Theoretical chemical physics is my primary field of interest, particularly as it applies to dynamically complex systems. I develop theoretical and computational methods, test theory with numerical calculations, and complement experimental studies of single molecules. Currently, dynamical systems, computational mechanics, and statistical mechanics are being investigated motivated by challenging questions in chemistry. This work on small, nanoscale systems far from equilibrium has further implications in physics and applied mathematics.

Molecular motion can be complex, characterized by dynamical chaos that hinders the accurate experimental measurement and control of chemical reactions. Numerical simulations of this motion, however, can provide insights that complement experiments, particularly those of isolated molecules, and have the potential to overcome current difficulties. Presently, comparison with experiment is difficult due to computational costs and limitations of some measures of chaos. One goal of my research is to overcome these challenges.

Dynamics examines the qualitative properties of a system's time history in phase space: taking the state of a system to be a point, the history is a trajectory curve whose patterns are within the purview of dynamics. Irregular patterns, sensitive to the initial state, result when the motion is chaotic. Furthermore, if an initial state is perturbed slightly and the evolution of the perturbed and unperturbed states is followed, their motion may be quite different. Consequently, two molecules with initially similar configurations may behave quite differently on the experimental time scale.

Exponentially fast divergence of two initially close trajectories, known as Lyapunov instability, is a better signature of chaos than their irregular patterns. Quantification of this instability using dynamics employs properties of Lyapunov vectors: intuitively, at each instant in time, a Lyapunov vector points from a trajectory of interest to another nearby. Chaos may then be quantified numerically by the rate at which a vector expands and the two trajectories separate. This rate is a Lyapunov exponent.

Improved measurements of isolated molecules require an understanding of their regular and irregular, chaotic behavior. The information needed to understand this motion is contained within the mathematical construct known as Lyapunov vectors. Just as the time-dependent Schroedinger equation is solved for wavefunctions in quantum dynamics, an equation of motion is solved for Lyapunov vectors in classical chaotic dynamics. This solution reveals characteristics that are essential to understanding the behavior of a chaotic system. Thus, Lyapunov vectors are a key to characterizing, predicting the behavior of, and ultimately controlling chaotic molecular motions through the complex potential landscapes of biologically interesting molecules.

Most physical systems found in nature are in motion, and yet the classical theoretical framework of non-equilibrium statistical mechanics is less established than that of equilibrium statistical mechanics. Exciting progress in non-equilibrium statistical mechanics has been made recently by accounting for the chaotic nature of microscopic dynamics. Central to these developments are measures of chaos, specifically Lyapunov exponents. Lyapunov exponents, however, are properties of objects still more fundamental to microscopic dynamics, namely Lyapunov vectors.

Past and future research will aid the design of procedures for controlling atomic motion. A consequence of chaotic dynamics is that atomic systems in an ensemble, initially prepared in a narrow region of reactant space, will not all follow the same reaction path and will not all lead to the same products. Lyapunov exponents quantify how quickly ensemble members converge or diverge in the possible Lyapunov vector directions and set limits on one's ability to predict future motion, given incomplete knowledge of the initial state. A bottleneck to new insights is the calculation of Lyapunov vectors; an obstacle we seek to overcome using novel calculation of both the desired Lyapunov vectors and their associated exponents. This work will therefore make direct connections with the generation of single molecule experiments that is already yielding dramatic new insight into the complexities of molecular dynamics.