J. Phys. Chem. 1996, 100, 12771-12800 12771 Current Status of Transition-State Theory Donald G. Truhlar* Department of Chemistry and Supercomputer Institute, UniVersity of Minnesota, Minneapolis, Minnesota 55455-0431 Bruce C. Garrett EnVironmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, 902 Battelle BouleVard, MS K1-96, Richland, Washington 99352 Stephen J. Klippenstein Department of Chemistry, Case Western ReserVe UniVersity, CleVeland, Ohio 44106-7078 ReceiVed: December 18, 1995; In Final Form: February 26, 1996X We present an overview of the current status of transition-state theory and its generalizations. We emphasize (i) recent improvements in available methodology for calculations on complex systems, including the interface with electronic structure theory, (ii) progress in the theory and application of transition-state theory to condensed- phase reactions, and (iii) insight into the relation of transition-state theory to accurate quantum dynamics and tests of its accuracy via comparisons with both experimental and other theoretical dynamical approximations. 1. Introduction further assumptions:19 (II) The reactants are equilibrated in a canonical (fixed-temperature) or microcanonical (fixed-total- Transition-state theory (TST) has a long history and a bright energy) ensemble; in the latter case one sometimes also takes future. The status of the theory was reviewed in this journal in account of conservation of total angular momentum. (III) The 19831 on the occasion of a special issue dedicated to Henry reaction is electronically adiabatic (i.e., the Born-Oppenheimer Eyring. The present status report will emphasize important separation of electronic motion from internuclear motions is developments since around that time. The reader is referred to valid) in the vicinity of the dynamical bottleneck. Within this a historical account of the origin of the theory2 and to several context we can identify several versions of transition-state theory books,3-10 pedagogical articles,11-13 handbook chapters,14-16 and and related theories. For example, conventional transition-state reviews17,18 for background. theory5 is distinguished by placing the dividing surface at the The organization of this chapter is as follows. Section 2 saddle point and equating the net rate coefficient to the one- reviews recent developments in the transition-state theory of way flux coefficient.11 Variational transition-state theory simple barrier reactions in the gas phase, the original and (VTST)20-23 is distinguished by varying the definition of the prototypical type of system on which the transition-state story dividing surface to minimize the one-way flux coefficient. has unfolded. Section 3 considers reactions without an intrinsic 24-33 barrier, i.e., reactions that have no barrier in the exoergic RRKM theory is a name for transition-state theory applied direction and whose barrier equals the endoergicity in the other to a microcanonical ensemble of unimolecular reactions, and direction. The most common examples are radical-radical and the theory also included a thermal average incorporating ion-molecule associations that do not involve curve crossing collisional effects in its original form. Using the RRKM name Downloaded via RADBOUD UNIV NIJMEGEN on January 30, 2019 at 19:29:25 (UTC). and their reverse simple bond dissociations. Section 4 considers for transition-state theory places emphasis on the reactant equilibrium assumption of Rice and Ramsperger34,35 and See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles. the theory and application of TST for reactions in condensed 36-38 phases and addresses the current “hot topic” of “environmental Kassel, which is consistent with assumption II (above) of effects” (i.e., solvent effects and phonon assistance) on reacting transition-state theory. (RRKM theory is a “quantum mechan- 28 species. ical transition-state reformulation of RRK theory”. ) In the ion dissociation literature, this theory is often termed the quasi- Throughout this article, we use “transition-state theory” as a 26 general name for any theory based in whole or in part on the equilibrium theory (QET). The transition-state theory of 24,25 fundamental assumption of transition-state theory or some unimolecular reactions was developed by Marcus and Rice, 26,27 39 quantum mechanical generalization of this assumption. Clas- Eyring and co-workers, Magee, Rabinovitch and co- 40 41-44 sically, the fundamental assumption19 is that there exists a workers, Bunker and co-workers, and a host of later 3,4,7-10,29-33 hypersurface (or surface, for brevity) in phase space with two researchers (for further references, see monographs properties: (1) it divides space into a reactant region and a on kinetics and unimolecular rate theory). product region, and (2) trajectories passing through this “divid- Transition-state theory is directed to the calculation of the ing surface” in the products direction originated at reactants one-way rate constant at equilibrium. It is usually assumed in and will not reach the surface again before being thermalized interpreting experimental data that the phenomenologically or captured in a product state. Part 2 of the fundamental defined and measured rate constants under ordinary laboratory assumption is often called the no-recrossing assumption or the conditions with reactants at translational and internal equilibrium dynamical bottleneck assumption. In addition to the funda- may be interpreted as being reasonably independent of the extent mental assumption, transition-state theory invariably makes two of chemical disequilibrium. Then the observed one-way rate constants should be well approximated by the one-way rate X Abstract published in AdVance ACS Abstracts, June 15, 1996. constants corresponding to chemical as well as translational S0022-3654(95)03748-8 CCC: $12.00 © 1996 American Chemical Society 12772 J. Phys. Chem., Vol. 100, No. 31, 1996 Truhlar et al. internal equilibrium. Once we impose the condition of reactant solvent and for which the solute molecule is sufficiently small equilibrium, the condition of transition-state equilibrium is not (few body) so that the solute itself cannot provide a heat bath an additional assumption; classically, it is a consequence of for equilibrium. Liouville’s theorem. In other words, a system with an equi- Because TST makes the equilibrium assumption, it can be librium distribution in one part of phase space evolves into a cast in a quasithermodynamic form. For example, it can be system with an equilibrium distribution in other parts of phase shown that VTST for a canonical ensemble is equivalent to space. However, the quasiequilibrium assumption of transition- minimizing the free energy of activation.71,72 state theory is that all forward crossing trajectories that A unifying element in several approaches to TST is the originated as reactants and that will proceed to products without adiabatic theory of reactions. In this theory vibrations and recrossing the dividing surface also constitute a population that rotations (as well as electronic motionssassumption III above) is in equilibrium with reactants. Although all phase points on are considered adiabatic as the system proceeds along the the dividing surface are in equilibrium, it is not necessarily true reaction coordinate. That the adiabatic assumption is related that this particular subset of all phase points is also in to the transition-state theory may be at first surprising since equilibrium. Since the dynamical and quasiequilibrium deriva- the adiabatic assumption involves global dynamics, but con- tions of transition-state theory are equivalent, the quasiequilib- nections were pointed out by many workers,73-82 and it was rium assumption breaks down whenever any recrossing occurs. shown about 15 years ago that the adiabatic theory of reactions In classical mechanics, TST provides an upper bound12,13,20-22 is identical to microcanonical variational transition-state theory to the rate constant if reactant equilibration replenishes reactant as far as overall (i.e., non-state-specified) rate constants are states fast enough (and this leads to the variational approach concerned.83,84 Perhaps more important for current thinking, by which the transition-state location is varied to minimize the though, is that local vibrational adiabaticity assumptions are calculated rate, as mentioned above). For bimolecular reactions useful for classifying variational and supernumerary transition in the gas phase, deviations from local equilibrium in the reactant states (see section 2) and making extensions of TST for state- states are usually considered small,45-50 whereas for uni- selected dynamics. molecular reactions in the gas phase one is almost always in One can distinguish the various transition-state theories by the “falloff” regime, where it is essential to consider competition the way in which quantal effects are incorporated, an area of between energy transfer repopulating reactive states and reactive considerable current interest. The choice of coordinates for depletion of those states.31 The enormous literature of the falloff representing the transition state dividing surface or the reaction problem is beyond our scope here, but we note that inclusion coordinate is another important distinguishing feature among of falloff effects is essential for using theory to predict the fate the various transition-state theories. Other classification ele- of