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

化学界的一个长期存在的问题是如何根据第一性原理可靠地计算化学反应和异构化速率。这样做的传统技术，称为过渡态理论，对潜在的动态过程的性质进行统计假设。即使是对相对简单的系统的详细研究，如松散束缚电子的电离，也表明这一假设是不成立的，现在人们意识到，为了解决这些问题，必须考虑动力学的详细结构。利用最近发展起来的基于几何力学的技术(特别是利用约化理论和形状空间动力学)，再加上对动力学的详细了解(特别是规范形理论和不变流形管)，以及蒙特卡罗抽样方法(用于计算相空间中相交的反应出入口区域的体积)，这个问题将首先对于经历异构化的相对简单的分子得到解决，然后发展到更复杂的情况。这项研究对数学和计算科学的重大挑战之一做出了贡献--即根据第一性原理(即使用所谓的从头计算)准确计算重要的分子过程的能力，如化学反应速率和生物分子的动力学行为。使这成为可能的具体进展是动力学系统计算技术的最新发展，以及对控制反应和构象变化的出入口通道(或管)的复杂几何结构的更好理解。这些都是基础性的进展，当与降维和模型降维技术相结合时，将为更复杂的分子系统的研究提供强大的工具。

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.