General expressions for Stevens and Racah operator equivalents

Abstract Several definitions of the crystal-field (CF) Hamiltonian have been used over time and their variety has lead to many misunderstandings, in both theoretical and experimental literature. Two categories of definitions can be mentioned, the first being in terms of the operator equivalents introduced by Stevens in 1952 and the second being in terms of Racah’s spherical tensors, introduced by different authors from 1962. This paper aims at providing some clarification in this field. We first introduce various forms of the CF Hamiltonian. Then, we introduce several explicit ways to compute Racah spherical tensors, in terms of angular momentum quantum numbers j and m as well as in terms of J 2 and J z operators. We eventually give some connections between the parameters used to expand the CF Hamiltonian on Stevens operator equivalents and on spherical tensors. Lastly, we make usage of the coefficients of fractional parentage for the expression of the CF Hamiltonian on spherical tensors for the many-body states. Various computational codes, using different CF conventions, are also reviewed.