Abstract
The fundamental energy gap of a periodic solid distinguishes insulators from metals and characterizes low-energy single-electron excitations. However, 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 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 metageneralized 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.
Highlights
The fundamental energy gap of a periodic solid distinguishes insulators from metals and characterizes low-energy single-electron excitations
We argue that Eq 4 is valid within typical approximations in generalized KS theory (GKS) theory, as typically implemented, extending the argument of Williams and von Barth [9] from KS to GKS theory
Within xc approximations using the noninteracting density matrix, relaxing the KS demand for a multiplicative effective potential is a “practical” approximation with an unexpected benefit: It yields the interpretation of Eq 4 for the GKS band gap of a solid, explaining how meta-generalized gradient approximations (GGAs) and especially hybrids can improve the estimation of the fundamental energy gap of a solid: For a typical approximate functional, the GKS band gap g is the ground-state energy difference G
Summary
The resulting GKS potential is not a multiplication operator but is in practice continuous (does not change when one delocalized electron is added to or subtracted from a solid) and self-adjoint for differentiable functionals of the noninteracting density matrix It is an integral (Fock) operator [11] for hybrids but a differential operator [30, 31] for metaGGAs, the same operator for occupied and unoccupied oneelectron states. Within xc approximations using the noninteracting density matrix, relaxing the KS demand for a multiplicative effective potential is a “practical” approximation with an unexpected benefit: It yields the interpretation of Eq 4 for the GKS band gap of a solid, explaining how meta-GGAs and especially hybrids can improve the estimation of the fundamental energy gap of a solid: For a typical approximate functional, the GKS band gap g is the ground-state energy difference G. Improvements in G correlate at least roughly with other improvements in ground-state energy differences for integer electron numbers, relevant to atomization energies and lattice constants
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