Stochastic self-consistent second-order Green's function method for correlation energies of large electronic systems

TL;DR

This paper introduces a stochastic version of the GF2 method that significantly reduces computational scaling, enabling efficient and accurate correlation energy calculations for large electronic systems, demonstrated on hydrogen chains with up to 1000 electrons.

Contribution

The authors develop a stochastic GF2 approach that reduces the dominant computational step from fifth to quadratic scaling, making large-scale correlation energy calculations feasible.

Findings

The stochastic GF2 method is numerically stable and accurate for large systems.

It achieves a reduction in computational effort with errors below 0.1%.

The approach is applicable to finite and periodic systems at finite temperatures.

Abstract

The second-order Matsubara Green's function method (GF2) is a robust temperature dependent quantum chemistry approach, extending beyond the random-phase approximation. However, till now the scope of GF2 applications was quite limited as they require computer resources which rise steeply with system size. In each step of the self-consistent GF2 calculation there are two parts: the estimation of the self-energy from the previous step's Green's function, and updating the Green's function from the self-energy. The first part formally scales as the fifth power of the system size while the second has a much gentler cubic scaling. Here, we develop a stochastic approach to GF2 (sGF2) which reduces the fifth power scaling of the first step to merely quadratic, leaving the overall sGF2 scaling as cubic. We apply the method to linear hydrogen chains containing up to 1000 electrons, showing that the approach is numerically stable, efficient and accurate. The stochastic errors are very small, of the order of 0.1% or less of the correlation energy for large systems, with only a moderate computational effort. The first iteration of GF2 is an MP2 calculation that is done in linear scaling, hence we obtain an extremely fast stochastic MP2 (sMP2) method as a by-product. While here we consider finite systems with large band gaps where at low temperatures effects are negligible, the sGF2 formalism is temperature dependent and general and can be applied to finite or periodic systems with small gaps at finite temperatures.