Approximate Quantum Mechanical Methods for Rate Computation in Complex Systems

Abstract

The last 20 years have seen qualitative leaps in the complexity of chemical reactions that have been studied using theoretical methods. While methodologies for small molecule scattering are still of great importance and under active development [1], two important trends have allowed the theoretical study of the rates of reaction in complex molecules, condensed phase systems, and biological systems. First, there has been the explicit recognition that the type of state to state information obtained by rigorous scattering theory is not only not possible for complex systems, but more importantly, not meaningful. Thus, methodologies have been developed that compute averaged rate data directly from a Hamiltonian. Perhaps the most influential of these approaches has been the correlation function formalisms developed by Bill Miller et al. [2]. While these formal expressions for rate theories are certainly not the only correlation function descriptions of quantum rates [3, 4], these expressions of rates directly in terms of evolution operators, and in their coordinate space representations as Feynman Propagators, have lent themselves beautifully to complex systems because many of the approximation methods that have been devised are for Feynman propagator computation. This fact brings us to the second contributor to the blossoming of these approximate methods, the development of a wide variety of approximate mathematical methods to compute the time evolution of quantum systems. Thus the marriage of these mathematical developments has created the necessary powerful tools needed to probe systems of complexity unimagined just a few decades ago.