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: $G_0W_0$, $G_0W_0^tc$, 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.