Electric and magnetic dipole shielding constants for the ground state of the relativistic hydrogen-like atom: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function

TL;DR

This paper derives explicit formulas for electric and magnetic dipole shielding constants in the ground state of a relativistic hydrogen-like atom using the Sturmian expansion of the Dirac-Coulomb Green function, confirming previous results for magnetic shielding.

Contribution

It provides closed-form expressions for the shielding constants using Sturmian expansion, simplifying previous complex series representations.

Findings

Electric shielding constant $\sigma_E$ equals $Z^{-1}$.

Magnetic shielding constant $\sigma_M$ matches earlier analytical results.

Derived formulas are elementary and more straightforward than recent series-based expressions.

Abstract

The Sturmian expansion of the generalized Dirac-Coulomb Green function [R. Szmytkowski, J. Phys. B 30 (1997) 825; erratum 30 (1997) 2747] is exploited to derive closed-form expressions for electric ($\sigma_{\mathrm{E}}$) and magnetic ($\sigma_{\mathrm{M}}$) dipole shielding constants for the ground state of the relativistic hydrogen-like atom with a point-like and spinless nucleus of charge $Ze$. It is found that $\sigma_{\mathrm{E}}=Z^{-1}$ (as it should be) and $$\sigma_{\mathrm{M}}=-(2Z\alpha^{2}/27)(4\gamma_{1}^{3}+6\gamma_{1}^{2}-7\gamma_{1}-12) /[\gamma_{1}(\gamma_{1}+1)(2\gamma_{1}-1)],$$ where $\gamma_{1}=\sqrt{1-(Z\alpha)^{2}}$ ($\alpha$ is the fine-structure constant). This expression for $\sigma_{\mathrm{M}}$ agrees with earlier findings of several other authors, obtained with the use of other analytical techniques, and is elementary compared to an alternative one presented recently by Cheng \emph{et al.} [J. Chem. Phys. 130 (2009) 144102], which involves an infinite series of ratios of the Euler's gamma functions.