion integrals are adapted to multi-center integrals.