The interaction of the electronic spin and molecular vibrations mediated by spin-orbit coupling governs spin relaxation in molecular qubits. We derive a dynamic molecular spin Hamiltonian that includes both adiabatic and non-adiabatic spin-dependent interactions, and we implement the computation of its matrix elements using density functional theory. The dynamic molecular spin Hamiltonian contains a novel spin-vibronic interaction with non-adiabatic origin in addition to the conventional molecular Zeeman and dipolar spin interactions with adiabatic origin. The new spin-vibronic interaction represents a non-Abelian Berry curvature on the ground-state electronic manifold and generates an effective magnetic field for the ground-state electronic spin dynamics. We further develop a spin relaxation rate theory that estimates the spin relaxation time via the two-phonon Raman process and the molecular crystal vibrational dynamics. The application of the dynamic molecular spin Hamiltonian to S=1/2 molecular qubits demonstrates that the spin relaxation time at elevated temperatures is dominated by the non-adiabatic spin-vibronic interaction. The computed spin relaxation rate and its magnetic field orientation dependence are in agreement with experimental measurements. Analysis of the influence of the molecular electronic structure on the strength of the spin-vibronic interaction charts new avenues for the improvement of the spin-coherent properties of molecular systems.
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