All-electron quasi-particle self-consistent $GW$ band structures for SrTiO$_3$ including lattice polarization corrections in different phase

The electronic band structure of SrTiO$_3$ is investigated in the all-electron QS$GW$ approximation. Unlike previous pseudopotential based QS$GW$ or single-shot $G_0W_0$ calculations, the gap is found to be significantly overestimated compared to experiment. After putting in a correction for the underestimate of the screening by the random phase approximation in terms of a 0.8$Sigma$ approach, the gap is still overestimated. The 0.8$Sigma$ approach is discussed and justified in terms of various recent literature results including electron-hole corrections. Adding a lattice polarization correction (LPC) in the ${bf q}rightarrow0$ limit for the screening of $W$, agreement with experiment is recovered. The LPC is alternatively estimated using a polaron model. We apply our approach to the cubic and tetragonal phases as well as a hypothetical layered post-perovskite structure and find that the LDA (local density approximation) to $GW$ gap correction is almost independent of structure.