Geometric Mechanics and Dynamical Systems Approach to
the Theory and Computation of Chemical Reaction Rates
The objective of this NSF 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 largeamplitude 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 multichannel 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 Sciencethe dynamics of biomolecules, including predictive protein folding. Our long term vision is to develop some coarsegraining 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.
ACKNOWLEDGMENT

The project is funded by the Division of Mathematical Sciences of
the National Science Foundation (NSFDMS0505711), August 2005July 2008. It involves Jerrold E. Marsden (PI), Wang Sang Koon (CoPI), Tomohiro Yanao, and Katalin Grubits.

PUBLICATIONS
 A Nonequilibrium Rate Formula for Collective Motions of Complex Molecular Systems, Proceedings of ICNAAM, Numerical Analysis and Applied Mathematics, International Conference 2010.(Yanao, T., W.S. Koon, and J.E. Marsden,[2010].pdf)
 Intramolecular energy transfer and the driving mechanisms for largeamplitude collective motions of clusters, Journal of Chemical Physics, 130, 144111.(Yanao, T., W.S. Koon, and J.E. Marsden, [2009].pdf)
 GyrationRadius Dynamics in Structural Transitions of Atomic Clusters, Journal of Chemical Physics, 126, 124102. (Yanao, T., W.S. Koon, J.E. Marsden, and I.G. Kevrekidis [2007].pdf)
 Mass Effects and Internal Space Geometry in Triatomic Reaction Dynamics, Physical Review A, 73, 052704. (Yanao, T., W.S. Koon, and J.E. Marsden [2006]. pdf)
 Application of Tube Dynamics to Nonstatistical Reaction Processes, FewBody Systems, Volume 38, Numbers 24, pages 167  172. (Gabern F., W. S. Koon, J. E. Marsden, S. D. Ross, and T. Yanao, [2006]. pdf)
 MassRelated Dynamical Barriers in Triatomic Reactions, FewBody Systems, Volume 38, Numbers 24, pages 161  166. (Yanao, T., W.S. Koon, and J.E. Marsden [2006], pdf)
 Theory and Computation of NonRRKM Lifetime Distributions and Rates of Chemical Systems with Three and More Degrees of Freedom, Physica D, volume 211, issue 34, pages 391406. (Gabern, F., W.S. Koon, J.E. Marsden, and S.D. Ross [2005], pdf)
RELATED ARTICLES
 Effects of an intrinsic metric of molecular internal space on chemical reaction dynamics, Advances in Chemical Physics 130 B, 87128. (T. Yanao and K. Takatsuka [2005])
 Setoriented computation of transport rates in 3degree of freedom systems: the Rydberg atom in crossed fields, Regular and Chaotic Dynamics, 10, 173192. (Dellnitz, M., K. Grubits, J. E. Marsden, K. Padberg, and B. Thiere [2005], pdf)
 Kinematic effects associated with molecular frames in structural isomerization dynamics of clusters, Journal of Chemical Physics 120, 89248936. (T. Yanao and K. Takatsuka [2004], pdf)
 Collective coordinates and an accompanying metric force in structural isomerization dynamics of molecules, Physical Review A 68, 032714 (116). (T. Yanao and K. Takatsuka [2003], pdf)