Understanding Band Gaps of Solids in Generalized Kohn-Sham Theory

TL;DR

This paper proves a theorem explaining how generalized Kohn-Sham (GKS) theory can produce more accurate band gaps for solids than traditional Kohn-Sham methods, supported by numerical examples.

Contribution

It introduces a theorem linking GKS band gaps to fundamental gaps under specific conditions, clarifying why GKS functionals yield more realistic results.

Findings

GKS band gaps can match the fundamental gap when the potential is continuous and density change is delocalized.

The theorem explains the improved accuracy of meta-GGAs and hybrid functionals in predicting band gaps.

Numerical examples demonstrate the theorem's applicability to real materials.

Abstract

The fundamental energy gap of a periodic solid distinguishes insulators from metals and characterizes low-energy single-electron excitations. But the gap in the band-structure of the exact multiplicative Kohn-Sham (KS) potential substantially underestimates the fundamental gap, a major limitation of KS density functional theory. Here we give a simple proof of a new theorem: In generalized KS theory (GKS), the band gap of an extended system equals the fundamental gap for the approximate functional if the GKS potential operator is continuous and the density change is delocalized when an electron or hole is added. Our theorem explains how GKS band gaps from meta-generalized gradient approximations (meta-GGAs) and hybrid functionals can be more realistic than those from GGAs or even from the exact KS potential. The theorem also follows from earlier work. The band edges in the GKS one-electron spectrum are also related to measurable energies. A linear chain of hydrogen molecules, solid aluminum arsenide, and solid argon provide numerical illustrations.