Kohn-Sham potentials in exact density-functional theory at non-integer electron numbers

TL;DR

This paper investigates Kohn-Sham potentials at non-integer electron numbers using quantum Monte Carlo data, revealing a spatially constant discontinuity and nearly linear orbital energies and kinetic energies, leading to a new approximation method.

Contribution

It introduces a novel approach to accurately determine Kohn-Sham potentials at fractional electron numbers and proposes a simple approximation based on adjacent integer systems.

Findings

Reproduced the theoretically predicted discontinuity in KS potential at integer N.

Found that orbital energies and kinetic energies are nearly piecewise linear in N.

Proposed a simple approximation for KS potential using adjacent integer system data.

Abstract

Within exact electron density-functional theory, we investigate Kohn-Sham (KS) potentials, orbital energies, and non-interacting kinetic energies of the fractional ions of Li, C and F. We use quantum Monte Carlo densities as input, which are then fitted, interpolated at non-integer electron numbers $N$, and inverted to produce accurate KS potentials $v_s^N(r)$. We study the dependence of the KS potential on $N$, and in particular we numerically reproduce the theoretically predicted spatially constant discontinuity of $v_s^N(r)$ as $N$ passes through an integer. We further show that, for all the cases considered, the inner orbital energies and the non-interacting kinetic energy are nearly piecewise linear functions of $N$. This leads us to propose a simple approximation of the KS potential $v_s^N(r)$ at any fractional electron number $N$ which uses only quantities of the systems with the adjacent integer electron numbers.