  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.
