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

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.

LaTeX, 12 pages

References in corpus (3)

• Recurrence and differential relations for spherical spinors
• Electron shielding of the nuclear magnetic moment in hydrogen-like atom
• Comment on "Four-component relativistic theory for NMR parameters: Unified formulation and numerical assessment of different approaches" [J. Chem. Phys. 130, 144102 (2009)]

Cited by in corpus (8)

• Magnetic field-induced electric quadrupole moment in the ground state of the relativistic hydrogen-like atom: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function
• Magnetizabilities of relativistic hydrogenlike atoms in some arbitrary discrete energy eigenstates
• Magnetizability of the relativistic hydrogenlike atom in an arbitrary discrete energy eigenstate: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function
• Magnetic-field-induced electric quadrupole moments for relativistic hydrogenlike atoms: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function
• 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
• Magnetic-dipole-to-electric-quadrupole cross-susceptibilities for relativistic hydrogenlike atoms in some low-lying discrete energy eigenstates
• 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
• Nuclear magnetic shielding constants of Dirac one-electron atoms in some low-lying discrete energy eigenstates