Estimating memory time within the frameworks of generalized quantum master equation and transfer tensor methods

Simulating long-time nonadiabatic dynamics in condensed-phase systems is computationally demanding due to the inherent non-Markovianity of the electronic reduced density matrix evolution. While the generalized quantum master equation (GQME) and transfer tensor method (TTM) allow for the reconstruction of long-time dynamics from short-time projection-free inputs, their accuracy hinges on the rigorous estimation of the memory time, a parameter often determined by heuristic trial-and-error. In this work, we establish a comprehensive framework for estimating memory time and benchmarking propagation accuracy using semiclassical and numerical exact inputs on both standard spin-boson models and general multistate harmonic models. We develop an error estimation scheme that reveals a characteristic three-stage decay pattern in the non-Markovian propagation error: an initial transient drop, an exponential decay, and a saturation plateau. This estimator serves as a critical diagnostic tool for GQME and TTM, successfully distinguishing between converged predictions and reliability failures in complex systems, such as the carotenoid-porphyrin-fullerene triad. These findings provide a robust, quantitative protocol for validating memory-kernel-based simulations of nonadiabatic dynamics.