Path Integral Methods in Quantum Rate Theories

Abstract

In this thesis, we propose path integral approaches to include nuclear quantum effects such as tunneling, delocalization and zero point energy in a chemical simulation. Our methods exhibit a strong connection to semiclassical instanton (SCI) theory, which has been rigorously derived from first principles. SCI theory relies on computing the quantum mechanical rate constant by making a steepest-descent approximation for the quantum Boltzmann operator. However, our methods go beyond the semiclassical approximation, and evaluate this operator exactly by sampling over instanton-like paths and other paths in its vicinity. This allows us to treat systems with strong anharmonicity where zero-frequency modes cause higher-order quantum fluctuations to dominate the reaction rates. We present a set of methods to treat reactions characterized by asymmetric barriers at high and low temperatures for a challenging set of system parameters. Two of the methods are inspired by quantum transition-state theories, and include constraint functions in their formulations to enforce sampling of dominant tunneling paths. The final approach relies on evaluating the rate constant by the method of saddle-point approximation along the time variable. All these methods can be evaluated using path integral Monte Carlo or molecular dynamics techniques, and applied to compute the reaction rates in complex systems. Our approach to treat chemical reactions in the nonadiabatic limit has the correct classical limit, and is exact for a system of many crossed linear potentials. The method is tested on a wide variety of model systems, and predicts excellent rates even deep in the tunneling regime. A particularly appealing feature of this approach is that it is able to accurately describe the inverted regime of the spin-boson model without resorting to numerical extrapolation techniques. The method has also been applied by coworkers to study electron-transfer reactions in the aqueous ferrous-ferric system, where its spin-boson nature was confirmed in the classical limit. Likewise, our methods developed to treat reactions in the Born-Oppenheimer regime show a similar level of accuracy in predicting rates where nuclear quantum effects are seen to play a significant role. We show that, for example, our quantum instanton approach outperforms SCI theory in computing accurate rates for an asymmetric Eckart barrier. In presenting the theory and results for these methods, we also demonstrate that some of the other well-established methods that rely on the saddle-point approximation to evaluate the time correlation function break down in certain important regimes of chemical reactions. The deficiencies in these methods are explained and contrasted against our approaches.