Shadowing, rare events, and rubber bands. A variational Verlet algorithm for molecular dynamics

Abstract

We present a variational implementation of the Verlet algorithm for molecular dynamics which is both conceptually and computationally attractive. Given an approximate path, this variational Verlet algorithm computes an actual trajectory close to the path. Instead of initial conditions, we specify end-point conditions, which makes it possible to compute trajectories with fixed end points as well as trajectories that are exactly periodic (nonlinear modes of vibration). With a simple sign change, the method yields approximate reaction pathways. We present several applications which illustrate how this alternative approach to dynamics can yield unusual trajectories that are difficult to attain by conventional means. In our first model system, a simple double well, we examine the seemingly paradoxical shadowing lemma of classical mechanics which claims that, in spite of the cumulative numerical errors which cause rapid deviation from the correct path, a calculated trajectory can still be close to some true trajectory of the system over a relatively long time period. We explicitly calculate such so-called shadowing trajectories. We also study isomerization events in which trajectories cross from one potential well to another, obtaining guesses from simple line segments and from dynamics on a perturbed version of the double-well potential. A closed loop of briefly recurrent motion in a trajectory can serve as an initial guess in the computation of a large-amplitude nonlinear mode of vibration. Finally, we use a model AB+C reaction in rare-gas solution to demonstrate the feasibility of calculations on molecular systems with many degrees of freedom. A gas-phase reactive trajectory of the A, B, and C atoms is, with the variational Verlet algorithm, transformed into a similar solution-phase trajectory and vice versa. This calculation suggests that rare events, such as reactions in solution, may be obtained through refining initial paths of perturbed or decoupled systems where such events are more easily computable.