# Stochastic vertex corrections: linear scaling methods for accurate quasiparticle energies

## TL;DR

This paper introduces stochastic linear-scaling methods within many-body perturbation theory to compute quasiparticle energies accurately, incorporating vertex corrections and validated on various molecular systems.

## Contribution

It develops and verifies three stochastic approaches for quasiparticle energy calculations, including non-local vertex corrections, with linear scaling demonstrated for large molecules.

## Key findings

- Vertex corrections are essential for accurate unoccupied state description.
- All three stochastic methods scale linearly with the number of electrons.
- Methods are validated against deterministic results for small molecules.

## Abstract

New stochastic approaches for the computation of electronic excitations are developed within the many-body perturbation theory. Three approximations to the electronic self-energy are considered: $G_0W_0$, $G_0W_0^tc$, and $G_0W_0^{tc}\Gamma_x$. All three methods are formulated in the time domain and the latter two incorporate non-local vertex corrections. In case of $G_0W_0^{tc}\Gamma_x$, the vertex corrections are included both in the screened Coulomb interaction and in the expression for the self-energy. The implementation of the three approximations is verified by comparison to deterministic results for a set of small molecules. The performance fully stochastic implementation is tested on acene molecules, C$_{60}$ and PC$_{60}$BM. The vertex correction appears crucial for the description of unoccupied states. Unlike conventional (deterministic) approaches, all three stochastic methods scale linearly with the number of electrons.