NATPS: Nonadiabatic Transition Path Sampling Using Time-Reversible MASH Dynamics

Abstract

Rare nonadiabatic events play a central role in photochemistry but remain difficult to simulate because excited-state dynamics is computationally demanding and often stochastic. Here we introduce a deterministic and time-reversible implementation of nonadiabatic dynamics that enables the application of transition path sampling (TPS) to excited-state processes. Our approach builds on the Mapping Approach to Surface Hopping (MASH) and establishes the conditions required for path ensemble sampling, in particular time reversibility and detailed balance. Combining this dynamics with the TPS framework yields a new method, termed nonadiabatic transition path sampling (NATPS). Using a model system of electronically coupled potential energy surfaces, we demonstrate that NATPS efficiently generates ensembles of reactive trajectories and provides mechanistic insight into nonadiabatic pathways. Compared with brute-force trajectory simulations and forward-flux sampling approaches, NATPS substantially reduces the computational effort required to obtain reactive trajectories.

Figures (6)

• Figure 1: a) Illustration of the analytic potential along the nuclear coordinate $q$. b) Reactive trajectory connecting basins $B$ and $A$. c) Corresponding time evolution of the spin-vector component $S_z$. Red segments indicate portions of the trajectory evolving on the excited state. The discontinuities in $S_z$ near the equator is a result of the choice of trajectory frame bookkeeping and does not represent a physical jump in $S_z$.
• Figure 2: a) Lineage history of trajectories during ten successive Monte Carlo steps in path space. Earlier trajectories are shown as faint thin lines, while the most recent trajectories are plotted as solid thick lines. Shooting points, from which new trajectories are sampled, are highlighted with a diamond. b) Autocorrelation function associated with the transition time in MC path space. On average, a statistically independent path emerges after every 9 random walks, where this is visually hinted from a), judging from the similarities among the trajectories.
• Figure 3: Distribution of transition times at a) $T=12000$ K and b) $T=1000$ K. The average transition time $\langle \tau \rangle$ is indicated by a red line. The golden line in a) indicates the mean transition time obtained from the brute-force MASH simulation, which matches with the TPS-accelerated result. c) Distribution of hopping positions at $12000$ K. Hops are mainly concentrated symmetrically near the crossing seam where $q=1.0$. Any apparent asymmetries arise from the stochastic noises due to finite sampling. A kernel density envelope (KDE) is placed over the distribution to provide visual guidance on the general trend.
• Figure 4: Temperature dependence of TPS ensemble statistics: (a) mean transition time $\langle \tau \rangle$ and mean number of hops $\langle N_\mathrm{hop} \rangle$, and (b) the standard deviation of transition times $\langle \sigma_\tau \rangle$. Simulated for $V_c = 0.2\epsilon$. Dependence of (c) mean transition time $\langle \tau \rangle$ and mean number of hops $\langle N_\mathrm{hop} \rangle$ and (d) standard deviation of the hopping time $\langle \sigma_\tau \rangle$ on the electronic coupling $V_c$; simulated at 10,000 K. (e) analysis of the average transition time for adiabatic and nonadiabatic paths seperately.
• Figure 5: Distribution of hopping positions for (a) varying temperature and (b) varying electronic coupling and state-resolved path densities at (c) $300$ K, (d) $2500$ K, and (e) $30,000$ K.