Exact exchange-correlation potentials for calculating the fundamental gap with a fixed number of electrons

TL;DR

This paper demonstrates that the fundamental gap in density functional theory can be obtained without explicit discontinuities in the exchange-correlation potential by using an ensemble approach and analyzing the potential's shifts.

Contribution

The study provides an exact analytical interpretation of the discontinuous shift as a potential shift within the $N$-centered ensemble DFT framework, challenging the necessity of discontinuities in approximate functionals.

Findings

Exact xc potentials for a 1D system were calculated.

Discontinuous shifts can be interpreted as potential shifts in ensemble DFT.

Local functionals' inability to capture discontinuities is less problematic due to potential relocation.

Abstract

Capturing the discontinuous shift by $\Delta$ in the exact exchange-correlation (xc) potential is the standard proposal for calculating the fundamental gap, $E_\mathrm{g}$, from the Kohn-Sham (KS) gap, $\varepsilon_\mathrm{g}$, within KS density functional theory (DFT), as $E_\mathrm{g} = \varepsilon_\mathrm{g} + \Delta$, yet this discontinuity is absent from existing approximations. The '$N$-centered' formulation of ensemble DFT artificially maintains a total electron number, $N$, in order to yield $E_\mathrm{g}$ not through a discontinuous shift in the xc potential but via the ensemble-weight derivative of the xc energy. Within the $N$-centered approach we calculate exact xc potentials for a one-dimensional finite system and show analytically that $\Delta$ can in fact be interpreted as a discontinuous shift in the exact $N$-centered ensemble xc potential, thereby extending to charged excitations an exact property of uncharged excitations. We show that applying the Levy-Zahariev 'shift-in-potential' procedure in this context relocates the discontinuous shift to the unimportant periphery of the system, so that the exact xc potential in effect is free of discontinuities and thus the inability of a local functional to capture discontinuous behavior is inconsequential.