Electrostatically correlated crystal-field interaction through the $B_0^0$ term of ${\mathrm{Gd}}^{3+}$ in Ca${\mathrm{F}}_2$, Sr${\mathrm{F}}_2$, and Ba${\mathrm{F}}_2$

The second-order correction to the crystal-field energy due to configuration interaction via the electron-electron Coulomb energy was calculated in the ${}_{}{}^8{}_{}{}^{}S$, ${}_{}{}^6{}_{}{}^{}(P,I,D)$ states of the $4f^7$ configuration for the $B_0^0$ term. The calculation was made possible by the inclusion of overlap of the ligand orbitals with the $4f$ and $5f$ ${\mathrm{Gd}}^{3+}$ orbitals. Center of gravity (c.g.) shifts were satisfactorily accounted for after subtracting out the shifts due to $J$ mixing. The calculated and observed shifts were found to follow the functional form $\frac{[a-(\frac{b}{R}\left)\right]}{R}$, where $R$ is the anion-cation separation in Ca${\mathrm{F}}_2$, Sr${\mathrm{F}}_2$, and Ba${\mathrm{F}}_2$ and $a$ and $b$ are positive. The calculated shift differences in the cubic site due to this interaction using "free-ion" eigenvectors obtained from La${\mathrm{Cl}}_3$:${\mathrm{Gd}}^{3+}$ spectra due to Judd, Crosswhite, and Crosswhite and Rajnak's $nf$ radial wave functions ranged between —190 and —520 ${\mathrm{cm}}^{-1\mathrm{}}$ for the excited c.g.'s relative to the ${}_{}{}^8{}_{}{}^{}S$ ground level with the observed range being about one-fourth as large. The observed shifts in the trigonal and tetragonal sites were also found to be consistent with this calculation. Shifts of the corresponding c.g.'s in the free-ion host were found to be between —100 and —150 ${\mathrm{cm}}^{-1\mathrm{}}$. The results clearly show the influence of the Madelung energy through configuration interaction on the crystal-field levels. It is proposed that this influence can be accounted for by a single new parameter if adequate radial wave functions are available.