Closed-form expression for the magnetic shielding constant of the relativistic hydrogenlike atom in an arbitrary discrete energy eigenstate: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function

TL;DR

This paper derives a closed-form analytical expression for the magnetic shielding constant of a relativistic hydrogen-like atom in any discrete energy state, using Sturmian expansion and hypergeometric functions.

Contribution

It provides the first general closed-form formula for the magnetic shielding constant applicable to any discrete state of a Dirac one-electron atom.

Findings

Derived an elementary closed-form expression for the shielding constant.

Confirmed the formula's consistency with previous specific-state results.

Applied Sturmian expansion and hypergeometric functions in the derivation.

Abstract

We present analytical derivation of the closed-form expression for the dipole magnetic shielding constant of a Dirac one-electron atom being in an arbitrary discrete energy eigenstate. The external magnetic field, by which the atomic state is perturbed, is assumed to be weak, uniform and time independent. With respect to the atomic nucleus we assume that it is pointlike, spinless, motionless and of charge Ze. Calculations are based on the Sturmian expansion of the generalized Dirac- Coulomb Green function [R. Szmytkowski, J. Phys. B 30, 825 (1997); erratum 30, 2747 (1997)], combined with the theory of hypergeometric functions. The final result is of an elementary form and agrees with corresponding formulas obtained earlier by other authors for some particular states of the atom.