Hamilton-Jacobi equation for the least-action/least-time dynamical path based on fast marching method

Classical dynamics can be described with Newton's equation of motion or, totally equivalently, using the Hamilton-Jacobi equation. Here, the possibility of using the Hamilton-Jacobi equation to describe chemical reaction dynamics is explored. This requires an efficient computational approach for constructing the physically and chemically relevant solutions to the Hamilton-Jacobi equation; here we solve Hamilton-Jacobi equations on a Cartesian grid using Sethian's fast marching method [J. A. Sethian, Proc. Natl. Acad. Sci. USA 93, 1591 (1996)]. Using this method, we can-starting from an arbitrary initial conformation-find reaction paths that minimize the action or the time. The method is demonstrated by computing the mechanism for two different systems: a model system with four different stationary configurations and the H+H-2-->H-2+H reaction. Least-time paths (termed brachistochrones in classical mechanics) seem to be a suitable chioce for the reaction coordinate, allowing one to determine the key intermediates and final product of a chemical reaction. For conservative systems the Hamilton-Jacobi equation does not depend on the time, so this approach may be useful for simulating systems where important motions occur on a variety of different time scales. (C) 2004 American Institute of Physics.