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Vibronic coupling has a dramatic influence over a large number of molecularprocesses, ranging from photo-chemistry, to spin relaxation and electronictransport. The simulation of vibronic coupling with multi-referencewavefunction methods has been largely applied to organic compounds, and onlyearly efforts are available for open-shell systems such as transition metal andlanthanide complexes. In this work, we derive a numerical strategy todifferentiate the molecular electronic Hamiltonian in the context ofmulti-reference ab initio methods and inclusive of spin-orbit coupling effects.We then provide a formulation of open quantum system dynamics able to predictthe time evolution of the electrons' density matrix under the influence of aMarkovian phonon bath up to fourth-order perturbation theory. We apply ourmethod to Co(II) and Dy(III) molecular complexes exhibiting long spinrelaxation times and successfully validate our strategy against the use of aneffective spin Hamiltonian. Our study shed light on the nature of vibroniccoupling, the importance of electronic excited states in spin relaxation, andthe need for high-level computational chemistry to quantify it.

Spin-vibronic dynamics in open-shell systems beyond the spin Hamiltonian formalism