Spin-dependent crystal field on f-electrons as a consequence of spin–orbit coupling renormalization due to exchange-covalent bonds with ligands

The problem of renormalization of the spin–orbit interaction operator of electrons of partially filled nf-shells due to exchange-covalent bonds with surrounding ligands has been solved. It is found that along with the change of the standard spin–orbit interaction parameter, new energy operators are generated, which can be interpreted as a spin-dependent crystal field operator. Simple formulas are obtained that allows to calculate its parameters via covalence parameters and overlap integrals. Numerical evaluations have been performed for multiplets \(Tb^{3 + } (f^{8} ){}^{7}F_{6}\) $T{b}^{3+}(f^8)_{}^7{F}_6$ and \(Er^{3 + } (f^{11} ){}^{4}I_{15/2}\) $E{r}^{3+}(f^{11})_{}^4{I}_{15/2}$ in fluorides with cubic symmetry. Calculated parameters were found to be of the order of 10% relative to the standard crystal field parameters.