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Current Status of Transition-State Theory

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Current Status of Transition-State Theory 2664 J. Phys. Chem. 1983, 87,2664-2682 we cannot properly identify the reaction coordinate; this for any calculations, of solvent effects, relative rates of raises particular problems for the treatment of kinetic- similar processes, kinetic-isotope ratios, pressure influences, isotope effects. With all of these criticisms Eyring was in and a host of other important effects. full agreement. Porter’s assessment in 196290of transition-state theory The second standpoint from which we must judge makes the point very well: “On the credit side, transi- transition-state theory is: To what extent does it provide tion-state theory has an indestructable argument in its us with a conceptual framework with the aid of which favour. Since its inception, it has provided the basis of experimental chemists (and others) can gain some insight chemical kinetic theory; imperfect as it may be, it is un- into how chemical processes occur? On this score the doubtedly the most useful theory that we possess. During theory must receive the highest marks; for nearly half a the last twenty-five years its greatest success has been not century it has been a valuable working tool for those who in the accurate prediction of the rates even of the simplest are not concerned with the calculation of absolute rates reactions, but in providing a framework in terms of which but are helped by gaining some insight into chemical and even the most complicated reactions can be better physical processes. The theory provides both a statisti- understood. Most of us would agree that these comments cal-mechanical and a thermodynamic insight-one can remain valid today. take one’s choice or use both formulations. It leads to extremely useful qualitative predictions, without the need (90) G. Porter, ref 86, p 2. Current Status of Transition-State Theory Donald G. Truhlar,” Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455 Wllllam L. Hase, Department of Chemistry, Wayne State University, Detroit, Michigan 48202 and James T. Hynes Department of Chemistry, University of Colorado, Boulder, Colorado 80309 (Received: Aprii 20, 1983) We review the current status of transition-state theory. We focus on the validity of its basic assumptions and of corrections to and improvements of conventional transition-state theory. The review is divided into sections concerned in turn with bimolecular reactions in the gas phase, unimolecular reactions in the gas phase, and isomerizations and atom-transfer reactions in liquid-phase solutions. Some aspects that are emphasized are variational transition-state theory, tunneling, the assumption of an equilibrium distribution of reactants, and the frictional effects of solvent molecules. 1. Introduction simple reactions. Furthermore, it may be necessary in Transition-state theory has achieved widespread ac- many cases to calculate global regions of the potential ceptance as a tool for the interpretation of chemical re- surface, not just barrier heights, in order, for example, to action rates. The theory has, however, been less successful accurately treat systems with large tunneling effects or in its original goal, the calculation of absolute reaction systems with temperature-dependent or energy-dependent rates. A difficulty with the calculation of absolute reaction dynamical bottlenecks. rates is that it requires very accurate knowledge of po- Despite the difficulties, many workers remain enthusi- tential energy surfaces. For example, an error of 1 astic about the value of absolute rate calculations for a kcal/mol in an activation barrier causes an error of a factor variety of reasons. First of all, the methods of electronic of 5.6 in a rate calculation at room temperature, and an structure theory and the capabilities of computers are still error of 2 kcal/mol causes an error of a factor of 31. Even improving rapidly. Accurate ab initio prediction of acti- though very remarkable advances have been made in the vation barriers for simple reactions and then for increas- calculation of potential energy surfaces, it is still very hard ingly complicated ones may be “just around the cornerr. to calculate activation barriers this accurately even for However, when ab initio methods are still insufficiently 0 1983 American Chemical Society Current Status of Transltion-State Theory The Journal of phvsicai Chemistry, Vol. 87, No. 15, 1983 2665 accurate, there is always hope for empirical or semi- of the various testa constitute the current status of tran- empirical correlations, especially when these are carried sition-state theory in physical chemistry, and these are the out in the convenient transition-state-theory formalism of areas with which the present article is concerned. enthalpy of activation, entropy of activation, and heat Transition-state concepts are so widely used in chemistry capacity of activation.' Furthermore, in some applications, that complete coverage of all current approaches and ap- as for kinetic isotope effects and high-temperature rate plications in one article seems impossible. This fact itself constants, the sensitivity of the calculated results to the may be the greatest tribute to the theory and ita founders. barrier height of the potential energy surface is reduced. Recent extensions of the theory exhibit very encouraging For other cases, as in choosing between postulated mech- vitality, however, so we expect that future successes of anisms for some reactions of complicated species, definite transition-state theory and closely related models de- conclusions can be drawn even in the face of large un- scended from it will continue to make the task of complete certainties in the absolute rate constants. In still other coverage harder and harder and make our debt to the early cases, e.g., reactions of mechanistic intermediates or re- work described in the preceding paper even greater. actions under extreme conditions, there may not be a good Having stated up front that it is impossible to be complete, way to obtain information on rate coefficients by labora- we hope there will not be any misimpressions in this re- tory experiments, so theory may be the only recourse. gard, and we will spend the rest of this article summarizing Thus, absolute rate calculations can often be very useful, a few aspects of the current status of transition-state theory and we may expect them to become more and more useful that are relevant to the general applicability of the theory. as techniques for both the structural and dynamic parts We will give representative examples rather than ex- of the calculation improve. Transition-state theory, despite haustive references. its age, is still the method of choice for most attempts at Before starting on the present review we wish to single absolute rate calculations. Furthermore, it appears to be out three recent reviews of transition-state theory by Pe- almost without competition for more qualitative discus- chukas." All are very well written and are recommended sions of a wide variety of important questions in chemical strongly to anybody interested in the subject of the present kinetics, such as solvent effects. The reduction of the review. dynamics problem to the consideration of a single structure provides unique opportunities for qualitative considera- 2. Dynamical Foundations and Key Concepts tions; e.g., can this fast a rate constant or these relative A good reference to mark the beginning of the modern rates possibly be consistent with this transition-state era of transition-state theory is the classic paper of Eliason structure? and Hirschfelder.' They formulated the equilibrium rate As discussed very clearly in the preceding article by constant in terms of a Maxwell-Boltzmann average over Laidler and King: the calculation of absolute reaction rates state-to-state reaction cross sections, and in a very stim- from potential energy surfaces was cast in a particularly ulating appendix they considered more rigorously than had appealing form by Eyring in 193L3 Because of the sim- been done previously how one may make a series of ap- plicity and elegance of the resulting equations, they became proximations leading to transition-state theory. They widely used and are still widely used in their original 1935 considered state-dependent transition states and proposed form. Transition-state theory is based on a quasiequili- the idea of locating a generalized transition state to max- brium hypothesis with a simple physical interpretation. imize the free energy of activation. Later Marcus6 was to Most importantly from a practical standpoint, this qua- utilize the idea of state-dependent transition states very siequilibrium hypothesis greatly reduces the computational fruitfully, and Laidler and co-workersgand Szwarc'O fur- requirements of the theory. The reduced computational ther developed the idea of maximizing the free energy of requirements plus the ability of the theory to explain activation. These ideas form the basis, resepdively, of the general trends in preexponential factors and kinetic isotope adiabatic theory of reactionsl1*l2and canonical variational effects for bimolecular reactions and the energy depen- theory,12 two important approaches in current work on dence of rate constants for unimolecular reactions have generalized transition-state theory. Reexpression of the been the major reasons for its general popularity. equilibrium rate constant as a thermal average of reaction Modern work, however, is not so constrained by the cross sections plays an essential role in the use of modern requirement to minimize computing, and a major goal of collision theory13J4to calculate rate constants without the much current research on transition-state theory is to re- assumptions of transition-state theory, and thereby to fine and improve or correct the theory, even at increased computational expense. This is the general goal of the rapidly widening field of generalized transition-state the- (4) P. Pechhin 'Dynamics of Molecular Collisions, Part B", W. H. ories. Most attempts to improve the theory are based on Miller, Ed., Plenum Press, New York, 1976, p 239.
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