Computing tunneling paths with the Hamilton–Jacobi equation and the fast marching method

Abstract

We present a new method for computing the most probable tunneling paths based on the minimum imaginary action principle. Unlike many conventional methods, the paths are calculated without resorting to an optimization (minimization) scheme. Instead, a fast marching method coupled with a back-propagation scheme is used to efficiently compute the tunneling paths. The fast marching method solves a Hamilton–Jacobi equation for the imaginary action on a discrete grid where the action value at an initial point (usually the reactant state configuration) is known in the beginning. Subsequently, a back-propagation scheme uses a steepest descent method on the imaginary action surface to compute a path connecting an arbitrary point on the potential energy surface (usually a state in the product valley) to the initial state. The proposed method is demonstrated for the tunneling paths of two different systems: a model 2D potential surface and the collinear reaction. Unlike existing methods, where the tunneling path is based on a presumed reaction coordinate and a correction is made with respect to the reaction coordinate within an ‘adiabatic’ approximation, the proposed method is very general and makes no assumptions about the relationship between the reaction coordinate and tunneling path.

Keywords:

• Ab initio
• Electronic structure
• Quantum chemistry
• Computational chemistry

Acknowledgements

The authors thank NSERC and the Canada Research Chairs for research support. BKD would like to thank E. K. U. Gross for valuable discussions on semiclassical mechanics. BKD also acknowledges support from a Humboldt fellowship and by the DPG Sonderforshungs-bereich 450 at the Free University of Berlin, where part of this work was performed.

Additional information

Notes on contributors

Bijoy K. Dey

Current address: Institut fur Theoretische Physik, Freie Universitat Berlin, Arnimalle 14, Berlin, Germany 14195.