Band-gap energy in the random-phase approximation to density-functional theory

We calculate the interacting bandgap energy of a solid within the random-phase approximation (RPA) to density functional theory (DFT). The interacting bandgap energy is defined as $E_g=E^{\mathrm{RPA}}(N+1)+E^{\mathrm{RPA}}(N-1)-2E^{\mathrm{RPA}}\left(N\right)$, where $E^{\mathrm{RPA}}\left(N\right)$ is the total DFT-RPA energy of the $N$-electron system. We compare the interacting bandgap energy with the Kohn-Sham bandgap energy, which is the difference between the conduction and valence band edges in the Kohn-Sham band structure. We show that they differ by an unrenormalized “$G_0{W}_0$” self-energy correction (i.e., a $GW$ self-energy correction computed using Kohn-Sham orbitals and energies as input). This provides a well-defined and meaningful interpretation to $G_0{W}_0$ quasiparticle bandgap calculations, but questions the physics behind the renormalization factors in the expression of the bandgap energy. We also separate the kinetic from the Coulomb contributions to the DFT-RPA bandgap energy, and discuss the related problem of the derivative discontinuity in the DFT-RPA functional. Last we discuss the applicability of our results to other functionals based on many-body perturbation theory.