Self-consistent solution of Hedin's equations: semiconductors/insulators

The band gaps of a few selected semiconductors/insulators are obtained from the self-consistent solution of the Hedin's equations. Two different schemes to include the vertex corrections are studied: (i) the vertex function of the first-order (in the screened interaction $W$) is applied in both the polarizability $P$ and the self-energy $\Sigma$, and (ii) the vertex function obtained from the Bethe-Salpeter equation is used in $P$ whereas the vertex of the first-order is used in $\Sigma$. Both schemes show considerable improvement in the accuracy of the calculated band gaps as compared to the self-consistent $GW$ approach (sc$GW$) and to the self-consistent quasi-particle $GW$ approach (QS$GW$). To further distinguish between the performances of two vertex-corrected schemes one has to properly take into account the effect of the electron-phonon interaction on the calculated band gaps which appears to be of the same magnitude as the difference between schemes i) and ii).