The Calculation for the Splitting Chain of Crystal-field Energy Levels of Alexandrite with Two Crystal-field Parameter Fitting Method

For the Cr³⁺ ion being at the site of C_s symmetry in crystal BeAl₂O₄, We have determined its approximate crystal field with symmetry of O_h and C_3v group, respectively. Using two crystal-field parameter fitting method we have calculated the energy levels and the wave functions of Cr³⁺ ion (in BeAl₂O₄) in three different symmetric crystal field, and obtain the energy level splitting chain corresponding to the crystal field group chain O_h⊃C_3v⊃C_s. The results can be used to discuss the problems such as d-d transition intensities due to static low-symmetry crystal-field and electron-phonon interactions.
We found that the stop from O_h to C_3v Symmetry in the chain is relatively small but the step from C_3v to C_s is very important. So in the calculation of d-d transition intensities due to static low-symmetry crystal field, we must treat C_s-symmetry crystal field, and when we calculate optical intensities caused by electron-phonon interaction, only O_h symmetry is enough.