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Fermi's golden rule (FGR) offers an empirical framework for understanding the dynamics of spin-lattice relaxation in magnetic molecules, encompassing mechanisms like direct (one-phonon) and Raman (two-phonon) processes. These principles effectively model experimental longitudinal relaxation rates, denoted as $T_1^{-1}$. However, under scenarios of increased coupling strength and nonlinear spin-lattice interactions, FGR's applicability may diminish. This paper numerically evaluates the exact spin-lattice relaxation rate kernels, employing the extended dissipaton equation of motion (DEOM) formalism. Our calculations reveal that when quadratic spin-lattice coupling is considered, the rate kernels exhibit a free induction decay-like feature, and the damping rates depend on the interaction strength. We observe that the temperature dependence predicted by FGR significantly deviates from the exact results since FGR ignores the non-Markovian nature of spin-lattice relaxation. Our methods can be readily applied to other systems with nonlinear spin-lattice interactions and provide valuable insights into the temperature dependence of $T_1$ in molecular qubits., Comment: 10 pages, 5 figures

Molecular dynamics with electronic friction (MDEF) approach can describe nonadiabatic effects accurately at metal surfaces in the weak nonadiabatic limit. That being said, MDEF treats nuclear motion classically, such that the nuclear quantum effects are missing completely in the approach. To address this limitation, we combine electronic friction with Ring Polymer Molecular Dynamics (RPMD). In particular, we apply the averaged electronic friction from the metal surface to the centroid mode of the ring polymer. We benchmark our approach against quantum dynamics to show that electronic friction with RPMD (EF-RPMD) can capture zero-point energy as well as transition dynamics accurately. In addition, we show EF-RPMD can correctly predict the electronic transfer rate near metal surfaces in the tunneling limit as well as the barrier crossing limit. We expect our approach will be very useful to study nonadiabatic dynamics near metal surface when nuclear quantum effects become essential.