  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.
