Microscopic spin Hamiltonian approaches for 3d⁸ and 3d² ions in a trigonal crystal field - perturbation theory methods versus complete diagonalization methods

In this paper, we critically review the existing microscopic spin Hamiltonian (MSH) approaches, namely the complete diagonalization method (CDM) and the perturbation theory method (PTM), for 3d⁸(3d²) ions in a trigonal (C_3v, D₃, D_3d) symmetry crystal field (CF). A new CDM is presented and a CFA/MSH computer package based on our crystal-field analysis (CFA) package for 3d^N ions is developed for numerical calculations. Our method takes into account the contribution to the SH parameters (D, g_∥ and g_⊥) from all 45 CF states for 3d⁸(3d²) ions and is based on the complete diagonalization of the Hamiltonian including the electrostatic interactions, the CF terms (in the intermediate CF scheme) and the spin-orbit coupling. The CFA/MSH package enables us to study not only the CF energy levels and wavefunctions but also the SH parameters as functions of the CF parameters (B₂₀, B₄₀ and B₄₃ or alternatively Dq, v and v') for 3d⁸(3d²) ions in trigonal symmetry. Extensive comparative studies of other MSH approaches are carried out using the CFA/MSH package. First, we check the accuracy of the approximate PTM based on the `quasi-fourth-order' perturbation formulae developed by Petrosyan and Mirzakhanyan (PM). The present investigations indicate that the PM formulae for the g-factors (g_∥ and g_⊥) indeed work well, especially for the cases of small v and v' and large Dq, whereas the PM formula for the zero-field splitting (ZFS) exhibits serious shortcomings. Earlier criticism of the PM approach by Zhou et al (Zhou K W, Zhao S B, Wu P F and Xie J K 1990 Phys. Status Solidi b 162 193) is then revisited. Second, we carry out an extensive comparison of the results of the present CFA/MSH package and those of other CDMs based on the strong- and weak-CF schemes. The CF energy levels and the SH parameters for 3d² and 3d⁸ ions at C_3v symmetry sites in several crystals are calculated and analysed. Our investigations reveal serious inconsistencies in the CDM results of Zhou et al and Li (Li Y 1995 J. Phys.: Condens. Matter 7 4075) based on the strong-CF scheme for Ni²⁺ ions in LiNbO₃ crystals. The correctness of our CFA/MSH package is verified by comparing our results with the predictions of Ma et al (Ma D P, Ma N, Ma X D and Zhang H M 1998 J. Phys. Chem. Solids 59 1211, Ma D P, Ma X D, Chen J R and Liu Y Y 1997 Phys. Rev. B 56 1780) and Macfarlane (Macfarlane R M 1964 J. Chem. Phys. 40 373) for α-Al₂O₃ : V³⁺(3d²) and MgO : Ni²⁺(3d⁸). It appears that the two independent approaches show perfect agreement with our approach, unlike those of Zhou et al and Li, which turn out to be unreliable. Our results reveal that the contributions to the ZFS parameter from the higher excited states cannot be neglected; also, the ZFS parameter is very sensitive to slight changes of the crystal structure. Hence our CFA/MSH package, which takes into account the contributions to the ZFS parameter from the higher excited states, can provide reliable results and proves to be a useful tool for the studies of the effect of the lattice distortions, defects and structural disorder on the spectroscopic properties of 3d² and 3d⁸ ions at trigonal symmetry sites in crystals.