The reaction path intrinsic reaction coordinate method and the Hamilton-Jacobi theory

The definition and location of an intrinsic reaction coordinate path is of crucial importance in many areas of theoretical chemistry. Differential equations used to define the path hitherto are complemented in this study with a variational principle of Fermat type, as Fukui [Int. J. Quantum Chem., Quantum Chem. Symp. 15, 633 (1981)] reported in a more general form some time ago. This definition is more suitable for problems where initial and final points are given. The variational definition can naturally be recast into a Hamilton-Jacobi equation. The character of the variational solution is studied via the Weierstrass necessary and sufficient conditions. The characterization of the local minima character of the intrinsic reaction coordinate is proved. Such result leads to a numerical algorithm to find intrinsic reaction coordinate paths based on the successive minimizations of the Weierstrass E-function evaluated on a guess curve connecting the initial and final points of the desired path.