Foundations of TST

Consider a system of particles moving in a box at thermal equilibrium, under their mutual interactions. In the absence of any external forces the system will be homogenous, characterized by the equilibrium particle density. From the Maxwell velocity distribution for the particles, we can easily calculate the equilibrium flux in any direction inside the box, say in the positive x direction, Jx = p(vx), where p is the density of particles and dvxVx exp(—fimv /2 ). 

Obviously, this quantity has no relation to the kinetic processes observed in the corresponding nonequilibrium system. For example, if we disturb the homogeneous distribution of particles, the rate ofthe resulting diffusion process is associated with the net particle flux (difference between fluxes in opposing directions) which is zero at equilibrium. 

The reaction process does not disturb the thermal distribution within the reactant and product wells. This assumption is based on the observation that the timescale of processes characterized by high banders is much longer than the timescale of achieving local thermal equilibrium in the reactants and products regions. The only quantity which remains in a nonequilibrium state on this long timescale is the relative concentrations of reactants and products. 

This is far less simple than it sounds after a particle has crossed the barrier, say from left to right, its fate is not yet determined. It is only after subsequent relaxation leads it toward the bottom of the right well that its identity as a product is established. If this happens before the particle is reflected back to the left, that crossing is reactive. Assumption (2) in fact states that all equilibrium trajectories crossing the barrier are reactive, that is, they go from reactants to products without being reflected. For this to be a good approximation to reality two conditions should be satisfied  

The barrier region should be small relative to the mean free path of the particles along the reaction coordinate, so that their transition from a well defined left to a well defined right is undisturbed and can be calculated from the thermal velocity. 

Once a particle crosses the barrier, it relaxes quickly to the final equilibrium state before being reflected to its well of origin. 

These conditions cannot be satisfied exactly. Indeed, they are incompatible with each other the fast relaxation required by the latter implies that the mean free path is small, in contrast to the requirement of the former. In fact, assumption (2) must fail for processes without barrier. Such processes proceed by diffusion, which is defined over length scales large relative to the mean free path of the diffusing 

---

Further investigations into the classical foundations of TST that helped elucidate relationships between classical trajectories and TST. 

With this replacement of the strong collider assumption now commonplace, the term RRKM theory has become largely synonymous with quantum TST for unimolecular reactions, and we use this terminology here. The foundations of RRKM theory have been tested in depth with a wide variety of inventive theoretical and experimental studies [9]. While these tests have occasionally indicated certain limitations in its applicability, for example to timescales of a picosecond or longer, the primary conclusion remains that RRKM theory is quantitatively valid for the vast majority of conditions of importance to chemical kinetics. The H + O2 HO2 OH + O reaction is an example of an important reaction where deviations from RRKM predictions are significant [10, 11]. The foundations of RRKM theory and TST have been aptly reviewed in various places [7, 9, 12-15]. Thus, the present chapter begins with only a brief... 

Advancements in TST have been well documented in the literature over the past 23 years [9-16]. Much of the work on TST has focused on understanding the dynamical foundations of the theory and the extension of the theory to allow for quantitatively accurate estimates of rate constants. Advancements in these areas can be attributed to the fact that the TST expression for the classical equilibrium rate constant can be formulated by making a single approximation, Wigner s fundamental assumption. 

In 1981 Pechukas wrote [12] Transition state theory (TST) is 50 years old, and it is a tribute to the power and subtlety of the theory that work on the foundations of it is still a respectable and popular activity . To a large extent, this is still true today. The development and advancement of TST can be credited to many workers over the years. H. Eyring, M. Polanyi, and E. Wigner... 

We thank the authors of the numerous rosette publications, which names are on the reference list, theEU for the MCR Grant to Dr. J. J. Garcia-L6pez and the Dutch Technology Foundation for financial support (project number TST 4624) to Dr. J.M.C.A. Kerckhoffs. Dr. M. Crego-Calama s research has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Science. 

Over the years, TST has been modified and corrected for kinetic effects of tunneling, barrier recrossing and medium viscosity, yet, developing a theory that will explain such a phenomenon is an on-going challenge. The next section describes attempts to lay a general foundation for such a theory. 

We can be confident that various incarnations of Eyring s TST will continue to provide the foundation for our qualitative and quantitative understanding of chemical reaction rates throughout the next century. 

In atomic scale simulations, there is often a clear separation of timescales. The rate of rare events, e.g., chemical reactions, in a system coupled to a heat bath can be estimated by evaluating the free energy barriers for the transitions. Transition State Theory (TST) [9] is the foundation for this approach. Due to the large difference in time scale between atomic vibrations and typical thermally induced processes such as chemical reactions or diffusion, this would require immense computational power to directly simulate dynamical trajectories for a sufficient period of time to include these rare events. Identification of transition states is often the critical step in assessing rates of chemical reactions and path techniques like the nudged elastic band method is often used to identify these states [10-12,109]. 

---