Spin Purity and Spin Mixing in Lanthanide-Cyclooctatetraene Complexes: SO-MCQDPT2 and Axial-Field Analysis

Despite strong relativistic effects in organolanthanide systems, the origin and extent of spin-state mixing within the resulting electronic manifolds remain not fully resolved. In this article, we present a systematic analysis of Ln(COT) (Ln = Ce, Pr, Nd; COT = cyclooctatetraene) by mapping second-order spin-orbit multiconfiguration quasi-degenerate perturbation theory (SO-MCQDPT2)-calculated energies onto the Stevens operator formalism. The approximately axial C8v ligand field imposes symmetry constraints that govern spin-orbit interactions. We demonstrate that direct spin mixing is restricted to states sharing the same J and Ω, whereas indirect mixing arises from ligand-field-induced J-mixing followed by spin-orbit coupling. J is the total electronic angular momentum quantum number associated with the lanthanide ion, and Ω is the unsigned projection of J onto the principal molecular C8 symmetry axis. This analysis accounts for the coexistence of spin-pure and spin-mixed states. By combining multiconfigurational relativistic theory with axial-field analysis, this work provides a framework for understanding spin-state character in organometallic f-block systems. The computed results are in close quantitative agreement with available high-resolution spectroscopic measurements.