Fractional charge perspective on the band-gap in density-functional theory

TL;DR

This paper investigates the band-gap prediction issues in density-functional theory (DFT), attributing inaccuracies mainly to fractional charge errors, and proposes improved functionals for better gap estimation.

Contribution

It introduces a fractional charge perspective to understand and improve band-gap predictions in DFT, deriving formulas and demonstrating a functional with enhanced fractional charge behavior.

Findings

Improved functional yields better band-gap predictions.

Fractional charge errors are the main source of inaccuracies.

Eigenvalues can accurately predict the band-gap with suitable functionals.

Abstract

The calculation of the band-gap by density-functional theory (DFT) methods is examined by considering the behavior of the energy as a function of number of electrons. It is found that the incorrect band-gap prediction with most approximate functionals originates mainly from errors in describing systems with fractional charges. Formulas for the energy derivatives with respect to number of electrons are derived which clarify the role of optimized effective potentials in prediction of the band-gap. Calculations with a recent functional that has much improved behavior for fractional charges give a good prediction of the energy gap and also $\epsilon_{{\rm homo}}\simeq-I$ for finite systems. Our results indicate it is possible, within DFT, to have a functional whose eigenvalues or derivatives accurately predict the band-gap.