Calculation of Point-to-Point Short-Time and Rare Trajectories with Boundary Value Formulation

Sampling rare, short-time, and reactive trajectories is of considerable interest in molecular simulations. These trajectories, which are also called "activated", hop between stable states separated by energy or entropy barriers. Simulations of activated trajectories with random sampling of initial conditions are inefficient, since most initial conditions lead to trajectories that do not pass the barrier in short times. A boundary value formulation is proposed that selects these rare trajectories, making the sampling of point-to-point reactive trajectories more effective. Earlier boundary value formulations by one of us focused on computations of approximate trajectories. In the proposed method, trajectories are accurate even when we employ a relatively large integration step (by a factor of about 100 compared to initial value methods). The boundary value solutions to short-time reactive trajectories tend to be unique and have significant statistical weights compared to other reactive trajectories of the microcanonical ensemble. Three numerical examples are considered:  a transition in the Mueller potential, a conformational change in alanine dipeptide, and an isomerization in a Lennard-Jones cluster.