On the Electronic Path-Integral Normal Modes of the Spin-Mapping Representation of Nonadiabatic Dynamics

Abstract

Nonadiabatic Matsubara dynamics has been proposed in both the Meyer-Miller-Stock-Thoss (MMST) and spin-mapping representations, whereby the dynamics are truncated in the higher path-integral normal modes of the nuclear degrees of freedom, but no truncation is performed in the electronic variables. In contrast to single-surface Matsubara dynamics, these methods do not appear to conserve the Quantum Boltzmann Distribution (QBD) for general systems. Recently, it was shown that truncating in the electronic path-integral normal modes of the MMST representation does not lead to conservation of the QBD for a single trajectory or accurate computation of a correlation function. Here, the electronic path-integral normal modes in the spin-mapping representation are investigated. We focus on a two-level system although we expect our findings to be applicable to any number of electronic states. We find, perhaps surprisingly, that the higher normal modes of the spin-vector $z$-component, the difference in state populations, are constrained by the QBD, that the time-evolved state population observable is a function of only the spin-mapping centroid, and that the full $N$-bead computational results are replicated using only the spin-mapping centroid, leading to accurate, QBD conserving dynamics. While at present these results are not a generally-applicable method, we believe this may explain the recent success of spin-mapping approaches compared to MMST approaches, %To our knowledge, this is the first method obtaining conservation of the QBD when truncating in electronic normal modes and should aid the future derivation of highly accurate and Quantum Boltzmann conserving nonadiabatic dynamics methods.