# Evaluating two-electron-repulsion integrals over arbitrary orbitals using Zero Variance Monte Carlo: Application to Full Configuration Interaction calculations with Slater-type orbitals

## TL;DR

This paper introduces a universal Monte Carlo method for accurately evaluating complex two-electron integrals over various orbitals, enabling large-scale full CI calculations with Slater-type orbitals.

## Contribution

It presents a simple, parallelizable Monte Carlo approach with zero-variance estimators for two-electron integrals over arbitrary orbitals, improving accuracy and applicability in quantum chemistry.

## Key findings

- Achieved high-accuracy two-electron integrals for Slater orbitals.
- Performed large-scale FCI calculations with 18 electrons and 90 orbitals.
- Demonstrated the method's potential for extensive molecular simulations.

## Abstract

A Monte Carlo method for evaluating multi-center two-electron-repulsion integrals over any type of orbitals (Slater, Sturmian, finite-range, numerical, etc.) is presented. The approach is based on a simple and universal (orbital-independent) gaussian sampling of the two-electron configuration space and on the use of efficient zero-variance Monte Carlo estimators. Quite remarkably, it is shown that the high level of accuracy required on two-electron integrals to make Hartree-Fock (HF) and configuration interaction (CI) calculations feasible can be achieved. A first zero-variance estimator is built by introducing a gaussian approximation of the orbitals and by evaluating the two-electron integrals using a correlated sampling scheme for the difference between exact and approximate orbitals. A second one is based on the introduction of a general coordinate transformation. The price to pay for this simple and general Monte Carlo scheme is the high computational cost required. However, we argue that the great simplicity of the algorithm, its embarrassingly parallel nature, its ideal adaptation to modern computational platforms and, most importantly, the possibility of using more compact and physically meaningful basis sets make nevertheless the method attractive. HF and near full CI (FCI) calculations using Slater-type orbitals (STO) are reported for Be, CH4 and [H2N(CH)NH2]^+ (a simple model of cyanine). To the best of our knowledge, our largest FCI calculation involving 18 active electrons distributed among 90 orbitals for the cyanine molecule, is the most extensive molecular calculation performed so far using pure STO orbitals (no gaussian approximation, even for the challenging four-center two-electron integrals).