The generalization of the extended Stevens operators to higher ranks and spins, and a systematic review of the tables of the tensor operators and their matrix elements

Spherical (S) and tesseral (T) tensor operators (TOs) have been extensively used in, for example, EMR and optical spectroscopy of transition ions. To enable a systematic review of the published tables of the operators and their matrix elements (MEs) we have generated the relevant tables by computer, using Mathematica programs. Our review reveals several misprints/errors in the major sources of TTOs - the conventional Stevens operators (CSOs - components q ≥ 0) and the extended ones (ESOs - all q) for rank k = 2, 4, and 6 - as well as of some STOs with k ≤ 8. The implications of using incorrect operators and/or MEs for the reliability of EMR-related programs and interpretation of experimental data are discussed. Studies of high-spin complexes like Mn₁₂ (S = 10) and Fe₁₉ (S = 33/2) require operator and ME listings up to k = 2S, which are not presently available. Using the algorithms developed recently by Ryabov, the generalized ESOs and their MEs for arbitrary rank k and spin S are generated by computer, using Mathematica. The extended tables enable simulation of the energy levels for arbitrary spin systems and symmetry cases. Tables are provided for the ESOs not available in the literature, with odd k = 3, 5, and 7 for completeness; however, for the newly generalized ESOs with the most useful even rank k = 8, 10, and 12 only, in view of the large listings sizes. General source codes for the generation of the ESO listings and their ME tables are available from the authors.