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Electronic structure of Na, K, Si, and LiF from self-consistent solution of Hedin's equations including vertex corrections

A few self-consistent schemes to solve the Hedin equations are presented. They include vertex corrections of different complexity. Commonly used quasiparticle approximation for the Green function and static approximation for the screened interaction are avoided altogether. Using alkali metals Na and K as well as semiconductor Si and wide gap insulator LiF as examples, it is shown that both the vertex corrections in the polarizability P and in the self energy $Σ$ are important. Particularly, vertex corrections in $Σ$ with proper treatment of frequency dependence of the screened interaction always reduce calculated band widths/gaps, improving the agreement with experiment. The complexity of the vertex included in P and in $Σ$ can be different. Whereas in the case of polarizability one generally has to solve the Bethe-Salpeter equation for the corresponding vertex function, it is enough (for the materials in this study) to include the vertex of the first order in the self energy. The calculations with appropriate vertices show remarkable improvement in the calculated band widths and band gaps as compared to the self-consistent GW approximation as well as to the self-consistent quasiparticle GW approximation.