Asymptotic nodal planes in the electron density and the potential in the effective equation for the square root of the density

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.