Asymptotic Behavior of the Hartree-exchange and Correlation Potentials for Fractional Electron Numbers in Atoms

We report on previously unnoticed features of the exact Hartree-exchange and correlation potentials for atoms with fractional electron numbers. We show that these potentials, when treated separately, can reach non-vanishing asymptotic constant values in the outer region of spherical, spin unpolarized atoms. In the next leading order, the potentials resemble Coulomb potentials created by effective charges which have the peculiarity of not behaving as piecewise constants as a function of the electron number. We provide analytical derivations and complement them with numerical results using the inversion of the Kohn-Sham equations for interacting densities obtained by accurate quantum Monte Carlo calculations. The present results expand on the knowledge of crucial exact properties of
Kohn-Sham systems, which can guide development of advanced exchange-correlation approximations.