Accurate GW₀ band gaps and their phonon-induced renormalization in solids

Recent years, huge progress of first-principles methods has been witnessed in calculating the quasiparticle band gaps, with many-body perturbation theory in the GW approximation being the standard choice, where G refers to Green's function and W denotes the dynamically screened Coulomb interaction. Numerically, the completeness of the basis set has been extensively discussed, but in practice far from carefully addressed. Beyond the static description of the nuclei, the electron-phonon interactions (EPIs) are ubiquitous, which cause zero-point renormalization (ZPR) of the band gaps. Therefore, to obtain high quality band gaps, one needs both accurate quasiparticle energies and accurate treatments of EPIs. In this article, we review methods on this. The completeness of the basis set is analyzed in the framework of linearized augmented plane waves, by adding high-energy local orbitals (HLOs). The electron-phonon matrix elements and self-energy are discussed, followed by the temperature dependence of the band gaps in both perturbative and non-perturbative methods. Applications of such an analysis on bulk wurtzite BeO and monolayer honeycomb BeO are given. Adding HLOs widens their GW₀ band gaps by ～ 0.4 eV while ZPR narrows them by similar amount. These influences cancel each other, which explains the fortuitous agreement between experiment and theory when the basis set is incomplete and the EPIs are absent. The phonon-induced renormalization, a term often neglected in calculations of the band gaps, is also emphasized by its large magnitude.