Why Do Hybrid Density Functionals and Meta-GGAs Improve the Band Gaps of Solids in Generalized Kohn-Sham Theory?*

The fundamental gap of a solid is not only an excitation energy but also a ground-state second energy difference (ionization energy minus electron affinity). For an approximate functional constructed from a non-interacting density matrix, the gap between the occupied and unoccupied one-electron energies, computed within a generalized Kohn-Sham (GKS) scheme in which the energy-minimizing exchange-correlation potential is continuous and not constrained to be a multiplication operator, has been shown [1] to equal the ground-state second energy difference within the same approximation, so long as the created electron and hole delocalize fully over one, two, or three dimensions. Thus improvements of the ground-state second energy difference from local density and generalized gradient approximations (GGAs) to meta-GGAs and then to hybrids explain the correspondingly improved band-structure gaps, which are found only in GKS and not [2] in KS theory.
[1] J.P. Perdew, W. Yang, K. Burke, Z. Yang, E.K.U. Gross, M. Scheffler, G.E. Scuseria, T.M. Henderson, I.Y. Zhang, A. Ruzsinszky, H. Peng, J. Sun, E. Trushin, and A. Goerling, Proc. Nat. Acad. Sci. (USA) 114, 2801 (2017).
[2] Z. Yang et al., Phys. Rev. B 93, 205205 (2016).