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 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.
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The fact that Eq. (2) includes contributions only from the N0- and the N0 + 1-states relies on the conjecture that the series E(N0) for
$N_0 \in \mathbb {N}$
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We draw special attention to the assumption underlying the derivation leading to Eq. (19): the states with N0, N0 − 1, and N0 + 1 electrons have to be pure, i.e., non-degenerate, states. Note that some degeneracies can be trivially removed: Degeneracy between energy levels in the two spin channels is removed by introducing an infinitesimal magnetic field; degeneracy between symmetric k-points in a periodic crystal is removed by an infinitesimal spatial distortion of the crystal structure. However, non-trivial degeneracies, e.g., a simultaneous fractional occupation of d- and s-bands, requires a non-infinitesimal perturbation for their removal. For these cases, the ensemble approach should be generalized to include more than two components in Eqs. (2) and (8), which may affect the expression obtained for Δ. Therefore, for cases of, e.g., metals and semi-metals a more general treatment is needed.
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