Determining the Mechanism and Rate of Transitions in Atomic Systems: Searching for Saddle Points on High-Dimensional Surfaces

Computer simulations of classical dynamics of atoms and molecules have lead to valuable insight and improved understanding in many diverse areas of science, including materials design, drug discovery and protein structure. Such simulations are, however, limited to extremely short time scales, about a nanosecond or less with ca. a week of CPU time. This represents a very severe limitation on the types of phenomena that can be studied by direct classical dynamics simulations. Most chemical reactions, diffusion events and conformational changes in macromolecules are `rare events', in that the atoms typically vibrate about their optimal position multiple times, on the order of 10^10 times, in between `reactive' events. For example, the activation energy for many relevant processes is on the order of 0.5 eV and an `event' may occur on average every millisecond or so at room temperature. A direct classical dynamics simulation long enough to be likely to include one such event would take on the order of 10^5 years of CPU time on the fastest present day CPU! It is clearly of great importance to find ways to carry out molecular level simulations on a much longer timescale than presently can be done. This is one of the important challenges in theoretical chemistry, materials science and condensed matter physics.

Transition state theory often gives very good estimates of transition rates of, for example, chemical reactions and diffusion if the transition state is defined sufficiently well. For low temperature, the harmonic approximation to transition state theory typically works well enough. The problem then becomes finding saddle points on the multidimensional (~10^3 degrees of freedom) potential energy surface. In many cases, it is too expensive to evaluate the second derivatives of the potential energy during the search. Two recent methods for approaching this problem using only the potential energy and its first derivative will be discussed, as well as applications to surface and bulk diffusion of atoms.