Piecewise Linearity of Approximate Density Functionals Revisited: Implications for Frontier Orbital Energies

Abstract

In the exact Kohn-Sham density-functional theory (DFT), the total energy versus the number of electrons is a series of linear segments between integer points. However, commonly used approximate density functionals produce total energies that do not exhibit this piecewise-linear behavior. As a result, the ionization potential theorem, equating the highest occupied eigenvalue with the ionization potential, is grossly disobeyed. Here, we show that, contrary to conventional wisdom, most of the required piecewise-linearity of an arbitrary approximate density functional can be restored by careful consideration of the ensemble generalization of DFT. Furthermore, the resulting formulation introduces the desired derivative discontinuity to any approximate exchange-correlation functional, even one that is explicitly density-dependent. This opens the door to calculations of the ionization potential and electron affinity even without explicit electron removal or addition. All these advances are achieved while neither introducing empiricism nor changing the underlying functional form. The power of the approach is demonstrated on benchmark systems using the local density approximation as an illustrative example.

Minor correction to Eqs. (10) and (III.7), information added to the Supplementary Material, typos corrected

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• Polaronic hole localization and multiple hole binding of acceptors in oxide wide-gap semiconductors
• Koopmans' condition for density-functional theory
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• A self-interaction-free local hybrid functional: Accurate binding energies vis-à-vis accurate ionization potentials from Kohn-Sham eigenvalues
• Kohn-Sham potentials in exact density-functional theory at non-integer electron numbers
• Exact ensemble density functional theory for excited states in a model system: investigating the weight dependence of the correlation energy
• Theoretical unification of hybrid-DFT and DFT+U methods for the treatment of localized orbitals
• Deviations from piecewise linearity in the solid-state limit with approximate density functionals
• Transport through correlated systems with density functional theory
• Frozen density embedding with non-integer subsystems' particle numbers