Curvature and Frontier
Orbital Energies in Density
Functional Theory
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Abstract
Perdew et al. discovered two different properties of
exact Kohn–Sham
density functional theory (DFT): (i) The exact total energy versus
particle number is a series of linear segments between integer electron
points. (ii) Across an integer number of electrons, the exchange-correlation
potential “jumps” by a constant, known as the derivative
discontinuity (DD). Here we show analytically that in both the original
and the generalized Kohn–Sham formulation of DFT the two properties
are two sides of the same coin. The absence of a DD dictates deviation
from piecewise linearity, but the latter, appearing as curvature,
can be used to correct for the former, thereby restoring the physical
meaning of orbital energies. A simple correction scheme for any semilocal
and hybrid functional, even Hartree–Fock theory, is shown to
be effective on a set of small molecules, suggesting a practical correction
for the infamous DFT gap problem. We show that optimally tuned range-separated
hybrid functionals can inherently minimize <i>both</i> DD
and curvature, thus requiring no correction, and that this can be
used as a sound theoretical basis for novel tuning strategies