The direct role of nuclear motion in spin-orbit coupling in strongly correlated spin systems

The interaction between the magnetic moment of an electron and the magnetic field generated by a moving charge is one component of the spin-orbit interaction. The nuclei in a molecule or solid are charged, are generally in vibrational motion, and so contribute to this interaction, but the direct coupling between nuclear momentum and electron spin is normally ignored in discussions of spin-forbidden phenomena such as transitions between states of different spin, even when the nuclei are recognized as playing a fundamental role (spin-vibronic coupling). Here, we investigate the spin-orbit interaction in a Heisenberg model interacting with vibrating point charges representing nearby bridging ligands. To reach the model, we apply second order perturbation theory to the Hubbard model with the spin-orbit interaction. In contrast to the other components of the spin-orbit interaction, the part that directly couples the momentum of the charge and electron spin appears at first order as an effective magnetic field at each site. We find that the inclusion of this nuclear-motion induced spin-orbit coupling can increase the rate of otherwise spin-forbidden transitions between different spin states of the Heisenberg model by many orders of magnitude. This overlooked interaction may, therefore, play a significant role in spin-forbidden phenomena such as spin relaxation in coupled spin-qubits.