On the electronic normal modes of the Meyer-Miller-Stock-Thoss representation of non-adiabatic dynamics

Accurate and efficient simulation of non-adiabatic dynamics is highly desirable for understanding charge and energy transfer in complex systems. A key criterion for obtaining an accurate method is conservation of the Quantum Boltzmann Distribution (QBD). For a single surface, Matsubara dynamics is known to conserve the QBD, as a consequence of truncating the dynamics in the higher normal modes of the imaginary-time path integral. Recently, a Non-Adiabatic Matsubara (NA-Mats) dynamics has been proposed (J. Chem. Phys., 2021, 154, 124124) which truncates in the normal modes of the nuclear variables but not in the electronic variables, which are described with the Meyer-Miller-Stock-Thoss (MMST) representation. Surprisingly, this NA-Mats method does not appear to conserve the QBD for a general system. This poses the question of the effect of truncating the higher normal modes of the electronic variables in the MMST representation. In this article, we present what we believe is the first study of electronic normal modes of the MMST representation. We find that observables are not usually a function of a finite number of normal modes and that the higher normal modes are not constrained by the distribution, unlike in conventional nuclear normal modes. While for an ensemble of trajectories, truncation in MMST normal modes appears to numerically conserve the QBD, this is not the case for a single trajectory. When considering a correlation function, all MMST normal modes are required to obtain reasonable dynamics. Overall, this suggests that MMST variables are not optimal for obtaining an accurate, QBD conserving nonadiabatic dynamics method.