  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.
