Abstract
It is known that the asymptotic decay (|r|→∞) of the electron density n(r) outside a molecule is informative about its first ionization potential I0. 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 [P. Gori-Giorgi et al., Mol. Phys. 114, 1086 (2016)]. Here we investigate the consequences of such a HOMO nodal plane for the effective potential in the Schrödinger-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 different asymptotic behavior on the nodal plane.
Highlights
IntroductionBoth the (square root of the) density and the KS orbitals ψk(r) obey Schrodinger-type equations,
Both the density and the KS orbitals ψk(r) obey Schrodinger-type equations, − ∇2 +vext(r) veff (r) n(r) = −I0 n(r), (1)vHxc(r) ψk(r) = kψk(r), (2)where the sum of the external and the Hartree-exchangecorrelation potentials constitutes the KS potential, vs =vext + vHxc
We briefly review a few aspects of the asymptotic behavior of the exact density based on the analysis of reference [10]
Summary
Both the (square root of the) density and the KS orbitals ψk(r) obey Schrodinger-type equations,. The decay in the HNP of the second Dyson orbital (and of the density) is not governed by d0√but by d1 with asymptotic behavior according to exp[− 2I1 rp] This is what has been called Case 1 in reference [10]. The same exponential divergence has been observed [10] for the effective potential for the square root of the density for an exact density like the minimal model Such a density with different decay in a particular plane than elsewhere has been called Case 1 [10] (fast decay according to I1 in the plane, slower decay according to I0 everywhere else). These expectations are not borne out if there is a HNP, see below
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