  Using a simplified one-dimensional model of a diatomic molecule, the
associated interacting density and corresponding Kohn-Sham potential have been
obtained analytically for all fractional molecule occupancies $N$ between 0 and
2. For the homonuclear case, and in the dissociation limit, the exact Kohn-Sham
potential builds a barrier at the midpoint between the two atoms, whose
strength increases linearly with $N$, with $1 < N \leq 2$. In the heteronuclear
case, the disociating KS potential besides the barrier also exhibits a plateau
around the atom with the higher ionization potential, whose size (but not its
strength) depends on $N$. An anomalous zero-order scaling of the Kohn-Sham
potential with regards to the strength of the electron-electron repulsion is
clearly displayed by our model; without this property both the unusual barrier
and plateau features will be absent.
