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

摘要

The ground state of the Dirac one-electron atom, placed in a weak, staticelectric field of definite $2^{L}$-polarity, is studied within the framework ofthe first-order perturbation theory. The Sturmian expansion of the generalizedDirac-Coulomb Green function [R. Szmytkowski, J. Phys. B 30 (1997) 825,erratum: 30 (1997) 2747] is used to derive closed-form analytical expressionsfor various far-field and near-nucleus static electric multipolesusceptibilities of the atom. The far-field multipole susceptibilities --- thepolarizabilities $\alpha_{L}$, electric-to-magnetic cross-susceptibilities$\alpha_{\mathrm{E}L\to\mathrm{M}(L\mp1)}$ and electric-to-toroidal-magneticcross-susceptibilities $\alpha_{\mathrm{E}L\to\mathrm{T}L}$ --- are found to beexpressible in terms of one or two non-terminating generalized hypergeometricfunctions ${}_{3}F_{2}$ with the unit argument. Counterpart formulas for thenear-nucleus multipole susceptibilities --- the electric nuclear shieldingconstants $\sigma_{\mathrm{E}L\to\mathrm{E}L}$, near-nucleuselectric-to-magnetic cross-susceptibilities$\sigma_{\mathrm{E}L\to\mathrm{M}(L\mp1)}$ and near-nucleuselectric-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 andhexadecapole 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 thenuclear charge number, are derived for all types of the far-field andnear-nucleus susceptibilities considered in the paper.