How to improve the accuracy of semiclassical and quasiclassical dynamics with and without generalized quantum master equations

Semi- and quasi-classical (SC) theories can handle anharmonic interactions and are thus well-suited to predict atomistic quantum dynamics in condensed phases that encode energy and charge transport, spectroscopic responses, and chemical reactivity. However, SC theories can be computationally expensive and inaccurate. When combined with generalized quantum master equations (GQMEs), the resulting SC-GQMEs can enhance the efficiency and accuracy of SC dynamics. Yet, while the origin of improved efficiency is clear, the mechanism that improves accuracy remains elusive. Even worse, SC-GQMEs can yield unphysical dynamics in challenging parameter regimes-a shortcoming that might be avoided if the mechanism of accuracy improvement were understood. Here, we uncover this mechanism. We leverage short-time analyses to prove that exact, "left-handed" time-derivatives delay the onset of SC inaccuracy, even without the GQME. However, these derivatives are a double-edged sword: while offering greater short-time accuracy, they become unphysical in challenging parameter regimes. Because short-lived SC-GQME kernels combine short-time accuracy with long-time stability, we develop a protocol to unambiguously determine the memory kernel cutoff, even in challenging cases where previous treatments had failed. Our protocol employs only SC calculations and combines self-consistency with mixed-accuracy auxiliary kernels to triangulate a propitious kernel cutoff, yielding SC-GQMEs with greater accuracy than SC theory alone, while remaining physical and accurate over arbitrary times. Our insights into accuracy improvement, identification of when the SC-GQME is advantageous, and kernel cutoff protocol are general and can be expected to apply to complex systems that go beyond simple models.