Performance of Different Spin–Orbit Operators for the Splitting of 4f-Multiplets in Trivalent Lanthanides

The spin-orbit (SO) coupling is a key topic in lanthanide chemistry and physics since it is directly relevant to essential physicochemical properties of rare-earth materials, especially in spectroscopic, optical and magnetic domains. In this work, we systematically study different SO operators of relativistic Hamiltonians within both all-electron and pseudopotential (i.e., effective core potential) frameworks, and their impact on the microstate energy levels of the 4f-multiplets of trivalent lanthanide ions (Ln³⁺, Ln = Ce-Yb). It is found that some SO operators adapted for the relativistic all-electron Hamiltonian can also replicate the experimental SO splitting of 4f-multiplets for Ln³⁺ with fairly reasonable accuracy when combined with the small-core pseudopotential method, such as Breit-Pauli and screened-nuclear SO operators. The feasibility of the perturbative SO coupling scheme for lanthanides is further discussed. We anticipate that these SO coupling methodologies will be applicable to extended lanthanide-doped material systems, enabling comprehensive investigations of their spectroscopic, optical, and magnetic properties.