GWΓ approximation for electron self-energies in semiconductors and insulators

The widely used GW approximation for the self-energy operator of a system of interacting electrons may, in principle, be improved using an approximate vertex correction Γ. We estimate Γ using the local-density approximation. We report the results of a comparable series of GW calculations for the band structure of silicon, in which such a vertex correction is (i) excluded entirely, (ii) included only in the screened Coulomb interaction W, and (iii) included in both W and the expression for the self-energy. We also discuss the symmetry properties of the exact vertex correction and how they may be retained in further improvements.