The d2 and d8 Noncubic Ligand Field Spectrum. I. The Complete Theory of Quadrate, Trigonal, and Cylindrical Ligand Fields

The complete theory of Liehr and Ballhausen for d2 and d8 electronic configurations immersed in cubic fields has been extended to include noncubic ligand fields of quadrate, trigonal, and cylindrical symmetry. The complete set of symmetry adapted eigenvectors for the three symmetries have been derived in various coupling schemes in which the spin-orbit interaction, electron cor-relation, and ligand field in turn are varied from minor to dominant perturbations. The cor-responding energy matrices as a function of the parameters of the ligand field, electron correlation, and spin-orbit constant have been constructed in all the representations. Unitary transformations connecting different formalisms were obtained. The energy matrices have been solved for representative sets of parametric values and energy diagrams have been plotted in all the symmetries as well as in the square planar limit of the quadrate crystalline field. The secular determinants, the eigenfunctions, the energy diagrams, and the unitary transformations presented here are extremely useful in the study of the various aspects of spectroscopic, magnetic, and other properties of appropriate systems. The theory is applicable to quadrately distorted or substituted, trigonally distorted or substituted, octahedral and tetrahedral complexes and to compounds of cylindrical symmetry of d2 and d8 electronic configurations.