AMELI: Angular Matrix Elements of Lanthanide Ions

Abstract

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. 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 $f^N$ 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 the 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 to classic tables from Judd and Carnall are presented, and application examples are demonstrated using the open-source Python reference implementation YALIP.