On the electronic path integral normal modes of the Meyer-Miller-Stock-Thoss representation of nonadiabatic dynamics

TL;DR

The paper investigates whether electronic normal modes in the Meyer-Miller-Stock-Thoss (MMST) representation can yield a Quantum Boltzmann Distribution ($QBD$)-conserving nonadiabatic Matsubara method by studying an electronic-only system. It derives the electronic Liouvillian and correlation functions in MMST normal modes and demonstrates that, unlike nuclear Matsubara modes, MMST electronic modes do not constrain higher modes and do not provide $QBD$ conservation for a single trajectory when truncated. The results show that [C^{[N]}_{AB}(t)] depends on all MMST modes, the higher modes do not narrow in distribution, and truncation in normal modes degrades single-trajectory dynamics, though ensemble averaging can mask violations. The findings suggest MMST electronic normal modes are not optimal for constructing a practical, $QBD$-conserving NA dynamics method, and point to the need for alternative electronic metrics (e.g., spin-mapping) to realize a robust nonadiabatic Matsubara framework.

Abstract

Accurate and efficient simulation of nonadiabatic 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 nonadiabatic 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 path integral 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. Furthermore, truncating the dynamics in MMST normal modes leads to inaccurate correlation functions and while the QBD appears conserved for an ensemble of trajectories, it is not for a single trajectory. Overall, this suggests that MMST path integral normal modes are not optimal for obtaining an accurate, QBD conserving nonadiabatic dynamics method.

Figures (7)

• Figure 1: Schematic diagram illustrating bead and normal mode truncation of the path integral, where the blue arrows indicate time propagation. The dashed path integral is time-evolved. Bead truncation is where the first $j$ beads are propagated. Normal mode truncation is where the lowest $k$ normal modes are propagated. All unpropagated beads and normal modes are left at their initial time ($t=0$) values.
• Figure 2: Correlation function for the electronic population of the first state with the exact Kubo-Transformed result (KT, black) compared with a full 8 bead calculation with truncation in both beads (dotted circle) and normal modes (NM, dashed) for; 8 (green), 5 (magenta), 3 (red) and 1 (orange). When truncating, for example with 5 beads/normal modes, the first 5 beads or the lowest 5 normal modes are propagated, with the remaining beads/normal modes kept at their initial time values. While the result improves with more beads/normal modes included, all need to be included to obtain an accurate result.
• Figure 3: The propagated Boltzmann term against time for a single trajectory with a full 8 bead calculation with truncation in both beads (dotted circle) and normal modes (dashed) for; 8 (green), 5 (magenta), 3 (red) and 1 (orange). The 8 beads and normal modes are the only lines that are flat, indicating QBD conservation.
• Figure 4: Correlation function for the conservation of electronic population of the first state with the exact Kubo-Transformed result (KT, black) compared with a full 8 bead calculation with truncation in both beads (dotted circle) and normal modes (NM, dashed) for; 8 (green), 5 (magenta), 3 (red) and 1 (orange). All lines oscillate quite close to the Kubo-Transformed result (note the y-axis scale) so there appears to be an averaging effect that conserves the QBD for an ensemble of trajectories.
• Figure 5: Correlation function for the conservation of electronic population of the first state with the exact Kubo-Transformed result (KT, black) compared with a full 8 bead calculation with different numbers of trajectories; 1 (purple), 10 (magenta), $10^3$ (blue), $10^6$ (light blue), $10^7$ (green) and $2\times 10^7$ (yellow). A general trend is observed where increasing the number of trajectories improves the accuracy of the correlation function and the amplitude of oscillations decreases.