Derivative discontinuity with localized Hartree-Fock potential

TL;DR

This paper extends the localized Hartree-Fock potential to fractional particle numbers, demonstrating it reproduces the necessary derivative discontinuities with computational efficiency, making it valuable for applications involving the fundamental gap.

Contribution

The paper introduces a fractional particle number extension of the localized Hartree-Fock potential that accurately captures derivative discontinuities while maintaining computational simplicity.

Findings

Reproduces derivative discontinuities close to Hartree-Fock results

Maintains a 'direct-energy' property linking energy to eigenvalues

Satisfies a specific condition for spin-component discontinuities

Abstract

The localized Hartree-Fock potential has proven to be a computationally efficient alternative to the optimized effective potential, preserving the numerical accuracy of the latter and respecting the exact properties of being self-interaction free and having the correct $-1/r$ asymptotics. In this paper we extend the localized Hartree-Fock potential to fractional particle numbers and observe that it yields derivative discontinuities in the energy as required by the exact theory. The discontinuities are numerically close to those of the computationally more demanding Hartree-Fock method. Our potential enjoys a "direct-energy" property, whereby the energy of the system is given by the sum of the single-particle eigenvalues multiplied by the corresponding occupation numbers. The discontinuities $c_\uparrow$ and $c_\downarrow$ of the spin-components of the potential at integer particle numbers $N_\uparrow$ and $N_\downarrow$ satisfy the condition $c_\uparrow N_\uparrow+c_\downarrow N_\downarrow=0$. Thus, joining the family of effective potentials which support a derivative discontinuity, but being considerably easier to implement, the localized Hartree-Fock potential becomes a powerful tool in the broad area of applications in which the fundamental gap is an issue.