Crystal field parameters with Wannier functions: Application to rare-earth aluminates

A method to calculate the crystal field parameters is proposed and applied to trivalent rare-earth impurities in yttrium aluminate and to Tb${}^{3+}$ ion in TbAlO${}_{3}$. To determine crystal field parameters local Hamiltonian expressed in the basis of Wannier functions is expanded in a series of spherical tensor operators. Wannier functions are obtained by transforming the Bloch functions calculated using the density functional theory based program. The results show that the crystal field is continuously decreasing as the number of $4f$ electrons increases and that the hybridization of $4f$ states with the states of oxygen ligands is important. The method contains a single adjustable parameter characterizing the $4f$-ligand charge transfer. Theory is confronted with experiment for Nd${}^{3+}$ and Er${}^{3+}$ ions in the YAlO${}_{3}$ matrix and for the Tb${}^{3+}$ ion in TbAlO${}_{3}$, and a good agreement within a few meV is found.