# Recurrence relations for four-electron integrals over Gaussian basis functions

## TL;DR

This paper develops recurrence relations for efficient computation of four-electron integrals over Gaussian basis functions, extending previous three-electron algorithms to handle more complex integrals in quantum chemistry.

## Contribution

It introduces a generalized recursive approach for four-electron integrals, applicable to various operators in explicitly-correlated F12 methods, enhancing computational efficiency.

## Key findings

- Derived vertical, transfer, and horizontal recurrence relations
- Applicable to a broad class of four-electron operators
- Facilitates accurate and efficient integral calculations

## Abstract

In the spirit of the Head-Gordon-Pople algorithm, we report vertical, transfer and horizontal recurrence relations for the efficient and accurate computation of four-electron integrals over Gaussian basis functions. Our recursive approach is a generalization of our algorithm for three-electron integrals [J.~Chem.~Theory Comput.~12, 1735 (2016)]. The RRs derived in the present study can be applied to a general class of multiplicative four-electron operators. In particular, we consider various types of four-electron integrals that may arise in explicitly-correlated F12 methods.