  It is known that the asymptotic decay of the electron density outside a
molecule is informative about its first ionization potential. It has recently
become clear that the special circumstance that the Kohn-Sham (KS)
highest-occupied molecular orbital (HOMO) has a nodal plane that extends to
infinity may give rise to different cases for the asymptotic behavior of the
exact density and of the exact KS potential [Mol. Phys. 114 (2016) 1086]. Here
we investigate the consequences of such a HOMO nodal plane for the effective
potential in the Schr\"odinger-like equation for the square root of the
density, showing that for atoms and molecules it will usually diverge
asymptotically on the plane, either exponentially or polynomially, depending on
the coupling between Dyson orbitals. We also analyze the issue in the etxernal
harmonic potential, reporting an example of an exact analytic density for a
fully interacting system that exhibits a different asymptotic behavior on the
nodal plane.
