Charge density of semiconductors in the $\mathrm{GW}$ approximation

We present a method to calculate the electronic charge density of periodic solids in the $\mathrm{GW}$ approximation, using the space-time approach. We investigate, for the examples of silicon and germanium, to what extent the $\mathrm{GW}$ approximation is charge conserving and how the charge density compares with experimental values. We find that the $\mathrm{GW}$ charge density is close to experiment and charge is practically conserved. We also discuss how using a Hartree potential consistent with the level of approximation affects the quasiparticle energies and find that the common simplification of using the local-density approximation Hartree potential is very well justified.