Self-screening corrections beyond the random-phase approximation: Applications to band gaps of semiconductors

The self-screening error in the random-phase approximation (RPA) and the $GW$ approximation (GWA) is a well-known issue and has received attention in recent years with several methods for a correction being proposed. We here apply two of these, a self-screening and a so-called "self-polarization" correction scheme, to model calculations to examine their applicability. We also apply an explicit self-screening correction to \textit{ab-initio} calculations of real materials. We find indications for the self-polarization scheme to be the more appropriate choice of correction for localized states, and additionally we observe that it suffers from causality violations in the strongly correlated regime. The self-screening correction used in this work on the other hand significantly improves the description in more delocalized states. It provides a notable reduction in the remaining GWA error when calculating the band gaps of several semiconductors, indicating a physical explanation for a part of the remaining discrepancy in one-shot $GW$ compared to experiment, while leaving the localized semicore $d$ states mostly unaffected.