Tensorial spin-phonon relaxation reveals mode-selective relaxation pathways in a single-molecule qubit

Understanding and controlling spin relaxation in molecular qubits is essential for developing chemically tunable quantum information platforms. We present a first-principles-parametrized analytical framework for evaluating spin relaxation dynamics in vanadyl phthalocyanine (VOPc) and its oxygenated derivative, VOPc(OH)8. By expanding the spin Hamiltonian in vibrational normal modes and computing both linear and quadratic spin-phonon coupling tensors via finite differences of the g-tensor, we construct a relaxation tensor that enters a Lindblad-type master equation, capturing both direct (one-phonon) and Raman (two-phonon) processes. A mode-resolved analysis reveals that relaxation is funneled through only a handful of low-frequency vibrations: in VOPc, three out-of-plane distortions of the phthalocyanine ring and V-O unit dominate, whereas in VOPc(OH)8, the additional oxygens shift these modes downward and suppress two of them, leaving a single strongly coupled mode as the main decoherence pathway. Both longitudinal (T1) and transverse (T2) relaxation are governed by this same set of vibrational modes, indicating that coherence loss is controlled by a common microscopic mechanism. This mode-selective picture offers a design strategy for engineering longer-lived molecular qubits.