  The interaction of electronic spin and molecular vibrations mediated by
spin-orbit coupling governs spin relaxation in molecular qubits. I derive an
extended molecular spin Hamiltonian that includes both adiabatic and
non-adiabatic spin-dependent interactions, and I implement the computation of
its matrix elements using state-of-the-art density functional theory. The new
molecular spin Hamiltonian contains a novel spin-vibrational orbit interaction
with non-adiabatic origin together with the traditional molecular Zeeman and
zero-field splitting interactions with adiabatic origin. The spin-vibrational
orbit interaction represents a non-Abelian Berry curvature on the ground-state
electronic manifold and corresponds to an effective magnetic field in the
electronic spin dynamics. I further develop a spin relaxation rate model that
estimates the spin relaxation time via the two-phonon Raman process. An
application of the extended molecular spin Hamiltonian together with the spin
relaxation rate model to Cu(II) porphyrin, a prototypical $S=1/2$ molecular
qubit, demonstrates that the spin relaxation time at elevated temperatures is
dominated by the non-adiabatic spin-vibrational orbit interaction. The computed
spin relaxation rate and its magnetic field orientation dependence are in
excellent agreement with experimental measurements.
