Mixed Stochastic-Deterministic Approach for Many-Body Perturbation Theory Calculations

We present an approach for $GW$ calculations of quasiparticle energies with quasiquadratic scaling by approximating high-energy contributions to the Green's function in its Lehmann representation with effective stochastic vectors. The method is easy to implement without altering the $GW$ code, converges rapidly with stochastic parameters, and treats systems of various dimensionality and screening response. Our calculations on a 5.75^∘{} twisted MoS₂ bilayer show how large-scale $GW$ methods include geometry relaxations and electronic correlations on an equal basis in structurally nontrivial materials.