First-principles calculations of the amplitudes of the electron transition from a ligand to the $5d$ shell of Yb$^{3+}$: KZnF$_3$

The expressions describing the amplitudes of a transition of an electron from a ligand to vacant shells of the central ion of an impurity center are derived. The amplitudes of a transition of an electron from a ligand to the $5d$ shell of a rare-earth ion are calculated from first principles. The calculations are performed in the basis set of orbitals taken in the form of $5s$, $5p$, $4f$, $5d$, and $6s$ electron shells of the central ion and $2s$ and $2p$ electron shells of the ligands. No expansion in terms of overlap integrals is employed in the calculations. The matrix elements of the $(I+S)^{-1}$ matrix are determined in the chosen basis set of orbitals. The amplitudes thus calculated are in good agreement with those obtained by fitting of the experimental data.