Right band gaps for the right reason at low computational cost with a meta-GGA

In density functional theory, traditional explicit density functionals such as the local density approximation and generalized gradient approximations cannot accurately predict the band gap of solids for a fundamental reason: They lack the exchange-correlation derivative discontinuity. By comparing Kohn-Sham and generalized Kohn-Sham calculations, we here show that the nonempirical meta-generalized-gradient-approximation (meta-GGA) TASK from Aschebrock and K\"ummel [Phys. Rev. Res. 1, 033082 (2019)] predicts the right gaps for the right reason, i.e., as a combination of a proper Kohn-Sham gap and a substantial derivative discontinuity contribution. For many materials from small-gap semiconductors to large-gap insulators, the proper band gap is thus obtained. We further study a group of metal-halide perovskites for which the band gap is notoriously hard to predict. For these materials, TASK yields band gaps very similar to the nonlocal screened hybrid Heyd-Scuseria-Ernzerhof functional, yet at a fraction of the hybrid functional's computational cost. We discuss the influence of correlation functionals, and open questions in the comparison of calculated band gaps with experimental ones.