Density-functional energy gaps of solids demystified

The fundamental energy gap of a solid is a ground-state second energy difference. Can one find the fundamental gap from the gap in the band structure of Kohn–Sham density functional theory? An argument of Williams and von Barth (WB), 1983, suggests that one can. In fact, self-consistent band-structure calculations within the local density approximation or the generalized gradient approximation (GGA) yield the fundamental gap within the same approximation for the energy. Such a calculation with the exact density functional would yield a band gap that also underestimates the fundamental gap, because the exact Kohn–Sham potential in a solid jumps up by an additive constant when one electron is added, and the WB argument does not take this effect into account. The WB argument has been extended recently to generalized Kohn–Sham theory, the simplest way to implement meta-GGAs and hybrid functionals self-consistently, with an exchange–correlation potential that is a non-multiplication operator. Since this operator is continuous, the band gap is again the fundamental gap within the same approximation, but, because the approximations are more realistic, so is the band gap. What approximations might be even more realistic?