Analytic Expressions for Most $f^n$ Valence Multiplet Eigenvalues

TL;DR

This paper derives analytical expressions for eigenvalues of $f^n$ valence multiplet states, correcting previous errors and providing new results to aid experimental and theoretical studies in x-ray spectroscopy of transition metal ions.

Contribution

It presents the first comprehensive analytical solutions for most $f^n$ multiplet eigenstates, correcting past inaccuracies and enhancing tools for spectroscopic analysis.

Findings

Analytical expressions for $f^n$ multiplet eigenvalues are derived.

Previous misprints in literature are corrected.

New results for $f$-electron configurations are provided.

Abstract

It is well know that many full atomic multiplet codes are available for experimentalists to check x-ray absorption or emission spectra against known valence materials to identify effect valence configuration of transition metal ions as well as their ligands. This has grown recently with the continued development of resonant inelastic x-ray scattering as a general tool that can characterize orbital, spin, charge, and lattice excitations in quantum materials. In this note, I show that all multiplet eigenstates for most $f^n$ configurations can be obtained analytically. I correct prior misprints in the literature and present new results for $f$-electrons. These results can serve as checks against new and developed numerical codes, and further provide experimentalists with deeper insights into the many configurations probed by advanced x-ray spectroscopies.