Electronic Structure of Transition Metals. I. Quantum Defects and Model Potential

Recent applications of the concept of quantum defects in setting up model pseudopotentials for simple or transition-metal ions presuppose that the atomic-spectroscopy data of such metals can be expressed in terms of certain quantum defects. Experience indicates, however, that the old quantum-defect idea applies only to group I and II metals at best, but not to metals of higher chemical valence ($z\ge 3$, $z$ being the nominal valence given by the group number, $\mathrm{I}A$ or $B$, $\mathrm{II}A$ or $B$, etc. in the Periodic Table). In this paper, a generalized quantum-defect law applicable to elements having $z\ge 2$ is deduced empirically from an extensive study of the spectroscopic data of the first six rows of the Periodic Table. The empirical law states that the energy levels $E_{nl}$ of a single electron in the field of the positive ions of elements having the same inert-gas core, e.g., the elements of the isoelectric sequence, ${\mathrm{Li}}^{+}$, ${\mathrm{Be}}^{2+}$,...,$F^{7+}$, which are given (in the Heine-Abarenkov model-potential method) by the spectroscopic term value of the ion plus one electron, i.e., by the term values of Li, ${\mathrm{Be}}^{+}$,...,${\mathrm{F}}^{6+}$, by the term values of Li, ${\mathrm{Be}}^{+}$,..., ${\mathrm{F}}^{7+}$, obey the relation $E_{\mathrm{nl}}=-\frac{z^2}{{(n-{\delta }_{\mathrm{nl}})}^2}+{\Delta }_{\mathrm{nl}}$, $z\ge 2$, for the same quantum defects (${\delta }_{\mathrm{nl}},{\Delta }_{\mathrm{nl}}$). The old quantum defect (${\delta }_{\mathrm{nl}}$) and the new quantum defect (${\Delta }_{\mathrm{nl}}$), equivalent to an "atomic core shift") thus represent the deviation of the atomic potential of a given inert-gas configuration from a Coulombic potential due to a nuclear charge $z\left|e\right|\mathrm{}$. On the basis of this empirical law, the parameters of a transition-metal model potential of the Heine-Abarenkov type, adjusted to the energies ${\mathcal{E}}_{\mathrm{nl}}\equiv E_{\mathrm{nl}}-{\Delta }_{\mathrm{nl}}$, have been calculated for all 30 group-$B$ (excepting rare-earth) metals of the Periodic Table; and it has been found that the $l=2\mathrm{}$ model potential parameter $A_2$ reflects the Ziman-Heine-Hubbard resonance model of $s-d$ hybridization through its strong energy dependence of the form ${(\mathcal{E}-{\mathcal{E}}_{3d})}^{-1\mathrm{}}$ for the $3d$ series, and similarly for the $4d$ and $5d$ series. The application of the new model potential to the calculation of the various aspects of the electronic structure of solids will be presented in the next and subsequent papers of this series.