# Three- and four-electron integrals involving Gaussian geminals: fundamental integrals, upper bounds and recurrence relations

## TL;DR

This paper develops methods for efficiently calculating three- and four-electron integrals involving Gaussian geminals, including fundamental integrals, recurrence relations, and upper bounds, to improve explicitly-correlated F12 quantum chemistry calculations.

## Contribution

It introduces a comprehensive approach with fundamental integrals, recurrence relations, and upper bounds for three- and four-electron integrals involving Gaussian geminals, optimizing computational efficiency.

## Key findings

- Derived straightforward methods for fundamental integrals.
- Established recurrence relations for angular momentum buildup.
- Provided upper bounds to identify significant integrals.

## Abstract

We report the three main ingredients to calculate three- and four-electron integrals over Gaussian basis functions involving Gaussian geminal operators: fundamental integrals, upper bounds, and recurrence relations. In particular, we consider the three- and four-electron integrals that may arise in explicitly-correlated F12 methods. A straightforward method to obtain the fundamental integrals is given. We derive vertical, transfer and horizontal recurrence relations to build up angular momentum over the centers. Strong, simple and scaling-consistent upper bounds are also reported. This latest ingredient allows to compute only the $\order{N^2}$ significant three- and four-electron integrals, avoiding the computation of the very large number of negligible integrals.