A long-standing problem in the chemistry community has been to reliably compute chemical reaction and isomerization rates from first principles. The traditional technique for doing this, known as transition state theory, makes statistical assumptions on the nature of the underlying dynamical processes. A detailed examination of even relatively simple systems, such as the ionization of loosely bound electrons, shows that this assumption is not valid, and it is now realized that in order to solve these problems one must take into account the detailed structure of the dynamics. Using recently developed techniques based on geometric mechanics (especially the use of reduction theory and shape space dynamics), together with a detailed knowledge of the dynamics (especially normal form theory and invariant manifold tubes), as well as Monte Carlo sampling methods (for computing volumes of intersecting reaction entrance and exit regions in phase space), this problem will be solved first for relatively simple molecules undergoing isomerization and then progressing to more complex situations.
This research contributes to one of the grand challenges of the mathematical and computational sciences -- namely the ability to accurately compute from first principles (that is, using so-called ab initio calculations) important molecular processes such as chemical reaction rates and the dynamical behavior of biomolecules. The specific advances that make this possible are recent developments in computational techniques for dynamical systems as well as improved understanding of the complex geometry of the exit and entrance channels (or tubes) that govern reactions and conformation changes. These are fundamental advances that, when combined with dimension and model reduction techniques, will provide a powerful tool for the study of more complex molecular systems.
