Non-isotropic cusp conditions and regularity of the electron density of molecules at the nuclei

TL;DR

This paper studies the regularity and cusp conditions of molecular electron densities near nuclei, deriving explicit representations and proving sharp regularity results, including non-isotropic cusp conditions that generalize classical results.

Contribution

It introduces a new representation of electron density near nuclei, establishing sharp regularity results and generalizing Kato's cusp conditions to non-isotropic cases.

Findings

Representation rho(x)=mu(x)*e^{F(x)} with explicit F

Sharp regularity results: mu in C^{1,1} and C^{2,α} under symmetry

Non-isotropic cusp conditions generalizing Kato's classical result

Abstract

We investigate regularity properties of molecular one-electron densities rho near the nuclei. In particular we derive a representation rho(x)=mu(x)*(e^F(x)) with an explicit function F, only depending on the nuclear charges and the positions of the nuclei, such that mu belongs to C^{1,1}(R^3), i.e., mu has locally essentially bounded second derivatives. An example constructed using Hydrogenic eigenfunctions shows that this regularity result is sharp. For atomic eigenfunctions which are either even or odd with respect to inversion in the origin, we prove that mu is even C^{2,\alpha}(R^3) for all alpha in (0,1). Placing one nucleus at the origin we study rho in polar coordinates x=r*omega and investigate rho'(r,omega) and rho''(r,omega) for fixed omega as r tends to zero. We prove non-isotropic cusp conditions of first and second order, which generalize Kato's classical result.