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 DFT-LDA Green's function. The vertex appears in two places -- in the screened Coulomb interaction, W, and in the self-energy, \Sigma -- and we obtain a systematic discrimination of these two effects by turning the vertex in \Sigma on and off. We also make comparisons to standard GW results within the usual random-phase approximation (RPA), which omits the vertex from both. When a vertex is included for closed-shell atoms, both ground-state and excited-state properties demonstrate only limited improvements over standard GW. For jellium we observe marked improvement in the quasiparticle band width when the vertex is included only in W, whereas turning on the vertex in \Sigma 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 \Sigma is unphysical, and points the way to development of improved vertices for ab initio electronic structure calculations.