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

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.

Nuclear magnetic shielding constants of Dirac one-electron atoms in some low-lying discrete energy eigenstates

“…In Ref. [16] we have proved analytically that the above result is valid for an arbitrary atomic state. However, in the aforementioned article, there are no tables with the numerical values of the relativistic shielding factors of hydrogenic ions.…”

“…In Refs. [16][17][18][19][20] we have considered an arbitrary discrete energy eigenstate of relativistic hydrogenlike atom, in order to obtain analytical expressions (and further -the numerical values) for some properties illustrating the influence of the external electromagnetic perturbations on the atom. In particular, in Ref.…”

“…In particular, in Ref. [16] one can find a detailed derivation of the formula for the magnetic shielding constant of Dirac one-electron atom being in an arbitrary energy eigenstate. The quantum state of the electron is generally characterized by the set of quantum numbers {n, κ, µ}, in which n denotes the radial quantum number, the Dirac quantum number κ is an integer different form zero, whereas µ = −|κ| + 1 2 , −|κ| + 3 2 , .…”

“…The final closed-form expression for the shielding factor for such a state we have arrived at in Ref. [16] reads as follows:…”

Magnetizabilities of relativistic hydrogenlike atoms in some arbitrary discrete energy eigenstates

“…In calculations of such general formula we have used the perturbation theory combined with the Green function technique. To derive the explicit expressions for dia-and paramagnetic terms, we have utilized the expansion of the firstorder generalized Dirac-Coulomb Green function (DCGF) in the Sturmian basis [16]; this form of Green's function was widely used by us in various calculations of electromagnetic properties of the Dirac oneelectron atom [17][18][19][20][21][22] (for other applications of this representation of DCGF, see Refs. [23][24][25][26][27][28]).…”