Bounds for the second and the third derivatives of the electron density at the nucleus

TL;DR

This paper derives bounds for the second and third derivatives of the spherically averaged electron density at the nucleus, providing new theoretical limits applicable to atomic ground and excited states.

Contribution

It introduces new bounds for the derivatives of electron density at the nucleus, extending previous results and including bounds valid for excited states.

Findings

Established lower bound for second derivative of electron density at nucleus.

Derived upper bound for third derivative of electron density at nucleus.

Provided explicit examples with one-electron atomic ions.

Abstract

Lower bound for ${\bar \rho}''(0)$, the second derivative of the spherically averaged atomic electronic density at the nucleus, and upper bound for ${\bar \rho}'''(0)$, the third derivative, are obtained respectively. It is shown that, for the ground state, ${\bar \rho}''(0) \ge {10/3} Z^2 \rho(0)$ and ${\bar \rho}'''(0) \le -{14/3}Z^3 \rho(0)$ where $Z$ is the charge of the nucleus, and $\rho(0)$ is the electron density at the nucleus. Tighter bounds for ${\bar \rho}''(0)$ and ${\bar \rho}'''(0)$ are also reported which are valid for both the ground state and excited states. Explicit illustration with the example of one-electron atomic ions is given.