Full versus quasi-particle self consistency in vertex corrected GW approaches

Using seven semiconductors/insulators with band gaps covering the range from 1 eV to 10 eV we systematically explore the performance of two different variants of self-consistency associated with famous Hedin's system of equations: the full self-consistency and the so called quasi-particle approximation to it. The pros and cons of these two variants of self-consistency are sufficiently well documented in literature for the simplest GW approximation to the Hedin's equations. Our study, therefore, aims primarily at the level of theory beyond GW approximation, i.e. at the level of theory which includes vertex corrections. Whereas quasi-particle self-consistency has certain advantages at GW level (well known fact), the situation becomes quite different when vertex corrections are included. In the variant with full self-consistency, vertex corrections (both for polarizability and for self energy) systematically reduce the calculated band gaps making them closer to the experimental values. In the variant with quasi-particle self-consistency, however, an inclusion of the same diagrams has considerably larger effect and calculated band gaps become severely underestimated. Different effect of vertex corrections in two variants of self-consistency can be related to the Z-factor cancellation which plays positive role in quasi-particle self-consistency at GW level of theory but appears to be destructive for the quasi-particle approximation when higher order diagrams are included. Second result of our study is that we were able to reproduce the results obtained with the Questaal code using our FlapwMBPT code when the same variant of self-consistency (quasi-particle) and the same level of vertex corrections (for polarizability only, static approximation for screened interaction, and Tamm-Dancoff approximation for the Bethe-Salpeter equation) are used.