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

TL;DR

The note delivers analytic expressions for the multiplet eigenvalues of all major $f^n$ configurations, correcting longstanding misprints and extending analytic coverage to new $f$-electron cases. Central to the approach is a Coulomb-interaction framework expressed through Gaunt coefficients and Slater integrals, with diagonalization performed in fixed $(L_z,S_z)$ subspaces and eigenvalues recovered via trace powers and Newton identities. Where feasible, closed-form solutions from cubic and quartic polynomials (Viète-based) are provided; in higher-dimensional cases, the work relies on power-sum moments and root-finding, supplemented by a Python tool to generate matrix elements. The comprehensive tables for $p^n$, $d^n$, and especially $f^n$ ($n=2$ to $7$) enable rigorous cross-checks against numerical codes and offer experimentalists direct analytic benchmarks for interpreting x-ray spectroscopic data, including RIXS and XAS measurements.

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.