  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.
