Dynamical corrections to quantum transition state theory

Reaction rate theory: what it was, where is it today, and where is it going?

A brief history is presented, outlining the development of rate theory during the past century. Starting from Arrhenius [Z. Phys. Chem. 4, 226 (1889)], we follow especially the formulation of transition state theory by Wigner [Z. Phys. Chem. Abt. B 19, 203 (1932)] and Eyring [J. Chem. Phys. 3, 107 (1935)]. Transition state theory (TST) made it possible to obtain quick estimates for reaction rates for a broad variety of processes even during the days when sophisticated computers were not available. Arrhenius' suggestion that a transition state exists which is intermediate between reactants and products was central to the development of rate theory. Although Wigner gave an abstract definition of the transition state as a surface of minimal unidirectional flux, it took almost half of a century until the transition state was precisely defined by Pechukas [Dynamics of Molecular Collisions B, edited by W. H. Miller (Plenum, New York, 1976)], but even this only in the realm of classical mechanics. Eyring, considered by many to be the father of TST, never resolved the question as to the definition of the activation energy for which Arrhenius became famous. In 1978, Chandler [J. Chem. Phys. 68, 2959 (1978)] finally showed that especially when considering condensed phases, the activation energy is a free energy, it is the barrier height in the potential of mean force felt by the reacting system. Parallel to the development of rate theory in the chemistry community, Kramers published in 1940 [Physica (Amsterdam) 7, 284 (1940)] a seminal paper on the relation between Einstein's theory of Brownian motion [Einstein, Ann. Phys. 17, 549 (1905)] and rate theory. Kramers' paper provided a solution for the effect of friction on reaction rates but left us also with some challenges. He could not derive a uniform expression for the rate, valid for all values of the friction coefficient, known as the Kramers turnover problem. He also did not establish the connection between his approach and the TST developed by the chemistry community. For many years, Kramers' theory was considered as providing a dynamic correction to the thermodynamic TST. Both of these questions were resolved in the 1980s when Pollak [J. Chem. Phys. 85, 865 (1986)] showed that Kramers' expression in the moderate to strong friction regime could be derived from TST, provided that the bath, which is the source of the friction, is handled at the same level as the system which is observed. This then led to the Mel'nikov-Pollak-Grabert-Hanggi [Mel'nikov and Meshkov, J. Chem. Phys. 85, 1018 (1986); Pollak, Grabert, and Hanggi, ibid. 91, 4073 (1989)] solution of the turnover problem posed by Kramers. Although classical rate theory reached a high level of maturity, its quantum analog leaves the theorist with serious challenges to this very day. As noted by Wigner [Trans. Faraday Soc. 34, 29 (1938)], TST is an inherently classical theory. A definite quantum TST has not been formulated to date although some very useful approximate quantum rate theories have been invented. The successes and challenges facing quantum rate theory are outlined. An open problem which is being investigated intensively is rate theory away from equilibrium. TST is no longer valid and cannot even serve as a conceptual guide for understanding the critical factors which determine rates away from equilibrium. The nonequilibrium quantum theory is even less well developed than the classical, and suffers from the fact that even today, we do not know how to solve the real time quantum dynamics for systems with "many" degrees of freedom.

Path integral evaluation of the quantum instanton rate constant for proton transfer in a polar solvent

The quantum instanton approximation for thermal rate constants, a type of quantum transition state theory (QTST), is applied to a model proton transfer reaction in liquid methyl chloride developed by Azzouz and Borgis. Monte Carlo path integral methods are used to carry out the calculations, and two other closely related QTST's, namely, the centroid-density and Hansen-Andersen QTST, are also evaluated for comparison using the present path integral approach. A technique is then introduced that calculates the kinetic isotope effect directly via thermodynamic integration of the rate with respect to hydrogen mass, which has the practical advantage of avoiding costly evaluation of the activation free energy. The present application to the Azzouz-Borgis problem shows that the above three types of QTST provide very similar results for the rate, within 30% of each other, which is nontrivial considering the totally different derivations of these QTSTs; the latter rates are also in reasonable agreement with some other previous results (e.g., obtained via molecular dynamics with quantum transitions), within a factor of approximately 2(7) for the H(D) transfer, thus significantly diminishing the possible range of the exact rates. In addition, it is revealed that a small but nonnegligible inconsistency exists in the parametrization of the Azzouz-Borgis model employed in previous studies, which resulted in the large apparent discrepancy in the calculated rates.