'The Nonlinear Dynamical Foundations of Transition State Theory in Systems with Three or More Degrees-of-Freedom'

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NSF Award Abstract - DMS-0071338Mathematical Sciences: The Nonlinear Dynamical Foundations of Transition State Theory in Systems with Three or More Degrees of FreedomAbstract0071338 WigginsThis research project is concerned with understanding the nonlinear dynamical foundations of transition state theory in systems with three or more degrees of freedom. While there has been much progress along these lines for two degree-of-freedom systems, there has been little progress of a similar nature for systems with three or more degrees-of-freedom. The geometrical point of view of dynamical systems theory offers a framework for discovering the types of higher dimensional phase space structures that govern reaction rates and energy flow in molecules. We will apply this approach to several problems: three dimensional Rydberg atoms in crossed electric and magnetic fields, the London-Eyring-Polanti-Sato potential, rotation-vibration interaction in formaldehyde, and a study of energy flow through resonances in carbonyl sulphide (OCS). In each case we will show that there are higher dimensional phase space structures that act as a "transition state." Indeed, it may even be possible to generalize the very useful idea of a periodic orbit dividing surface to systems with three or more degrees of freedom. Our research will also focus on developing computational approaches to realize these geometric structures in these specific problems. We will also develop a wavelet-based frequency map analysis tool for studying energy flow in resonances. This work will be carried out in collaboration with chemists and physicists, as well as computational scientists.This project investigates fundamental questions concerning chemical reactions. The underlying mathematical questions involve the nonlinear dynamical foundations of transition state theory in systems with three or more degrees of freedom. This is the fundamental theory that allows predictions concerning chemical reaction dynamics. While there is a fairly complete understanding of this theory for simple, low dimensional, systems, there is as yet no analogous theoretical framework for more physically realistic high dimensional systems. Such a theory will involve mathematical description of surfaces in dimensions larger than four and will require development of computational tools for realizing and visualizing such surfaces. This will result in a detailed understanding of the dynamics of how molecules break up and of how atoms combine to form molecules. The research is interdisciplinary, involving the collaboration of chemists, physicists, and computational scientists.