Geometric Mechanics and Dynamical Systems Approach to
the Theory and Computation of Chemical Reaction Rates

The objective of this project is to advance the theory and computation of chemical reaction rates by combining the techniques of geometric mechanics with the methods of theoretical and computational dynamical systems and invariant manifolds.

We have developed new techniques merging tube dynamics with Monte Carlo methods that enable computation of reaction rates in systems with three or more degrees of freedom. The main model problem is the isomerization of polyatomic molecules, which involves complicated and large-amplitude collective motions. In Phase I, a cluster of three atoms, called the modified M3 system (since one of the atomic masses differs from the other two), will be studied in detail. In Phase II, more complex polyatomic systems (from 4 to 7 atoms) will be considered using the experience acquired in Phase I. The polyatomic cluster is a prototype to investigate not only the mass effects but also multi-channel chemical reactions; it is worth scrutinizing its isomerization dynamics in terms of ``reaction rate" and ``branching ratio". This system is appropriate for studying the mechanism of breakdown of transition state theory (TST). It can also serve as an excellent model to construct a comprehensive theory of chemical reactions and to develop efficient computational tools for reaction rate calculations in three or more degrees of freedom systems that takes into account both geometric effects and phase space structures.

A main goal is to contribute some key stepping stones towards one of the grand challenges in Science---the dynamics of biomolecules, including predictive protein folding. Our long term vision is to develop some coarse-graining methods that can link (i) the statistical methods which have been used to probe the dynamics in high dimensional systems and (ii) the geometric methods which provide detailed insight into the dynamics of low dimensional systems.

The project is funded by the Division of Mathematical Sciences of the National Science Foundation (NSF-DMS-0505711), August 2005-July 2008. It involves Jerrold E. Marsden (PI), Wang Sang Koon (Co-PI), Tomohiro Yanao, Frederic Gabern, Katalin Grubits, and Paul Skerritt.