# Compact Formulae for Three-Center Nuclear Attraction Integrals Over Exponential Type Functions

## TL;DR

This paper presents new analytical formulas for three-center nuclear attraction integrals over exponential type functions, significantly improving computational efficiency in molecular orbital calculations.

## Contribution

It introduces compact analytical expressions and an efficient evaluation method for three-center nuclear attraction integrals, enhancing accuracy and speed in quantum chemistry computations.

## Key findings

- Substantial efficiency gains over existing methods
- Effective use of rational minimax approximants
- Accurate double precision evaluations

## Abstract

In any ab initio molecular orbital (MO) calculations, the major task involves the computation of the so-called molecular multi-center integrals. Multi-center integral calculations is a very challenging mathematical problem in nature. Quantum mechanics only determines which integrals we evaluate, but the techniques employed for their evaluations are entirely mathematical. The three-center nuclear attraction integrals occur in a very large number even for small molecules and are among of the most difficult molecular integrals to compute efficiently to a high pre-determined accuracy. In the present contribution, we report analytical expressions for the three-center nuclear attraction integrals over exponential type functions. We describe how to compute the formula to obtain an efficient evaluation in double precision arithmetic. This requires the rational minimax approximants that minimize the maximum error on the interval of evaluation. The numerical tests show a substantial gain in efficiency over the state-of-the-art.