Gordon decomposition of the magnetizability of a Dirac one-electron atom in an arbitrary discrete energy state
We present a Gordon decomposition of the magnetizability of a Dirac one-electron atom in an arbitrary discrete energy eigenstate, with a pointlike, spinless, and motionless nucleus of charge $Ze$. The external magnetic field, by which the atomic state is perturbed, is assumed to be weak, static, and uniform. Using the Sturmian expansion of the generalized Dirac--Coulomb Green function proposed by Szmytkowski in 1997, we derive a closed-form expressions for the diamagnetic ($χ_{d}$) and paramagnetic ($χ_{p}$) contributions to $χ$. Our calculations are purely analytical; the received formula for $χ_{p}$ contains the generalized hypergeometric functions ${}_3F_2$ of the unit argument, while $χ_{d}$ is of an elementary form. For the atomic ground state, both results reduce to the formulas obtained earlier by other author. This work is a prequel to our recent article, where the numerical values of $χ_{d}$ and $χ_{p}$ for some excited states of selected hydrogenlike ions with $1 \leqslant Z \leqslant 137$ were obtained with the use of the general formulas derived here.