Crystal-field parameters in rare-earth compounds: Extended charge contribution

The Coulombic interaction between the extended charge distributions of a rare-earth $4f$ electron and its ligands' closed electronic shells is exactly calculated utilizing the development by Buehler and Hirschfelder. The results are presented in terms of crystal-field parameters acting within the rare-earth orbitals basis set. A comparison is made with respect to point-charge values. The higher the $k$ rank of the $B_q^k$ parameter, the more it is reduced with respect to the point-charge value, the latter being the limit attained for large internuclear distances. For usual nearest-neighbor distances, the $B_0^0$ value is not much changed with respect to the point-charge limit whereas the $B_q^6$ values are greatly reduced and may even display an opposite sign. The calculations are performed with use of analytical radial wave functions given in the literature for nine pairs of rare-earth ligands: ${\mathrm{Nd}}^{3+}$, ${\mathrm{Eu}}^{3+}$, ${\mathrm{Tm}}^{3+}$ ${/\mathrm{C}\mathrm{l}}^{-}$,${\mathrm{O}}^{2-}$, and ${\mathrm{F}}^{-}$ The values obtained are fitted with an analytical function of the distance so as to be utilized in a subsequent LCAO calculation.