The zero-field splitting and local structure for V²⁺ ions in CsMgX₃ $(X = Cl, Br, I)$ crystals are theoretically investigated using complete diagonalization method (CDM) for a $3d^{3}$ ion in trigonal symmetry. Spin–spin (SS) and spin–other-orbit (SOO) interactions are taken into account in addition to the general spin–orbit (SO) interaction. On this basis, using ligand ion displacement model, we find that the ligand ions move away from $C_{3}$-axis, and therefore the local angles in the $V^{2+}$ centres are larger than the angles in the hosts. The results show good agreement with the observed values.

The EPR spectra for VO²⁺ in CaO–Al₂O₃–SiO₂ system are calculated using complete diagonalization method (CDM) and perturbation theory method (PTM). The calculated results are in good agreement with the observed values. By comparing the calculated results by CDM and PTM in a wide range of crystal field parameters, the validity of the PTM formulas has been studied. It is shown that, the PTM formulas are reasonable and reliable in a wide range of CF parameters.