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

TL;DR

This paper demonstrates that the piecewise linearity of total energy versus electron number in DFT can be restored for approximate functionals through ensemble generalization, improving predictions of frontier orbital energies without altering functional forms.

Contribution

It introduces a method to recover piecewise linearity and derivative discontinuity in approximate density functionals via ensemble generalization, enhancing the accuracy of ionization potential predictions.

Findings

Restores piecewise linearity in approximate DFT functionals.

Introduces derivative discontinuity to approximate functionals.

Enables calculation of ionization potentials without explicit electron removal.

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.