Current Status of Transition-State Theory

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1 J. Phys. Chem. 1996, 100, Current Status of Transition-State Theory Donald G. Truhlar* Department of Chemistry and Supercomputer Institute, UniVersity of Minnesota, Minneapolis, Minnesota Bruce C. Garrett EnVironmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, 902 Battelle BouleVard, MS K1-96, Richland, Washington Stephen J. Klippenstein Department of Chemistry, Case Western ReserVe UniVersity, CleVeland, Ohio ReceiVed: December 18, 1995; In Final Form: February 26, 1996 X 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 condensedphase 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 Transition-state theory (TST) has a long history and a bright future. The status of the theory was reviewed in this journal in on the occasion of a special issue dedicated to Henry Eyring. The present status report will emphasize important developments since around that time. The reader is referred to a historical account of the origin of the theory 2 and to several books, 3-10 pedagogical articles, handbook chapters, and reviews 17,18 for background. The organization of this chapter is as follows. Section 2 reviews recent developments in the transition-state theory of simple barrier reactions in the gas phase, the original and prototypical type of system on which the transition-state story has unfolded. Section 3 considers reactions without an intrinsic barrier, i.e., reactions that have no barrier in the exoergic direction and whose barrier equals the endoergicity in the other direction. The most common examples are radical-radical and ion-molecule associations that do not involve curve crossing and their reverse simple bond dissociations. Section 4 considers the theory and application of TST for reactions in condensed phases and addresses the current hot topic of environmental effects (i.e., solvent effects and phonon assistance) on reacting species. Throughout this article, we use transition-state theory as a general name for any theory based in whole or in part on the fundamental assumption of transition-state theory or some quantum mechanical generalization of this assumption. Classically, the fundamental assumption 19 is that there exists a hypersurface (or surface, for brevity) in phase space with two properties: (1) it divides space into a reactant region and a product region, and (2) trajectories passing through this dividing surface in the products direction originated at reactants and will not reach the surface again before being thermalized or captured in a product state. Part 2 of the fundamental assumption is often called the no-recrossing assumption or the dynamical bottleneck assumption. In addition to the fundamental assumption, transition-state theory invariably makes two X Abstract published in AdVance ACS Abstracts, June 15, S (95) CCC: $12.00 further assumptions: 19 (II) The reactants are equilibrated in a canonical (fixed-temperature) or microcanonical (fixed-totalenergy) ensemble; in the latter case one sometimes also takes account of conservation of total angular momentum. (III) The reaction is electronically adiabatic (i.e., the Born-Oppenheimer separation of electronic motion from internuclear motions is valid) in the vicinity of the dynamical bottleneck. Within this context we can identify several versions of transition-state theory and related theories. For example, conventional transition-state theory 5 is distinguished by placing the dividing surface at the saddle point and equating the net rate coefficient to the oneway flux coefficient. 11 Variational transition-state theory (VTST) is distinguished by varying the definition of the dividing surface to minimize the one-way flux coefficient. RRKM theory is a name for transition-state theory applied to a microcanonical ensemble of unimolecular reactions, and the theory also included a thermal average incorporating collisional effects in its original form. Using the RRKM name for transition-state theory places emphasis on the reactant equilibrium assumption of Rice and Ramsperger 34,35 and Kassel, which is consistent with assumption II (above) of transition-state theory. (RRKM theory is a quantum mechanical transition-state reformulation of RRK theory. 28 ) In the ion dissociation literature, this theory is often termed the quasiequilibrium theory (QET). 26 The transition-state theory of unimolecular reactions was developed by Marcus and Rice, 24,25 Eyring and co-workers, 26,27 Magee, 39 Rabinovitch and coworkers, 40 Bunker and co-workers, and a host of later researchers (for further references, see monographs 3,4,7-10,29-33 on kinetics and unimolecular rate theory). Transition-state theory is directed to the calculation of the one-way rate constant at equilibrium. It is usually assumed in interpreting experimental data that the phenomenologically defined and measured rate constants under ordinary laboratory conditions with reactants at translational and internal equilibrium may be interpreted as being reasonably independent of the extent of chemical disequilibrium. Then the observed one-way rate constants should be well approximated by the one-way rate constants corresponding to chemical as well as translational 1996 American Chemical Society

2 12772 J. Phys. Chem., Vol. 100, No. 31, 1996 Truhlar et al. internal equilibrium. Once we impose the condition of reactant equilibrium, the condition of transition-state equilibrium is not an additional assumption; classically, it is a consequence of Liouville s theorem. In other words, a system with an equilibrium distribution in one part of phase space evolves into a system with an equilibrium distribution in other parts of phase space. However, the quasiequilibrium assumption of transitionstate theory is that all forward crossing trajectories that originated as reactants and that will proceed to products without recrossing the dividing surface also constitute a population that is in equilibrium with reactants. Although all phase points on the dividing surface are in equilibrium, it is not necessarily true that this particular subset of all phase points is also in equilibrium. Since the dynamical and quasiequilibrium derivations of transition-state theory are equivalent, the quasiequilibrium assumption breaks down whenever any recrossing occurs. In classical mechanics, TST provides an upper bound 12,13,20-22 to the rate constant if reactant equilibration replenishes reactant states fast enough (and this leads to the variational approach by which the transition-state location is varied to minimize the calculated rate, as mentioned above). For bimolecular reactions in the gas phase, deviations from local equilibrium in the reactant states are usually considered small, whereas for unimolecular reactions in the gas phase one is almost always in the falloff regime, where it is essential to consider competition between energy transfer repopulating reactive states and reactive depletion of those states. 31 The enormous literature of the falloff problem is beyond our scope here, but we note that inclusion of falloff effects is essential for using theory to predict the fate of reaction intermediates in complex mechanisms. For example, the dissociation rate of CH 2 ClO in N 2 gas at 1 atm and 600 K is an order of magnitude less than the infinite-pressure TST value. 51 There is another reason why reactant equilibrium states might be out of equilibrium with transition states in unimolecular gas-phase reactions, namely that the reactant phase space might have internal bottlenecks or be nonergodic and in particular metrically decomposable, which again would violate the fundamental assumptions of TST. 4,41,42,44,52-56 Similar nonequilibrium issues arise for reactions in solution. In order for transition-state theory to be valid, the coupling between the solvent and the reacting solute molecules must be sufficiently strong to maintain an equilibrium population of reactants. If the coupling is too small, the rate of reaction is limited by the flow of energy into the reaction coordinate from the environment. This energy diffusion regime in liquids is similar to the low-pressure falloff region of unimolecular rate theory. This region of low-to-moderate coupling has been the subject of much research dating back to the seminal work of Kramers. 57 Recent efforts have sought to obtain a unified theory of the low- and high-friction theories. Since energy diffusion is not the emphasis of this review, we just provide a few references We note that in some cases weak coupling between solute and solvent can lead to seemingly anomalous behavior when analyzed in terms of effective one-dimensional models such as Kramer s theory. When the coupling between the solute reaction coordinate and the environment is modeled in terms of just the solute-solvent coupling, the theory predicts that the system is in the energy diffusion regime. However, strong coupling between the reaction coordinate and internal modes of the solute can lead to much faster dissipation of energy and fast equilibration (i.e., the system is not in the energy diffusion regime). This can be included in the theory if the internal modes of the solute are included in modeling the coupling between the reaction coordinate and environment. This implies that energy diffusion should be rate limiting only for those systems with weak coupling between the solute and solvent and for which the solute molecule is sufficiently small (few body) so that the solute itself cannot provide a heat bath for equilibrium. Because TST makes the equilibrium assumption, it can be cast in a quasithermodynamic form. For example, it can be shown that VTST for a canonical ensemble is equivalent to minimizing the free energy of activation. 71,72 A unifying element in several approaches to TST is the adiabatic theory of reactions. In this theory vibrations and rotations (as well as electronic motionssassumption III above) are considered adiabatic as the system proceeds along the reaction coordinate. That the adiabatic assumption is related to the transition-state theory may be at first surprising since the adiabatic assumption involves global dynamics, but connections were pointed out by many workers, and it was shown about 15 years ago that the adiabatic theory of reactions is identical to microcanonical variational transition-state theory as far as overall (i.e., non-state-specified) rate constants are concerned. 83,84 Perhaps more important for current thinking, though, is that local vibrational adiabaticity assumptions are useful for classifying variational and supernumerary transition states (see section 2) and making extensions of TST for stateselected dynamics. One can distinguish the various transition-state theories by the way in which quantal effects are incorporated, an area of considerable current interest. The choice of coordinates for representing the transition state dividing surface or the reaction coordinate is another important distinguishing feature among the various transition-state theories. Other classification elements include the recognition of multiple dynamical bottlenecks that reflect flux through one another 73,85-89 or other approximations for estimating how much flux recrosses the assumed dynamical bottleneck. As one begins to include such transmission coefficients, generalized transition-state theory begins to approach accurate dynamics; the precise border where a theory stops being a generalized transition-state theory and becomes something else, i.e., between what one calls a statistical transition-state theory and what one calls a dynamical theory, is certainly a matter of taste and semantics that cannot be resolved here. It is worthwhile to note, though, that a better pair of names for what is usually meant by this distinction would be local dynamics and global dynamics theories since the essence of transition-state theory is that it is based on the local dynamics at the dynamical bottleneck. Moving into the condensed phase, one finds yet other classifying elements: How is solute separated from solvent? Is friction included by letting solvent enter into the reactant coordinate, or is it added on? Does one assume linear response of the solvent to solute motions? And so forth. Recognizing these generalizations, extensions, and distinctions, we are left with a host of closely related transition-state theories. Their usefulness for analyzing, correlating, predicting, and understanding a wide variety of rate processes and dynamical behaviors is beyond a doubt. What, though, is the current status of the theory? 2. Simple Barrier Reactions In 1983, one might have written: Transition-state theory is basically a classical theory because the fundamental assumption of transition-state theory for either bimolecular or unimolecular reactions (that one can define a phase space surface dividing reactants from products such that trajectories passing through this surface do not pass through it a second time on the time scale between thermalizing collisions with third bodies) is intrinsically classical. To apply transition-state theory in a quantal world, we combine classical assumptions from transi-

3 Current Status of Transition-State Theory J. Phys. Chem., Vol. 100, No. 31, Figure 1. Typical example of a flux autocorrelation function through a transition state as a function of time. tion-state theory with quantal or semiclassical prescriptions for quantizing bound degrees of freedom and for including tunneling and nonclassical reflection of the reaction coordinate motion. Transition-state theory is relevant to the total flux from reactants to products; and only for modes with a well-understood adiabatic or diabatic correlation between reactants and a transition state or between a transition state and products does transition-state theory provide state-selective information. The goal of extending the theory is to find suitable ways to merge classical and quantal concepts and to exploit adiabatic connections when possible. Today one might agree with most of this but look at it differently and say: Transition states are quantum mechanical metastable states with intrinsic energies, widths, transmission coefficients, and partial widths. Their widths (in energy space) are due to (or control, depending on the point of view) their lifetime and the extent of tunneling. Their partial widths (in the language of resonance scattering theory 90,91 ) give information about state-selective processes, and their transmission coefficients give a measure of their contribution to total reactivity. To some extent the properties of transition states can be understood in classical terms, but classical approximations that do not correspond closely enough to the true quantal nature of the transition states will often predict qualitatively incorrect results. Our goal is to develop efficient and accurate ways to calculate the properties of the quantized transition states or appropriate classical analogs when degrees of freedom are classical enough. In contrast to the above, a 1984 review 92 of the roles played by metastable states in chemistry did not even mention barrier states. This shifting perspective indicates a new appreciation for the presence of quantized transition-state features in accurate quantal reaction probabilities. A shifting paradigm raises new questions (and eyebrows!), and many of these questions are not answered yet. Thus, it is an exciting and challenging time. The difficulty of incorporating quantum effects into transitionstate theory essentially results from the classical nature of the concept of one-way flux through a surface. The direction of motion that takes the system through the surface (i.e., the coordinate orthogonal to the surface) will be called the reaction coordinate s. Specifying a one-way flux through a surface requires simultaneous specification of the precise value of s and of the sign of its conjugate momentum p s. But when s is specified precisely, we are forbidden by the uncertainty principle from knowing anything about p s and in particular from knowing its sign. The uncertainty principle applies to any pair of noncommuting quantum mechanical variables, 93 and all attempts to translate the local one-way flux concept that is intrinsic to transition-state theory into quantal language result in noncommuting variables 94 and thus are uncertain. Although transition-state theory as originally formulated is fundamentally a time-independent theory of stationary-state reaction processes, considerable insight can be gained by formulating it in time-dependent language. Both languages played an important role in the 1930s with Eyring s focus on the static picture of time-independent quasiequilibrium between reactants and transition states 95 and Wigner s focus 19 on the fundamental no-recrossing dynamical assumption discussed in the Introduction. The quasiequilibrium language has dominated textbooks, but for at least 25 years advances in TST methodology have been dominated by the dynamical picture. The timedependent approach to reaction rates was cast as a flux-flux time correlation function C f (t) ) F(t)0) F(t), where F is the net flux through a dividing surface at time t, by Yamamoto, 96 who also explained the relation of this formulation to transitionstate theory. The idea was further explicated in classical mechanical terms by various workers. 71,97,98 Miller et al. 99,100 proposed a quantum mechanical method to calculate Yamamoto s time-correlation function. This leads to accurate reaction rates if fully converged, but it also affords the possibility of a short-time approximation that might be considered a quantal generalization of the transition-state approximation. A typical time dependence for the quantal fluxflux correlation function is shown in Figure 1. The accurate rate is obtained by integrating C f (t) from t ) 0 to t ) ; the proposed transition-state approximation 100,101 is to stop the integral at t ) t 0 where t 0 denotes the first time at which C f (t) ) 0. We will call this short-time quantum TST. Hansen and Anderson 102 have suggested another approximation to the quantum mechanical flux-flux correlation function in which the flux-flux correlation function for a parabolic barrier is fitted so that it reproduces the actual correlation function and its second derivative at t ) 0. Clearly, at times t ) t 0 + ɛ, where ɛ is small, recrossing is dominating over flux moving in the initial direction of the wave packet, but due to the uncertainty principle, wave packets necessarily have a spread of momenta and coordinates, and the front or fastest edge of the wave packet may have already recrossed the dividing surface before t ) t 0, whereas other parts of the wave packet have not yet revealed to what extent they will react or reflect. Thus, the short-time approximation is not a direct analog of the classical norecrossing assumption. Furthermore, once one has the apparatus to calculate F(t)0) F(t) from t to t 0, it is often possible to continue on to convergence Day and Truhlar 107 proposed a variational version of the short-time quantum TST (V-ST- QTST) in which they varied the location of the dividing surface to minimize the calculated rate. Comparing the results to accurate quantum dynamics for the O + HD, H + OD, and D + OH reactions, they found typical differences of less than 10% with, however, an error of a factor of 1.8 for D + OH at 200 K. Such calculations are very powerful, but since they intrinsically involve global dynamics, they do not provide the same conceptual picture or applicability to complex systems as transition-state theory. Park and Light 103 and Seideman, Miller, and Manthe have provided other formulations of the exact rate constant based on the flux concept, and thus these formulations are related to transition-state theory with global dynamics extensions. Another perspective on the time-correlation function approach is offered in the work of Voth et al. 113 This leads to an expression for the rate constant in which the dynamical effects are delineated from the energetic factors. In separate work, Voth et al. 114 proposed a Feynman path integral 115,116 (PI) formulation of quantum mechanical transition-state theory (QTST) based

4 12774 J. Phys. Chem., Vol. 100, No. 31, 1996 Truhlar et al. on earlier work of Gillan Path-integral quantum transition-state theory (PI-QTST) has recently been reviewed by Voth, 120,121 including a review of applications and extensions of the method. A similar approach has been presented by Stuchebrukhov. 122 These path-integral-based methods offer a convenient way to include quantum mechanical effects for all modes of the system on an equal footing and also provide a means for including anharmonic effects of the potential energy surface. To date, the application of the method to gas-phase reactions has been limited to the collinear H + H 2 reaction, 114 and the advantages of the method have been more fully exploited for treating condensed-phase systems (see section 4). The concept of a quantum state provides an alternative (and perhaps more natural) way to incorporate the inevitable broadening associated with the uncertainty principle. Thus, whereas a classical equilibrium state of a bound oscillator has x ) x e and p x ) 0, adding zero-point motion prevents violation of the uncertainty principle. This same concept can be used for transition-state levels. (We will say levels in this context because transition-state states involves using the classical and quantal meanings of state too close to each other.) Transitionstate levels, however, are unbound states. Because unbound states form a continuous spectrum, it has not been clear how to relate the discrete levels ( states ) in the quantized transitionstate partition function ( sum over states ) to a continuum of unbound quantal states. Recently, though, it has been pointed out that transition-state levels are associated with complexenergy poles of the scattering matrix (S matrix); such poles are variously called metastable states, resonances, decaying states, quasibound states, or Siegert states. (Poles of the S matrix with real energies are the ordinary bound states of a system.) The whole theory of metastable quantal states (quantum mechanical resonance theory, a branch of quantum mechanical scattering theory) immediately becomes available when the S matrix pole identification is made, and this provides a powerful tool for analysis and interpretation of transition-state phenomena with, however, one rather serious limitationsnamely that transitionstate levels are overlapping, broad resonances rather than isolated, narrow resonances (INRs), whereas many of the theoretical results 90,91 of resonance theory apply quantitatively only to INRs. Nevertheless, resonance theory has a lot to offer for conceptualizing transition states, and it does provide important quantitative guidance; e.g., it allows us to assign lifetimes to transition states, and it provides much needed insight into the role of initial rotational excitation enhancing or inhibiting reactivity. 126 The discussion of transition-state levels as quantized discrete states whose effects on reactivity can be seen individually was initiated a few years ago by the analysis of accurate quantum scattering calculations on H + H 2, O + H 2, and X + HX where X is a halogen. 123,125,126, (The interpretation using the concepts of resonance theory arose as one aspect of these studies.) Individual quantized-transition-state energy levels have now been seen and discussed for several reactions, namely H + H 2, 123,125,130 D + H 2, 132 O + H 2, 128,129,131 F + H 2, Cl + H 2, 132,135 H + XH, 125 He + H 2+, 136,137 Ne + H 2+, H + O 2, Li + HF, 145 and O + HCl. 146 The subject has been reviewed very recently. 132,147 A highlight of this kind of analysis is the ability to sort out state-to-state-to-state reaction probabilities, i.e., from a specified level of the reactants to a specified level of the transition state to a specified level of the products. 123,126,131 This provides new insight into the origins of the effective threshold energies for reactions of individual vibrational-rotational states and how these depend on the quantum numbers. 124,127,132 Sadeghi and Skodje 148 have found the quantized transitionstate resonances associated with the reaction barriers for the H + H 2 and D + H 2 reactions by using spectral quantization techniques. The resonance wave functions clearly demonstrated the localization of probability densities on the variational transition state, and their procedures allow one to distinguish various kinds of transition-state resonances Their analysis of the time-dependent quantum mechanics of barrier resonances leads to an understanding of the line shape in terms of the sequence of poles responsible for the barrier resonances. 152 Experimental observation of quantized transition-state structure is impeded by the difficulty of isolating contributions from individual total angular momenta J since the structure is likely to be harder to resolve or even unresolvable if spectra corresponding to various J are added together. Nevertheless, quantized structure associated with transition states or other states with significant amplitude in the transition-state region does appear to have been observed in anion photodetachment spectra, photofragmentation spectra, and photoisomerization experiments. 166 Even when discrete levels of the transition state are not observed, anion photodetachment spectroscopy provides a powerful probe of the transition-state regions. 167 In the most definitive of the experimental observations, Kim et al. 160 and Green et al. 158 point out that, for the photodissociation of triplet ketene, observing the j ) 2 product state of CO gives a signal with extra structure in the first several hundred cm -1 above threshold, which they interpret as an enhanced role of flux through the CCO bend excited states of the transition state. This is fully consistent with theoretical analysis 131 which shows that state-selected reaction probabilities often provide a less clustered view of certain levels of the quantized transition state. Kim et al. and Green et al. interpreted the spectrum as showing a 250 cm -1 spacing for the CCO bend, in excellent agreement with ab initio electronic structure calculations 168 that predict 252 or 282 cm -1 for this spacing, depending on the level of theory. Electronic structure theory also predicts levels at , , and cm -1 that were not observed. In order to express the usual rate constant k(t) in terms of quantized transition states, we start with the contributions of individual total energies E and total angular momenta J. First, we note that the canonical-ensemble rate constant k(t) for a temperature T is 83 where β ) (k B T) -1, k B is Boltzmann s constant, F R (E) is the density of reactant states (per unit energy and volume for bimolecular reactions, per unit energy for unimolecular reactions), Q R (T) is the partition function of reactants (per unit volume for bimolecular reactions and unitless for unimolecular reactions), and k(e) is the rate constant in a microcanonical ensemble with energy E. The latter is given by where F R (E,J) is the density of states of a given total angular momentum, and k(e,j) is the rate constant for the fixed-j microcanonical ensemble. Note that and k(t) ) de e -βe F R (E) k(e) Q R (T) k(e) ) (2J + 1)F R (E,J) k(e,j) J F R (E) (1) (2) F R (E) ) (2J + 1)F R (E,J) (3) J

5 Current Status of Transition-State Theory J. Phys. Chem., Vol. 100, No. 31, thus k(t) ) Q R (T) ) de e -βe F R (E) (4) de e -βe (2J + 1)F R (E,J) k(e,j) J de e -βe (2J + 1)F R (E,J) J Finally, for a bimolecular reaction, the E and J resolved rate constant is given by k(e,j) ) N(E,J) hf R (E,J) where h is Planck s constant, and N(E,J) is the cumulative reaction probability defined as 86,130,169 In eq 7, P ifj (E,J) is a fully state-resolved quantal transition probability from state i to state j at total energy E and total angular momentum J, and the sums run over all energetically accessible states i of the reactants R and all energetically accessible states j of the products P. Finally putting (6) into (1) yields k(t) ) Equations 1-8 are exact. 78,170 In TST, N(E,J) becomes the number of energy states of the transition state with total angular momentum J and energy below E, and the numerator of eq 8 becomes the canonical partition function. Chatfield et al. 130 interpreted the results of accurate quantal dynamics results using the relation where P R (E,J) is the probability of reaction at energy E associated with a quantized transition state R of total angular momentum J with a particular value M J of the component of total angular momentum along an arbitrary space-fixed axis. This relation may be understood most easily by comparing it to earlier expressions based on adiabatic 78,84,171 or separable 169 theory or later work based on resonance theory. 172,173 In the latter approach, the R and J quantum numbers specify a transition-state resonance. Because the resonances are not isolated, the transmission coefficients depend quantitatively on the effects of additional poles of the S matrix, farther away in energy or farther off the real axis. (This complexity is an intrinsic feature of interpreting the dynamics with overlapping, broad resonances.) Putting (9) into (8) yields k(t) ) Note that eq 10 multiplies by (2J + 1) rather than summing over M J. Note that, as a function of total energy E, P R (E,J) typically rises from 0 to some finite value κ R (e1) in the vicinity of the resonance energy E R for state R; κ R is the level-specific transmission coefficient for transition-state level R. (5) (6) N(E,J) ) P ifj (E,J) (7) i R j P de e -βe (2J + 1)N(E,J) J hq R (T) (8) N(E,J) ) P R (E,J) (9) R de e -βe (2J + 1) P R (E,J) J R hq R (T) (10) Equation 10 takes a form that allows a connection to be made between the resonance-state approach to QTST and path-integral QTST. It is convenient to define an aggregate quantum number τ ) (R, J, M J ). First define a Boltzmann-weighted transmission probability for quantized transition state τ by γ τ ) dɛ e -βɛ P τ (E τ +ɛ,j)/k B T (11) [For a narrow resonance, P τ is a step function, κ τ θ(ɛ), where θ is a Heaviside function, and γ τ ) κ τ ]. Since P τ is independent of M J, we can write eq 10 as k B T k(t) ) exp(-βe τ )γ τ (12) hq R (T) τ Equations 11 and 12 are identical to the resonance-state version of quantum TST (RS-QTST) proposed previously. 172 In that theory the transmission coefficient γ τ is evaluated by assuming that P τ increases from 0 to κ τ in the vicinity of transition-state level E τ with an energy dependence determined by the properties of the metastable barrier state, and we make the transition-state assumption that κ τ equals unity. If, instead, we consider a unimolecular reactant and write the complex energies of the reactant resonances by E res (τ) ) E τ - iγ τ /2 (13) and assume that Γ τ is small, then eqs 10 and 12 with γ τ replaced by Γ τ /k B T take the form of the rate constant from the imaginary free energy (Im F) method k(t) ) 2 exp(-βe τ ) Im E res (τ) hq R (T) τ 2k B T/hQ R Im Q (14) where Q is the partition function for the entire system. In principle, the partition function Q is real and should diverge for a dissociative system, but complex values are obtained by analytical continuation. For a unimolecular reaction Q R and Q both refer to the whole system, and eq 14 becomes k(t) 2 Im F R /h (15) where F R denotes the reactant free energy. Makarov and Topaler 177 have demonstrated that the centroid-constraint relationship in the PI-QTST method can be derived from the Im F method. The Im F formulation can be used for describing escape from a metastable potential, and as shown above, it has a formal resemblance to the RS-QTST, which can be used to describe either unimolecular or bimolecular reactions. However, the analogy is only formal since the Im F formulation is based on the properties of the metastable states of a unimolecular reactant that decays into a continuum, while RS-QTST is based on the properties of the metastable states 123,172 at the barrier top independent of the nature of the reactant. It would be interesting to make further connections between these viewpoints. It is instructive to compare eq 10 to the most widely employed version of variational transition-state theory (VTST), namely canonical variational theory (CVT) with multidimensional ground-state transmission coefficients κ(t). 15,84,181 In this theory (to be denoted VTST/MT for VTST with multidimensional tunneling), one defines a reaction path and a coordinate s denoting progress along this path. Then we consider a sequence of trial transition states with s fixed at various points along the path. Since s is fixed, these transition states are systems with dimensionality one less than the full dimensionality of the

6 12776 J. Phys. Chem., Vol. 100, No. 31, 1996 Truhlar et al. system. For each value of s, we calculate the energy levels E R (s,j) of the trial transition state, and we find the location that minimizes the transition-state canonical partition function; we call the s value at the location of this variational transition state s * (T). In the absence of quantization, this procedure would be equivalent to finding the dividing surface with minimum one-way flux coefficient. The rate constant calculated from the partition functions at s * (T) is called k CVT (T); it corresponds to the overbarrier contribution to the rate. We also calculate an approximate probability of reaction P G (E) associated with the ground state of the variational transition state to include the effects of tunneling at energies below the effective barrier height and nonclassical reflection (diffraction by the barrier top) at energies above the effective barrier. In the most reliable version of such theories, the tunneling calculation is multidimensional, and the transmission coefficient may be denoted MT (multidimensional tunneling) or MTG (multidimensional tunneling based on the ground state). Finally, the composite rate expression is where k VTST/MT (T) = κ MTG (T) k CVT (T) (16) κ MTG (T) ) de e -βe P G (E) E* G de e -βe ) (k B T) -1 e βe * 0 de e -βe P G (E) k CVT (T) ) k B T J R (2J + 1)e -βe * hq R (T) R (J) (17) (18) E* R (J) is shorthand notation for E R [s * (T),J], and E * G is shorthand for E* 0 (J)0). Combining (16)-(18) yields k VTST/MT (T) ) ) de e -βe (2J + 1) P G (E) e -β[e * J R hq R (T) R (J)-E * G ] de e -βe (2J + 1) P G [E - E* R (J) + E G * ] J R hq R (T) (19) Comparing (19) to (10) shows that variational transition-state theory with a ground-state transmission coefficient is equivalent to assuming that P R (E,J) ) P G [E - {E* R (J) - E* 0 (J)0)}] (20) i.e., the transmission probability for each successive transition state is just shifted from that of the ground state by the excitation energy evaluated at the variational transition state. For gas-phase bimolecular reactions, accurate quantum rate constants can be obtained by solving the Schrödinger equation by scattering theory, e.g., by expanding the scattering wave function in a basis set and converging the calculation with respect to the basis set and all numerical parameters. At the time of the previous status report, 1 VTST/MT theory had been tested against accurate quantal results for 33 cases (30 reactions in a collinear world and 3 in a three-dimensional world). For the 33 cases, conventional transition-state theory was accurate within a factor of 2 in only 8 cases and within a factor of 5 in only 23 cases, whereas VTST/MT theory was accurate within 54% in 27 cases and within a factor ranging from 0.32 to 1.81 in all cases. Since then, VTST/MT has been tested even more widely against accurate quantal results 23,135, with many more tests in three dimensions. 135,182,185, These tests confirm the previous conclusions, and the most recent tests, for D + H and Cl + H in 3D, show very high accuracy over a wide range of temperature. For D + H 2 VTST/MT calculations 192 of k(t) agreed with accurate quantal calculations 191 performed on the same potential energy surface eight years later within 17% or better for the whole temperature range of K. Agreement with experiment, 193 which also tests the potential energy surface, 194 is 32% or better over this temperature range. The Cl + H 2 reaction provides another test of VTST/MT methods; for this reaction the average absolute deviation of VTST/MT calculations from accurate quantum dynamics is only 10% over the K range. 135 The Cl + H 2 reaction also provides 135 a very clear example of a supernumerary transition state 131 of the first kind. Such a transition state may be interpreted as a secondary maximum on a vibrationally adiabatic potential curve for some level, say β. If a system in a lower level, say R, passes the location of the highest maximum (the variational transition state) of the curve for level β, it may then undergo a vibrationally nonadiabatic R f β transition and reflect from the supernumerary transition state of level β. Thus, such nonconventional transition states provide a detailed explanation of why the transmission coefficients are less than unity in some cases. The O + H 2 reaction has also provided a prototype test case for VTST/MT. VTST/MT calculations have been tested successfully against accurate quantal dynamics, 185,187,189 and they have been very successful at interpreting experimental kinetic isotope effects. 195,196 O + H 2 (V)1) f OH + H and OH(V)0,1) + H 2 (V)0,1) f H 2 O + H, where V denotes a vibrational quantum number, have been used to test the extension of VTST concepts to predict state-selected rates for high-frequency mode excitation. 187, The first analysis of quantized transitionstate structure in a cumulative reaction probability based on accurate quantal calculations 128 was reported for the O + H 2 system by Bowman, 129 who noted the existence of structure due to a bend excited transition state. Most recently, this system provided the most fertile ground to date for analyzing the detailed state-to-state dynamics of a chemical reaction in terms of variational and supernumerary transition states observed in accurate quantum dynamics calculations. 131 A critical element in the success of VTST/MT theory is the accuracy of the methods used to calculate P G (E). In particular, the most accurate calculations are based on multidimensional semiclassical methods for estimating the low-e tunneling tails that are critical for the thermal rate constants because of the e -βe factor in eq 19. To be satisfactory, multidimensional tunneling methods must account both for zero-point variations along the reaction path and for corner cutting on the concave side of the reaction path. 170,186, The current status is that convenient and accurate multidimensional tunneling methods are available for both small and large curvature of the reaction path, namely the centrifugal-dominant small-curvature semiclassical adiabatic ground-state (CD-SCSAG) method 211,212 in the former case, and the large-curvature ground-state method, version-3 (LCG3) 211, in the latter. These methods differ in the extent to which corner-cutting tunneling occurs, as discussed elsewhere. 186,216 For atom-diatom reactions, a leastaction method 208 has been used to optimize the choice of tunneling paths more finely between these limits whereas for

7 Current Status of Transition-State Theory J. Phys. Chem., Vol. 100, No. 31, polyatomic reactions it has been considered sufficiently accurate to just optimize the tunneling path by choosing between the CD-SCSAG or LCG3 algorithms whichever predicts the greater amount of tunneling. 215,217 Small-curvature tunneling calculations, like VTST itself, require a knowledge of the reaction path and the energies and frequencies along it; however, this information may be required over a longer section of the path than is required for VTST, especially at low temperature, where tunneling through the lower, wider part of the barrier may contribute significantly to the rate. Large-curvature tunneling requires, in addition, some information about the potential in a wider region (termed the reaction swath) on the concave side of the reaction path. 184,209,210, The extension of validated multidimensional tunneling approximations to arbitrary polyatomic systems will allow a considerable range of applications in the future. Some recent examples based on electronic structure calculations for bimolecular reactions are the reaction of CF 3 with CH 4, 215 the reactions of OH with H 2, 200 CH 4, 221 CD 4, 222 C 2 H 6, 223 and NH 3, 224 and the reactions of H with H 2 O, 225 NH 3, 226,227 and CH In the CD 4 case the predicted CH 4 /CD 4 KIE at 416 K was 4.5, in poor agreement with the only available measured value of 11. However, a new measurement yielded 4, in excellent agreement with the prediction. For the first six examples mentioned above, the transmission coefficients at 300 K are calculated to be 67, 5.9, 8.7, 7.6, 3.2, and 5.6, respectively. For the unimolecular [1,5] sigmatropic rearrangement of cis-1,3-pentadiene (an internal hydrogen transfer), the transmission coefficient for H transfer is 6.5 at 470 K, leading to a KIE for H vs D transfer of 4.9, 212 in excellent agreement with the experimental value of 5.2. The role of tunneling in this system had been very controversial prior to this full 36-dimensional VTST/MT calculation. Miller, Hernandez, and co-workers have considered a transition-state-like approximation to the rate constant by using semiclassical theory in action-angle variables. A difficulty in applying this theory is that good global action-angle variables will seldom be available, if they even exist. The theory has been implemented using second-order perturbation theory. The disadvantages of this approach are that for tunneling, secondorder perturbation theory is not very accurate for representing corner-cutting effects, 207,233 and for overbarrier processes it is not very accurate for representing large deviations of the variational transition-state location from the saddle point. 172 If, however, one has a situation where the second-order theory is adequate, the theory has the advantage that it is a convenient way 234 to include mode-mode coupling. Recently, 235 a method of treating anharmonicity in modes perpendicular to the reaction coordinate has been developed for cases where they may exhibit wide-amplitude motion; this method may provide improvements in cases where anharmonicity in the perpendicular modes is more important than their coupling to the reaction coordinate and than reaction-coordinate anharmonicity (the latter two features being responsible for variational displacements of the transition state from the saddle point). Conventional (traditional) TST was concerned entirely with the properties of the saddle point (the highest internal energy point on the minimum-energy path), and indeed that myopic view of the potential energy surface was and is a strength of the theory because of the resulting low demand for structural and energetic information about the system. Modern generalized transition-state theories are still relatively modest in their needs for such input, but they typically require information along a considerable segment of the one-dimensional reaction path rather than at just two points. The local quadratic approximation along this path plays a central role in the theories as does the concept of generalized normal modes, where the generalization refers to defining such modes in a subspace orthogonal to the reaction coordinate at a nonstationary point (a point where the gradient of the potential does not vanish, as it does at a minimum or a saddle point). Even at the harmonic level, there are open research questions about coordinate systems and vibrational energy levels of generalized transition states. At stationary points (i.e., potential minima and saddle points), which are the only points at which information is required for conventional transition-state theory without tunneling, vibrational frequencies at the harmonic level are independent of coordinate system. However, for nonstationary points, even for a given choice of reaction path, harmonic frequencies depend on the coordinate system; in particular, they depend on the definition of the reaction coordinate for points off the reaction path Thus, it becomes very important to choose a physically appropriate coordinate system. Most reaction-path calculations to date use rectilinear coordinates, i.e., coordinates writeable as linear combinations of Cartesians, whereas curvilinear coordinates (such as bond stretches, valence bends, and dihedral angles) are more physical. One manifestation of the inadequacy of rectilinear coordinates is the frequent appearance of imaginary frequencies for bound modes in reaction-path calculations. Recently, a general formalism for calculating reaction-path frequencies in curvilinear coordinates has been presented and shown to eliminate (at least in the cases considered) the problem with unphysical imaginary frequencies. 239,240 Even when the frequencies are not imaginary, they may be inaccurate when computed with rectilinear coordinates; this may, for example, lead to an overestimate of the tunneling. It has been known for a long time that anharmonicity and vibration-rotation coupling can have an important effect on TST calculations, even for tight transition states; 182,231,233,241,242 however, progress in devising practical general methods for including anharmonicity of the generalized normal modes has been slow. Three promising approaches include second-order perturbation theory, 231, especially in curvilinear internal coordinates 182,218,242,249,250 where interaction terms are much smaller than in rectilinear coordinates, a classical configurational integral method, 251 and Monte Carlo methods for quantum mechanical path integrals. 246 A convenient approximation for one-dimensional hindered rotations has been presented. 215,252 Anharmonicity is even more important for loose transition states, and further discussion of anharmonic partition functions and numbers and densities of states is provided in section 3. A promising avenue for future development is the unified dynamical theory, in which recrossing corrections to VTST/ MT calculations are evaluated from trajectories beginning at a quantized variational transition state. 15, To the extent that short-time dynamics in the vicinity of a localized dynamical bottleneck determines the rate, this includes quantum effects and recrossing in a very effective way. If a particular vibrational mode is adiabatic (i.e., conserves its quantum number) from reactants to the dynamical bottleneck, or from the dynamical bottleneck to products, TST can be extended to predict state-selected rate constants for that mode. 123,131,182,192,187,193, , In some cases one can predict state-selected rates due to state-specific tunneling processes. 184 Transition-state theory has traditionally been not only the primary tool for interpreting kinetic isotope effects (KIEs) but practically the only tool. 262 The standard interpretation of KIEs involves using conventional TST to infer detailed information about transition-state structure. The modern perspective differs from the conventional one in two ways: 222, (i) VTST indicates that the geometry (and hence the force field) can be

8 12778 J. Phys. Chem., Vol. 100, No. 31, 1996 Truhlar et al. quite different for isotopic versions of the same reaction. 264 (ii) Tunneling effects are often very significant even when the KIE does not exceed the maximum allowed by conventional models that neglect tunneling. Recent progress in developing reliable multidimensional methods for including tunneling contributions in TST has assisted in identifying 189,195,209,212,215 the role of tunneling much more definitively than in the past where an Occam s razor approach often underestimated its role. Particularly noteworthy in recent work is a more sophisticated treatment of secondary kinetic isotope effects; factor analyses of partition function ratios and tunneling calculations, both based on full-dimension transition-state models and electronic structure calculations, have provided a better appreciation of the contributions to the KIEs of each kind of motion at the transition state. 217, These studies have tested the traditional view that secondary KIEs mainly reflect the position of the transition state on the loose-tight continuum of transition-state structure; a quick summary of the conclusions is that this factor is important, but high-frequency modes and tunneling are no less significant in the general case. Another general conclusion from the recent work is that real-world KIEs are much more complicated than the enticingly simplesand even more enticingly successfulspopular models would have led us to believe. Nevertheless, TST interpretations of experimental KIEs remain one of the most powerful methods for testing models of dynamical bottlenecks and for inferring transition-state structures and properties. Another kind of successful application 279 of VTST/MT to organic chemistry is provided by calculations on hydrogen 1,2- shifts in singlet carbenes. Here s-dependent zero-point effects on the effective barrier for tunneling were critical in calculating a qualitatively correct entropy of activation. Transition-state theory is most directly applicable to reactions which proceed on a single electronically adiabatic (i.e., Born- Oppenheimer) potential energy surface, although in some cases, for example, O( 3 P) + H 2 f OH + H, it has been applied to reactions proceeding on multiple surfaces. 196 A situation that occurs commonly is where the degeneracy of the single surface on which reaction occurs is smaller than the degeneracy of the reactants. This occurs, for example, in the Br reaction with H 2. The conventional treatment in the case where state splitting of occupied reactant states is small compared to k B T is to assume that the ratio of reactant to transition-state electronic degeneracy determines the average fraction of collisions that proceed on the reactive surface. 280 More generally, if all reactive surfaces are similar, one can use the ratio of the transition-state electronic partition function, counting only reactive surfaces, to the product of the reactants electronic partition functions, counting all surfaces. A recent development is that advances in quantum scattering theory have allowed this kind of assumption to be tested, and Schatz 281 has presented such a test for the reaction Cl( 2 P 3/2,1/2 ) + H*Cl f HCl + *Cl, where *Cl is a tagged Cl. Schatz concludes that the approximation is good to better than 20%. If reaction proceeds on more than one surface, and the surfaces have significantly different properties, a reasonable assumption in the context of TST is to add the rates for the different surfaces. 190 Reactions with a barrier preceded by a well provide a more complicated scenario than simple barrier reactions. Ionmolecule reactions like the Cl a- + CH 3 Cl b, Cl - + CH 3 Br, and F - (H 2 O) + CH 3 Cl S N 2 reactions fall into this more complicated class of reaction. Trajectory calculations on the former two reactions indicate that multiple crossings of the barrier region may occur, leading to a breakdown of the fundamental assumption of TST. 282,283 It is not clear to what extent these recrossing effects might be due to the classical nature of the calculations, the way in which initial phase space was sampled, the choice of dividing surface, or the nature of the assumed potential energy functionsor to real recrossing effects in the actual ion-molecule reactions. The authors note that several aspects of the dynamical behavior are explained as being due to weak coupling between the relative translational motions of reactants (or products) and the reaction coordinate (asymmetric stretch) at the transition state. 284 This raises the question of whether additional degrees of freedom (e.g., C 2 H 5 instead of CH 3 or the presence of solvent or microhydration) would significantly increase this coupling. Interestingly, ab initio TST calculations 285 of the CH 3 /CD 3 and H 2 O/D 2 O KIEs for the microhydrated F - (H 2 O) + CH 3 Cl reaction are in good agreement with experiment. There are two recent studies of the KIE for Cl - + CH 3 Br, and they reach different conclusions about agreement with experiment. 285,286 However, combining the largest-basis-set harmonic KIE (0.95) from the latter study 285 with the effect of anharmonicity (0.64/ 0.79) from the former 286 yields a KIE of 0.77, in reasonable agreement with experiment ( ) when one considers the difficulty of the experiments, of converging the electronic structure calculations, and of estimating the anharmonic effect. Gas-phase S N 2 reactions illustrate some of the major unknowns in the current status of TST Transition-state theory has had many names over the years, two of which are activated complex theory (ACT) and absolute reaction rate theory (ARRT). Nowadays, TST and ACT are considered to be exact synonyms, and ARRT has fallen out of favor. The disappearance of the ARRT moniker seems reasonable since it is much less descriptive than the other two names of the type of physical approximation involved, but actually that is probably not the main reason for the ARRT name to have fallen out of favor. When TST was first called ARRT, there was tremendous enthusiasm for combining it with electronic structure theory to predict absolute reaction rates entirely from theory. However, by the end of the 1930s it was certainly clear that the goal of predicting chemically accurate potential energy surfaces or even barrier heights was a much more difficult challenge than originally anticipated. The challenge survives today, largely unmet, but nowsin the 1990ssit seems that a bit of cautious optimism may be in order, at least for few-body reactions. Two reasons for this may be advanced. First is the realization by a wider group of practitioners that very complete basis sets cannot be avoided when reliable barrier heights are desired, e.g., the use of a single set of d functions on C, N, O, or F, which was once considered a good basis set, is now recognized as at best semiquantitiative and but a small first step toward completeness. Second is the development of practical size-consistent treatments of electron correlation that include the dominant effects of double, triple, and higher excitations. This array of techniques includes Møller-Plesset fourth-order (MP4) perturbation theory 289 sas used in the Gaussian-2 (G2) semiempirical calculations, 290 coupled cluster theory with single and double excitations and perturbative inclusion of unlinked triples [CCSD(T)], 291 and quadratic configuration interaction with singles and doubles and perturbative inclusion of unlinked triples [QCISD(T)]. 292 Note that both CCSD(T) and QCISD(T) include the dominant effects of quadruples, and CCSD(T) has the extremely important potential to make up for a poor reference set of orbitals. A non-sizeconsistent alternative that is competitive for very small systems is multireference configuration interaction (MRCI) with single and double excitations from a full-valence complete-active-space self-consistent-field (CASSCF) reference state, 267, but this method appears to have dimmer prospects for extendibility to larger systems. Complete-active-space second-order perturbation theory (CASPT2) may play an important role in filling that

9 Current Status of Transition-State Theory J. Phys. Chem., Vol. 100, No. 31, niche Empirical correction and extrapolation schemes for the high-level results, such as the bond-addivitiy-corrected MP4 (BAC-MP4), 226, scaling external correlation (SEC), 194,267,304,305 and scaling all correlation (SAC) 222,223,227, methods, are also very useful. For simple enough systems the use of such ab initio calculations has replaced older techniques of estimating entropies and energies of activation by empirical group contributions, 312,313 but the empirical procedures still play an important role for larger species and for the correlation of experimental data. TST practitioners also have considerable interest in density functional theory, which has had some notable successes for transition states but which still appears to be basically unreliable for barrier heights. 321 Density functional theory (like many other levels of electronic structure theory) appears to be more accurate for transition-state and reaction-path geometries than for absolute energies. 322,323 Many other issues in the reliability of electronic structure calculations of various types for transition states are also at the forefront of TST research, for example, the reliability of unrestricted Hartree-Fock reference functions in comparison to restricted open-shell Hartree-Fock (or in comparison to experiment). A very important aspect of using the above theories constructively is the ability to calculate analytical gradients 324 and even analytical hessians. The former capability is essential to making geometry optimizations feasible, where optimization refers to the process of finding the zero-gradient structure of a reagent or a saddle point. Geometry optimization of transition states is particularly important for comparative evaluation of possible reaction paths in complex systems, e.g., those where complexed water molecules participate in the reaction. 325,326 At present, geometry optimization is possible with some but not all of the methods mentioned above, and we can anticipate that further advances in this area will have dramatic impacts on TST applications. The reader should keep in mind that the overall level and reliability of electronic structure calculations depends not only on the level of treatment of electron correlation, as just discussed, but also on the completeness of the one-electron basis set. Thus, an encouraging development is the use of systematic basis set studies to explore the convergence of transition-state barrier heights with respect to the one-electron basis. 327 Another computational issue of great importance for VTST calculations is the efficient calculation of the reaction path itself. Many algorithms have been advanced for this. 15, A critical area of current research is designing new and more efficient ways to interface electronic structure calculations with dynamics. The goal of such work is to find ways to calculate the reaction-path and swath information needed for VTST and tunneling calculations from a minimum of electronic structure information. Two promising approaches are interpolated VTST (IVTST) 343 and VTST with interpolated corrections (VTST- IC, also called dual-level dynamics). 344 In IVTST, one carries out high-level electronic structure calculations for reactants, products, the saddle point, and perhaps one or two additional points near the saddle point; then all required reaction-path information is interpolated. (Interpolation can also be carried out by power series 345 or Shepard interpolation 346,347 methods.) In the IVTST-IC method, the high-level input is similar, but interpolation is aided by the presence of an approximate potential energy function or additional lower-level electronic structure calculations that are carried out for a longer segment of the reaction path and in the swath region. The goal of dual-level variational transition-state theory is not just to use high-level electronic structure calculations to correct the energies along a reaction path calculated at a lower level and to correct the vibrational frequencies, but also to provide improved data for optimized multidimensional tunneling calculations including reaction-path curvature and tunneling paths that deviate from the minimum-energy path by more than can be predicted by a quadratic expansion about this path and/or more than the radius of curvature. Recent successful examples of these interpolatory approaches are provided by the calculations, mentioned above, on OH + CH 4, 309 CD 4, 222 and C 2 H (IVTST) and on OH + NH (VTST-IC). IVTST and IVTST-IC are examples of direct dynamics, which is the calculation of dynamical attributes from electronic structure calculations without the intermediacy of fitting a potential energy function. These methods involve interpolation, but straight direct dynamics has also been used for VTST calculations. 200,212,215, , ,279,303,310, ,322,323,344, In straight direct dynamics, one carries out an electronic structure calculation directly every time that the algorithm requires an energy, gradient, or hessian. Recent examples of combining high-level electronic structure theory with transition-state theory for practical applications to larger systems are provided by the work of Page and co-workers on the reactions CH 3 O f CH 2 O + H 353 (a dissociation reaction with a barrier and a tight transition state), H + HNO, 354 O + NH 2, 355 and H + NH and the reactions of H, OH, and NH 2 with N 2 H In each case the reaction path was calculated by the CASSCF method, and single-point calculations with multireference configuration interaction were used to improve the energetics along these paths. These data were used as input for variational transition-state theory calculations and estimates of tunneling corrections based on the zero-curvature vibrationally adiabatic approximation for the transmission coefficient. Reaction-path methods and the interface of electronic structure theory with chemical kinetics using these methods for variational transition-state theory and semiclassical tunneling calculations have recently been reviewed. 220, As TST is being applied in recent years to complicated reactions, a question that comes up is the treatment of competing pathways for a single set of reactants In general, if both reactions are slow, and their pathways have no part in common, TST can handle this just as it handles relative rates of different reactants. When the competitive pathways share a common intermediate or when one or both of the reactions are fast, one must make additional assumptions. The simplest assumption, used in statistical phase space theory, 365,366 is that the scattering matrix is a random unitary matrix within a subset of the channels specified by some state-specific extension of TST, 367,368 e.g., those channels with accessible orbiting transition states. 3. Reactions without an Intrinsic Potential Barrier 3.1. Theory. In the VTST treatment of unimolecular reactions 33,369 the rate constant at energy E and total angular momentum J is given by k(e,j) ) N * (E,J) hf R (E,J) (21) where N * (E,J) is the number of energetically available quantum states at the variational transition state, and F R (E,J) represents the reactant density of states. In conventional TST, N * (E,J) is replaced by N (E,J), which is evaluated at the saddle point, if there is one. The detailed aspects of applying VTST to unimolecular reactions with well-defined saddle point are essentially identical to those described in section 2 for the corresponding bimolecular reactions. However, the implementation of VTST for unimolecular dissociations and their reverse

10 12780 J. Phys. Chem., Vol. 100, No. 31, 1996 Truhlar et al. associations in cases where the association process is barrierless raises new issues as do other processes without an intrinsic barrier or with very flat potentials near the dynamical bottlenecks. For such barrierless reactions, it is especially important to provide a proper treatment of the variation in location of the transition state, due to its possible wide variation from interfragment separations R as large as tens of angstroms at low energies, down to 2-3 Å at higher energies. (The extent of variation depends of course on the reaction, being larger, for example, for a typical ion-molecule reaction than for a typical radical-radical reaction. For example, for the CH 2 + H f CH 3 reaction, CASSCF-MRCI calculations indicate that the canonical variational transition state moves from a C-H distance of 3.7 Å at 159 K to 2.7 Å at 2850 K. 370 ) The separation of modes into the vibrations of the fragments, termed the conserved modes, and the rotational and orbital motions of the fragments, termed the transitional modes, has provided the basis for much of the theoretical work in this area. This separation of modes is particularly meaningful at large subsystem separations, where the reaction coordinate is well described by the distance between the centers of mass of the two fragments, and the transition state is loose ; i.e., the interacting fragments have almost free rotation. Phase space theory (PST) 365,366, is a version of TST that focuses on the energetics of the separated fragments at a completely loose transition state (i.e., rotations of the fragments are completely unhindered), and it often provides an accurate treatment when the transition state is at large separations, as in ion-molecule reactions. The simplest algorithms are based on locating the transition state at the centrifugal barriers for spherically symmetric R -n potentials. This is sometimes called the orbiting transition-state model; 378 for n ) 4 it is the model of Langevin 379 and Gioumousis and Stevenson, 380 and for n ) 6 it is the model of Gorin. 381,382 These models can be invalidated in some cases by asymmetries in the long-range potential, arising, for example, from ion-dipole and other longrange dipole interactions. (Variational transition-state theory 383 and the adiabatic channel model 384,385 have provided an accurate understanding of such effects.) Within PST, N * (E,J) is given by the total number of asymptotic rovibronic states whose effective centrifugal barriers are below the available energy, and the total angular momentum is explicitly conserved via the consideration of the vector sums of the angular momentum of each of the fragments and of the orbital motion. An alternative version of PST, as reviewed by Peslherbe et al., 378 is to locate the transition state at R ), with properties identical to those of the completely separated collision partners or products. A nonstatistical phase space theory, called the intermediate coupling probability matrix approach, includes effects of weak energy transfer in associative or dissociative half-collisions by postulating a finite but less than 100% probability of forming a complex in the former case or of coupling to a final state in the latter. 386,387 In this theory these probabilities depend on the rate of energy transfer and the half-collision duration for a nonenergy-mixing half-collision; the latter in turn depends on the asymptotic relative translational energy and the orbital angular momentum. Recognition of the intermediate-coupling nature of the system has a significant effect on the temperature dependence of association rate constants. 387 In the following, we will focus on the description of the deviations from PST-type capture rate constants due to shortrange interactions, which are critical for association of neutral molecules. At shorter separations where some of the rotational degrees of freedom have become internal rotations, librations, or bending vibrations, the intermolecular motions become increasingly strongly coupled, resulting in the breakdown of various aspects of the PST assumptions. The development of accurate and efficient procedures for treating these breakdowns has received considerable attention in recent years. The statistical adiabatic channel model (SACM) 79,80, provided an early and widely applied approach for treating such deviations. Though SACM is called an adiabatic theory, 74-80,83,84 it is based on a reference assumption of diabatic modes coupled with a statistical distribution of energy among the modes. Diagonalizing the Hamiltonian to convert the diabatic energies to adiabatic ones would not have a large effect on the sums over states under typical non-state-selected conditions, though, so the adiabatic-diabatic distinction is not generally important. N * (E,J) is then approximated in terms of the number of channels whose diabatic effective barriers are below the available energy. The primary difference of SACM from PST is the implicit implementation of asymmetries in the intermolecular potential and also the inclusion of R-dependent variations in the fragment rovibronic energies. Although VTST for a dissociation reaction may be based on a very different reference assumption of a complete randomization of all motions of reactants, its estimates of N * (E,J) are identical to the SACM estimates, when implementing the same estimates for the energetics and using equivalent symmetry factors. 83,84,396 However, the difference of adiabatic/diabatic assumptions from statistical ones does result in major differences in estimates for the product state distributions, 396 which are really beyond the realm of ordinary TST, as mentioned in section 2. Comparisons 164, with experiment, which are still more limited than one would like, suggest that a combination of adiabatic/diabatic assumptions for the conserved vibrational modes with a statistical assumption for the transitional modes is most appropriate, 400 although for larger molecules than those considered to date, one might expect some low-frequency conserved modes to couple significantly with the transitional modes. The applications of SACM have generally focused on the interpolation of energy levels from reactants to products, although some of the more recent studies have explicitly considered the energetics of long-range potentials. 392,394,395, Similar interpolations of the energetics have also been incorporated within a VTST-based method. 404,405 However, the explicit consideration of the fragment-fragment potential energies of interaction, as in many of the recent VTST studies, provides more meaningful tests of the validity of the underlying TST assumptions and also provides a direct means for making a priori predictions. The implementations of VTST for barrierless reactions are based on one of two alternative procedures. 406,407 The first alternative is based on the determination of a minimum-energy reaction path (RP) and the corresponding normal-mode harmonic frequencies and moments of inertia along the reaction path. 378,405, The transition state partition function is then generally determined on the basis of rigid rotor harmonic oscillator (RRHO) type assumptions for the overall complex employing classical expressions for the rotational motions and quantum expressions for the harmonic vibrations. The questionable validity of harmonic oscillator assumptions for the intermolecular bending motions has also led to the use of hindered rotor expressions. Unfortunately, the rigid rotor assumptions are of equal uncertainty and are difficult to remove within the RP approach. The second alternative is based on an assumed decoupling of the conserved and transitional modes with the quantity N * (E,J) evaluated via the convolution of a classical-phase-spaceintegral-based evaluation of the transitional mode contribution

11 Current Status of Transition-State Theory J. Phys. Chem., Vol. 100, No. 31, with a direct quantum sum for the conserved mode contribution In the original version of this approach, the reaction coordinate is taken as the separation between the centers of mass of the two reacting fragments. The classical treatment of the transitional mode contributions is entirely satisfactory due to their low-frequency nature as confirmed in a variety of studies (Note that tunneling is usually expected to be of negligible importance for barrierless reactions due to the typically large masses and widths for the effective centrifugal barriers, in which case quantum corrections for the reaction coordinate are also unimportant.) The ability to directly and accurately incorporate quantum mechanical effects for the highfrequency modes provides one of the key advantages of TST methods over classical-trajectory-based methods. In some cases, this phase-space-integral-based VTST (PSI-VTST) gives similar results to the RP approach. 378 An important feature of the PSI-VTST approach is its classically accurate treatment of the interfragment couplings, low-frequency-mode anharmonicities, and low-frequency-mode vibration-rotation couplings while conserving J. Recent advances in the methodology, involving analytic integrations over the momentum portions of the phase space integrals, have yielded algorithms of sufficient efficiency to be widely applicable. 414, Furthermore, simplified expressions providing extremely efficient approximate estimates have also been presented. 421 Unfortunately, the implicit assumption of a centerof-mass separation distance reaction coordinate breaks down at shorter separation distances, particularly for those reactions where at least one of the atoms involved in the reactive bond is well separated from the fragment center of mass. 396 A recently developed approach has its basis in the PSI methodology but explicitly considers the variation in the form of the reaction coordinate In particular, a variable reaction coordinate (VRC) is defined in terms of the distance between two variably located fixed points, with one fixed point in each of the two fragments. An optimization is then carried out not only of the value of the reaction coordinate, as in the most popular version of VTST for bimolecular reactions, but also of the definition of the reaction coordinate in terms of the location of the two fixed points. This VRC-TST approach provides a better representation of the reaction coordinate at close separations, as in the reaction-path VTST method, while retaining a classically accurate PSI-based treatment of the transitional mode contributions. Analytic integrations over the momentum components of the integrals again yield an approach which is of sufficient efficiency to be widely applicable, 425,426 while simplifying approximations provide an even more efficient procedure. 427 The precise location predicted for the transition state in VTST is a key physical quantity since this location broadly determines the reaction rate. A primary importance of this location is, for example, in directing the focus of quantum chemical estimates of the interaction energies. Furthermore, the ability to predict this location [and correspondingly N * (E,J)] on the basis of relatively limited potential energy surface information is another key advantage of VTST over more dynamical methods such as classical trajectory simulations. (Another advantage is the ability to quantize high-frequency modes, which can be crucial for predicting accurate threshold energies and thermal rates.) Of course, this transition-state location depends not only on the methods employed in the state counting but also on the details of the potential energy surface employed. The implementation of high-level quantum chemical data in the formulation of such potentials, as in a number of recent VTST studies, is of the utmost importance due to the difficulty of obtaining information about the potential in the strong interaction region from spectroscopic probes. Unfortunately, such data are more difficult to obtain than for the corresponding equilibrium situation due to the general occurrence of near degeneracies in the electronic states in the transition-state region. One interesting byproduct of the recent developments in methodology for counting states was an indication of just how rapidly the phase space integrals converge with number of sampling points. This rapid convergence suggests the feasibility of bypassing the analytic representation of the interfragment interaction potential via a direct ab initio quantum chemical evaluation of the interaction energy for each phase space point sampled in the integrationssimilar to the direct dynamics methods for bimolecular reactions discussed in section 2. A first demonstration of the feasibility and validity of this direct sampling approach has recently been provided for the CH 2 CO dissociation. 352,428 The energy and angular momentum resolved density of states F R (E,J) for the reactant also plays a key role in the determination of the rate constant. Evaluations of F R (E,J) based on RRHO expressions for the energetics provide a good first approximation to the density of states. The Beyer-Swinehart algorithm, 429,430 which is unfortunately limited in application to separable expressions for the energy levels, provides the standard procedure for such evaluations. Improvements beyond the RRHO level are becoming more and more important as both the experimental results and the other aspects of the theoretical methodology become increasingly accurate. The direct summation over the nonseparable energy levels has been found to be sufficiently efficient for the evaluation of the density of states for molecules as large as CH 2 CO, 352,397,431 and simplified treatments 432,433 are also useful. Alternatively, a method based on the random sampling of the quantum numbers should be particularly effective for larger molecules. 434,435 However, in many instances accurate expressions for the underlying energy levels are not available, in which case alternative procedures based on the integration of classical phase space integrals provide useful means for estimating the anharmonic effects. 284,378, Again, the rapid convergence properties of recently developed phase space integration methods suggest the possibility of a direct sampling of the potential in place of its analytic representation. Adiabatic switching has been used to compute the anharmonic density of states for Al 3, and values times larger than the harmonic result have been obtained. 378 However, the anharmonic density for this system determined by the phase space integration method is very sensitive to the assumed phase space boundary of the reactant. 378 Peslherbe et al. 284 have studied the effect of anharmonicity on the RRKM rate constant for the unimolecular dissociation of Cl - CH 3 Cl. They find that anharmonicity increases the reactant density of states by a factor of 2-3. This would decrease the rate constant to the extent it is not canceled by transition-state anharmonicity. An alternative Monte Carlo random sampling based methodology directly couples the evaluation of the reactant density of states with the transition state number of states In this method the rate constant is evaluated as the average velocity through the dividing surface for a random sampling over all available phase space. Unfortunately, the need to numerically evaluate the delta function in the reaction coordinate makes this approach somewhat inefficient. Furthermore, the present implementation of this methodology is restricted to a completely classical description for even the conserved vibrational modes. Another important advance in treating barrierless reactions has been a coupling of the basic VTST methodology with other aspects of the reaction kinetics beyond simply evaluating k(e,j) and/or the high-pressure thermal rate constant k (T). For example, the VTST methodology has been combined with both

12 12782 J. Phys. Chem., Vol. 100, No. 31, 1996 Truhlar et al. the standard RRKM formalism and the master equation approach in order to obtain a description of the pressure dependence of the reaction kinetics Also, the coupling of the VRC- TST methodology with quantum chemical estimates of the radiative relaxation rate provides a novel route to the estimation of complex dissociation energies via comparison with experimental measurements of the zero-pressure radiative association rate. 454,455 The VTST calculations of N * (E,J) have also provided meaningful predictions for the product-state distributions via the hybrid assumption of vibrational adiabaticity and rotational mixing. 164, The relation between transition-state theory and accurate quantum dynamics has also been pursued for unimolecular reactions and the related reverse recombination processes , For example, Bowman has presented a description of the canonical rate constants in terms of thermal averages over the trace of the Smith collision lifetime matrix and has also illustrated the relation to VTST. 179 Bowman and Wagner and co-workers have derived and applied an isolated resonance version of TST theory to the H + CO recombination/ dissociation process ,463 Miller and co-workers derived and applied a random matrix based formulation of TST which further explored the relation between scattering resonances and TST. 180, Most recently, Miller has provided a formulation of recombination rate constants in terms of flux correlation functions. 470 This formulation demonstrates how TST becomes exact for recombination processes in the high-pressure limit and furthermore makes it clear how to evaluate deviations from the TST limit quantum mechanically. Berblinger and Schlier 471 tested classical RRKM theory with numerical phase space integration (i.e., no harmonic approximation) against classical trajectory calculations for the reaction HD + 2 f D + + HD and H + + D 2. They considered total energies ev above the energetic threshold and total angular momenta 0-50 p. Their results illustrate that there is an interesting theoretical subtlety in discussing unimolecular decay due to the rapid dissociation, prior to equilibration, of some trajectories from an initially defined reactant ensemble. 472 After that occurs, decay may be more statistical. In fact, statistical theories like TST do not apply to the direct, shorttime component at all. In practical applications one cannot always separate these effects, but for DH 2+ this was possible. The direct trajectories caused TST to underestimate the unimolecular decay rate by up to 40%. Removing these trajectories, one finds the encouraging result that TST overestimates the rate constant by only about 3% due to recrossing. In certain instances, such as the dissociation of van der Waals molecules, there are substantial failings of the common formulations of TST. These failures are primarily the result of bottlenecks to the redistribution of energy 42 within the molecular complex. Considerable progress in understanding these failures has been made via analyses of the phase space structure of these reactions In fact, one recognizes two (or more) separate bottlenecks (or transition states), and the kinetics can be adequately modeled in terms of the statistical rates for crossing each of them. In related work, Dumont has developed a generalized flux renewal model which in addition to the slow intramolecular energy flow also considers the effect of direct components to the unimolecular decay process. 491 Bohigas et al. 492 and Tang et al. 489 have discussed the relation of this work to a quantum mechanical picture. A related concern regards the extent of randomization and sharing of the rotational energy with vibrational modes. The uncertainty in treating the rotational energy arises from the fact that the quantum number K, denoting the component of the angular momentum along a body-fixed axis z, is not conserved when one includes vibration-rotation coupling. Within applications of TST, the motion in the coordinate associated with K has generally been treated as either active, i.e., available to be shared with the vibrational degrees of freedom, or inactive. 81,407 An indication of the variation in the predictions obtained from these limiting cases was presented by Hase and co-workers for the Cl + C 2 H 2 reaction. 82 Gray and Davis have also presented a classical trajectory study of the extent of conservation of K over a picosecond time scale for formaldehyde at moderate energies. 493 General reviews of the role of angular momentum in unimolecular reactions have been presented. 407, Applications. The first application of the reaction-path VTST methodology to barrierless reactions was to the O + OH f HO 2 f H + O 2 reaction. 405 Deviations as large as a factor of 5 were observed between quasiclassical trajectory and VTST results, and the importance of accurate estimates of the interaction energies in the inner transition-state region from 2.5 to 5.5 Å was noted for the first time. Furthermore, this study provided the first a priori indication of the presence of two wellseparated transition states. Various other aspects of this reaction system have also been the subject of recent detailed VTST calculations. 144, Broadening the scope beyond VTST, Duchovic and Pettigrew 498 have recently compiled 100 references for the reverse H + O 2 f OH + H reaction, including over 40 theoretical studies. The quantitative validity of VTST for the dissociation of HO 2 has been demonstrated via direct comparison with averaged quantum scattering estimates. 144 Meanwhile, VRC-TST results for the reverse bimolecular reaction are found to overestimate both quantum scattering theory and classical trajectory simulations by a factor of Similar comparisons for the He and Ne + H 2+ reactions also indicated a factor of 2 overestimate by VRC-TST calculations when an appropriate symmetry correction factor is included. 137,139 (Note that, as described in a previous paper, 497 a symmetry correction factor of 2 was neglected in these studies which leads to the currently stated discrepancy between the TST and scattering theory results.) Such deviations appear to be the result of the direct redissociation of a substantial fraction of the incoming trajectories as a result of an incomplete coupling in the complex. This incomplete coupling is in turn related to the generally short lifetime ( 0.1 ps) and low density of states for the HO 2 complex. While numerous comparisons between theory and experiment for this reaction system have also been presented, such comparisons are unfortunately clouded by uncertainties in the potential energy surface and also in the contribution from excited electronic states. The first detailed applications of the PSI-VTST approach were to the recombinations of CH 3 with CH and with H For both reactions satisfactory agreement with the experimentally determined canonical rate constants was obtained while employing ab initio based potentials. Furthermore, the VTST predictions for the H + CH 3 recombination were in good agreement with subsequent trajectory simulations. 410 However, for the latter reaction the D isotope effect of 1.4 predicted by VTST does not agree with the experimentally observed value of 2.5. Overall, the detailed VTST studies for these two recombinations have provided an important testing ground for both simplified VTST models 421, and also for some of the inherent assumptions. 409,414,505 Interestingly, earlier harmonic oscillator and hindered rotor based implementations of the reaction-path VTST method 409 differ very little from the PSI-VTST results for the H atom recombination and also for the association reactions of Li + with H 2 O and (CH 3 ) 2 O Detailed comparisons for these reactions suggest that explicit variation of the transition-state location at the E- and J-resolved level typically leads to an improvement by about 20% over its

13 Current Status of Transition-State Theory J. Phys. Chem., Vol. 100, No. 31, consideration at only the E-resolved level. The methyl recombination reactions have also provided an important testing ground for studies of the pressure dependence of the reaction kinetics 450,451 with subsequent applications made to the CH + H and CH CH 3 CN reaction kinetics. 452 The detailed understanding of the CH 3 plus H association has played an important role in the development of models for the association of alkyl radicals with H atoms, which in turn is of key importance in the understanding of the kinetics of diamond formation. 516 The NCNO reaction is one of the first reactions for which wide-ranging energy-resolved dissociation rate constants were determined experimentally. 517 Such data provide a more stringent test for the theoretical predictions due in part to the absence of any need to consider the collisional energy transfer process. An initial model-potential-based PSI-VTST application indicated the occurrence of a transition from a long-range transition state to an inner transition state as the energy rises above 100 cm -1, with only the ground singlet electronic state contributing at the shorter separations. 397 Subsequent comparisons of related VRC-TST calculations with local trajectory propagations suggest the quantitative validity of VTST, particularly when a unified statistical treatment 73,85-89 of the two transition states is employed. 518 Related indications of at least the qualitative validity of the unified statistical treatment were found in a PSI-VTST study of the CH 2 CO product-state distributions 399 and also in a VRC-TST study of the Li + HF reaction. 145 The NCNO reaction also provided the first test of the VRC-TST approach with the optimization of the reaction coordinate yielding a reduction in the rate constant by a factor of about Similar estimates of the effect of optimizing the form of the reaction coordinate have been obtained for a variety of related reactions, including the reaction of NC with O and the dissociations of NO 2, 164 CH 2 CO, 352,416,428 and C 6 H These studies further suggest that the bond length of the reacting bond generally provides a good first approximation to the reaction coordinate in the inner transition-state region corresponding to atom-atom separations of about 2-3 Å. For each of these reactions the VRC-TST predicted rate constants have been found to be in reasonable agreement with experimental predictions. The experimental studies of the singlet dissociation of CH 2 - CO provide a detailed and wide ranging set of data for barrierless reaction dynamics. 158, In the latest VRC-TST application, both N * (E,J) and F R (E,J) were obtained while employing potentials based on high-level quantum chemical investigations, including direct evaluations of the potential for both bond length and center-of-mass separation distance reaction coordinates. 352,428 Furthermore, the indirect kinetic coupling between the conserved and transitional modes, as modulated by the reaction coordinate, was explicitly treated. The resulting nonempirical estimates were found to quantitatively (i.e., within 35% or better) reproduce the experimentally observed energy dependence for the dissociation rate constant. In addition, PST, a RRHO-based implementation of VTST, and even the centerof-mass-separation-distance-based PSI implementation of VTST are each in error by factors of 2 or greater. For ionic reactions, the stronger long-range interactions generally lead to a better validity of the PST-type estimates. However, for large enough molecules and/or weak enough attractions the short-range repulsions will still lead to a reduction in the flux. The recent VTST applications are beginning to address the point at which one needs to consider such shortrange repulsions. For the dissociation of C 6 H 6+ into C 6 H 5+ + H, model potential based VRC-TST calculations indicate a reduction by a factor of about 6 as compared to PST based estimates. 521 In contrast, sample applications to the Cl - + CH 3 Cl 282, and Li + + H 2 O or (CH 3 ) 2 O reactions provide no indication of any reduction in the reactive flux due to short-range interactions, and VTST appears to provide an accurate description of the initial association process. However, quasiclassical calculations indicate that the overall reaction kinetics for the Cl - + CH 3 Cl reaction deviates from TST expectations as a result of the redissociation of the initially formed complex prior to a randomization of the energy, 282, perhaps related once again to a short lifetime for the complex. The dissociation of C 6 H 5 Br + into C 6 H 5+ + Br provides an interesting intermediate example where the importance of shortrange interactions was shown to depend on the particular parametrization of the potential employed. 531 The interesting question of the occurrence of two separate transition states for ionic proton transfer reactions has also been studied on the basis of the PSI-VTST methodology while employing model potentials. 532,533 Unimolecular reactions provide some examples of very flat potential energy surfaces where variational transition-state theory is invaluable even for determining which structures are the intermediates and which are the activated complexes. Examples are the tetramethylene 534,535 and trimethylene 348 rearrangements. The latter species occurs as an intermediate in the cis-trans isomerization of cyclopropane. In many cases, accurate ab initio data for the transition-state region is unavailable. For such cases, simplified Gorin model type representations of the potential may prove useful as illustrated in a series of PSI-VTST studies of the reactions of CH 3 with CH 3 or OH and the dissociation of C(CH 3 ) 4 and neopentane The temperature dependence of the rate constant for the reactions of HCO + NO and of OH with HO have been similarly predicted. A number of comparisons between the Monte Carlo VTST methodology mentioned above and trajectory simulations, also employing empirical potentials for the transition-state region, have been presented. While good agreement was found for the dissociation of SiH 2, 447 the comparisons for the dissociations of Si 2 H 447,537 6 and C 2 H 4 F 538,539 2 were not very favorable. The large deviation observed in the Si 2 H 6 study is somewhat surprising in light of the excellent agreement observed between VTST and experiment for the closely related C 2 H 6 dissociation. One point worth noting is that the requirement of short propagation times (i.e., 10 ps or less) within the trajectory simulations means that the comparisons must be made for higher excess energies than are generally considered in thermal studies. Furthermore, the empirical potential employed in these studies for the dynamically important inner transition-state region appears to be substantially more attractive than the values obtained in ab initio calculations for related reactions. For example, the CC bond strength in C 2 H 4 F 2 is estimated to be 0.67 and 0.13 ev at R CC ) 3 and 4 Å, respectively, whereas ab initio calculations 501 predict values of 0.34 and 0.03 ev at the same separations for the similar C 2 H 6 dissociation. A detailed examination of the extent of the failure for these reactions for more realistic representations of the potential energy surface in the transition-state region will be an important issue for future studies. A detailed picture of the factors affecting both neutral and ionic barrierless associations and their reverse dissociations is gradually emerging from these studies. Of particular note, for neutral reactions, is the general occurrence of a dominant inner transition state at separations of 2-3 Å between the atoms involved in the reactive bond. Overall, these applications suggest that the more sophisticated versions of VTST can generally be expected to describe the reactive flux within a given potential energy surface to within about a factor of 2. Further-

14 12784 J. Phys. Chem., Vol. 100, No. 31, 1996 Truhlar et al. more, when the complex lifetime is on the order of a nanosecond or longer, even better agreement might be expected. An interesting procedure for making improved predictions for the reaction rate, and whose usefulness needs to be explored in greater detail, involves a coupling of quantum mechanically based VTST estimates with classical trajectory based estimates of nonstatistical effects. As for reactions with tight transition states, the greatest current uncertainty in VTST estimates for the reaction rates of systems with loose transition states involves the uncertainty in the underlying potential energy surfaces. Thus, an important aspect of future work will involve the continued development of accurate descriptions of the potential energy surfaces, both for sample reactions and ultimately for larger classes of reactions. The availability of high-level quantum chemical data for the 2-3 Å region of separations will be of the utmost importance in these developments. Related quantum chemical data will be of use in providing more accurate descriptions of the reactant density of states. Many association reactions involve radicals with low-lying excited states and/or for which the electronic state at infinity splits into several electronic states upon interaction with the association partner. Thus, another large uncertainty in the a priori prediction of the rate constants regards the contribution from excited electronic states. At short separations, where the transition state generally lies at higher energies, the electronic states are often widely spaced so that an important contribution is expected from only the ground electronic state. However, at larger separations electronic degeneracies often arise, and the proper description is then uncertain, depending on the strengths of the couplings among the electronic states. Importantly, an accurate a priori description of the temperature dependence (near room temperature and lower) of the radical-radical association process will generally require an accurate description of the switching of the transition state from large separations to shorter separations and thereby the changing contribution of the excited electronic states. 4. Reactions in the Condensed Phase Transition-state theory has been widely used for the calculation and analysis of rate constants for chemical reactions in a variety of condensed-phase systems such as liquids, solids, and gas-solid interfaces. The use of TST in its simplest form for reactions in condensed phases dates back to the work of Evans and Polanyi 539 for liquids and Wert and Zener 540 and Vineyard 541 for solids. Although there is a long history of applying TST to condensed-phase systems, the accurate prediction of rate constants in condensed phases still presents a major challenge because of the complexity of including the extended nature of the system in the rate constant calculation as well as the difficulty of accurately evaluating the interaction energies for the extended systems. The condensed medium can profoundly influence the reaction dynamics, for example, by inducing recrossings of the transition-state dividing surface that lead to a breakdown of the fundamental assumption of TST. A computational issue that is still not fully solved is the inclusion of multidimensional quantum mechanical effects when they are important in these dissipative systems. There has been great progress in recent years in addressing these challenges. In this section we review the advances made in the theoretical development and applications of the theory to reactions in liquids and to molecular processes in solids and on surfaces Reactions in Liquids. At the most basic level, TST for reactions in solution is based on the equilibrium solvation approach. This involves evaluating the free energy of activation in solution, e.g., by adding the difference in free energy of solvation between the saddle point and reactants to the gasphase free energy of activation. 542 At the next level of sophistication one should include nonequilibrium effects of the dynamics of the solvent; the analytical theory of Kramers 57,543 and the transmission coefficient expression of Grote and Hynes (G-H) 544 have been particularly well studied. Several reviews 1,16,70, of solution-phase reactions are available that include discussions of TST and of these models. Within the past decade, the state of the theory has advanced considerably to include complex aspects not present in the earlier work. The applications of the newer theories are just beginning to appear. In this section we will review recent work with the discussion organized as follows. First we consider classical mechanical models, then we review quantum mechanical generalizations to include important bound-state-quantization and tunneling effects, and finally we consider recent applications, with an emphasis on attempts to treat to realistic systems Classical Theory and Application. A formal derivation of classical TST for reaction in liquids was presented by Chandler. 71,555,557 The new element in the liquid phase is that collisions of solvent molecules with the reacting solute molecules can lead to recrossings of the dividing surface that do not occur in the gas phase and, therefore, to a breakdown of the fundamental dynamical assumption of TST. Since, as mentioned in the Introduction, reactant activation and equilibration are not the subject of this review, we proceed to consider the case where coupling energy into the reactants is not rate limiting. One approach to approximating the influence of extended condensed-phase systems on the solute reaction dynamics is by separating static and dynamic effects of the condensed phase (solvent); such effects are often called equilibrium and nonequilibrium effects, respectively, and we will sometimes follow this convention even though these overworked terms sometimes lead to confusion. The concept of nonequilibrium solvation is old; modern work dates back at least to the models of proton transfer developed by Kurz and Kurz, 558 and the concepts have been refined in more recent work. 1,545,546,548,555, Equilibrium solvation provides a good starting point for treating the reaction energetics of the solute. The surrounding condensed-phase molecules change (dress) the effective force field of the reacting molecules. The resulting mean-field potential for the reacting molecule(s) is obtained from an equilibrium ensemble average over configurations of the other molecules in the condensed phase. Since this mean-field potential is obtained from an equilibrium ensemble average for fixed configurations of the reacting molecules, the equilibrium solvation assumption implies that the solvating molecules instantaneously equilibrate to each new configuration of the reacting molecules. In the thermodynamic formulation of TST, 5,16,71,539 the effect of the condensed phase on the reaction energetics is included by the free energy of solvation that is obtained from equilibrium ensemble averages (a mean-field approach) and is therefore an equilibrium solvent effect. This is the most common approach used for including solvation effects in rate constants. 542 Over the past several years there has been increased interest in developing methods for calculating the equilibrium free energy of solvation as required to include equilibrium solvation effects in TST. In one approach the free energy of solvation of a rigid solute complex is computed from model solute-solvent and solvent-solvent potentials using classical ensemble averaging and/or statistical perturbation theory and added onto the gas-phase potential energy profile obtained from electronic structure calculations (The solute and solvent molecules may be treated as either electronically inert or polarizable when solvent is explicit, but in most

15 Current Status of Transition-State Theory J. Phys. Chem., Vol. 100, No. 31, applications so far at least the solvent molecules are assumed nonpolarizable.) Other approaches are based on representing the solvent bath as a three-dimensional continuum; in such models the electrostatic effects are treated by solving the Poisson equation for a dielectric medium or by an equivalent algorithm for putting in the same physics, and nonelectrostatic effects are either ignored or modeled based on the solvent-exposed atomic surface areas. Most of this work treats the case where the solvent polarization is assumed to be in the linear response regime, although in some cases nonlinear effects in the first solvation shell (e.g., dielectric saturation) are included by empirical atomic surface tensions. 580,581,604,605 Equilibrium solvation neglects any dynamical influence the condensed phase may have on the reaction dynamics resulting from fluctuations of the solvent around equilibrium. Nonequilibrium or dynamic solvent effects can be separated into local and collective effects. Local effects involve only a limited number of solvent molecules such as can be included in a cluster model. Transition-state theory can be applied to solution-phase reactions by separating the system into a cluster model that contains the part of the system undergoing reaction and the solvent that is treated in an approximate manner. The cluster model can include a finite collection of solvent molecules as well as the reactants or solute molecules. The effects of microsolvation 217,269,270,273,285,303,343,560,578,598, on reaction dynamics in small clusters have been studied using TST and VTST/MT in several papers; a review is available. 616 The VTST calculations have included only one or two solvent molecules; in principle, this approach can be extended to larger clusters, but very quickly (as the number of floppy degrees of freedom increases) the potential surface starts to exhibit multiple low-energy pathways that are more appropriately treated by condensed-phase methods such as those discussed in the following paragraphs. When solvent molecules are treated explicitly, 285,560,578,585,586, ,612,614 their motion can be included in the reaction coordinate, and this can influence the geometry of the solute and the reaction energetics along the reaction path and thereby the transition-state structure. Transition-state geometries in solution can also be optimized by continuum solvation methods. 579,586,589, ,602,603 Both explicit-solvent and continuum-solvent treatments show that the structure of the solute at the liquid-phase transition state may be quite different from the gas-phase transition-state structure or even from any point along the gas-phase reaction path. 599 The approach of including one or a few solvent molecules explicitly can account for nonequilibrium solvation effects both of the solvent caging type, in which the solvent molecules must move out of the way of the reacting molecules, and of the type where solvent molecules participate in the reaction as either a reactant or a catalyst. 303 Collective effects (including long-range electric polarization of the solvent dielectric) involve cooperative motions of the molecules in the condensed phase and are more difficult to include by explicit few-body methods because of the large size of the system required. The larger number of equilibrium geometries and transition states in liquid-phase reactions and even in medium-size clusters also motivates a transition from few-body gas-phase methods to many-body condensed-phase methods. Carter et al. 617 have formulated an efficient molecular dynamics method for evaluating the classical TST rate constant in terms of a constrained-reaction-coordinatedynamics ensemble, 618 which is applicable to any general definition of the reaction coordinate. Harris and Stillinger 619 discuss the application of VTST to the full coupled system of the solute and solvent and conclude that identifying the optimal dividing surface in the full space of the system may be impractical. They suggest an approach based on an inherentstructure formalism in which the total rate constant is obtained from a sum over TST rate constants for saddle points separating reactant basins from product basins (the inherent structures in the liquid). Another approach, complementary to the explicit-solvent and continuum models, is to approximate the collective effects using models of reduced dimensionality. Inclusion of collective effects of nonequilibrium solvation in a TST framework finds its origin in the seminal work of Kramers. 57,543 In the Kramers theory, the reaction is treated in a highly simplified manner: the reacting solute is treated as a single reaction coordinate, and the rest of the system is treated as a bath in terms of a Langevin equation of motion. Takeyama 620 used this model to derive a transmission coefficient that accounts for the leading effect of friction when all friction components vanish except the reaction-coordinate one. Grote and Hynes, 544 retaining the one-dimensional reaction coordinate, obtained a more general result for the transmission coefficient via a more realistic treatment of the bath using a generalized Langevin equation (GLE). The GLE recognizes that friction does not act instantaneously (as assumed in the original Langevin equation, sometimes called the assumption of ohmic friction), and thus its effect may be reduced for narrow barriers that can be crossed rapidly. The GLE for motion along a one-dimensional reaction coordinate can be recast into Hamilton s equation of motion for a system in which the reaction coordinate is linearly coupled to a bath of harmonic oscillators; 621,622,626,627 we will call this the GLE model. Notice that in the GLE model, as in the original Langevin model, all potentials involving the bath are quadratic or bilinear, and thus all forces on the solvent are linear. (Therefore, this model is a special case of treating the solvent by linear response theory.) TST with the Grote-Hynes transmission coefficient (TST/G-H) is equivalent to classical VTST for the GLE model in which the potential along the reaction coordinate is a parabolic barrier. 553, For this purely quadratic potential, VTST gives the exact result. The G-H expression may be interpreted as giving the effect of friction (or microscopic viscosity) on the barrier traversal rate. Since the G-H equation can be derived by mixing bath modes into the reaction coordinate, ,636 we see that friction and nonequilibrium solvation provide two different ways of looking at the same physical effect. Furthermore, since friction causes recrossing of a transition state that is defined without solvent participation in the reaction coordinate, we see that for such a reaction coordinate nonequilibrium solvation is an example of the general phenomenon that was mentioned in section 1 by which recrossing can lead to a breakdown of the quasiequilibrium assumption of TST. The breakdown can be corrected by including solvent friction or by letting the solvent participate in the reaction coordinate. The regime where the coupling of solute to solvent is strong enough that energy diffusion and reactant equilibration need not be considered (but friction and nonequilibrium solvation might be important) is sometimes called the spatial diffusion regime, and it is the subject of the rest of this section. Nonlinearities in the forces along the reaction coordinate can cause the optimum dividing surface to be different from the approximate one obtained in the parabolic barrier approximation. Variational TST has been widely used to treat this problem of a nonquadratic reaction-coordinate potential that is linearly coupled to a harmonic bath (i.e., the GLE model). Canonical and microcanonical VTST theories have been applied to the GLE model to elucidate such effects Microcanonical and canonical VTST provide a significant improvement over TST/G-H for intermediate friction, where TST/G-H under-

16 12786 J. Phys. Chem., Vol. 100, No. 31, 1996 Truhlar et al. estimates the effect of friction due to its neglect of nonlinear potential forces. 641,644 Berezhkovskii et al. 645 have used variational TST with planar dividing surfaces to develop an improved approximation for the case of nonlinear potential forces; an analytical expression for the rate constant has been obtained for the quartic double-well potential and compared with accurate numerical results. 644 Frishman et al. 646 have further extended this method to allow for bent planar dividing surfaces that are needed for improved descriptions of some asymmetric potentials. Classical theories for reactions in solution that are based on TST typically have as a goal either the minimization of the solvent-induced recrossing by optimizing the dividing surface or estimating the recrossing of a given dividing surface. Pollak and Talkner 647 have developed a statistical theory based upon the unified statistical model 73,85-89 that relates the average number of recrossings of the dividing surface to the reactive flux. 89 Another method for including recrossing effects is to explicitly calculate the reaction dynamics near the dividing surface. Pollak and Talkner 648 have developed a dynamical VTST in which approximate dynamical corrections are included for an optimized dividing surface that is a function of the reaction coordinate and bath coordinates but goes through the saddle point. The dynamical corrections are approximated by a perturbation approach similar to those used 61 to describe the energy-diffusion regime. Kramers theory was extended to include spatially dependent friction by Carmeli and Nitzan using an approach based on the Fokker-Planck equation. 649 Since then there have been many studies of spatially dependent friction, especially friction that is dependent upon the location along a reaction coordinate Straus et al have considered a spatially dependent friction that is modeled by a Hamiltonian in which the coupling between the reaction coordinate and the harmonic bath is a function of the reaction coordinate. They compared approximate VTST results with accurate results from simulations and found that the VTST expression was quite accurate. Voth 653 has presented an effective Grote-Hynes method that provides a simple procedure to include spatially dependent friction. Haynes et al. 656 have applied the VTST approach of Berezhkovskii et al. 645 to a spatially dependent and time-correlated friction model. As in the previous study, the method was applied to a quartic double-well potential. The method is in excellent agreement with exact simulations. This approach has been further extended to allow for curved dividing surfaces. 657 Pollak has extended the variational transition-state theory approach to treat condensed-phase reactions in which the system and bath forces are both of a general nonlinear form. 658,659 Two orthogonal collective modes, an effective reaction coordinate and a collective solvent mode, are defined that are linear combinations of the bath and system coordinates, and canonical VTST is applied to the two-degrees-of-freedom problem. 637 This approach reduces to the TST/G-H approximation for the case of a harmonic bath linearly coupled to a parabolic barrier. The Kramers and G-H models are for a one-dimensional system (solute) coupled to a multidimensional bath. Multidimensional effects were discussed within the context of these one-dimensional theories by Nitzan and Schuss. 660 Berezhkovskii et al. 645,661 have developed a VTST with planar dividing surfaces for systems with two degrees of freedom coupled to a GLE model. Multidimensional effects of the solute motion, such as the effects of reaction path curvature, have also been explored; 635 this work showed that at low-to-intermediate coupling between the solute and the bath the reaction-path curvature can induce recrossing of the transition-state dividing surface which is not well described by the simple harmonic model underlying the Kramers and G-H expressions. Nonequilibrium solvation in charge transfer systems was studied by van der Zwan and Hynes. 559,628,629 In these studies, the nonequilibrium polarization effect was treated by a GLE and the rate constant was calculated by the TST/G-H method. Lee and Hynes 662,663 extended the treatment to include anharmonic effects by defining an explicit solvation coordinate and an effective Hamiltonian for studying the reaction dynamics. An expression for the rate constant was obtained for the twodimensional solution reaction-path Hamiltonian using variational TST and applied to a model S N 2 reaction. The work of Lee and Hynes stimulated further work in this area to develop alternative definitions of the solvent polarization coordinates and calculate rate constants for multidimensional systems with methods based on TST Some calculations 667 indicate the possibility of very large nonequilibrium effects, even as large as a factor of 6, although we expect that nonequilibrium effects are typically smaller in classical systems. van der Zwan and Hynes 670 have examined nonequilibrium solvation effects on a model dipolar isomerization reaction in an electrolytic solution using TST/G-H Quantum Mechanical Theory. Quantization and tunneling effects are neglected in the classical approaches. The major contribution to a classical reaction rate at temperature T typically comes from energies about k B T above the barrier height, whereas accurate quantum mechanical reaction probabilities are typically very small at such total energies because zero-point energy requirements in modes transverse to the reaction coordinate are much greater than k B T. Quantitative studies that do not enforce quantization conditions on transverse modes, at least approximately, have little relevance to the physical world unless there is a fortuitous cancellation of reactant and transition-state quantization effects. Classical methods also assume that reactive trajectories must surmount barriers, whereas quantum mechanically the system can tunnel through barriers, and this can be the dominant mode of reaction, especially for systems where hydrogenic motions participate in the reaction coordinate. The standard approach for including quantum mechanical effects in TST for condensed-phase reactions is the same as for the gas phasesan approximate, ad hoc procedure in which the classical partition functions are replaced by quantum mechanical ones, and a factor is included to correct for quantum mechanical motion along a reaction coordinate (e.g., quantum mechanical tunneling for energies below the classical barrier and nonclassical reflection above the barrier). One way to implement this scheme would be (i) to quantize some of the solute modes but treat the low-frequency modes of the environment classically (e.g., the solvation effect on the free energy of activation is included classically, but bound modes of the solute that change appreciably in going from the reactants to saddle point are quantized) and (ii) to treat tunneling by a transmission coefficient. In recent years, significant progress has been made in developing consistent approaches for including quantum mechanical effects in this way. A quantum mechanical analog of the classical Kramers and G-H theories was first derived by Wolynes. 671 Dakhnovskii and Ovchinnikov 630 and Pollak 672,673 showed that applying quantized TST with a parabolic tunneling correction factor to the GLE model underlying these approximationssthat of a reaction coordinate linearly coupled to harmonic oscillators representing the bathsreproduced the Wolynes expression. In this quantum version of the model, tunneling is approximated by the result for the one-dimensional parabolic barrier (which diverges at low temperature and that fails to capture the physics

17 Current Status of Transition-State Theory J. Phys. Chem., Vol. 100, No. 31, of multidimensional tunneling at other temperatures 674 ). There has been much interest in developing methods for going beyond this simple approach to include tunneling effects, and reviews covering selected aspects of the tunneling problem are available. 16,70,176,675,676 In this section we will focus on two approximate quantum mechanical methods based on TST for treating reactions in solution. The first is quantized variational transition-state theory with semiclassical corrections for quantum mechanical effects on reaction coordinate motion. 15,16,84 Approaches to applying this method to reactions in solution have recently been described 561,677 and are discussed below. The second method is the path integral formulation of QTST 114, discussed briefly in section 2. This method has recently been reviewed by Voth 120,121 including applications and extensions of the method. Both the VTST/MT and PI-QTST approaches have been applied to the GLE model of a reaction in solution. 678 For the case of a parabolic barrier the VTST/MT approach reduces to the TST result already shown to reproduce Wolynes expression. The path integral method has also been show to reproduce this exact result. 114 These two approaches have been applied numerically to the problem of an Eckart barrier linearly coupled to a harmonic bath, and although they employ different approximations, the results from the two methods agree well with each other and with accurate benchmark results. 678 In further work along these lines, 679 it was found that accurate treatment of anharmonic effects is important in the VTST calculations for treating reactions in solution where lowfrequency modes of the bath enhance the effects of anharmonicity. Furthermore, it was found that, for the model solutionphase reaction, VTST/MT calculations are often limited more by the treatment of anharmonicity than by errors inherent in the approximations to the reaction dynamics. Garrett and Schenter 561 have described an approach for applying VTST to activated chemical reactions in liquids. In this approach the total system is separated into a cluster model that is treated explicitly and a bath that is treated with a reduceddimensionality model. Within an equilibrium solvation approach, the effective potential for the cluster is the potential of mean force as a function of the coordinates of the cluster. When nonequilibrium solvation is included within a GLE approach, the effective potential becomes the sum of the potential of mean force and a set of harmonic modes that are coupled to the cluster coordinates. The issue of how quantum mechanical tunneling is implemented in this VTST approach has been discussed by Truhlar et al. 677 A new solute-bath separation is presented based on tunneling through a canonically averaged mean-shaped potential which can be evaluated from the bath contribution to the potential of mean force. It is worth noting that the opposite point of view has been taken by Pollak, 673 who advocates a sudden approximation for the tunneling; this approach has been more fully developed by Levine et al. 680 The PI-QTST approach has also been used to provide a quantum mechanical generalization of the G-H model. 681,682 In this variational approach, a Gibbs-Bogoliubov-Feynman inequality 116 is used to derive an effective multidimensional parabolic model that can be solved analytically. The quantum mechanical generalization of the G-H recrossing factor takes the same form as the classical factor, but with the classical value of the imaginary barrier frequency replaced by an effective quantum mechanical frequency. In a similar approach, Voth and O Gorman 683 obtained a simple analytical theory in which an effective one-dimensional parabolic potential is used to effectively include the nonquadratic nature of the potential barrier and the influence of linear dissipation. Haynes and Voth 654 have examined the effect of spatially dependent friction within the path-integral QTST method. At low temperatures, where quantum mechanical effects are important, the nonlinear dissipation was seen to give large (order of magnitude) corrections to the results from a linear dissipation. Messina et al. 684 have suggested a generalization of the PI- QTST that allows for general dividing surfaces in phase space. Although this expression does not provide a rigorous upper bound to the quantum mechanical rate constant, in the same spirit as the quantum VTST approach, the optimum dividing surface is found variationally to minimize the rate constant. This approach has been applied to the model problem of an Eckart potential coupled to a bath of oscillators, 684 and a procedure for optimizing planar dividing surfaces in the path-integral formalism (that is analogous to the classical variational method of Berezhkovskii et al. 645 ) has also been presented. 685 Pollak has also discussed variationally determining the optimum dividing surface within PI-QTST. 686 This maximum free energy approach is formulated for the GLE model and for the general case of nonlinear coupling to the bath. The variational PI-QTST greatly improves low-temperature tunneling corrections for asymmetric barriers. 687 Schenter et al. 688 have shown how dynamical corrections, which are based on classical trajectories on an effective potential that includes quantum mechanical effects, can be included in the PI-QTST formalism. This work has similarities to the unified dynamical theory 15, discussed in section 2. Along the same lines, Sagnella et al. 689 have developed a semiclassical TST, which is based on the semiclassical formulation of Chapman et al., 690 that estimates dynamical recrossings from classical trajectories initiated at the transition-state dividing surface from a semiclassical phase-space distribution. The use of an effective potential that includes quantum mechanical effects is closely related to the approach used by Valone et al. 691,692 to study H diffusion on Cu surfaces (called EQP-TST in section 4.2.2). The original path-integral-based rate theory by Gillan formulated the problem in terms of the reversible work for moving the centroid of the quantum mechanical paths from the reactant region to the saddle point. This idea was extended to a reversible-work formulation that involves moving a generalized transition-state dividing surface from the reactant region to a generalized transition state in the interaction region. 693,694 When translations and rotations of the dividing surface are properly taken into account, this method is rigorously equivalent to the method of Voth et al. 114 The reversible-work formulation has the advantage over the earlier approach 114 that it requires evaluation of averages of forces rather than the centroid-density constrained partition function that is more difficult computationally. The theory developed by Marcus has played a central role in describing electron transfer reactions in polar solvents. In this approach the solvent reorganization is the rate-limiting process for the electron transfer, and a solvent reorganization energy often plays the role of the reaction coordinate. In the electronically adiabatic limit, this approach is equivalent to TST. For example, adiabatic electron transfer, in which the heavy particle motion of the solvent limits the rate of reaction, has been studied using a quantum mechanical TST approach for the GLE model. 699 Smith and Hynes 700 have formulated a rate expression for electron transfer in or near the electronically adiabatic regime based on the G-H approximation Applications to Reactions in Solution. Over the past decade, most theoretical studies of reaction in solution have stressed the importance of nonequilibrium solvation effects. However, in the analysis of real experimental data, it is often

18 12788 J. Phys. Chem., Vol. 100, No. 31, 1996 Truhlar et al. hard to deconvolute equilibrium from nonequilibrium solvation effects. For example, changing physical conditions such as the solvent density to affect the friction will often also change the potential of mean force of the system. 560,633, An understanding of the importance of nonequilibrium solvation effects for a given system often requires first the knowledge of equilibrium solvation effects. Ladanyi and Hynes 704 have used VTST to study equilibrium solvent effects on H atom transfer reactions and simple geometric isomerizations in model compressed rare gas solvents. Compared to gas-phase rate constants, in solution large enhancements were seen for the H atom transfers with much smaller enhancements for the isomerizations. Garrett and Schenter 703 have argued that, for systems in which the free energy of solvation is independent of solute mass, equilibrium solvation will not affect kinetic isotope effects (KIEs). Therefore, for these systems KIEs can be used to isolate nonequilibrium solvation effects. VTST with semiclassical tunneling corrections was used to study a model of the reaction of H isotopes, including muonium (Mu), with benzene in aqueous solution. Anomalous Mu KIEs that were observed experimentally could not be explained with an equilibrium solvation model. Including nonequilibrium solvation showed substantial suppression of the Mu rate constant compared to the H and D isotopes, in agreement with the experimental findings. The majority of applications of TST methods to reactions in solution are based upon the simple Langevin equation in Kramers theory or the GLE model in the G-H model. We will focus primarily on applications of TST or VTST to multidimensional systems, and we will mention important benchmark tests of the Kramers and G-H transmission coefficients for realistic systems. Reference to recent work in which these theories were used to analyze experimental data include Saltiel, Sun, and co-workers, 705,706 who explored the validity of Kramers model for isomerization reactions, Cho et al., 707 who review such analyses for three types of friction (mechanical, internal, dielectric), Schroeder and Troe, 65 who use pressure and temperature as independent variables in a single solvent to test Kramers theory, Sumi and Asano, 708,709 who discuss the difficulty of using the theory to understand isomerization reactions of DBNA in an alkane solvent, and Anderton and Kauffman, 710 who used Kramers expression to analyze dielectricdependent activation energies for isomerizations. S N 1 and S N 2 Charge Transfer Reactions. Bimolecular nucleophilic substitution (S N 2) reactions in polar solvents have provided fairly extensive tests of the G-H model. Extensive work on the Cl - + CH 3 Cl reaction provided benchmark tests against classical trajectory simulations of realistic solute models in molecular solvents. 634,711,712 In this work, the G-H model was found to be very accurate, whereas the Kramers model gave results that were much too low. Nonequilibrium effects reduced the rate constants by about a factor of 0.5. Similar results were also observed in comparisons of G-H model with trajectory simulations for the process of ion pair interconversion between contact ion pairs and solvent-separated ion pairs in polar solvents. 713 Aguilar et al. explored the effect of solvent lagging behind solute as the system proceeds along the reaction path (nonequilibrium solvation) for the F - + CH 3 F reaction; large effects on the effective barrier to reaction were predicted. 714 Various formulations 658,659,662,667 of VTST have been applied to S N 2 reactions to examine the nature and effect of solutesolvent coupling. 663,667,715,716 Transition-state theory has also been used to treat S N 1 ionic dissociations in polar solvents. Zichi and Hynes 717 have applied TST/G-H to a model for activated ionic dissociation with substantial charge rearrangement in polar solvents. They found that Kramers theory predicted frictional transmission coefficients of whereas the more realistic TST/G-H model yielded , much closer to unity, by taking account of the fact that the reaction time scale is too fast for the full friction to develop. For a narrow range of parameters leading to solute caging by the solvent polarization field, κ values were calculated to be as small as 0.1. Keirstead et al. 718 have studied a model S N 1 reaction in water and also found that the G-H transmission coefficient leads to excellent agreement with trajectory simulations. Kim and Hynes 719,720 used a nonlinear response treatment 574,575 with the solvent reaction path (minimum-energy path in the multidimensional space of solvent and solute coordinates) to develop an ionic dissociation model that was then used in calculations on nonequilibrium solvation effects based upon the method of van der Zwan and Hynes. 559,628,629 Mathis et al. 668 extended these studies to analyze ionization of tert-butyl halides. Mathis and Hynes 721 have studied anomalous behavior in ionization of alkyl iodides using a similar approach to develop the reaction model and then either harmonic TST (e.g., the method of van der Zwan and Hynes) or VTST where the barrier is not treated as parabolic. They have also studied the alkyl iodide ionization using a formulation of TST in which the reaction coordinate is assumed to be the solvent coordinate. 722 Although the above studies focus on dynamical issues related to the validity of transition-state theory and the additional approximations in the G-H model, one should keep in mind that the dynamical, frictional, and nonequilibrium aspects of the problem often change the equilibrium TST prediction by less than a factor of 2, whereas the precise value of the solute barrier height and equilibrium free energy of solvation of the transition state relative to reactants may have a much larger effect; e.g., errors of 1-2 kcal/mol lead to rate constant errors of factors of 5.5 and 30 at 295 K. Thus, just as emphasized above for gas-phase reactions, the interface of dynamics with electronic structure theory (or other methods of approximating solute potential energy surfaces and free energies of solvation) assumes a critical role in predicting rate constants. Chandresekhar et al. 723,724 calculated the free energy of solvation of the Cl - + CH 3 Cl reaction at various points along the reaction path by classical mechanical simulations involving hundreds of water molecules. They estimated a loss in equilibrium solvation energy of 23 kcal/mol on proceeding from reactants to the transition state, which led to good agreement with experiment. Later these studies were extended to OH - + H 2 CO. 725 Continuum solvation models and molecular orbital-molecular mechanics 563, methods go beyond these calculations in allowing the solute charge distribution to polarize in solution. The techniques are maturing, and this kind of calculation should become even more useful in the future. Electron and Proton Transfer. TST can be applied to electron transfer reactions, but such applications often involve additional assumptions to handle the two-electronic-state aspects and the issues of solvent-induced charge localization. In the limit of strong electronic coupling of the initial and final valence states, electron transfer becomes a single-surface electronically adiabatic reaction and standard TST methods become applicable. Straus et al. 726,727 have studied adiabatic heterogeneous electron transfer with both classical and quantum mechanical techniques. In the classical study, they compared the TST/G-H theory with trajectory simulations and found that, as in the S N 1 and S N 2 reactions, it accurately reproduces the trajectory results, and the recrossing factors are about a factor of 0.6. In a similar study, Rose and Benjamin 728 came to the same conclusions and found recrossing factors in the range In contrast, PI-QTST calculations on the same system indicate that the quantum

19 Current Status of Transition-State Theory J. Phys. Chem., Vol. 100, No. 31, mechanical effects are much larger than the classical recrossing effects. 726,727 Zichi et al. 729 and Smith et al. 730 also compared TST/G-H and trajectory calculations for electron transfer reactions. The model of slow solvent reorganization controlling charge transfer reactions appears in both electron transfer and proton transfer reactions. The use of an effective solvent coordinate as the reaction coordinate for these processes has been questioned recently. Path-integral QTST has been applied to this problem such that the proton tunneling coordinate and solvent activation were treated on equal footing. 731 An application to a realistic model of proton transfer in a polar fluid, including electronic polarization, has been presented by Lobaugh and Voth. 732,733 They conclude that solvent electronic polarization cannot be neglected and must be included quantum mechanically for quantitative accuracy of the proton tunneling rates. Azzouz and Borgis 734 have applied TST approaches to study an asymmetrical proton transfer model in liquid chloromethane. They compare results from a curve-crossing, transition-state rate constant with those of PI-QTST and with conventional quantized TST with a Bell tunneling correction factor. The agreement between the curve-crossing TST and PI-QTST results is fairly good, ranging from differences of about 25% to just over a factor of 2 for different systems. They conclude that conventional TST with parabolic tunneling is inadequate for these types of systems. It is noted that Warshel and Chu 735 and Hwang and Warshel 736 have also used PI-QTST for proton transfer, but based on a different reaction coordinate. Similarly, Hwang et al. 737 and Kong and Warshel 738 have applied PI-QTST with an energy-gap reaction coordinate to hydride transfer in enzymes and solution. Staib et al. 739 have carried out classical trajectory calculations for proton transfer in a hydrogen-bonded acid-base complex in methyl chloride. Transition-state theory within an equilibrium solvation model was compared with the trajectory results and also with TST/G-H. The full dynamical nonequilibrium solvation effect was calculated to be a factor of 0.8, and the G-H transmission coefficient theory reproduced this value. Casamassina and Huskey used experimental KIEs to conclude that motions of solvent hydrogens do not participate in the reaction coordinate for proton transfers from carbon acids (i.e., acids in which the proton is bonded to carbon) to methoxide in methanol or hydroxide in water. 740 An important lesson learned in gas-phase dynamics is that tunneling probabilities are very sensitive to the quantitative aspects of the barrier and the reaction path curvature. 207,741,742 We should keep this in mind in assessing the reliability of simulations in the condensed phase. Reactions of Uncharged Species. Solvation effects can be important for reactions of neutrals as well as ions, and the Claisen rearrangement, H 2 CdCH-O-CH 2 -CdCH 2 f OdCH- CH 2 -CH 2 -CHdCH 2, has served as a prototype for testing methods. 579,589, Gao calculated the potential of mean force for isomerization of dimethylformamide in water. 747 Solvent effects on the ring opening of cyclopropanones were studied in four solvents using statistical perturbation theory, and the resulting shifts in free energies of activation were in good agreement with experiment. 748 A critical issue in predicting reactivities and solvent effects even for neutral molecules is the set of values of atomic partial charges at the transition state. 589 Enzyme Catalysis. Enzymatic reactions provide a special case. Nearly 50 years ago it was hypothesized that enzymes act by binding to and stabilizing transition states. 749 Within this picture, knowledge of transition-state structures and charge distributions is crucial to designing transition-state inhibitors that bind tightly to active sites. Over the past several years there have been efforts to model transition states from heavyatom kinetic isotope effects; a review is available. 758 Warshel and co-workers have used transition-state concepts to discuss enzyme reactions in several studies, and Warshel et al. 762 have argued that reactions of substrates at enzyme active sites do not proceed by displacing solvent molecules to create a gas-phase environment but that enzymes are designed to solvate ionic transition states and act much like water in this respect. Free energy profiles have been computed for a few enzymatic reactions Molecular Processes in Solid-State Systems. The treatment of molecular processes in solid-state systems is somewhat simpler than the treatment of reactions in liquids because the relative rigidity of solid systems often allows simplifying approximations such as treating the solid as rigid or treating the phonon modes within a harmonic approximation. As a result, there have been many more applications of TST to solid-state systems than to liquid-phase ones. These include desorption/adsorption, diffusion, reactions, and surface reconstruction. The majority of these applications deal with diffusion and desorption/adsorption processes in which no chemical bonds are broken or made. The treatment of heterogeneous reactions is more complicated primarily because of the increased complexity in describing the potential energy surface Desorption from Surfaces. The recent body of work from Pitt et al provides an excellent discussion of the applicability of TST to thermal surface adsorption in the absence of an intrinsic barrier as well as a review of the relevant literature. They argue that the variationally optimized dividing surface need not necessarily be located at infinite separation from the surface as had been previously suggested in the literature. They have developed a classical microcanonical VTST approach that is valid for clean surfaces and for surfaces with partial coverage. The method has been applied to a model of Xe desorption from Pt. 766 Doren and Tully 767 have used classical TST and the unified statistical model 73,85-89 to study precursor-mediated adsorption and desorption of molecules on surfaces. They find that variational optimization of the dividing surface (inherent in the US model) can be very important, leading to order of magnitude changes in the Arrhenius prefactor. Nagai has used transition-state theory based upon a lattice gas model and the grand canonical ensemble to obtain a simple rate expression that depends upon lateral interactions between adsorbates. For systems without a saddle point, the dividing surface is placed far away from the crystal where the potential energy attains its maximum value and becomes flat. The validity of the model for the lateral interactions and the coverage dependence has been questioned 771,772 and defended. 773,774 In related work, Anton 775 attempted to include adsorbate coverage dependence in classical TST and tested the method for desorption reactions. Pitt et al. 765 argue that this derivation cannot be correct for barrierless adsorption. In a more empirical approach, the desorption of CO from metals has been modeled using harmonic RRKM theory in which the vibrational frequencies of the reactants and transition state were taken from experimental data. 776 Zhdanov has used a phenomenological lattice-gas TST model to look at the effect of coverage dependence on the generalized transition-state partition function 777 and to study the effect of surface reconstruction caused by adsorption on the desorption rate. 777, Diffusion on and in Solids. Classical Theories. Doll and Voter 779 reviewed theories of diffusion for solid-state

20 12790 J. Phys. Chem., Vol. 100, No. 31, 1996 Truhlar et al. systems, including methods based on TST. For systems with sufficiently large barriers, and strong enough adsorbatesubstrate coupling, so that diffusion can be viewed as a succession of uncorrelated hops between binding sites on the surface, the diffusion constant can be related to the rate constants for jumps out of the binding sites. In these cases, TST can be applied to calculate the unimolecular rate constants for the jumps, and these can be used to calculate diffusion coefficients. In its simplest form, the classical TST approximation to the single-hop rate constants is obtained from an effective onedimensional diffusion model obtained by moving the diffusing atom along a 1D reaction coordinate and letting all the other coordinates in the system relax adiabatically. 780 In this formulation the rate constant takes the form of an attempt frequency obtained from the one-dimensional model and a Boltzmann factor in the difference in energy between the saddle point and reactants for the one-dimensional model. Vineyard 541 proposed that the attempt frequency be given from a full classical harmonic TST prescription which yields an expression that is the ratio of the product of frequencies at the reactants to the product of saddle-point frequencies. For this pseudounimolecular reaction, the saddle point has one less bound frequency than reactants. Voter and Doll have developed a Monte Carlo procedure for accurate numerical evaluation of the multidimensional expression of the classical TST hopping rate constant without resorting to a harmonic analysis. 780 This approach has been extended to include dynamical corrections based on classical trajectories. 781 Guo and Thompson 782 compared a simple version of TST to full molecular dynamics for diffusion of C and H atoms in Au matrices and found agreement within a factor of 2. Although not explicitly recognized as such, a formulation of classical TST was presented in which the transition-state dividing surface is curved. 783,784 The curved dividing surface goes through the saddle point and is tangent to the conventional TST dividing surface at the saddle point. It is defined by a quadratic expansion of the dividing surface (not the potential) about the saddle point and includes anharmonic effects. Approximate dynamical corrections (short-term memory effects) have also been included in this formulation. 785 These methods have been applied to defect diffusion in solids (e.g., vacancy diffusion in metals). In a similar vein, the more conventional classical TST approach to diffusion in solids 541 was used to study kink diffusion in a model system. 786 Wahnström 787 discussed the influence of dissipation on surface diffusion and reviewed Kramers theory in the context of surface diffusion and its extension to treat diffusion in a periodic potential. Recent applications of Kramers theory and its extension have been made to treating hopping rates for diffusion in periodic potentials Quantum Mechanical Theories. For lower temperatures and diffusion of light masses such as hydrogen, quantum mechanical effects are often important. It is interesting that the application of quantized conventional TST to the diffusion problem was first proposed by Wert and Zener 540 before the classical TST theory of Vineyard. 541 It is only within the past decade that modern TST-based theories that include quantization of bound modes and tunneling effects have been applied to diffusion. Both VTST/MT and PI-QTST methods have been used. In addition, Valone et al. 691,692 have proposed a method in which classical transition-state theory is applied to an Gaussian-averaged effective potential energy surface that approximately included quantum mechanical effects. We will refer to this version of quantum mechanical TST as effective quantum potential TST (EQP-TST). Wahnström et al. 791,792 have suggested an approximation to the quantal flux-flux correlation functions in which the rate constant is approximated from the correlation function up to the point that it first goes through its first zero. Since classical TST can be viewed as the short-time limit of the classical flux-flux correlation function, 793 the authors have termed this approximate method a quantum mechanical TST. As noted in section 2, this approximate QTST was first described by Tromp and Miller. 100,101 In this review we refer to this version of TST as short-time QTST. Self-Diffusion of Metal Atoms on Metal Surfaces. Monte Carlo TST has been applied to self-diffusion on several metal surfaces and compared with classical trajectory simulations of the mean-squared displacement of the adatom. Effects of correlated hops and recrossing were studied, with dynamical effects accounting for changes from the TST activation energy of up to 6.5 kcal/mol. 780 Dynamical corrections to classical TST have been calculated for Rh diffusion on Rh(100) 781 and for adatom diffusion on the (111) face of a face-centered cubic (fcc) system model by a Lennard-Jones potential. 794 For the Rh system the TST results differed from the accurate, dynamically corrected ones by at most 6% in the temperature range K. Differences were larger for the model fcc (111) system. At low temperatures TST exhibited small overestimates (less than about 20%) of the accurate diffusion constants because of recrossing of the transition-state dividing surface. As the temperature increases, larger underestimates (greater than a factor of 2) of the diffusion constant were observed because of the importance of multiple hops in the dynamical simulations that are not included in the TST results. Atomic Diffusion in and on Solids. Zhang et al. 795 compared classical trajectory diffusion constants for hydrogen on the (100) face of Ni with classical TST calculations. The accuracy of TST for surface diffusion is limited by the neglect of hops to nonadjacent sites, or multiple hops, and by recrossings of the dividing surface. This study showed that multiple jumps can be important, increasing the diffusion constant by as much as a factor of 3 over that assuming only single jumps. Recrossing factors were found to be less important causing decreases of the diffusion constant by only about 25%. Engberg et al. 796 have examined the validity of classical TST for the diffusion of H in Pd at 800 K and showed that the distribution of transition-state configurations (i.e., the probability of finding a H atom at a transition state) determined from classical trajectory simulations is well reproduced by the TST approximation in terms of the Boltzmann factor of the potential of mean force. They emphasize that a diffusive jump event should be treated as a fluctuation in a many-body system at thermal equilibrium. They concluded that the coupled H-Pd fluctuations are adequately treated within the TST approximation. The diffusion of H atoms on Si(111) with partial hydrogen coverage has been studied by Raff and Thompson and coworkers 797,798 using a classical Monte Carlo canonical VTST method, 799,800 which is closely related to the microcanonical method employed by this group for gas-phase studies and discussed in section 3.1. These methods were also applied to the diffusion of Si atoms on the reconstructed Si(111)-(7 7) surface. 801,802 The diffusion of oxygen atoms in Ar and Xe matrices has also been studied with this Monte Carlo approach to classical variational transition-state theory. 803 An underestimate of the experimental diffusion constants by several orders of magnitude lead the authors to suggest that the experimentally observed diffusion is not for a perfect crystal, but must occur primarily along defect pathways in the lattice. Blöchl et al. 804 have studied proton diffusion in silicon using classical TST. The lattice is allowed to adjust adiabatically to

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