Fundamental gaps with approximate density functionals: the derivative discontinuity revealed from ensemble considerations

TL;DR

This paper reveals that all exchange-correlation functionals inherently possess a derivative discontinuity, which can be calculated from standard DFT results, significantly improving gap predictions for finite systems.

Contribution

It demonstrates that the derivative discontinuity exists in all functionals and can be derived from ensemble considerations without empiricism, enhancing fundamental gap predictions.

Findings

Adding the derivative discontinuity improves gap predictions in finite systems.

LDA's derivative discontinuity correction vanishes for solids, explaining its limitations.

The derivative discontinuity can be expressed in closed form from standard DFT calculations.

Abstract

The fundamental gap is a central quantity in the electronic structure of matter. Unfortunately, the fundamental gap is not generally equal to the Kohn-Sham gap of density functional theory (DFT), even in principle. The two gaps differ precisely by the derivative discontinuity, namely, an abrupt change in slope of the exchange-correlation (xc) energy as a function of electron number, expected across an integer-electron point. Popular approximate functionals are thought to be devoid of a derivative discontinuity, strongly compromising their performance for prediction of spectroscopic properties. Here we show that, in fact, all exchange-correlation functionals possess a derivative discontinuity, which arises naturally from the application of ensemble considerations within DFT, without any empiricism. This derivative discontinuity can be expressed in closed form using only quantities obtained in the course of a standard DFT calculation of the neutral system. For small, finite systems, addition of this derivative discontinuity indeed results in a greatly improved prediction for the fundamental gap, even when based on the most simple approximate exchange-correlation density functional - the local density approximation (LDA). For solids, the same scheme is exact in principle, but when applied to LDA it results in a vanishing derivative discontinuity correction. This failure is shown to be directly related to the failure of LDA in predicting fundamental gaps from total energy differences in extended systems.