Reinhard Caspary
Matrix elements of spherical tensor operators are fundamental to analyzing lanthanide spectra in both amorphous and crystalline host materials. This work presents a comprehensive framework for calculating angular matrix elements using a Slater determinant basis and their subsequent transformation to the traditional LS-coupling scheme using the classification introduced by Racah [Phys. Rev. 61, 186-197 (1942); Phys. Rev. 62, 438-462 (1942); Phys. Rev. 63, 367-382 (1943); and Phys. Rev. 76, 1352-1365 (1949)]. While computationally demanding, this direct product-state approach is more universally applicable than conventional methods and remains well within modern desktop computing capabilities. We provide a concise set of general rules to calculate angular matrix elements for virtually any spherical tensor operator within an fN configuration. Because these matrices are mathematical constants independent of the host environment, they need only be calculated once. A comprehensive set of calculated matrix elements for unit and angular momentum operators, alongside perturbation Hamiltonians, is made available in the open-access repository AMELI. By utilizing exact arithmetic, AMELI eliminates the numerical artifacts and rounding errors inherent to conventional floating-point representations. This takes full advantage of the selection rules and symmetry properties of each operator, resulting in a very compact data format due to high sparsity of the matrices and the small number of unique non-zero elements. While the evaluation of final physical observables requires subsequent numerical diagonalization, this foundational repository is intended to replace legacy tables currently used for semi-empirical calculations. Extensive quantitative comparisons with classic tables from Judd and Carnall are presented, and application examples are demonstrated using the open-source Python reference implementation YALIP.