^{1}and Peter Talkner

^{2}

### Abstract

A brief history is presented, outlining the development of rate theory during the past century. Starting from Arrhenius [Z. Phys. Chem.4, 226 (Year: 1889)], we follow especially the formulation of transition state theory by Wigner [Z. Phys. Chem. Abt. B19, 203 (Year: 1932)] and Eyring [J. Chem. Phys.3, 107 (Year: 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, Year: 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 (Year: 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 (Year: 1940)] a seminal paper on the relation between Einstein’s theory of Brownian motion [Einstein, Ann. Phys.17, 549 (Year: 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 (Year: 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–Hänggi [Mel'nikov and Meshkov, J. Chem. Phys.85, 1018 (Year: 1986); Pollak, Grabert, and Hänggi, J. Chem. Phys.91, 4073 (Year: 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 (Year: 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.

This work has been supported by grants of the Israel Science Foundation, the US–Israel Binational Science Foundation, and the Deutsche Forschungsgemeinschaft.

I. RATE THEORY IN THE FIRST HALF OF THE 20TH CENTURY: ARRHENIUS, WIGNER, EYRING AND KRAMERS A. Arrhenius and activated molecules B. Wigner and Eyring: The transition state method C. Kramers–Brownian motion in a field of force II. RATE THEORY IN THE SECOND HALF OF THE 20TH CENTURY A. Variational transition state theory B. What is the activated complex? C. Unification of the TS method with Kramers’ Brownian motion theory and solution of the Kramers turnover problem D. Quantum TST? E. Reactive flux method F. Numerically exact quantum methods G. Rates in nonequilibrium systems III. FUTURE: OPEN PROBLEMS A. Classical rate theory at equilibrium B. Classical rate theory away from equilibrium C. Quantum equilibrium rate theory D. Quantum rate theory away from equilibrium E. Concluding remarks

### Key Topics

- Transition state theory
- 77.0
- Chemical reaction theory
- 21.0
- Friction
- 19.0
- Semiclassical theories
- 16.0
- Activation energies
- 15.0

Data & Media loading...

Article metrics loading...

Full text loading...

Commenting has been disabled for this content