Analytic Evaluation of some 2-, 3- and 4- Electron Atomic Integrals Containing Exponentially Correlated Functions of $r_{ij}$

TL;DR

This paper presents a straightforward analytic method for evaluating complex multi-electron atomic integrals involving exponential correlations, avoiding traditional techniques, and derives closed-form solutions for various integrals.

Contribution

It introduces a novel Fourier-based approach to analytically evaluate multi-electron atomic integrals with exponential correlations, simplifying previous methods.

Findings

Derived closed-form expressions for 2-electron integrals.

Extended method to 3- and 4-electron integrals.

Avoided use of spherical harmonic addition theorem or Feynman technique.

Abstract

A simple method is outlined for analytic evaluation of the basic 2-electron atomic integral with integrand containing products of atomic s-type Slater orbitals and exponentially correlated function of the form $r_{ij} exp(-\lambda_{ij}r_{ij})$, by employing the Fourier representation of $exp(-\lambda_{ij}r_{ij})/r_{ij}$ without the use of either the spherical harmonic addition theorem or the Feynman technique. This method is applied to obtain closed-form expressions, in a simple manner, for certain other 2-,3- and 4-electron atomic integrals with integrands which are products of exponentially correlated functions and atomic s-type Slater orbitals.