A full ligand-field energy matrix diagonalization treatment for 3d¹ ions in tetragonal symmetry is developed on the basis of the two spin–orbit coupling parameter model, and the contributions of the spin–orbit coupling of the ligand ions to the optical and electron paramagnetic resonance spectra are included. Spin Hamiltonian parameters of the tetragonal V⁴⁺ center in K₂SO₄−Na₂SO₄−ZnSO₄ glass are calculated from the complete energy matrix diagonalization and the perturbation theory methods. The results calculated by both methods are not only close to each other but also in good agreement with the experimental values. Furthermore, the compressed defect structure of the (VO₆)⁸⁻ cluster is discussed. Copyright © 2011 Springer-Verlag .