A Four-Component Relativistic Perspective on the E ⊗ e Jahn–Teller Model

Most advances in electronic spin-dependent non-adiabatic dynamics focus on refining the underlying dynamics methods. In contrast, this work considers an improved description of spin-orbit coupling by explicitly accounting for its relativistic origins. To this end, we extend the standard one-electron triatomic Jahn–Teller model to the four-component relativistic domain. In our formulation, the electron is treated using the Dirac–Coulomb Hamiltonian, while the nuclei remain non-relativistic, departing from the conventional for- mulation that incorporates the Pauli–Breit spin-orbit coupling as a perturbation. The most striking difference between our relativistic model and the conventional one is the presence of vibronic coupling terms on the anti-diagonal of the diabatic potential matrix. These terms scale as 1/c2, and although nominally small, they can be amplified near avoided- crossings or in systems with heavy nuclei. The presence of these couplings implies that nuclear motion can affect the direction of the electron spin, a feature entirely absent in the conventional formulation. In the adiabatic representation, these additional couplings trans- late to non-Abelian characteristics of the non-adiabatic coupling matrix that persist even in the single-surface limit. Thus, our relativistic formulation reveals a more intricate interplay between the electronic spin and nuclear degrees of freedom. These model findings suggest that more spin-dependent effects may yet be uncovered through improved treatments of spin-orbit coupling.