Stochastic Vertex Corrections: Linear Scaling Methods for Accurate Quasiparticle Energies

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: G₀W₀, G₀W₀, and G₀W₀^tcΓ. All three methods are formulated in the time domain, and the latter two incorporate nonlocal vertex corrections. In the case of G₀W₀^tcΓ, 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 of the fully stochastic implementation is tested on acene molecules, C₆₀ and PC₆₀BM. 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.