Simple vertex correction improves GW band energies of bulk and two-dimensional crystals

The GW self-energy method has long been recognized as the gold standard for quasiparticle (QP) calculations of solids in spite of the fact that the neglect of vertex corrections and the use of a density-functional theory starting point lack rigorous justification. In this work we remedy this situation by including a simple vertex correction that is consistent with a local-density approximation starting point. We analyze the effect of the self-energy by splitting it into short-range and long-range terms which are shown to govern, respectively, the center and size of the band gap. The vertex mainly improves the short-range correlations and therefore has a small effect on the band gap, while it shifts the band gap center up in energy by around 0.5 eV, in good agreement with experiments. Our analysis also explains how the relative importance of short-and long-range interactions in structures of different dimensionality is reflected in their QP energies. Inclusion of the vertex comes at practically no extra computational cost and even improves the basis set convergence compared to GW. Taken together, the method provides an efficient and rigorous improvement over the GW approximation.