Revisiting shooting point Monte Carlo methods for transition path sampling

Rare event sampling algorithms are essential for understanding processes that occur infrequently on the molecular scale, yet they are important for the long-time dynamics of complex molecular systems. One of these algorithms, transition path sampling (TPS), has become a standard technique to study such rare processes since no prior knowledge on the transition region is required. Most TPS methods generate new trajectories from old trajectories by selecting a point along the old trajectory, modifying its momentum in some way, and then "shooting" a new trajectory by integrating forward and backward in time. In some procedures, the shooting point is selected independently for each trial move, but in others, the shooting point evolves from one path to the next so that successive shooting points are related to each other. To account for this memory effect, we introduce a theoretical framework based on an extended ensemble that includes both paths and shooting indices. We derive appropriate acceptance rules for various path sampling algorithms in this extended formalism, ensuring the correct sampling of the transition path ensemble. Our framework reveals the need for amended acceptance criteria in the flexible-length aimless shooting and spring shooting methods.