# A Stochastic Formulation of the Resolution of Identity: Application to Second Order M{\o}ller-Plesset Perturbation Theory

## TL;DR

This paper introduces a stochastic orbital approach to the resolution of identity for electron repulsion integrals, enabling efficient MP2 calculations with improved scaling and performance on water clusters.

## Contribution

It presents a novel stochastic RI method with multiple orbitals that reduces MP2 computational scaling and outperforms traditional methods on water clusters.

## Key findings

- Achieves $N^{2.4}$ scaling for water clusters
- Outperforms MP2 for clusters with 21 water molecules
- Demonstrates efficiency of stochastic RI-MP2 approach

## Abstract

A stochastic orbital approach to the resolution of identity (RI) approximation for 4-index 2-electron electron repulsion integrals (ERIs) is presented. The stochastic RI-ERIs are then applied to M\o ller-Plesset perturbation theory (MP2) utilizing a \textit{multiple stochastic orbital approach}. The introduction of multiple stochastic orbitals results in an $N^3$ scaling for both the stochastic RI-ERIs and stochastic RI-MP2. We demonstrate that this method exhibits a small prefactor and an observed scaling of $N^{2.4}$ for a range of water clusters, already outperforming MP2 for clusters with as few as 21 water molecules.