Yet another look at the steepest descent path

Abstract

The path integral formulation of Brownian trajectories is employed to rederive the steepest descent path (SDP). An analogy between the mechanics of overdamped trajectories and a Hamiltonian system is found and exploited. The SDP is a special path selected from the curves that connect two predetermined fixed configurations. One fixed configuration is the reactant and the second is the product. It is the path that minimizes a functional that we call the “scalar work”. The minimum of the scalar work between two stationary points of the potential energy surface is the absolute value of the usual mechanical work. A new numerical algorithm to calculate steepest descent paths is proposed and a computational example is provided. In the new formulation the coordinates of the reactants and the products are the input required to determine the path. This is in contrast to the usual definition of the SDP that relies on the intermediate saddle point. The different input is more convenient for computations.