Study of the discontinuity of the exchange-correlation potential in an exactly soluble case

TL;DR

This paper analytically investigates the discontinuity of the exact exchange-correlation potential in a two-electron system with harmonic confinement, providing explicit results without approximations.

Contribution

It extends previous analysis by deriving exact exchange and correlation potentials for a harmonic oscillator model with fluctuating particle number.

Findings

Exact exchange and correlation potentials obtained analytically.

Discontinuity in the potential observed at particle number crossing one.

Separate discussion of exchange and correlation effects.

Abstract

It was found by Perdew, Parr, Levy, and Balduz [Phys. Rev. Lett. {\bf 49}, 1691 (1982)] and by Sham and Schl\"uter [Phys. Rev. Lett. {\bf 51}, 1884 (1983)] that the exact Kohn-Sham exchange-correlation potential of an open system may jump discontinuosly as the particle number crosses an integer, with important physical consequences. Recently, Sagvolden and Perdew [Phys. Rev. A {\bf 77}, 012517 (2008)] have analyzed the discontinuity of the exchange-correlation potential as the particle number crosses one, with an illustration that uses a model density for the H$^-$ ion. In this work, we extend their analysis to the case in which the external potential is the simple harmonic confinement, choosing spring-constant values for which the two-electron hamiltonian has an analytic solution. This way, we can obtain the exact, analytic, exchange and correlation potentials for particle number fluctuating between zero and two, illustrating the discontinuity as the particle number crosses one without introducing any model or approximation. We also discuss exchange and correlation separately.