Effective Electrostatic Operators for Two Inequivalent Electrons

Parameter analyses of $f^{n}d$ configurations of rare-earth ions generally result in mean errors in excess of 200 ${\mathrm{cm}}^{-1\mathrm{}}$ when conventional Slater operators only are used. We introduce tensor operators whose mathematical structure is similar to those of the Slater angular functions $f_k$ and $g_k$, except that intermediate values of $k$ are allowed. We may therefore assume that, for the $\mathrm{fd}$ and $f^{13}d$ configurations, if we parametrize four new operators $f_1$, $f_3$, $g_2$, and $g_4$, we will get a complete description of the ten $\mathrm{LS}$ terms. The orthogonality properties of the $6-j$ symbols insure that the new operators are independent of the previous ones. We find a reduction of the mean error to less than 10 ${\mathrm{cm}}^{-1\mathrm{}}$ for Pr IV and Ce III, and to about 30-60 ${\mathrm{cm}}^{-1\mathrm{}}$ for La II, Yb III, and Th III. The effects of spin-spin and spin-other-orbit interactions are negligible.