  It is known that the asymptotic decay of the electron density $n(\br)$
outside a molecule is informative about its first ionization potential $I_0$,
$n(|\br|\to\infty) \sim \text{exp}(-2\sqrt{2I_0}\,r)$. This dictates the
orbital energy of the highest occupied Kohn-Sham (KS) molecular orbital (HOMO)
to be $\epsilon_H=-I_0$, if the KS potential goes to zero at infinity. However,
when the Kohn-Sham HOMO has a nodal plane, the KS density in that plane will
decay as $\exp{(-2\sqrt{-2\epsilon_{H-1}}\,r)}$. Conflicting proposals exist
for the KS potential: from exact exchange calculations it has been found that
the KS potential approaches a {\em positive} constant in the plane, but from
the assumption of isotropic decay of the exact (interacting) density it has
been concluded this constant needs to be {\em negative}. Here we show that
either 1) the exact density decays differently (according to the second
ionization potential $I_1$) in the HOMO nodal plane than elsewhere, and the KS
potential has a regular asymptotic behavior (going to zero everywhere) provided
that $\epsilon_{H-1}=-I_1$; or 2) the density does decay like
$\text{exp}(-2\sqrt{2I_0}\,r)$ everywhere but the KS potential exhibits
strongly irregular if not divergent behavior around (at) the nodal plane.11
pages, 5 figures
