Vertex corrections in localized and extended systems

Within many-body perturbation theory, we apply vertex corrections to various closed-shell atoms and to jellium, using a local approximation for the vertex consistent with starting the many-body perturbation theory from a Kohn-Sham Green's function constructed from density-functional theory in the local-density approximation. The vertex appears in two places—in the screened Coulomb interaction W and in the self-energy Σ—and we obtain a systematic discrimination of these two effects by turning the vertex in Σ on and off. We also make comparisons to standard GW results within the usual random-phase approximation, which omits the vertex from both. When a vertex is included for closed-shell atoms, both ground-state and excited-state properties demonstrate little improvement over standard GW. For jellium, we observe marked improvement in the quasiparticle bandwidth when the vertex is included only in W, whereas turning on the vertex in Σ leads to an unphysical quasiparticle dispersion and work function. A simple analysis suggests why implementation of the vertex only in W is a valid way to improve quasiparticle energy calculations, while the vertex in Σ is unphysical, and points the way to the development of improved vertices for ab initio electronic structure calculations.