Embedding vertex corrections in GW self-energy: theory, implementation, and outlook

The vertex function ($\Gamma$) within the Green's function formalism encapsulates information about all higher-order electron-electron interaction beyond those mediated by density fluctuations. Herein, we present an efficient approach that embeds vertex corrections in the one-shot $GW$ correlation self-energy for isolated and periodic systems. The vertex-corrected self-energy is constructed through the proposed separation-propagation-recombination procedure: the electronic Hilbert space is separated into an active space and its orthogonal complement denoted as the "rest"; the active component is propagated by a space-specific effective Hamiltonian different from the rest. The vertex corrections are introduced by a rescaled time-dependent non-local exchange interaction. The direct $\Gamma$ correction to the self-energy is further updated by adjusting the rescaling factor in a self-consistent post-processing circle. Our embedding method is tested mainly on donor-acceptor charge-transfer systems. The embedded vertex effects consistently and significantly correct the quasiparticle energies of the gap-edge states. The fundamental gap is generally improved by 1-3 eV upon the one-shot $GW$ approximation. Furthermore, we provide an outlook for applications of (embedded) vertex corrections in calculations of extended solids.