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

The calculation of the band gap by density-functional theory (DFT) is examined by considering the behavior of the energy as a function of number of electrons. It is explained 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 ${\epsilon }_{\mathrm{HOMO}}\simeq -I$ for finite systems. Our results indicate that it is possible, within DFT, to have a functional whose eigenvalues or derivatives accurately predict the band gap.

Figure 1

(Color online) Difference energy of the carbon atom $[\Delta E=E(N_0+\delta N)-E\left(N_0\right)]$ against number of electrons $(N_0+\delta N,N_0=6)$ with several different functionals using OEP and GKS. Dotted line follows the initial slope for the nonstraight functionals. The inset shows $6<N<7$ range in more detail.

Figure 2

(Color online) Comparison of MCY3 eigenvalues from GKS and OEP calculations and also including ${\Delta }_{\mathit{xc}}^f$ for the carbon atom.