Stochastic many-body perturbation theory for electron correlation energies

TL;DR

This paper introduces a stochastic Monte Carlo method based on Goldstone's time-dependent formulation of MBPT to efficiently compute high-order electron correlation energies, significantly reducing computational costs.

Contribution

It presents a novel Monte Carlo algorithm that extends previous MP2 and MP3 approaches to higher orders using a reformulation of MBPT.

Findings

Successfully extends Monte Carlo methods to higher perturbation orders

Achieves lower computational scaling for electron correlation energies

Demonstrates practical applicability through proof-of-concept calculations

Abstract

Treating electron correlation more accurately and efficiently is at the heart of the development of electronic structure methods. In the present work, we explore the use of stochastic approaches to evaluate high-order electron correlation energies, whose conventional computational scaling is unpleasantly steep, being $O(N^{n+3})$ with respect to the system size $N$ and the perturbation order $n$ for the M{\o}ller-Plesset (MP) series. To this end, starting from Goldstone's time-dependent formulation of \emph{ab initio} many-body perturbation theory (MBPT), we present a reformulation of MBPT, which naturally leads to an Monte Carlo scheme with $O(nN^2+n^2N+f(n))$ scaling at each step, where $f(n)$ is a function of $n$ depending on the specific numerical scheme. Proof-of-concept calculations demonstrate that the proposed quantum Monte Carlo algorithm successfully extends the previous Monte Carlo approaches for MP2 and MP3 to higher orders. Therefore, for the first time the Goldstone's time-dependent formulation is made useful numerically for electron correlation energies, not only being purely as a theoretical tool.