A Hamilton–Jacobi type equation for computing minimum potential energy paths

Abstract

A new method for computing minimum-energy reaction paths is presented. Unlike existing approaches (e.g. intrinsic reaction coordinate methods), our approach works for any reactant configuration: the structure of the transition state, reactive intermediates and product will be determined by the algorithm, and so need not be known beforehand. The method we have developed is based on solving a Hamilton–Jacobi type equation. Specifically, we introduce a speed function so that the ‘first arrival times’ from the Hamilton–Jacobi equation correspond to least-potentials. Then, adopting a back-tracing method, we can use the first arrival times to determine the minimum-energy path between any classically allowed molecular conformation and the initial (reactant) conformation. The method is illustrated by applying it to six different systems: (1) a model system with four different minima in the potential energy surface, (2) a model Muller–Brown potential, (3) the isomerization reaction of malonaldehyde using a fitting potential energy surface, (4) a model Minyaev–Quapp potential representative of con- and dis-rotations of two BH2 groups in the BH2–CH2–BH2 molecule, (5) the F + H2→FH + H reaction and (6) the H + FH → HF + H reaction. Our results demonstrate that the proposed method represents a robust alternative to existing techniques for finding chemical reaction paths.