Influence of Covalency upon Rare-Earth Ligand Field Splittings

Experimental results for the shift with uniaxial stress of the $({}_{}{}^2{}_{}{}^{}F_{\frac{5}{2}}^{}{}_{}{}^{},{\Gamma }_7)\to ({}_{}{}^2{}_{}{}^{}F_{\frac{7}{2}}^{}{}_{}{}^{},{\Gamma }_7)$ laser transition in ${\mathrm{Tm}}^{2+}$:Ca${\mathrm{F}}_2$ and ${\mathrm{Tm}}^{2+}$:Sr${\mathrm{F}}_2$ are presented. The results, 1.75 ${\mathrm{cm}}^{-1\mathrm{}}$ ${(\mathrm{d}\mathrm{y}\mathrm{n}/{\mathrm{cm}}^2)}^{-1\mathrm{}}$ and 1.78 ${\mathrm{cm}}^{-1\mathrm{}}$ ${(\mathrm{d}\mathrm{y}\mathrm{n}/{\mathrm{cm}}^2)}^{-1\mathrm{}}$, are used to calculate the radial dependence of the cubic ligand field splitting. The resulting dependence is somewhat larger than that predicted by the familiar electrostatic model for the splitting. Partially to determine its influence on the above result, we have considered the effect of covalency by means of a semiempirical molecular-orbital model. The overlap of the $4f$ orbitals with the neighboring fluoride ions was calculated using Hartree-Fock wave functions and known internuclear distances. The off-diagonal elements of the interaction Hamiltonian were obtained from the Wolfsberg-Helmholz approximation $H_{\mathrm{ij}}=\frac{2S_{\mathrm{ij}}(H_{\mathrm{ii}}+H_{\mathrm{jj}})}{2}$. A range of reasonable values for the diagonal elements were obtained by analogy with those necessary to explain iron-series splittings. The largest group overlap of the $4f$ wave function with ${\mathrm{F}}^{1-}$ ligands was found to be 3.6% and leads to a sizable (our best estimate in Ca${\mathrm{F}}_2$ is 50%) covalent contribution to the ligand field splitting. We have also investigated some of the consequences of a covalent contribution of this magnitude. The radial dependence of the covalent part of the energy is greater than for the electrostatic part. The resulting radial dependence is thus in better agreement with experiment. Transferred hyperfine effects are calculated and compared to experiment, but the extent of the agreement is hard to ascertain because of uncertainty of the sign of the experimental quantity and polarization effects. The calculated orbital reduction factor for ${\mathrm{Tm}}^{2+}$:Ca${\mathrm{F}}_2$ is found to be much smaller than is observed. We have also calculated the expected variation of the ${\mathrm{}\left(\mathrm{rare}\mathrm{earth}\right)\mathrm{}}^{3+}-F^{1-}$ overlap as a function of atomic number.