All-electron projector-augmented-wave $\mathrm{GW}$ approximation: Application to the electronic properties of semiconductors

The so-called $\mathrm{GW}$ approximation (GWA) based on an all-electron full-potential projector-augmented-wave method (PAW) has been implemented. For the screening of the Coulomb interaction W three different plasmon-pole model dielectric function models have been tested, and it is shown that the accuracy of the quasiparticle energies is not sensitive to the details of these models. For the decoupling of the valence and core electrons two different schemes produced quasiparticle energies that differ on average by less than 0.1 eV for Si. This method has been used to study the quasiparticle band structure of some small, medium, and wide-band-gap semiconductors: Si, GaAs, AlAs, InP, ${\mathrm{Mg}}_2\mathrm{Si},$ diamond, and the insulator LiCl. Special attention was devoted to the convergence of the self-energy with respect to both the $\mathbf{k}$ points in the Brillouin zone and to the number of reciprocal-space $\mathbf{G}$ vectors. The most important and surprising result is that although the all-electron GWA improves considerably the local-density approximation electronic structure of semiconductors, it does not always provide the correct energy band gaps for small- and medium-band-gap semiconductors as originally inferred from pseudopotential GWA calculations. The discrepancy between the all-electron and pseudopotential quasiparticle band gaps is mainly traced back to differences between the exchange-correlation matrix elements obtained by the two methods.