Nonlinear and Linear Shielding of Rare-Earth Crystal-Field Interactions

It was shown previously that significant nonlinear shielding of the rare-earth electrostatic crystalline-field interactions may occur, and hence that the standard parametrization scheme used to determine the crystalfield parameters, ${V_L}^M={A_L}^M〈r^L〉$, was questionable. These nonlinear effects arise from the same distortions of the closed shells in rare-earth ions which produce the (larger) linear shielding of the crystal field $V_{\mathrm{c}}$. In this paper, both nonlinear and linear shielding effects have been investigated for the rare-earth ions ${\mathrm{Ce}}^{3+}$, and ${\mathrm{Tb}}^{3+}$ through ${\mathrm{Yb}}^{3+}$ inclusive. The method of single substituted configurations is employed and is shown to avoid the difficulties associated with the traditional single-determinant perturbation-theory approach previously used exclusively by various authors. Results based on both methods are given and compared; significant differences in linear-shielding predictions are found to be introduced by the inexact traditional approach. Calculations for the nonlinear $5p\to 4f$ effects have been done, and results displayed, as a function of the strength of $V_{\mathrm{c}}$ (based on the point-charge model). The relevance of these effects to experiment is examined qualitatively.