Static electric multipole susceptibilities of the relativistic hydrogen-like atom in the ground state: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function

TL;DR

This paper derives analytical formulas for the static electric multipole susceptibilities of a relativistic hydrogen-like atom in its ground state using Sturmian expansion, providing exact and approximate results for various multipole responses.

Contribution

It introduces a new analytical approach employing Sturmian expansion to calculate relativistic electric multipole susceptibilities of hydrogen-like atoms.

Findings

Closed-form expressions for far-field susceptibilities involving hypergeometric functions.

Exact numerical values for susceptibilities of selected hydrogenic ions.

Second-order quasi-relativistic approximations for all susceptibilities.

Abstract

The ground state of the Dirac one-electron atom, placed in a weak, static electric field of definite $2^{L}$-polarity, is studied within the framework of the first-order perturbation theory. The Sturmian expansion of the generalized Dirac-Coulomb Green function [R. Szmytkowski, J. Phys. B 30 (1997) 825, erratum: 30 (1997) 2747] is used to derive closed-form analytical expressions for various far-field and near-nucleus static electric multipole susceptibilities of the atom. The far-field multipole susceptibilities --- the polarizabilities $\alpha_{L}$, electric-to-magnetic cross-susceptibilities $\alpha_{\mathrm{E}L\to\mathrm{M}(L\mp1)}$ and electric-to-toroidal-magnetic cross-susceptibilities $\alpha_{\mathrm{E}L\to\mathrm{T}L}$ --- are found to be expressible in terms of one or two non-terminating generalized hypergeometric functions ${}_{3}F_{2}$ with the unit argument. Counterpart formulas for the near-nucleus multipole susceptibilities --- the electric nuclear shielding constants $\sigma_{\mathrm{E}L\to\mathrm{E}L}$, near-nucleus electric-to-magnetic cross-susceptibilities $\sigma_{\mathrm{E}L\to\mathrm{M}(L\mp1)}$ and near-nucleus electric-to-toroidal-magnetic cross-susceptibilities $\sigma_{\mathrm{E}L\to\mathrm{T}L}$ --- involve terminating ${}_{3}F_{2}(1)$ series and for each $L$ may be rewritten in terms of elementary functions. Exact numerical values of the far-field dipole, quadrupole, octupole and hexadecapole susceptibilities are provided for selected hydrogenic ions. Analytical quasi-relativistic approximations, valid to the second order in $\alpha Z$, where $\alpha$ is the fine-structure constant and $Z$ is the nuclear charge number, are derived for all types of the far-field and near-nucleus susceptibilities considered in the paper.