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 (vertical bar r vertical bar -> infinity) of the electron density n(r) outside a molecule is informative about its fi rst ionization potential I-0. 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 in fi nity may give rise to di ff erent cases for the asymptotic behavior of the exact density and of the exact KS potential [P. Gori-Giorgi et al., Mol. Phys. 114, 1086 (2016)]. Here we investigate the consequences of such a HOMO nodal plane for the e ff ective potential in the Schrodinger-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 external harmonic potential, reporting an example of an exact analytic density for a fully interacting system that exhibits a di ff erent asymptotic behavior on the nodal plane.