Third derivative of the one-electron density at the nucleus

TL;DR

This paper proves the existence of the third derivative of the spherically averaged atomic density at the nucleus and establishes an optimal bound relating it to the nuclear charge and density value.

Contribution

It demonstrates the existence of the third derivative of atomic density at the nucleus and provides an optimal bound for eigenfunctions below the essential spectrum.

Findings

Existence of the third derivative of atomic density at the nucleus.

An optimal bound relating the third derivative to nuclear charge and density.

The bound applies to eigenfunctions with eigenvalues below the essential spectrum.

Abstract

We study electron densities of eigenfunctions of atomic Schroedinger operators. We prove the existence of rho~'''(0), the third derivative of the spherically averaged atomic density rho~ at the nucleus. For eigenfunctions with corresponding eigenvalue below the essential spectrum we obtain the bound rho~'''(0) \leq -(7/12)Z^3 rho~(0), where Z denotes the nuclear charge. This bound is optimal.