J. Phys. Chem. 1996, 100, 12771-12800 12771

Current Status of Transition-State Theory

Donald G. Truhlar* Department of Chemistry and Supercomputer Institute, UniVersity of Minnesota, Minneapolis, Minnesota 55455-0431

Bruce C. Garrett EnVironmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, 902 Battelle BouleVard, MS K1-96, Richland, Washington 99352

Stephen J. Klippenstein Department of Chemistry, Case Western ReserVe UniVersity, CleVeland, Ohio 44106-7078 ReceiVed: December 18, 1995; In Final Form: February 26, 1996X

We present an overview of the current status of transition-state theory and its generalizations. We emphasize (i) recent improvements in available methodology for calculations on complex systems, including the interface with electronic structure theory, (ii) progress in the theory and application of transition-state theory to condensed- phase reactions, and (iii) insight into the relation of transition-state theory to accurate quantum dynamics and tests of its accuracy via comparisons with both experimental and other theoretical dynamical approximations.

1. Introduction further assumptions:19 (II) The reactants are equilibrated in a canonical (fixed-temperature) or microcanonical (fixed-total- Transition-state theory (TST) has a long history and a bright energy) ensemble; in the latter case one sometimes also takes future. The status of the theory was reviewed in this journal in account of conservation of total angular momentum. (III) The 19831 on the occasion of a special issue dedicated to Henry reaction is electronically adiabatic (i.e., the Born-Oppenheimer Eyring. The present status report will emphasize important separation of electronic motion from internuclear motions is developments since around that time. The reader is referred to valid) in the vicinity of the dynamical bottleneck. Within this a historical account of the origin of the theory2 and to several context we can identify several versions of transition-state theory books,3-10 pedagogical articles,11-13 handbook chapters,14-16 and and related theories. For example, conventional transition-state reviews17,18 for background. theory5 is distinguished by placing the dividing surface at the The organization of this chapter is as follows. Section 2 saddle point and equating the net rate coefficient to the one- reviews recent developments in the transition-state theory of way flux coefficient.11 Variational transition-state theory simple barrier reactions in the gas phase, the original and (VTST)20-23 is distinguished by varying the definition of the prototypical type of system on which the transition-state story dividing surface to minimize the one-way flux coefficient. has unfolded. Section 3 considers reactions without an intrinsic 24-33 barrier, i.e., reactions that have no barrier in the exoergic RRKM theory is a name for transition-state theory applied direction and whose barrier equals the endoergicity in the other to a microcanonical ensemble of unimolecular reactions, and direction. The most common examples are radical-radical and the theory also included a thermal average incorporating ion-molecule associations that do not involve curve crossing collisional effects in its original form. Using the RRKM name Downloaded via RADBOUD UNIV NIJMEGEN on January 30, 2019 at 19:29:25 (UTC). and their reverse simple bond dissociations. Section 4 considers for transition-state theory places emphasis on the reactant equilibrium assumption of Rice and Ramsperger34,35 and

See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles. the theory and application of TST for reactions in condensed 36-38 phases and addresses the current “hot topic” of “environmental Kassel, which is consistent with assumption II (above) of effects” (i.e., solvent effects and phonon assistance) on reacting transition-state theory. (RRKM theory is a “quantum mechan- 28 species. ical transition-state reformulation of RRK theory”. ) In the ion dissociation literature, this theory is often termed the quasi- Throughout this article, we use “transition-state theory” as a 26 general name for any theory based in whole or in part on the equilibrium theory (QET). The transition-state theory of 24,25 fundamental assumption of transition-state theory or some unimolecular reactions was developed by Marcus and Rice, 26,27 39 quantum mechanical generalization of this assumption. Clas- Eyring and co-workers, Magee, Rabinovitch and co- 40 41-44 sically, the fundamental assumption19 is that there exists a workers, Bunker and co-workers, and a host of later 3,4,7-10,29-33 hypersurface (or surface, for brevity) in phase space with two researchers (for further references, see monographs properties: (1) it divides space into a reactant region and a on kinetics and unimolecular rate theory). product region, and (2) trajectories passing through this “divid- Transition-state theory is directed to the calculation of the ing surface” in the products direction originated at reactants one-way rate constant at equilibrium. It is usually assumed in and will not reach the surface again before being thermalized interpreting experimental data that the phenomenologically or captured in a product state. Part 2 of the fundamental defined and measured rate constants under ordinary laboratory assumption is often called the no-recrossing assumption or the conditions with reactants at translational and internal equilibrium dynamical bottleneck assumption. In addition to the funda- may be interpreted as being reasonably independent of the extent mental assumption, transition-state theory invariably makes two of chemical disequilibrium. Then the observed one-way rate constants should be well approximated by the one-way rate X Abstract published in AdVance ACS Abstracts, June 15, 1996. constants corresponding to chemical as well as translational S0022-3654(95)03748-8 CCC: $12.00 © 1996 American Chemical Society 12772 J. Phys. Chem., Vol. 100, No. 31, 1996 Truhlar et al. internal equilibrium. Once we impose the condition of reactant solvent and for which the solute molecule is sufficiently small equilibrium, the condition of transition-state equilibrium is not (few body) so that the solute itself cannot provide a heat bath an additional assumption; classically, it is a consequence of for equilibrium. Liouville’s theorem. In other words, a system with an equi- Because TST makes the equilibrium assumption, it can be librium distribution in one part of phase space evolves into a cast in a quasithermodynamic form. For example, it can be system with an equilibrium distribution in other parts of phase shown that VTST for a canonical ensemble is equivalent to space. However, the quasiequilibrium assumption of transition- minimizing the free energy of activation.71,72 state theory is that all forward crossing trajectories that A unifying element in several approaches to TST is the originated as reactants and that will proceed to products without adiabatic theory of reactions. In this theory vibrations and recrossing the dividing surface also constitute a population that rotations (as well as electronic motionssassumption III above) is in equilibrium with reactants. Although all phase points on are considered adiabatic as the system proceeds along the the dividing surface are in equilibrium, it is not necessarily true reaction coordinate. That the adiabatic assumption is related that this particular subset of all phase points is also in to the transition-state theory may be at first surprising since equilibrium. Since the dynamical and quasiequilibrium deriva- the adiabatic assumption involves global dynamics, but con- tions of transition-state theory are equivalent, the quasiequilib- nections were pointed out by many workers,73-82 and it was rium assumption breaks down whenever any recrossing occurs. shown about 15 years ago that the adiabatic theory of reactions In classical mechanics, TST provides an upper bound12,13,20-22 is identical to microcanonical variational transition-state theory to the rate constant if reactant equilibration replenishes reactant as far as overall (i.e., non-state-specified) rate constants are states fast enough (and this leads to the variational approach concerned.83,84 Perhaps more important for current thinking, by which the transition-state location is varied to minimize the though, is that local vibrational adiabaticity assumptions are calculated rate, as mentioned above). For bimolecular reactions useful for classifying variational and supernumerary transition in the gas phase, deviations from local equilibrium in the reactant states (see section 2) and making extensions of TST for state- states are usually considered small,45-50 whereas for uni- selected dynamics. molecular reactions in the gas phase one is almost always in One can distinguish the various transition-state theories by the “falloff” regime, where it is essential to consider competition the way in which quantal effects are incorporated, an area of between energy transfer repopulating reactive states and reactive considerable current interest. The choice of coordinates for depletion of those states.31 The enormous literature of the falloff representing the transition state dividing surface or the reaction problem is beyond our scope here, but we note that inclusion coordinate is another important distinguishing feature among of falloff effects is essential for using theory to predict the fate the various transition-state theories. Other classification ele- of reaction intermediates in complex mechanisms. For example, ments include the recognition of multiple dynamical bottlenecks that reflect flux through one another73,85-89 or other approxima- the dissociation rate of CH2ClO in N2 gas at 1 atm and 600 K is an order of magnitude less than the infinite-pressure TST tions for estimating how much flux recrosses the assumed value.51 There is another reason why reactant equilibrium states dynamical bottleneck. As one begins to include such transmis- might be out of equilibrium with transition states in unimolecular sion coefficients, generalized transition-state theory begins to gas-phase reactions, namely that the reactant phase space might approach accurate dynamics; the precise border where a theory have internal bottlenecks or be nonergodic and in particular stops being a generalized transition-state theory and becomes metrically decomposable, which again would violate the fun- something else, i.e., between what one calls a “statistical damental assumptions of TST.4,41,42,44,52-56 transition-state theory” and what one calls a “dynamical theory,” is certainly a matter of taste and semantics that cannot be Similar nonequilibrium issues arise for reactions in solution. resolved here. It is worthwhile to note, though, that a better In order for transition-state theory to be valid, the coupling pair of names for what is usually meant by this distinction would between the solvent and the reacting solute molecules must be be “local dynamics” and “global dynamics” theories since the sufficiently strong to maintain an equilibrium population of essence of transition-state theory is that it is based on the local reactants. If the coupling is too small, the rate of reaction is dynamics at the dynamical bottleneck. limited by the flow of energy into the reaction coordinate from Moving into the condensed phase, one finds yet other the environment. This “energy diffusion’” regime in liquids is classifying elements: How is solute separated from solvent? Is similar to the low-pressure falloff region of unimolecular rate “friction” included by letting solvent enter into the reactant theory. This region of low-to-moderate coupling has been the coordinate, or is it “added on”? Does one assume linear subject of much research dating back to the seminal work of response of the solvent to solute motions? And so forth. Kramers.57 Recent efforts have sought to obtain a unified theory Recognizing these generalizations, extensions, and distinc- of the low- and high-friction theories. Since energy diffusion tions, we are left with a host of closely related transition-state is not the emphasis of this review, we just provide a few theories. Their usefulness for analyzing, correlating, predicting, references.58-70 We note that in some cases weak coupling and understanding a wide variety of rate processes and dynami- between solute and solvent can lead to seemingly anomalous cal behaviors is beyond a doubt. What, though, is the current behavior when analyzed in terms of effective one-dimensional status of the theory? models such as Kramer’s theory. When the coupling between the solute reaction coordinate and the environment is modeled 2. Simple Barrier Reactions in terms of just the solute-solvent coupling, the theory predicts that the system is in the energy diffusion regime. However, In 1983, one might have written: Transition-state theory is strong coupling between the reaction coordinate and internal basically a classical theory because the fundamental assumption modes of the solute can lead to much faster dissipation of energy of transition-state theory for either bimolecular or unimolecular and fast equilibration (i.e., the system is not in the energy reactions (that one can define a phase space surface dividing diffusion regime). This can be included in the theory if the reactants from products such that trajectories passing through internal modes of the solute are included in modeling the this surface do not pass through it a second time on the time coupling between the reaction coordinate and environment. This scale between thermalizing collisions with third bodies) is implies that energy diffusion should be rate limiting only for intrinsically classical. To apply transition-state theory in a those systems with weak coupling between the solute and quantal world, we combine classical assumptions from transi- Current Status of Transition-State Theory J. Phys. Chem., Vol. 100, No. 31, 1996 12773 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 extend- ing 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 Figure 1. Typical example of a flux autocorrelation function through a transition state as a function of time. lifetime and the extent of tunneling. Their partial widths (in the language of resonance scattering theory90,91) give information state theory. The idea was further explicated in classical about state-selective processes, and their transmission coef- mechanical terms by various workers.71,97,98 ficients give a measure of their contribution to total reactivity. Miller et al.99,100 proposed a quantum mechanical method to To some extent the properties of transition states can be calculate Yamamoto’s time-correlation function. This leads to understood in classical terms, but classical approximations that accurate reaction rates if fully converged, but it also affords do not correspond closely enough to the true quantal nature of the possibility of a short-time approximation that might be the transition states will often predict qualitatively incorrect considered a quantal generalization of the transition-state results. Our goal is to develop efficient and accurate ways to approximation. A typical time dependence for the quantal flux- calculate the properties of the quantized transition states or flux correlation function is shown in Figure 1. The accurate appropriate classical analogs when degrees of freedom are rate is obtained by integrating Cf(t) from t ) 0 to t ) ∞; the classical enough. proposed transition-state approximation100,101 is to stop the 92 In contrast to the above, a 1984 review of the roles played integral at t ) t0 where t0 denotes the first time at which Cf(t) by metastable states in chemistry did not even mention barrier ) 0. We will call this short-time quantum TST. Hansen and states. This shifting perspective indicates a new appreciation Anderson102 have suggested another approximation to the for the presence of quantized transition-state features in accurate quantum mechanical flux-flux correlation function in which quantal reaction probabilities. A shifting paradigm raises new the flux-flux correlation function for a parabolic barrier is fitted questions (and eyebrows!), and many of these questions are not so that it reproduces the actual correlation function and its answered yet. Thus, it is an exciting and challenging time. second derivative at t ) 0. Clearly, at times t ) t0 + , where The difficulty of incorporating quantum effects into transition-  is small, recrossing is dominating over flux moving in the state theory essentially results from the classical nature of the initial direction of the wave packet, but due to the uncertainty concept of one-way flux through a surface. The direction of principle, wave packets necessarily have a spread of momenta motion that takes the system through the surface (i.e., the and coordinates, and the front or fastest edge of the wave packet coordinate orthogonal to the surface) will be called the reaction may have already recrossed the dividing surface before t ) t0, coordinate s. Specifying a one-way flux through a surface whereas other parts of the wave packet have not yet revealed requires simultaneous specification of the precise value of s and to what extent they will react or reflect. Thus, the short-time approximation is not a direct analog of the classical no- of the sign of its conjugate momentum ps. But when s is specified precisely, we are forbidden by the uncertainty principle recrossing assumption. Furthermore, once one has the apparatus to calculate 〈F(t 0) F(t)〉 from t to t , it is often possible to from knowing anything about p and in particular from knowing ) 0 s continue on to convergence.103-107 Day and Truhlar107 proposed its sign. The uncertainty principle applies to any pair of a variational version of the short-time quantum TST (V-ST- noncommuting quantum mechanical variables,93 and all attempts QTST) in which they varied the location of the dividing surface to translate the local one-way flux concept that is intrinsic to to minimize the calculated rate. Comparing the results to transition-state theory into quantal language result in noncom- accurate quantum dynamics for the O + HD, H + OD, and D muting variables94 and thus are uncertain. + OH reactions, they found typical differences of less than 10% Although transition-state theory as originally formulated is with, however, an error of a factor of 1.8 for D + OH at 200 fundamentally a time-independent theory of stationary-state K. Such calculations are very powerful, but since they reaction processes, considerable insight can be gained by intrinsically involve global dynamics, they do not provide the formulating it in time-dependent language. Both languages same conceptual picture or applicability to complex systems as played an important role in the 1930s with Eyring’s focus on transition-state theory. Park and Light103 and Seideman, Miller, the static picture of time-independent quasiequilibrium between and Manthe108-112 have provided other formulations of the exact reactants and transition states95 and Wigner’s focus19 on the rate constant based on the flux concept, and thus these fundamental no-recrossing dynamical assumption discussed in formulations are related to transition-state theory with global the Introduction. The quasiequilibrium language has dominated dynamics extensions. textbooks, but for at least 25 years advances in TST methodol- Another perspective on the time-correlation function approach ogy have been dominated by the dynamical picture. The time- is offered in the work of Voth et al.113 This leads to an dependent approach to reaction rates was cast as a flux-flux expression for the rate constant in which the dynamical effects time correlation function Cf(t) ) 〈F(t)0) F(t)〉, where F is the are delineated from the energetic factors. In separate work, Voth net flux through a dividing surface at time t, by Yamamoto,96 et al.114 proposed a Feynman path integral115,116 (PI) formulation who also explained the relation of this formulation to transition- of quantum mechanical transition-state theory (QTST) based 12774 J. Phys. Chem., Vol. 100, No. 31, 1996 Truhlar et al. on earlier work of Gillan.117-119 Path-integral quantum transi- state resonances associated with the reaction barriers for the H tion-state theory (PI-QTST) has recently been reviewed by + H2 and D + H2 reactions by using spectral quantization Voth,120,121 including a review of applications and extensions techniques. The resonance wave functions clearly demonstrated of the method. A similar approach has been presented by the localization of probability densities on the variational Stuchebrukhov.122 These path-integral-based methods offer a transition state, and their procedures allow one to distinguish convenient way to include quantum mechanical effects for all various kinds of transition-state resonances.149-151 Their analy- modes of the system on an equal footing and also provide a sis of the time-dependent quantum mechanics of barrier means for including anharmonic effects of the potential energy resonances leads to an understanding of the line shape in terms surface. To date, the application of the method to gas-phase of the sequence of poles responsible for the barrier resonances.152 114 reactions has been limited to the collinear H + H2 reaction, Experimental observation of quantized transition-state struc- and the advantages of the method have been more fully exploited ture is impeded by the difficulty of isolating contributions from for treating condensed-phase systems (see section 4). individual total angular momenta J since the structure is likely The concept of a quantum state provides an alternative (and to be harder to resolve or even unresolvable if spectra corre- perhaps more natural) way to incorporate the inevitable broad- sponding to various J are added together. Nevertheless, ening associated with the uncertainty principle. Thus, whereas quantized structure associated with transition states or other states with significant amplitude in the transition-state region a classical equilibrium state of a bound oscillator has x ) xe does appear to have been observed in anion photodetachment and px ) 0, adding zero-point motion prevents violation of the 153-156 157-165 uncertainty principle. This same concept can be used for spectra, photofragmentation spectra, and photo- 166 transition-state levels. (We will say levels in this context isomerization experiments. Even when discrete levels of the because “transition-state states” involves using the classical and transition state are not observed, anion photodetachment spec- quantal meanings of “state” too close to each other.) Transition- troscopy provides a powerful probe of the transition-state 167 state levels, however, are unbound states. Because unbound regions. states form a continuous spectrum, it has not been clear how to In the most definitive of the experimental observations, Kim 160 158 relate the discrete levels (“states”) in the quantized transition- et al. and Green et al. point out that, for the photodisso- ′ state partition function (“sum over states”) to a continuum of ciation of triplet ketene, observing the j ) 2 product state of unbound quantal states. Recently, though, it has been pointed CO gives a signal with extra structure in the first several hundred -1 out123-127 that transition-state levels are associated with complex- cm above threshold, which they interpret as an enhanced role energy poles of the scattering matrix (S matrix); such poles are of flux through the CCO bend excited states of the transition 131 variously called metastable states, resonances, decaying states, state. This is fully consistent with theoretical analysis which quasibound states, or Siegert states. (Poles of the S matrix with shows that state-selected reaction probabilities often provide a real energies are the ordinary bound states of a system.) The less clustered view of certain levels of the quantized transition state. Kim et al. and Green et al. interpreted the spectrum as whole theory of metastable quantal states (quantum mechanical -1 resonance theory, a branch of quantum mechanical scattering showing a 250 cm spacing for the CCO bend, in excellent agreement with ab initio electronic structure calculations168 that theory) immediately becomes available when the S matrix pole -1 identification is made, and this provides a powerful tool for predict 252 or 282 cm for this spacing, depending on the level of theory. Electronic structure theory also predicts levels at analysis and interpretation of transition-state phenomena with, -1 however, one rather serious limitationsnamely that transition- 150-154, 318-366, and 472-557 cm that were not observed. state levels are overlapping, broad resonances rather than In order to express the usual rate constant k(T) in terms of quantized transition states, we start with the contributions of isolated, narrow resonances (INRs), whereas many of the individual total energies E and total angular momenta J. First, theoretical results90,91 of resonance theory apply quantitatively we note that the canonical-ensemble rate constant k(T) for a only to INRs. Nevertheless, resonance theory has a lot to offer temperature T is83 for conceptualizing transition states, and it does provide important quantitative guidance; e.g., it allows us to assign -âE R lifetimes to transition states, and it provides much needed insight dE e F (E) k(E) k(T) (1) into the role of initial rotational excitation enhancing or ) R ∫ Q (T) inhibiting reactivity.126 -1 R The discussion of transition-state levels as quantized discrete where â ) (kBT) , kB is Boltzmann’s constant, F (E) is the states whose effects on reactivity can be seen individually was density of reactant states (per unit energy and volume for initiated a few years ago by the analysis of accurate quantum bimolecular reactions, per unit energy for unimolecular reac- R scattering calculations on H + H2,O + H2, and X + HX where tions), Q (T) is the partition function of reactants (per unit X is a halogen.123,125,126,128-131 (The interpretation using the volume for bimolecular reactions and unitless for unimolecular concepts of resonance theory arose as one aspect of these reactions), and k(E) is the rate constant in a microcanonical studies.) Individual quantized-transition-state energy levels have ensemble with energy E. The latter is given by now been seen and discussed for several reactions, namely H 123,125,130 132 128,129,131 132-134 R + H2, D + H2, O + H2, F + H2, Cl ∑(2J + 1)F (E,J) k(E,J) 132,135 125 + 136,137 + 138-140 + H2, H + XH, He + H2 , Ne + H2 , H J 141-144 145 146 k(E) ) (2) + O2, Li + HF, and O + HCl. The subject has R been reviewed very recently.132,147 A highlight of this kind of F (E) analysis is the ability to sort out “state-to-state-to-state” reaction where FR(E,J) is the density of states of a given total angular probabilities, i.e., from a specified level of the reactants to a momentum, and k(E,J) is the rate constant for the fixed-J specified level of the transition state to a specified level of the microcanonical ensemble. Note that products.123,126,131 This provides new insight into the origins of the effective threshold energies for reactions of individual FR(E) ) ∑(2J + 1)FR(E,J) (3) vibrational-rotational states and how these depend on the J quantum numbers.124,127,132 148 Sadeghi and Skodje have found the quantized transition- and Current Status of Transition-State Theory J. Phys. Chem., Vol. 100, No. 31, 1996 12775

QR(T) ) dE e-âEFR(E) (4) Equation 10 takes a form that allows a connection to be made ∫ between the resonance-state approach to QTST and path-integral thus QTST. It is convenient to define an aggregate quantum number τ ) (R, J, MJ). First define a Boltzmann-weighted transmission dE ∑e-âE(2J + 1)FR(E,J) k(E,J) probability for quantized transition state τ by J ∫ -â k(T) ) (5) γτ ) d e Pτ(Eτ+,J)/kBT (11) dE ∑e-âE(2J + 1)FR(E,J) ∫ [For a narrow resonance, Pτ is a step function, κτθ(), where θ J ∫ is a Heaviside function, and γτ ) κτ]. Since Pτ is independent Finally, for a bimolecular reaction, the E and J resolved rate of MJ, we can write eq 10 as constant is given by kBT k(T) ) ∑exp(-âEτ)γτ (12) N(E,J) R k(E,J) ) (6) hQ (T) τ hFR(E,J) Equations 11 and 12 are identical to the resonance-state where h is Planck’s constant, and N(E,J) is the cumulative version of quantum TST (RS-QTST) proposed previously.172 86,130,169 reaction probability defined as In that theory the transmission coefficient γτ is evaluated by assuming that Pτ increases from 0 to κτ in the vicinity of N(E,J) ) ∑ ∑P (E,J) (7) ifj transition-state level Eτ with an energy dependence determined i R j P ∈ ∈ by the properties of the metastable barrier state, and we make the transition-state assumption that κτ equals unity. If, instead, In eq 7, Pifj(E,J) is a fully state-resolved quantal transition probability from state i to state j at total energy E and total we consider a unimolecular reactant and write the complex angular momentum J, and the sums run over all energetically energies of the reactant resonances by accessible states i of the reactants R and all energetically E (τ) ) E - iΓ /2 (13) accessible states j of the products P. Finally putting (6) into res τ τ (1) yields and assume that Γτ is small, then eqs 10 and 12 with γτ replaced by Γτ/kBT take the form of the rate constant from the imaginary -âE dE e ∑(2J + 1)N(E,J) free energy (Im F) method174-177 ∫ J k(T) ) (8) 2 R hQ (T) k(T) ) ∑exp(-âEτ) Im Eres(τ) R hQ (T) τ 78,170 Equations 1-8 are exact. R In TST, N(E,J) becomes the number of energy states of the 2kBT/hQ Im Q (14) transition state with total angular momentum J and energy below where Q is the partition function for the entire system. In E, and the numerator of eq 8 becomes the canonical partition principle, the partition function Q is real and should diverge function. Chatfield et al.130 interpreted the results of accurate for a dissociative system, but complex values are obtained by quantal dynamics results using the relation analytical continuation. For a unimolecular reaction QR and Q both refer to the whole system, and eq 14 becomes N(E,J) ) ∑PR(E,J) (9) R k(T) 2 Im FR/h (15) where P (E,J) is the probability of reaction at energy E ≈ R where FR denotes the reactant free energy. Makarov and associated with a quantized transition state R of total angular Topaler177 have demonstrated that the centroid-constraint rela- momentum J with a particular value M of the component of J tionship in the PI-QTST method can be derived from the Im F total angular momentum along an arbitrary space-fixed axis. method. The Im F formulation can be used for describing This relation may be understood most easily by comparing it escape from a metastable potential, and as shown above, it has to earlier expressions based on adiabatic78,84,171 or separable169 a formal resemblance to the RS-QTST, which can be used to theory or later work based on resonance theory.172,173 In the describe either unimolecular or bimolecular reactions. However, latter approach, the R and J quantum numbers specify a the analogy is only formal since the Im F formulation is based transition-state resonance. Because the resonances are not on the properties of the metastable states178-180 of a unimolecular isolated, the transmission coefficients depend quantitatively on reactant that decays into a continuum, while RS-QTST is based the effects of additional poles of the S matrix, farther away in on the properties of the metastable states123,172 at the barrier energy or farther off the real axis. (This complexity is an top independent of the nature of the reactant. It would be intrinsic feature of interpreting the dynamics with overlapping, interesting to make further connections between these view- broad resonances.) Putting (9) into (8) yields points. -âE It is instructive to compare eq 10 to the most widely employed dE e ∑(2J + 1)∑PR(E,J) version of variational transition-state theory (VTST), namely J R k(T) ) ∫ (10) canonical variational theory (CVT) with multidimensional 15,84,181 hQR(T) ground-state transmission coefficients κ(T). In this theory (to be denoted VTST/MT for VTST with multidimensional Note that eq 10 multiplies by (2J + 1) rather than summing tunneling), one defines a reaction path and a coordinate s over MJ. Note that, as a function of total energy E, PR(E,J) denoting progress along this path. Then we consider a sequence typically rises from 0 to some finite value κR (e1) in the vicinity of trial transition states with s fixed at various points along the of the resonance energy ER for state R; κR is the level-specific path. Since s is fixed, these transition states are systems with transmission coefficient for transition-state level R. dimensionality one less than the full dimensionality of the 12776 J. Phys. Chem., Vol. 100, No. 31, 1996 Truhlar et al. system. For each value of s, we calculate the energy levels within a factor of 2 in only 8 cases and within a factor of 5 in ER(s,J) of the trial transition state, and we find the location that only 23 cases, whereas VTST/MT theory was accurate within minimizes the transition-state canonical partition function; we 54% in 27 cases and within a factor ranging from 0.32 to 1.81 call the s value at the location of this “variational transition in all cases. Since then, VTST/MT has been tested even more 23,135,182-191 state” s*(T). In the absence of quantization, this procedure widely against accurate quantal results with many would be equivalent to finding the dividing surface with more tests in three dimensions.135,182,185,187-191 These tests minimum one-way flux coefficient. The rate constant calculated confirm the previous conclusions, and the most recent tests, for CVT 191 135 from the partition functions at s*(T) is called k (T); it D + H2 and Cl + H2 in 3D, show very high accuracy corresponds to the overbarrier contribution to the rate. We also over a wide range of temperature. G 192 calculate an approximate probability of reaction P (E) associated For D + H2 VTST/MT calculations of k(T) agreed with with the ground state of the variational transition state to include accurate quantal calculations191 performed on the same potential the effects of tunneling at energies below the effective barrier energy surface eight years later within 17% or better for the height and nonclassical reflection (diffraction by the barrier top) whole temperature range of 200-1500 K. Agreement with at energies above the effective barrier. In the most reliable experiment,193 which also tests the potential energy surface,194 version of such theories, the tunneling calculation is multidi- is 32% or better over this temperature range. mensional, and the transmission coefficient may be denoted MT The Cl + H2 reaction provides another test of VTST/MT (multidimensional tunneling) or MTG (multidimensional tun- methods; for this reaction the average absolute deviation of neling based on the ground state). Finally, the composite rate VTST/MT calculations from accurate quantum dynamics is only expression is 135 10% over the 200-1000 K range. The Cl + H2 reaction also provides135 a very clear example of a supernumerary kVTST/MT(T) = κMTG(T) kCVT(T) (16) transition state131 of the first kind. Such a transition state may where be interpreted as a secondary maximum on a vibrationally adiabatic potential curve for some level, say â. If a system in dE e-âEPG(E) a lower level, say R, passes the location of the highest maximum MTG -1 âE0* -âE G (the variational transition state) of the curve for level â, it may κ (T) ) ) (kBT) e dE e P (E) ∫ -âE then undergo a vibrationally nonadiabatic R f â transition and G dE e ∫ E* ∫ (17) reflect from the supernumerary transition state of level â. Thus, such nonconventional transition states provide a detailed

-âER*(J) explanation of why the transmission coefficients are less than kBT∑∑(2J + 1)e J R unity in some cases. CVT k (T) ) (18) The O + H2 reaction has also provided a prototype test case R hQ (T) for VTST/MT. VTST/MT calculations have been tested suc- cessfully against accurate quantal dynamics,185,187,189 and they G have been very successful at interpreting experimental kinetic E*R(J) is shorthand notation for ER[s*(T),J], and E is shorthand 195,196 * isotope effects. O + H2(V)1) f OH + H and OH(V)0,1) for E*0(J)0). Combining (16)-(18) yields + H2(V)0,1) f H2O + H, where V denotes a vibrational G quantum number, have been used to test the extension of VTST -âE G -â[ER*(J)-E* ] dE e ∑(2J + 1)∑P (E) e concepts to predict state-selected rates for high-frequency mode J R 187,197-210 kVTST/MT(T) ) ∫ excitation. The first analysis of quantized transition- hQR(T) state structure in a cumulative reaction probability based on 128 accurate quantal calculations was reported for the O + H2 system by Bowman,129 who noted the existence of structure due -âE G G to a bend excited transition state. Most recently, this system dE e ∑(2J + 1)∑P [E - ER*(J) + E* ] ∫ J R provided the most fertile ground to date for analyzing the ) detailed state-to-state dynamics of a chemical reaction in terms hQR(T) of variational and supernumerary transition states observed in (19) accurate quantum dynamics calculations.131 A critical element in the success of VTST/MT theory is the Comparing (19) to (10) shows that variational transition-state accuracy of the methods used to calculate PG(E). In particular, theory with a ground-state transmission coefficient is equivalent the most accurate calculations are based on multidimensional to assuming that semiclassical methods for estimating the low-E tunneling tails that are critical for the thermal rate constants because of the G { } e-âE factor in eq 19. To be satisfactory, multidimensional PR(E,J) ) P [E - ER*(J) - E0*(J)0) ] (20) tunneling methods must account both for zero-point variations i.e., the transmission probability for each successive transition along the reaction path and for corner cutting on the concave state is just shifted from that of the ground state by the excitation side of the reaction path.170,186,201-210 The current status is that energy evaluated at the variational transition state. convenient and accurate multidimensional tunneling methods For gas-phase bimolecular reactions, accurate quantum rate are available for both small and large curvature of the reaction constants can be obtained by solving the Schro¨dinger equation path, namely the centrifugal-dominant small-curvature semiclas- by scattering theory, e.g., by expanding the scattering wave sical adiabatic ground-state (CD-SCSAG) method211,212 in the function in a basis set and converging the calculation with former case, and the large-curvature ground-state method, respect to the basis set and all numerical parameters. At the version-3 (LCG3)211,213-215 in the latter. These methods differ time of the previous status report,1 VTST/MT theory had been in the extent to which corner-cutting tunneling occurs, as tested against accurate quantal results for 33 cases (30 reactions discussed elsewhere.186,216 For atom-diatom reactions, a least- in a collinear world and 3 in a three-dimensional world). For action method208 has been used to optimize the choice of the 33 cases, conventional transition-state theory was accurate tunneling paths more finely between these limits whereas for Current Status of Transition-State Theory J. Phys. Chem., Vol. 100, No. 31, 1996 12777 polyatomic reactions it has been considered sufficiently accurate of generalized normal modes, where the generalization refers to just optimize the tunneling path by choosing between the to defining such modes in a subspace orthogonal to the reaction CD-SCSAG or LCG3 algorithms whichever predicts the greater coordinate at a nonstationary point (a point where the gradient amount of tunneling.215,217 Small-curvature tunneling calcula- of the potential does not vanish, as it does at a minimum or a tions, like VTST itself, require a knowledge of the reaction path saddle point). and the energies and frequencies along it; however, this Even at the harmonic level, there are open research questions information may be required over a longer section of the path about coordinate systems and vibrational energy levels of than is required for VTST, especially at low temperature, where generalized transition states. At stationary points (i.e., potential tunneling through the lower, wider part of the barrier may minima and saddle points), which are the only points at which contribute significantly to the rate. Large-curvature tunneling information is required for conventional transition-state theory requires, in addition, some information about the potential in a without tunneling, vibrational frequencies at the harmonic level wider region (termed the reaction swath) on the concave side are independent of coordinate system. However, for nonsta- of the reaction path.184,209,210,218-220 tionary points, even for a given choice of reaction path, harmonic The extension of validated multidimensional tunneling ap- frequencies depend on the coordinate system; in particular, they proximations to arbitrary polyatomic systems will allow a depend on the definition of the reaction coordinate for points considerable range of applications in the future. Some recent off the reaction path.236-238 Thus, it becomes very important examples based on electronic structure calculations for bi- to choose a physically appropriate coordinate system. Most 215 molecular reactions are the reaction of CF3 with CH4, the reaction-path calculations to date use rectilinear coordinates, i.e., 200 221 222 223 224 reactions of OH with H2, CH4, CD4, C2H6, and NH3, coordinates writeable as linear combinations of Cartesians, 225 226,227 228 and the reactions of H with H2O, NH3, and CH4. In whereas curvilinear coordinates (such as bond stretches, valence the CD4 case the predicted CH4/CD4 KIE at 416 K was 4.5, in bends, and dihedral angles) are more physical. One manifesta- poor agreement with the only available measured value of 11. tion of the inadequacy of rectilinear coordinates is the frequent However, a new measurement yielded 4, in excellent agreement appearance of imaginary frequencies for bound modes in with the prediction. For the first six examples mentioned above, reaction-path calculations. Recently, a general formalism for the transmission coefficients at 300 K are calculated to be 67, calculating reaction-path frequencies in curvilinear coordinates 5.9, 8.7, 7.6, 3.2, and 5.6, respectively. For the unimolecular has been presented and shown to eliminate (at least in the cases [1,5] sigmatropic rearrangement of cis-1,3-pentadiene (an considered) the problem with unphysical imaginary frequen- internal hydrogen transfer), the transmission coefficient for H cies.239,240 Even when the frequencies are not imaginary, they transfer is 6.5 at 470 K, leading to a KIE for H vs D transfer of may be inaccurate when computed with rectilinear coordinates; 4.9,212 in excellent agreement with the experimental value of this may, for example, lead to an overestimate of the tunneling. 5.2. The role of tunneling in this system had been very It has been known for a long time that anharmonicity and controversial prior to this full 36-dimensional VTST/MT vibration-rotation coupling can have an important effect on calculation. TST calculations, even for tight transition states;182,231,233,241,242 Miller, Hernandez, and co-workers229-232 have considered a however, progress in devising practical general methods for transition-state-like approximation to the rate constant by using including anharmonicity of the generalized normal modes has semiclassical theory in action-angle variables. A difficulty in been slow. Three promising approaches include second-order applying this theory is that good global action-angle variables perturbation theory,231,243-248 especially in curvilinear internal will seldom be available, if they even exist. The theory has coordinates182,218,242,249,250 where interaction terms are much been implemented using second-order perturbation theory. The smaller than in rectilinear coordinates, a classical configurational disadvantages of this approach are that for tunneling, second- integral method,251 and Monte Carlo methods for quantum order perturbation theory is not very accurate for representing mechanical path integrals.246 A convenient approximation for corner-cutting effects,207,233 and for overbarrier processes it is one-dimensional hindered rotations has been presented.215,252 not very accurate for representing large deviations of the Anharmonicity is even more important for loose transition states, variational transition-state location from the saddle point.172 If, and further discussion of anharmonic partition functions and however, one has a situation where the second-order theory is numbers and densities of states is provided in section 3. adequate, the theory has the advantage that it is a convenient A promising avenue for future development is the unified way234 to include mode-mode coupling. Recently,235 a method dynamical theory, in which recrossing corrections to VTST/ of treating anharmonicity in modes perpendicular to the reaction MT calculations are evaluated from trajectories beginning at a coordinate has been developed for cases where they may exhibit quantized variational transition state.15,253-255 To the extent that wide-amplitude motion; this method may provide improvements short-time dynamics in the vicinity of a localized dynamical in cases where anharmonicity in the perpendicular modes is bottleneck determines the rate, this includes quantum effects more important than their coupling to the reaction coordinate and recrossing in a very effective way. and than reaction-coordinate anharmonicity (the latter two If a particular vibrational mode is adiabatic (i.e., conserves features being responsible for variational displacements of the its quantum number) from reactants to the dynamical bottleneck, transition state from the saddle point). or from the dynamical bottleneck to products, TST can be Conventional (traditional) TST was concerned entirely with extended to predict state-selected rate constants for that the properties of the saddle point (the highest internal energy mode.123,131,182,192,187,193,197-199,256-261 In some cases one can point on the minimum-energy path), and indeed that myopic predict state-selected rates due to state-specific tunneling view of the potential energy surface was and is a strength of processes.184 the theory because of the resulting low demand for structural Transition-state theory has traditionally been not only the and energetic information about the system. Modern generalized primary tool for interpreting kinetic isotope effects (KIEs) but transition-state theories are still relatively modest in their needs practically the only tool.262 The standard interpretation of KIEs for such input, but they typically require information along a involves using conventional TST to infer detailed information considerable segment of the one-dimensional reaction path rather about transition-state structure. The modern perspective differs than at just two points. The local quadratic approximation along from the conventional one in two ways:222,264-267 (i) VTST this path plays a central role in the theories as does the concept indicates that the geometry (and hence the force field) can be 12778 J. Phys. Chem., Vol. 100, No. 31, 1996 Truhlar et al. quite different for isotopic versions of the same reaction.264 (ii) the way in which initial phase space was sampled, the choice Tunneling effects are often very significant even when the KIE of dividing surface, or the nature of the assumed potential energy does not exceed the maximum allowed by conventional models functionsor to real recrossing effects in the actual ion-molecule that neglect tunneling. Recent progress in developing reliable reactions. The authors note that several aspects of the dynamical multidimensional methods for including tunneling contributions behavior are explained as being due to weak coupling between in TST has assisted in identifying189,195,209,212,215 the role of the relative translational motions of reactants (or products) and tunneling much more definitively than in the past where an the reaction coordinate (asymmetric stretch) at the transition Occam’s razor approach often underestimated its role. Par- state.284 This raises the question of whether additional degrees ticularly noteworthy in recent work is a more sophisticated of freedom (e.g., C2H5 instead of CH3 or the presence of solvent treatment of secondary kinetic isotope effects; factor analyses or microhydration) would significantly increase this coupling. 285 of partition function ratios and tunneling calculations, both based Interestingly, ab initio TST calculations of the CH3/CD3 and - on full-dimension transition-state models and electronic structure H2O/D2O KIEs for the microhydrated F (H2O) + CH3Cl calculations, have provided a better appreciation of the contribu- reaction are in good agreement with experiment. There are two - tions to the KIEs of each kind of motion at the transition recent studies of the KIE for Cl + CH3Br, and they reach state.217,268-278 These studies have tested the traditional view different conclusions about agreement with experiment.285,286 that secondary KIEs mainly reflect the position of the transition However, combining the largest-basis-set harmonic KIE (0.95) state on the loose-tight continuum of transition-state structure; from the latter study285 with the effect of anharmonicity (0.64/ a quick summary of the conclusions is that this factor is 0.79) from the former286 yields a KIE of 0.77, in reasonable important, but high-frequency modes and tunneling are no less agreement with experiment (0.80-0.88) when one considers significant in the general case. Another general conclusion from the difficulty of the experiments, of converging the electronic the recent work is that real-world KIEs are much more structure calculations, and of estimating the anharmonic effect. complicated than the enticingly simplesand even more en- Gas-phase SN2 reactions illustrate some of the major unknowns ticingly successfulspopular models would have led us to in the current status of TST.282-288 believe. Nevertheless, TST interpretations of experimental KIEs Transition-state theory has had many names over the years, remain one of the most powerful methods for testing models two of which are activated complex theory (ACT) and absolute of dynamical bottlenecks and for inferring transition-state reaction rate theory (ARRT). Nowadays, TST and ACT are structures and properties. considered to be exact synonyms, and ARRT has fallen out of Another kind of successful application279 of VTST/MT to favor. The disappearance of the ARRT moniker seems reason- organic chemistry is provided by calculations on hydrogen 1,2- able since it is much less descriptive than the other two names shifts in singlet carbenes. Here s-dependent zero-point effects of the type of physical approximation involved, but actually on the effective barrier for tunneling were critical in calculating that is probably not the main reason for the ARRT name to a qualitatively correct entropy of activation. have fallen out of favor. When TST was first called ARRT, Transition-state theory is most directly applicable to reactions there was tremendous enthusiasm for combining it with which proceed on a single electronically adiabatic (i.e., Born- electronic structure theory to predict absolute reaction rates Oppenheimer) potential energy surface, although in some cases, entirely from theory. However, by the end of the 1930s it was 3 for example, O( P) + H2 f OH + H, it has been applied to certainly clear that the goal of predicting chemically accurate reactions proceeding on multiple surfaces.196 A situation that potential energy surfaces or even barrier heights was a much occurs commonly is where the degeneracy of the single surface more difficult challenge than originally anticipated. The chal- on which reaction occurs is smaller than the degeneracy of the lenge survives today, largely unmet, but nowsin the 1990ssit reactants. This occurs, for example, in the Br reaction with seems that a bit of cautious optimism may be in order, at least H2. The conventional treatment in the case where state splitting for few-body reactions. Two reasons for this may be advanced. of occupied reactant states is small compared to kBT is to assume First is the realization by a wider group of practitioners that that the ratio of reactant to transition-state electronic degeneracy very complete basis sets cannot be avoided when reliable barrier determines the average fraction of collisions that proceed on heights are desired, e.g., the use of a single set of d functions the reactive surface.280 More generally, if all reactive surfaces on C, N, O, or F, which was once considered a good basis set, are similar, one can use the ratio of the transition-state electronic is now recognized as at best semiquantitiative and but a small partition function, counting only reactive surfaces, to the product first step toward completeness. Second is the development of of the reactants’ electronic partition functions, counting all practical size-consistent treatments of electron correlation that surfaces. A recent development is that advances in quantum include the dominant effects of double, triple, and higher scattering theory have allowed this kind of assumption to be excitations. This array of techniques includes Møller-Plesset tested, and Schatz281 has presented such a test for the reaction fourth-order (MP4) perturbation theory289sas used in the 2 Cl( P3/2,1/2) + H*Cl f HCl + *Cl, where *Cl is a tagged Cl. Gaussian-2 (G2) semiempirical calculations,290 coupled cluster Schatz concludes that the approximation is good to better than theory with single and double excitations and perturbative 20%. If reaction proceeds on more than one surface, and the inclusion of unlinked triples [CCSD(T)],291 and quadratic surfaces have significantly different properties, a reasonable configuration interaction with singles and doubles and pertur- assumption in the context of TST is to add the rates for the bative inclusion of unlinked triples [QCISD(T)].292 Note that different surfaces.190 both CCSD(T) and QCISD(T) include the dominant effects of Reactions with a barrier preceded by a well provide a more quadruples, and CCSD(T) has the extremely important potential complicated scenario than simple barrier reactions. Ion- to make up for a poor reference set of orbitals. A non-size- - - molecule reactions like the Cla + CH3Clb, Cl + CH3Br, and consistent alternative that is competitive for very small systems - F (H2O) + CH3Cl SN2 reactions fall into this more complicated is multireference configuration interaction (MRCI) with single class of reaction. Trajectory calculations on the former two and double excitations from a full-valence complete-active-space reactions indicate that multiple crossings of the barrier region self-consistent-field (CASSCF) reference state,267,293-297 but this may occur, leading to a breakdown of the fundamental assump- method appears to have dimmer prospects for extendibility to tion of TST.282,283 It is not clear to what extent these recrossing larger systems. Complete-active-space second-order perturba- effects might be due to the classical nature of the calculations, tion theory (CASPT2) may play an important role in filling that Current Status of Transition-State Theory J. Phys. Chem., Vol. 100, No. 31, 1996 12779 niche.298-300 Empirical correction and extrapolation schemes reaction path calculated at a lower level and to correct the for the high-level results, such as the bond-addivitiy-corrected vibrational frequencies, but also to provide improved data for MP4 (BAC-MP4),226,301-303 scaling external correlation optimized multidimensional tunneling calculations including (SEC),194,267,304,305 and scaling all correlation (SAC)222,223,227,306-311 reaction-path curvature and tunneling paths that deviate from methods, are also very useful. For simple enough systems the the minimum-energy path by more than can be predicted by a use of such ab initio calculations has replaced older techniques quadratic expansion about this path and/or more than the radius of estimating entropies and energies of activation by empirical of curvature. Recent successful examples of these interpolatory group contributions,312,313 but the empirical procedures still play approaches are provided by the calculations, mentioned above, 309 222 223 an important role for larger species and for the correlation of on OH + CH4, CD4, and C2H6 (IVTST) and on OH + 224 experimental data. NH3 (VTST-IC). TST practitioners also have considerable interest in density IVTST and IVTST-IC are examples of direct dynamics, functional theory, which has had some notable successes for which is the calculation of dynamical attributes from electronic transition states314-320 but which still appears to be basically structure calculations without the intermediacy of fitting a unreliable for barrier heights.321 Density functional theory (like potential energy function. These methods involve interpolation, many other levels of electronic structure theory) appears to be but straight direct dynamics has also been used for VTST cal- more accurate for transition-state and reaction-path geometries culations.200,212,215,224-228,269-271,279,303,310,314-316,322,323,344,348-352 than for absolute energies.322,323 In straight direct dynamics, one carries out an electronic structure Many other issues in the reliability of electronic structure calculation directly every time that the algorithm requires an calculations of various types for transition states are also at the energy, gradient, or hessian. forefront of TST research, for example, the reliability of Recent examples of combining high-level electronic structure unrestricted Hartree-Fock reference functions in comparison theory with transition-state theory for practical applications to to restricted open-shell Hartree-Fock (or in comparison to larger systems are provided by the work of Page and co-workers 353 experiment). on the reactions CH3O f CH2O + H (a dissociation reaction with a barrier and a tight transition state), H + HNO,354 O + A very important aspect of using the above theories con- NH ,355 and H + NH 296 and the reactions of H, OH, and NH structively is the ability to calculate analytical gradients324 and 2 2 2 with N H .356 In each case the reaction path was calculated by even analytical hessians. The former capability is essential to 2 2 the CASSCF method, and single-point calculations with mul- making geometry optimizations feasible, where “optimization” tireference configuration interaction were used to improve the refers to the process of finding the zero-gradient structure of a energetics along these paths. These data were used as input reagent or a saddle point. Geometry optimization of transition for variational transition-state theory calculations and estimates states is particularly important for comparative evaluation of of tunneling corrections based on the zero-curvature vibra- possible reaction paths in complex systems, e.g., those where tionally adiabatic approximation for the transmission coefficient. complexed water molecules participate in the reaction.325,326 At Reaction-path methods and the interface of electronic structure present, geometry optimization is possible with some but not theory with chemical kinetics using these methods for variational all of the methods mentioned above, and we can anticipate that transition-state theory and semiclassical tunneling calculations further advances in this area will have dramatic impacts on TST have recently been reviewed.220,357-361 applications. As TST is being applied in recent years to complicated The reader should keep in mind that the overall level and reactions, a question that comes up is the treatment of competing reliability of electronic structure calculations depends not only pathways for a single set of reactants.362-364 In general, if both on the level of treatment of electron correlation, as just reactions are slow, and their pathways have no part in common, discussed, but also on the completeness of the one-electron basis TST can handle this just as it handles relative rates of different set. Thus, an encouraging development is the use of systematic reactants. When the competitive pathways share a common basis set studies to explore the convergence of transition-state intermediate or when one or both of the reactions are fast, one 327 barrier heights with respect to the one-electron basis. must make additional assumptions. The simplest assumption, Another computational issue of great importance for VTST used in statistical phase space theory,365,366 is that the scattering calculations is the efficient calculation of the reaction path itself. matrix is a random unitary matrix within a subset of the channels Many algorithms have been advanced for this.15,328-341 specified by some state-specific extension of TST,367,368 e.g., A critical area of current research is designing new and more those channels with accessible orbiting transition states. efficient ways to interface electronic structure calculations with dynamics. The goal of such work is to find ways to calculate 3. Reactions without an Intrinsic Potential Barrier the reaction-path and swath information needed for VTST and 3.1. Theory. In the VTST treatment of unimolecular tunneling calculations from a minimum of electronic structure reactions33,369 the rate constant at energy E and total angular information. Two promising approaches are interpolated VTST momentum J is given by (IVTST)343 and VTST with interpolated corrections (VTST- IC, also called dual-level dynamics).344 In IVTST, one carries N*(E,J) out high-level electronic structure calculations for reactants, k(E,J) ) (21) products, the saddle point, and perhaps one or two additional hFR(E,J) points near the saddle point; then all required reaction-path information is interpolated. (Interpolation can also be carried where N*(E,J) is the number of energetically available quantum out by power series345 or Shepard interpolation346,347 methods.) states at the variational transition state, and FR(E,J) represents In the IVTST-IC method, the high-level input is similar, but the reactant density of states. In conventional TST, N*(E,J) is interpolation is aided by the presence of an approximate potential replaced by N‡(E,J), which is evaluated at the saddle point, if energy function or additional lower-level electronic structure there is one. The detailed aspects of applying VTST to calculations that are carried out for a longer segment of the unimolecular reactions with well-defined saddle point are reaction path and in the swath region. The goal of dual-level essentially identical to those described in section 2 for the variational transition-state theory is not just to use high-level corresponding bimolecular reactions. However, the implemen- electronic structure calculations to correct the energies along a tation of VTST for unimolecular dissociations and their reverse 12780 J. Phys. Chem., Vol. 100, No. 31, 1996 Truhlar et al. associations in cases where the association process is barrierless increasingly strongly coupled, resulting in the breakdown of raises new issues as do other processes without an intrinsic various aspects of the PST assumptions. The development of barrier or with very flat potentials near the dynamical bottle- accurate and efficient procedures for treating these breakdowns necks. has received considerable attention in recent years. The For such “barrierless” reactions, it is especially important to statistical adiabatic channel model (SACM)79,80,388-395 provided provide a proper treatment of the variation in location of the an early and widely applied approach for treating such devia- transition state, due to its possible wide variation from inter- tions. Though SACM is called an adiabatic theory,74-80,83,84 it fragment separations R as large as tens of angstroms at low is based on a reference assumption of diabatic modes coupled energies, down to 2-3 Å at higher energies. (The extent of with a statistical distribution of energy among the modes. variation depends of course on the reaction, being larger, for Diagonalizing the Hamiltonian to convert the diabatic energies example, for a typical ion-molecule reaction than for a typical to adiabatic ones would not have a large effect on the sums radical-radical reaction. For example, for the CH2 + H f over states under typical non-state-selected conditions, though, CH3 reaction, CASSCF-MRCI calculations indicate that the so the adiabatic-diabatic distinction is not generally important. canonical variational transition state moves from a C-H distance N*(E,J) is then approximated in terms of the number of channels of 3.7 Å at 159 K to 2.7 Å at 2850 K.370) The separation of whose diabatic effective barriers are below the available energy. modes into the vibrations of the fragments, termed the conserved The primary difference of SACM from PST is the implicit modes, and the rotational and orbital motions of the fragments, implementation of asymmetries in the intermolecular potential termed the transitional modes, has provided the basis for much and also the inclusion of R-dependent variations in the fragment of the theoretical work in this area. This separation of modes rovibronic energies. is particularly meaningful at large subsystem separations, where Although VTST for a dissociation reaction may be based on the reaction coordinate is well described by the distance between a very different reference assumption of a complete randomiza- the centers of mass of the two fragments, and the transition tion of all motions of reactants, its estimates of N*(E,J) are state is “loose”; i.e., the interacting fragments have almost free identical to the SACM estimates, when implementing the same rotation. estimates for the energetics and using equivalent symmetry 83,84,396 Phase space theory (PST)365,366,371-377 is a version of TST factors. However, the difference of adiabatic/diabatic that focuses on the energetics of the separated fragments at a assumptions from statistical ones does result in major differences 396 completely loose transition state (i.e., rotations of the fragments in estimates for the product state distributions, which are really are completely unhindered), and it often provides an accurate beyond the realm of ordinary TST, as mentioned in section 2. 164,397-399 treatment when the transition state is at large separations, as in Comparisons with experiment, which are still more ion-molecule reactions. The simplest algorithms are based on limited than one would like, suggest that a combination of locating the transition state at the centrifugal barriers for adiabatic/diabatic assumptions for the conserved vibrational spherically symmetric R-n potentials. This is sometimes called modes with a statistical assumption for the transitional modes 400 the orbiting transition-state model;378 for n ) 4 it is the model is most appropriate, although for larger molecules than those of Langevin379 and Gioumousis and Stevenson,380 and for n ) considered to date, one might expect some low-frequency 6 it is the model of Gorin.381,382 These models can be conserved modes to couple significantly with the transitional invalidated in some cases by asymmetries in the long-range modes. potential, arising, for example, from ion-dipole and other long- The applications of SACM have generally focused on the range dipole interactions. (Variational transition-state theory383 interpolation of energy levels from reactants to products, and the adiabatic channel model384,385 have provided an accurate although some of the more recent studies have explicitly 392,394,395,401-403 understanding of such effects.) Within PST, N*(E,J) is given considered the energetics of long-range potentials. by the total number of asymptotic rovibronic states whose Similar interpolations of the energetics have also been incor- effective centrifugal barriers are below the available energy, and porated within a VTST-based method.404,405 However, the the total angular momentum is explicitly conserved via the explicit consideration of the fragment-fragment potential consideration of the vector sums of the angular momentum of energies of interaction, as in many of the recent VTST studies, each of the fragments and of the orbital motion. An alternative provides more meaningful tests of the validity of the underlying version of PST, as reviewed by Peslherbe et al.,378 is to locate TST assumptions and also provides a direct means for making the transition state at R ) ∞, with properties identical to those a priori predictions. of the completely separated collision partners or products. A The implementations of VTST for barrierless reactions are nonstatistical phase space theory, called the intermediate based on one of two alternative procedures.406,407 The first coupling probability matrix approach, includes effects of weak alternative is based on the determination of a minimum-energy energy transfer in associative or dissociative half-collisions by reaction path (RP) and the corresponding normal-mode harmonic postulating a finite but less than 100% probability of forming frequencies and moments of inertia along the reaction a complex in the former case or of coupling to a final state in path.378,405,408-410 The transition state partition function is then the latter.386,387 In this theory these probabilities depend on the generally determined on the basis of rigid rotor harmonic rate of energy transfer and the half-collision duration for a non- oscillator (RRHO) type assumptions for the overall complex energy-mixing half-collision; the latter in turn depends on the employing classical expressions for the rotational motions and asymptotic relative translational energy and the orbital angular quantum expressions for the harmonic vibrations. The ques- momentum. Recognition of the intermediate-coupling nature tionable validity of harmonic oscillator assumptions for the of the system has a significant effect on the temperature intermolecular bending motions has also led to the use of dependence of association rate constants.387 hindered rotor expressions. Unfortunately, the rigid rotor In the following, we will focus on the description of the assumptions are of equal uncertainty and are difficult to remove deviations from PST-type capture rate constants due to short- within the RP approach. range interactions, which are critical for association of neutral The second alternative is based on an assumed decoupling molecules. At shorter separations where some of the rotational of the “conserved” and “transitional” modes with the quantity degrees of freedom have become internal rotations, librations, N*(E,J) evaluated via the convolution of a classical-phase-space- or bending vibrations, the intermolecular motions become integral-based evaluation of the transitional mode contribution Current Status of Transition-State Theory J. Phys. Chem., Vol. 100, No. 31, 1996 12781 with a direct quantum sum for the conserved mode contri- spectroscopic probes. Unfortunately, such data are more bution.411-413 In the original version of this approach, the difficult to obtain than for the corresponding equilibrium reaction coordinate is taken as the separation between the centers situation due to the general occurrence of near degeneracies in of mass of the two reacting fragments. The classical treatment the electronic states in the transition-state region. One interest- of the transitional mode contributions is entirely satisfactory ing byproduct of the recent developments in methodology for due to their low-frequency nature as confirmed in a variety of counting states was an indication of just how rapidly the phase studies.414-416 (Note that tunneling is usually expected to be space integrals converge with number of sampling points. This of negligible importance for barrierless reactions due to the rapid convergence suggests the feasibility of bypassing the typically large masses and widths for the effective centrifugal analytic representation of the interfragment interaction potential barriers, in which case quantum corrections for the reaction via a direct ab initio quantum chemical evaluation of the coordinate are also unimportant.) The ability to directly and interaction energy for each phase space point sampled in the accurately incorporate quantum mechanical effects for the high- integrationssimilar to the direct dynamics methods for bi- frequency modes provides one of the key advantages of TST molecular reactions discussed in section 2. A first demonstration methods over classical-trajectory-based methods. In some cases, of the feasibility and validity of this direct sampling approach 352,428 this phase-space-integral-based VTST (PSI-VTST) gives similar has recently been provided for the CH2CO dissociation. results to the RP approach.378 The energy and angular momentum resolved density of states An important feature of the PSI-VTST approach is its FR(E,J) for the reactant also plays a key role in the determination classically accurate treatment of the interfragment couplings, of the rate constant. Evaluations of FR(E,J) based on RRHO low-frequency-mode anharmonicities, and low-frequency-mode expressions for the energetics provide a good first approximation vibration-rotation couplings while conserving J. Recent to the density of states. The Beyer-Swinehart algorithm,429,430 advances in the methodology, involving analytic integrations which is unfortunately limited in application to separable over the momentum portions of the phase space integrals, have expressions for the energy levels, provides the standard proce- yielded algorithms of sufficient efficiency to be widely dure for such evaluations. Improvements beyond the RRHO applicable.414,417-420 Furthermore, simplified expressions pro- level are becoming more and more important as both the viding extremely efficient approximate estimates have also been experimental results and the other aspects of the theoretical presented.421 Unfortunately, the implicit assumption of a center- methodology become increasingly accurate. The direct sum- of-mass separation distance reaction coordinate breaks down mation over the nonseparable energy levels has been found to at shorter separation distances, particularly for those reactions be sufficiently efficient for the evaluation of the density of states 352,397,431 where at least one of the atoms involved in the reactive bond is for molecules as large as CH2CO, and simplified well separated from the fragment center of mass.396 treatments432,433 are also useful. Alternatively, a method based A recently developed approach has its basis in the PSI on the random sampling of the quantum numbers should be methodology but explicitly considers the variation in the form particularly effective for larger molecules.434,435 However, in of the reaction coordinate.422-424 In particular, a variable many instances accurate expressions for the underlying energy reaction coordinate (VRC) is defined in terms of the distance levels are not available, in which case alternative procedures between two variably located fixed points, with one fixed point based on the integration of classical phase space integrals in each of the two fragments. An optimization is then carried provide useful means for estimating the anharmonic out not only of the value of the reaction coordinate, as in the effects.284,378,436-441 Again, the rapid convergence properties most popular version of VTST for bimolecular reactions, but of recently developed phase space integration methods suggest also of the definition of the reaction coordinate in terms of the the possibility of a direct sampling of the potential in place of location of the two fixed points. This VRC-TST approach its analytic representation. Adiabatic switching has been used provides a better representation of the reaction coordinate at to compute the anharmonic density of states for Al3, and values close separations, as in the reaction-path VTST method, while 2.5-2.9 times larger than the harmonic result have been retaining a classically accurate PSI-based treatment of the obtained.378 However, the anharmonic density for this system transitional mode contributions. Analytic integrations over the determined by the phase space integration method is very momentum components of the integrals again yield an approach sensitive to the assumed phase space boundary of the “reac- which is of sufficient efficiency to be widely applicable,425,426 tant”.378 Peslherbe et al.284 have studied the effect of anhar- while simplifying approximations provide an even more efficient monicity on the RRKM rate constant for the unimolecular - procedure.427 dissociation of Cl ‚‚‚CH3Cl. They find that anharmonicity The precise location predicted for the transition state in VTST increases the reactant density of states by a factor of 2-3. This is a key physical quantity since this location broadly determines would decrease the rate constant to the extent it is not canceled the reaction rate. A primary importance of this location is, for by transition-state anharmonicity. example, in directing the focus of quantum chemical estimates An alternative Monte Carlo random sampling based meth- of the interaction energies. Furthermore, the ability to predict odology directly couples the evaluation of the reactant density 442-449 this location [and correspondingly N*(E,J)] on the basis of of states with the transition state number of states. In relatively limited potential energy surface information is another this method the rate constant is evaluated as the average velocity key advantage of VTST over more dynamical methods such as through the dividing surface for a random sampling over all classical trajectory simulations. (Another advantage is the available phase space. Unfortunately, the need to numerically ability to quantize high-frequency modes, which can be crucial evaluate the delta function in the reaction coordinate makes this for predicting accurate threshold energies and thermal rates.) approach somewhat inefficient. Furthermore, the present imple- Of course, this transition-state location depends not only on the mentation of this methodology is restricted to a completely methods employed in the state counting but also on the details classical description for even the conserved vibrational modes. of the potential energy surface employed. The implementation Another important advance in treating barrierless reactions of high-level quantum chemical data in the formulation of such has been a coupling of the basic VTST methodology with other potentials, as in a number of recent VTST studies, is of the aspects of the reaction kinetics beyond simply evaluating k(E,J) utmost importance due to the difficulty of obtaining information and/or the high-pressure thermal rate constant k∞(T). For about the potential in the strong interaction region from example, the VTST methodology has been combined with both 12782 J. Phys. Chem., Vol. 100, No. 31, 1996 Truhlar et al. the standard RRKM formalism and the master equation approach when one includes vibration-rotation coupling. Within ap- in order to obtain a description of the pressure dependence of plications of TST, the motion in the coordinate associated with the reaction kinetics.450-453 Also, the coupling of the VRC- K has generally been treated as either active, i.e., available to TST methodology with quantum chemical estimates of the be shared with the vibrational degrees of freedom, or inac- radiative relaxation rate provides a novel route to the estimation tive.81,407 An indication of the variation in the predictions of complex dissociation energies via comparison with experi- obtained from these limiting cases was presented by Hase and 82 mental measurements of the zero-pressure radiative association co-workers for the Cl + C2H2 reaction. Gray and Davis have 454,455 rate. The VTST calculations of N*(E,J) have also provided also presented a classical trajectory study of the extent of meaningful predictions for the product-state distributions via conservation of K over a picosecond time scale for formaldehyde the hybrid assumption of vibrational adiabaticity and rotational at moderate energies.493 General reviews of the role of angular mixing.164,397-400 momentum in unimolecular reactions have been presented.407,494 The relation between transition-state theory and accurate 3.2. Applications. The first application of the reaction-path quantum dynamics has also been pursued for unimolecular VTST methodology to barrierless reactions was to the O + OH 405 reactions and the related reverse recombination pro- f HO2 f H + O2 reaction. Deviations as large as a factor cesses.178-180,456-470 For example, Bowman has presented a of 5 were observed between quasiclassical trajectory and VTST description of the canonical rate constants in terms of thermal results, and the importance of accurate estimates of the averages over the trace of the Smith collision lifetime matrix interaction energies in the inner transition-state region from 2.5 and has also illustrated the relation to VTST.179 Bowman and to 5.5 Å was noted for the first time. Furthermore, this study Wagner and co-workers have derived and applied an isolated provided the first a priori indication of the presence of two well- resonance version of TST theory to the H + CO recombination/ separated transition states. Various other aspects of this reaction dissociation process.459-461,463 Miller and co-workers derived system have also been the subject of recent detailed VTST and applied a random matrix based formulation of TST which calculations.144,497-499 Broadening the scope beyond VTST, further explored the relation between scattering resonances and Duchovic and Pettigrew498 have recently compiled 100 refer- 180,465-467 TST. Most recently, Miller has provided a formulation ences for the reverse H + O2 f OH + H reaction, including of recombination rate constants in terms of flux correlation over 40 theoretical studies. The quantitative validity of VTST 470 functions. This formulation demonstrates how TST becomes for the dissociation of HO2 has been demonstrated via direct exact for recombination processes in the high-pressure limit and comparison with averaged quantum scattering estimates.144 furthermore makes it clear how to evaluate deviations from the Meanwhile, VRC-TST results for the reverse bimolecular TST limit quantum mechanically. reaction are found to overestimate both quantum scattering Berblinger and Schlier471 tested classical RRKM theory with theory and classical trajectory simulations by a factor of 2.497 + numerical phase space integration (i.e., no harmonic approxima- Similar comparisons for the He and Ne + H2 reactions also tion) against classical trajectory calculations for the reaction indicated a factor of 2 overestimate by VRC-TST calculations + + + 137,139 HD2 f D + HD and H + D2. They considered total when an appropriate symmetry correction factor is included. energies 0.5-1.5 eV above the energetic threshold and total (Note that, as described in a previous paper,497 a symmetry angular momenta 0-50 p. Their results illustrate that there is correction factor of 2 was neglected in these studies which leads an interesting theoretical subtlety in discussing unimolecular to the currently stated discrepancy between the TST and decay due to the rapid dissociation, prior to equilibration, of scattering theory results.) Such deviations appear to be the result some trajectories from an initially defined reactant ensemble.472 of the direct redissociation of a substantial fraction of the After that occurs, decay may be more statistical. In fact, incoming trajectories as a result of an incomplete coupling in statistical theories like TST do not apply to the direct, short- the complex. This incomplete coupling is in turn related to the time component at all. In practical applications one cannot generally short lifetime ( 0.1 ps) and low density of states for + ∼ always separate these effects, but for DH2 this was possible. the HO2 complex. While numerous comparisons between The direct trajectories caused TST to underestimate the uni- theory and experiment for this reaction system have also been molecular decay rate by up to 40%. Removing these trajecto- presented, such comparisons are unfortunately clouded by ries, one finds the encouraging result that TST overestimates uncertainties in the potential energy surface and also in the the rate constant by only about 3% due to recrossing. contribution from excited electronic states. In certain instances, such as the dissociation of van der Waals The first detailed applications of the PSI-VTST approach were 499-501 502-505 molecules, there are substantial failings of the common formula- to the recombinations of CH3 with CH3 and with H. tions of TST. These failures are primarily the result of For both reactions satisfactory agreement with the experimen- bottlenecks to the redistribution of energy42 within the molecular tally determined canonical rate constants was obtained while complex. Considerable progress in understanding these failures employing ab initio based potentials. Furthermore, the VTST has been made via analyses of the phase space structure of these predictions for the H + CH3 recombination were in good reactions.473-491 In fact, one recognizes two (or more) separate agreement with subsequent trajectory simulations.410 However, bottlenecks (or transition states), and the kinetics can be for the latter reaction the D isotope effect of 1.4 predicted by adequately modeled in terms of the statistical rates for crossing VTST does not agree with the experimentally observed value each of them. In related work, Dumont has developed a of 2.5. Overall, the detailed VTST studies for these two “generalized flux renewal model” which in addition to the slow recombinations have provided an important testing ground for intramolecular energy flow also considers the effect of direct both simplified VTST models421,506-512 and also for some of components to the unimolecular decay process.491 Bohigas et the inherent assumptions.409,414,505 Interestingly, earlier har- al.492 and Tang et al.489 have discussed the relation of this work monic oscillator and hindered rotor based implementations of to a quantum mechanical picture. the reaction-path VTST method409 differ very little from the A related concern regards the extent of randomization and PSI-VTST results for the H atom recombination and also for + 513-515 sharing of the rotational energy with vibrational modes. The the association reactions of Li with H2O and (CH3)2O. uncertainty in treating the rotational energy arises from the fact Detailed comparisons for these reactions suggest that explicit that the “quantum number” K, denoting the component of the variation of the transition-state location at the E- and J-resolved angular momentum along a body-fixed axis z, is not conserved level typically leads to an improvement by about 20% over its Current Status of Transition-State Theory J. Phys. Chem., Vol. 100, No. 31, 1996 12783 consideration at only the E-resolved level. The methyl recom- estimates.521 In contrast, sample applications to the Cl- + 282,527-530 + 513-515 bination reactions have also provided an important testing CH3Cl and Li + H2O or (CH3)2O reactions ground for studies of the pressure dependence of the reaction provide no indication of any reduction in the reactive flux due kinetics450,451 with subsequent applications made to the CH + to short-range interactions, and VTST appears to provide an 217 + 452 H2 and CH3 + CH3CN reaction kinetics. The detailed accurate description of the initial association process. However, understanding of the CH3 plus H association has played an quasiclassical calculations indicate that the overall reaction - important role in the development of models for the association kinetics for the Cl + CH3Cl reaction deviates from TST of alkyl radicals with H atoms, which in turn is of key expectations as a result of the redissociation of the initially importance in the understanding of the kinetics of diamond formed complex prior to a randomization of the energy,282,527-530 formation.516 perhaps related once again to a short lifetime for the complex. + + The NCNO reaction is one of the first reactions for which The dissociation of C6H5Br into C6H5 + Br provides an wide-ranging energy-resolved dissociation rate constants were interesting intermediate example where the importance of short- determined experimentally.517 Such data provide a more range interactions was shown to depend on the particular stringent test for the theoretical predictions due in part to the parametrization of the potential employed.531 The interesting absence of any need to consider the collisional energy transfer question of the occurrence of two separate transition states for process. An initial model-potential-based PSI-VTST application ionic proton transfer reactions has also been studied on the basis indicated the occurrence of a transition from a long-range of the PSI-VTST methodology while employing model poten- transition state to an inner transition state as the energy rises tials.532,533 above 100 cm-1, with only the ground singlet electronic state Unimolecular reactions provide some examples of very flat contributing at the shorter separations.397 Subsequent compari- potential energy surfaces where variational transition-state theory sons of related VRC-TST calculations with local trajectory is invaluable even for determining which structures are the propagations suggest the quantitative validity of VTST, par- intermediates and which are the activated complexes. Examples ticularly when a unified statistical treatment73,85-89 of the two are the tetramethylene534,535 and trimethylene348 rearrangements. transition states is employed.518 Related indications of at least The latter species occurs as an intermediate in the cis-trans the qualitative validity of the unified statistical treatment were isomerization of cyclopropane. found in a PSI-VTST study of the CH2CO product-state In many cases, accurate ab initio data for the transition-state distributions399 and also in a VRC-TST study of the Li + HF region is unavailable. For such cases, simplified Gorin model reaction.145 The NCNO reaction also provided the first test of type representations of the potential may prove useful as the VRC-TST approach with the optimization of the reaction illustrated in a series of PSI-VTST studies of the reactions of coordinate yielding a reduction in the rate constant by a factor CH3 with CH3 or OH and the dissociation of C(CH3)4 and of about 2-3.422-424 Similar estimates of the effect of optimiz- neopentane.510-512 The temperature dependence of the rate 363 ing the form of the reaction coordinate have been obtained for constant for the reactions of HCO + NO2 and of OH with 536 a variety of related reactions, including the reaction of NC with HO2 have been similarly predicted. A number of comparisons 519 164 352,416,428 O2 and the dissociations of NO2, CH2CO, and between the Monte Carlo VTST methodology mentioned + 520 C6H6 . These studies further suggest that the bond length above442-449 and trajectory simulations, also employing empiri- of the reacting bond generally provides a good first approxima- cal potentials for the transition-state region, have been presented. 447 tion to the reaction coordinate in the inner transition-state region While good agreement was found for the dissociation of SiH2, 447,537 corresponding to atom-atom separations of about 2-3 Å. For the comparisons for the dissociations of Si2H6 and 538,539 each of these reactions the VRC-TST predicted rate constants C2H4F2 were not very favorable. The large deviation have been found to be in reasonable agreement with experi- observed in the Si2H6 study is somewhat surprising in light of mental predictions. the excellent agreement observed between VTST and experiment The experimental studies of the singlet dissociation of CH2- for the closely related C2H6 dissociation. One point worth CO provide a detailed and wide ranging set of data for noting is that the requirement of short propagation times (i.e., barrierless reaction dynamics.158,521-526 In the latest VRC-TST 10 ps or less) within the trajectory simulations means that the R application, both N*(E,J) and F (E,J) were obtained while comparisons must be made for higher excess energies than are employing potentials based on high-level quantum chemical generally considered in thermal studies. Furthermore, the investigations, including direct evaluations of the potential for empirical potential employed in these studies for the dynamically both bond length and center-of-mass separation distance reaction important inner transition-state region appears to be substantially coordinates.352,428 Furthermore, the indirect kinetic coupling more attractive than the values obtained in ab initio calculations between the conserved and transitional modes, as modulated for related reactions. For example, the CC bond strength in by the reaction coordinate, was explicitly treated. The resulting C2H4F2 is estimated to be 0.67 and 0.13 eV at RCC ) 3 and 4 nonempirical estimates were found to quantitatively (i.e., within Å, respectively, whereas ab initio calculations501 predict values 35% or better) reproduce the experimentally observed energy of 0.34 and 0.03 eV at the same separations for the similar C2H6 dependence for the dissociation rate constant. In addition, PST, dissociation. A detailed examination of the extent of the failure a RRHO-based implementation of VTST, and even the center- for these reactions for more realistic representations of the of-mass-separation-distance-based PSI implementation of VTST potential energy surface in the transition-state region will be are each in error by factors of 2 or greater. an important issue for future studies. For ionic reactions, the stronger long-range interactions A detailed picture of the factors affecting both neutral and generally lead to a better validity of the PST-type estimates. ionic barrierless associations and their reverse dissociations is However, for large enough molecules and/or weak enough gradually emerging from these studies. Of particular note, for attractions the short-range repulsions will still lead to a reduction neutral reactions, is the general occurrence of a dominant inner in the flux. The recent VTST applications are beginning to transition state at separations of 2-3 Å between the atoms address the point at which one needs to consider such short- involved in the reactive bond. Overall, these applications + + range repulsions. For the dissociation of C6H6 into C6H5 + suggest that the more sophisticated versions of VTST can H, model potential based VRC-TST calculations indicate a generally be expected to describe the reactive flux within a given reduction by a factor of about 6 as compared to PST based potential energy surface to within about a factor of 2. Further- 12784 J. Phys. Chem., Vol. 100, No. 31, 1996 Truhlar et al. more, when the complex lifetime is on the order of a nanosecond solvation between the saddle point and reactants to the gas- or longer, even better agreement might be expected. An phase free energy of activation.542 At the next level of interesting procedure for making improved predictions for the sophistication one should include nonequilibrium effects of the reaction rate, and whose usefulness needs to be explored in dynamics of the solvent; the analytical theory of Kramers57,543 greater detail, involves a coupling of quantum mechanically and the transmission coefficient expression of Grote and Hynes based VTST estimates with classical trajectory based estimates (G-H)544 have been particularly well studied. Several of nonstatistical effects. reviews1,16,70,546-556 of solution-phase reactions are available that As for reactions with tight transition states, the greatest current include discussions of TST and of these models. Within the uncertainty in VTST estimates for the reaction rates of systems past decade, the state of the theory has advanced considerably with loose transition states involves the uncertainty in the to include complex aspects not present in the earlier work. The underlying potential energy surfaces. Thus, an important aspect applications of the newer theories are just beginning to appear. of future work will involve the continued development of In this section we will review recent work with the discussion accurate descriptions of the potential energy surfaces, both for organized as follows. First we consider classical mechanical sample reactions and ultimately for larger classes of reactions. models, then we review quantum mechanical generalizations The availability of high-level quantum chemical data for the to include important bound-state-quantization and tunneling 2-3 Å region of separations will be of the utmost importance effects, and finally we consider recent applications, with an in these developments. Related quantum chemical data will emphasis on attempts to treat to realistic systems. be of use in providing more accurate descriptions of the reactant 4.1.1. Classical Theory and Application. A formal derivation density of states. of classical TST for reaction in liquids was presented by Many association reactions involve radicals with low-lying Chandler.71,555,557 The new element in the liquid phase is that excited states and/or for which the electronic state at infinity collisions of solvent molecules with the reacting solute mol- splits into several electronic states upon interaction with the ecules can lead to recrossings of the dividing surface that do association partner. Thus, another large uncertainty in the a not occur in the gas phase and, therefore, to a breakdown of priori prediction of the rate constants regards the contribution the fundamental dynamical assumption of TST. from excited electronic states. At short separations, where the Since, as mentioned in the Introduction, reactant activation transition state generally lies at higher energies, the electronic and equilibration are not the subject of this review, we proceed states are often widely spaced so that an important contribution to consider the case where coupling energy into the reactants is expected from only the ground electronic state. However, at is not rate limiting. One approach to approximating the larger separations electronic degeneracies often arise, and the influence of extended condensed-phase systems on the solute proper description is then uncertain, depending on the strengths reaction dynamics is by separating static and dynamic effects of the couplings among the electronic states. Importantly, an of the condensed phase (solvent); such effects are often called accurate a priori description of the temperature dependence (near equilibrium and nonequilibrium effects, respectively, and we room temperature and lower) of the radical-radical association will sometimes follow this convention even though these process will generally require an accurate description of the overworked terms sometimes lead to confusion. The concept switching of the transition state from large separations to shorter of nonequilibrium solvation is old; modern work dates back at separations and thereby the changing contribution of the excited least to the models of proton transfer developed by Kurz and electronic states. Kurz,558 and the concepts have been refined in more recent work.1,545,546,548,555,559-561 4. Reactions in the Condensed Phase Equilibrium solvation provides a good starting point for Transition-state theory has been widely used for the calcula- treating the reaction energetics of the solute. The surrounding tion and analysis of rate constants for chemical reactions in a condensed-phase molecules change (dress) the effective force variety of condensed-phase systems such as liquids, solids, and field of the reacting molecules. The resulting mean-field gas-solid interfaces. The use of TST in its simplest form for potential for the reacting molecule(s) is obtained from an reactions in condensed phases dates back to the work of Evans equilibrium ensemble average over configurations of the other and Polanyi539 for liquids and Wert and Zener540 and Vineyard541 molecules in the condensed phase. Since this mean-field for solids. Although there is a long history of applying TST to potential is obtained from an equilibrium ensemble average for condensed-phase systems, the accurate prediction of rate fixed configurations of the reacting molecules, the equilibrium constants in condensed phases still presents a major challenge solvation assumption implies that the solvating molecules because of the complexity of including the extended nature of instantaneously equilibrate to each new configuration of the the system in the rate constant calculation as well as the reacting molecules. In the thermodynamic formulation of difficulty of accurately evaluating the interaction energies for TST,5,16,71,539 the effect of the condensed phase on the reaction the extended systems. The condensed medium can profoundly energetics is included by the free energy of solvation that is influence the reaction dynamics, for example, by inducing obtained from equilibrium ensemble averages (a mean-field recrossings of the transition-state dividing surface that lead to approach) and is therefore an equilibrium solvent effect. This a breakdown of the fundamental assumption of TST. A is the most common approach used for including solvation computational issue that is still not fully solved is the inclusion effects in rate constants.542 Over the past several years there of multidimensional quantum mechanical effects when they are has been increased interest in developing methods for calculating important in these dissipative systems. There has been great the equilibrium free energy of solvation as required to include progress in recent years in addressing these challenges. In this equilibrium solvation effects in TST. In one approach the free section we review the advances made in the theoretical energy of solvation of a rigid solute complex is computed from development and applications of the theory to reactions in liquids model solute-solvent and solvent-solvent potentials using and to molecular processes in solids and on surfaces. classical ensemble averaging and/or statistical perturbation 4.1. Reactions in Liquids. At the most basic level, TST theory and added onto the gas-phase potential energy profile for reactions in solution is based on the equilibrium solvation obtained from electronic structure calculations.562-570 (The approach. This involves evaluating the free energy of activation solute and solvent molecules may be treated as either electroni- in solution, e.g., by adding the difference in free energy of cally inert or polarizable when solvent is explicit, but in most Current Status of Transition-State Theory J. Phys. Chem., Vol. 100, No. 31, 1996 12785 applications so far at least the solvent molecules are assumed impractical. They suggest an approach based on an inherent- nonpolarizable.) Other approaches are based on representing structure formalism in which the total rate constant is obtained the solvent bath as a three-dimensional continuum;571-605 in such from a sum over TST rate constants for saddle points separating models the electrostatic effects are treated by solving the Poisson reactant basins from product basins (the inherent structures in equation for a dielectric medium or by an equivalent algorithm the liquid). for putting in the same physics, and nonelectrostatic effects are Another approach, complementary to the explicit-solvent and either ignored or modeled based on the solvent-exposed atomic continuum models, is to approximate the collective effects using surface areas. Most of this work treats the case where the models of reduced dimensionality. Inclusion of collective solvent polarization is assumed to be in the linear response effects of nonequilibrium solvation in a TST framework finds regime, although in some cases nonlinear effects in the first its origin in the seminal work of Kramers.57,543 In the Kramers’ solvation shell (e.g., dielectric saturation) are included by theory, the reaction is treated in a highly simplified manner: empirical atomic surface tensions.580,581,604,605 the reacting solute is treated as a single reaction coordinate, Equilibrium solvation neglects any dynamical influence the and the rest of the system is treated as a bath in terms of a condensed phase may have on the reaction dynamics resulting Langevin equation of motion. Takeyama620 used this model to from fluctuations of the solvent around equilibrium. Nonequi- derive a transmission coefficient that accounts for the leading librium or dynamic solvent effects can be separated into local effect of friction when all friction components vanish except and collective effects. Local effects involve only a limited the reaction-coordinate one. Grote and Hynes,544 retaining the number of solvent molecules such as can be included in a cluster one-dimensional reaction coordinate, obtained a more general model. Transition-state theory can be applied to solution-phase result for the transmission coefficient via a more realistic reactions by separating the system into a cluster model that treatment of the bath using a generalized Langevin equation621-627 contains the part of the system undergoing reaction and the (GLE). The GLE recognizes that friction does not act instan- solvent that is treated in an approximate manner. The cluster taneously (as assumed in the original Langevin equation, model can include a finite collection of solvent molecules as sometimes called the assumption of ohmic friction), and thus well as the reactants or solute molecules. The effects of its effect may be reduced for narrow barriers that can be crossed microsolvation217,269,270,273,285,303,343,560,578,598,606-615 on reaction rapidly. The GLE for motion along a one-dimensional reaction dynamics in small clusters have been studied using TST and coordinate can be recast into Hamilton’s equation of motion VTST/MT in several papers; a review is available.616 The for a system in which the reaction coordinate is linearly coupled VTST calculations have included only one or two solvent to a bath of harmonic oscillators;621,622,626,627 we will call this molecules; in principle, this approach can be extended to larger the GLE model. Notice that in the GLE model, as in the original clusters, but very quickly (as the number of floppy degrees of Langevin model, all potentials involving the bath are quadratic freedom increases) the potential surface starts to exhibit multiple or bilinear, and thus all forces on the solvent are linear. low-energy pathways that are more appropriately treated by (Therefore, this model is a special case of treating the solvent condensed-phase methods such as those discussed in the by linear response theory.) TST with the Grote-Hynes following paragraphs. transmission coefficient (TST/G-H) is equivalent to classical When solvent molecules are treated explicit- VTST for the GLE model in which the potential along the ly,285,560,578,585,586,598-600,612,614 their motion can be included in reaction coordinate is a parabolic barrier.553,628-635 For this the reaction coordinate, and this can influence the geometry of purely quadratic potential, VTST gives the exact result. the solute and the reaction energetics along the reaction path The G-H expression may be interpreted as giving the effect and thereby the transition-state structure. Transition-state of friction (or microscopic viscosity) on the barrier traversal geometries in solution can also be optimized by continuum rate. Since the G-H equation can be derived by mixing bath solvation methods.579,586,589,595-599,602,603 Both explicit-solvent modes into the reaction coordinate,628-631,636 we see that and continuum-solvent treatments show that the structure of the “friction” and “nonequilibrium solvation” provide two different solute at the liquid-phase transition state may be quite different ways of looking at the same physical effect. Furthermore, since from the gas-phase transition-state structure or even from any friction causes recrossing of a transition state that is defined point along the gas-phase reaction path.599 without solvent participation in the reaction coordinate, we see The approach of including one or a few solvent molecules that for such a reaction coordinate nonequilibrium solvation is explicitly can account for nonequilibrium solvation effects both an example of the general phenomenon that was mentioned in of the solvent caging type, in which the solvent molecules must section 1 by which recrossing can lead to a breakdown of the move out of the way of the reacting molecules, and of the type quasiequilibrium assumption of TST. The breakdown can be where solvent molecules participate in the reaction as either a corrected by including solvent friction or by letting the solvent reactant or a catalyst.303 Collective effects (including long-range participate in the reaction coordinate. The regime where the electric polarization of the solvent dielectric) involve cooperative coupling of solute to solvent is strong enough that energy motions of the molecules in the condensed phase and are more diffusion and reactant equilibration need not be considered (but difficult to include by explicit few-body methods because of friction and nonequilibrium solvation might be important) is the large size of the system required. The larger number of sometimes called the spatial diffusion regime, and it is the equilibrium geometries and transition states in liquid-phase subject of the rest of this section. reactions and even in medium-size clusters also motivates a Nonlinearities in the forces along the reaction coordinate can transition from few-body gas-phase methods to many-body cause the optimum dividing surface to be different from the condensed-phase methods. Carter et al.617 have formulated an approximate one obtained in the parabolic barrier approximation. efficient molecular dynamics method for evaluating the classical Variational TST has been widely used to treat this problem of TST rate constant in terms of a constrained-reaction-coordinate- a nonquadratic reaction-coordinate potential that is linearly dynamics ensemble,618 which is applicable to any general coupled to a harmonic bath (i.e., the GLE model). Canonical definition of the reaction coordinate. Harris and Stillinger619 and microcanonical VTST theories have been applied to the discuss the application of VTST to the full coupled system of GLE model to elucidate such effects.637-644 Microcanonical the solute and solvent and conclude that identifying the optimal and canonical VTST provide a significant improvement over dividing surface in the full space of the system may be TST/G-H for intermediate friction, where TST/G-H under- 12786 J. Phys. Chem., Vol. 100, No. 31, 1996 Truhlar et al. estimates the effect of friction due to its neglect of nonlinear surface which is not well described by the simple harmonic potential forces.641,644 Berezhkovskii et al.645 have used varia- model underlying the Kramers and G-H expressions. tional TST with planar dividing surfaces to develop an improved Nonequilibrium solvation in charge transfer systems was approximation for the case of nonlinear potential forces; an studied by van der Zwan and Hynes.559,628,629 In these studies, analytical expression for the rate constant has been obtained the nonequilibrium polarization effect was treated by a GLE for the quartic double-well potential and compared with accurate and the rate constant was calculated by the TST/G-H method. numerical results.644 Frishman et al.646 have further extended Lee and Hynes662,663 extended the treatment to include anhar- this method to allow for bent planar dividing surfaces that are monic effects by defining an explicit solvation coordinate and needed for improved descriptions of some asymmetric potentials. an effective Hamiltonian for studying the reaction dynamics. Classical theories for reactions in solution that are based on An expression for the rate constant was obtained for the two- TST typically have as a goal either the minimization of the dimensional “solution reaction-path Hamiltonian” using varia- solvent-induced recrossing by optimizing the dividing surface tional TST and applied to a model SN2 reaction. The work of or estimating the recrossing of a given dividing surface. Pollak Lee and Hynes stimulated further work in this area to develop and Talkner647 have developed a statistical theory based upon alternative definitions of the solvent polarization coordinates the unified statistical model73,85-89 that relates the average and calculate rate constants for multidimensional systems with number of recrossings of the dividing surface to the reactive methods based on TST.664-669 Some calculations667 indicate flux.89 Another method for including recrossing effects is to the possibility of very large nonequilibrium effects, even as large explicitly calculate the reaction dynamics near the dividing as a factor of 6, although we expect that nonequilibrium effects surface. Pollak and Talkner648 have developed a “dynamical are typically smaller in classical systems. van der Zwan and 670 VTST” in which approximate dynamical corrections are in- Hynes have examined nonequilibrium solvation effects on a cluded for an optimized dividing surface that is a function of model dipolar isomerization reaction in an electrolytic solution the reaction coordinate and bath coordinates but goes through using TST/G-H. the saddle point. The dynamical corrections are approximated 4.1.2. Quantum Mechanical Theory. Quantization and by a perturbation approach similar to those used61 to describe tunneling effects are neglected in the classical approaches. The the energy-diffusion regime. major contribution to a classical reaction rate at temperature T Kramers’ theory was extended to include spatially dependent typically comes from energies about kBT above the barrier friction by Carmeli and Nitzan using an approach based on the height, whereas accurate quantum mechanical reaction prob- abilities are typically very small at such total energies because Fokker-Planck equation.649 Since then there have been many zero-point energy requirements in modes transverse to the studies of spatially dependent friction, especially friction that reaction coordinate are much greater than k T. Quantitative is dependent upon the location along a reaction coordinate.650-656 B studies that do not enforce quantization conditions on transverse Straus et al.651-655 have considered a spatially dependent friction modes, at least approximately, have little relevance to the that is modeled by a Hamiltonian in which the coupling between physical world unless there is a fortuitous cancellation of the reaction coordinate and the harmonic bath is a function of reactant and transition-state quantization effects. Classical the reaction coordinate. They compared approximate VTST methods also assume that reactive trajectories must surmount results with accurate results from simulations and found that barriers, whereas quantum mechanically the system can tunnel the VTST expression was quite accurate. Voth653 has presented through barriers, and this can be the dominant mode of reaction, an “effective Grote Hynes” method that provides a simple - especially for systems where hydrogenic motions participate in procedure to include spatially dependent friction. Haynes et the reaction coordinate. al.656 have applied the VTST approach of Berezhkovskii et al.645 to a spatially dependent and time-correlated friction model. As The standard approach for including quantum mechanical effects in TST for condensed-phase reactions is the same as for in the previous study, the method was applied to a quartic the gas phasesan approximate, ad hoc procedure in which the double-well potential. The method is in excellent agreement classical partition functions are replaced by quantum mechanical with exact simulations. This approach has been further extended ones, and a factor is included to correct for quantum mechanical to allow for curved dividing surfaces.657 motion along a reaction coordinate (e.g., quantum mechanical Pollak has extended the variational transition-state theory tunneling for energies below the classical barrier and nonclas- approach to treat condensed-phase reactions in which the sical reflection above the barrier). One way to implement this “system” and bath forces are both of a general nonlinear scheme would be (i) to quantize some of the solute modes but 658,659 form. Two orthogonal collective modes, an effective treat the low-frequency modes of the environment classically reaction coordinate and a collective solvent mode, are defined (e.g., the solvation effect on the free energy of activation is that are linear combinations of the bath and system coordinates, included classically, but bound modes of the solute that change and canonical VTST is applied to the two-degrees-of-freedom appreciably in going from the reactants to saddle point are problem.637 This approach reduces to the TST/G-H ap- quantized) and (ii) to treat tunneling by a transmission coef- proximation for the case of a harmonic bath linearly coupled ficient. In recent years, significant progress has been made in to a parabolic barrier. developing consistent approaches for including quantum me- The Kramers and G-H models are for a one-dimensional chanical effects in this way. “system” (solute) coupled to a multidimensional bath. Multi- A quantum mechanical analog of the classical Kramers and dimensional effects were discussed within the context of these G-H theories was first derived by Wolynes.671 Dakhnovskii one-dimensional theories by Nitzan and Schuss.660 Berezhk- and Ovchinnikov630 and Pollak672,673 showed that applying ovskii et al.645,661 have developed a VTST with planar dividing quantized TST with a parabolic tunneling correction factor to surfaces for systems with two degrees of freedom coupled to a the GLE model underlying these approximationssthat of a GLE model. Multidimensional effects of the solute motion, reaction coordinate linearly coupled to harmonic oscillators such as the effects of reaction path curvature, have also been representing the bathsreproduced the Wolynes expression. In explored;635 this work showed that at low-to-intermediate this quantum version of the model, tunneling is approximated coupling between the solute and the bath the reaction-path by the result for the one-dimensional parabolic barrier (which curvature can induce recrossing of the transition-state dividing diverges at low temperature and that fails to capture the physics Current Status of Transition-State Theory J. Phys. Chem., Vol. 100, No. 31, 1996 12787 of multidimensional tunneling at other temperatures674). There Voth654 have examined the effect of spatially dependent friction has been much interest in developing methods for going beyond within the path-integral QTST method. At low temperatures, this simple approach to include tunneling effects, and reviews where quantum mechanical effects are important, the nonlinear covering selected aspects of the tunneling problem are avail- dissipation was seen to give large (order of magnitude) able.16,70,176,675,676 corrections to the results from a linear dissipation. In this section we will focus on two approximate quantum Messina et al.684 have suggested a generalization of the PI- mechanical methods based on TST for treating reactions in QTST that allows for general dividing surfaces in phase space. solution. The first is quantized variational transition-state theory Although this expression does not provide a rigorous upper with semiclassical corrections for quantum mechanical effects bound to the quantum mechanical rate constant, in the same on reaction coordinate motion.15,16,84 Approaches to applying spirit as the quantum VTST approach, the optimum dividing this method to reactions in solution have recently been surface is found variationally to minimize the rate constant. This described561,677 and are discussed below. The second method approach has been applied to the model problem of an Eckart is the path integral formulation of QTST114,117-122 discussed potential coupled to a bath of oscillators,684 and a procedure briefly in section 2. This method has recently been reviewed for optimizing planar dividing surfaces in the path-integral by Voth120,121 including applications and extensions of the formalism (that is analogous to the classical variational method method. of Berezhkovskii et al.645) has also been presented.685 Pollak Both the VTST/MT and PI-QTST approaches have been has also discussed variationally determining the optimum applied to the GLE model of a reaction in solution.678 For the dividing surface within PI-QTST.686 This maximum free energy case of a parabolic barrier the VTST/MT approach reduces to approach is formulated for the GLE model and for the general the TST result already shown to reproduce Wolynes’ expression. case of nonlinear coupling to the bath. The variational PI-QTST The path integral method has also been show to reproduce this greatly improves low-temperature tunneling corrections for 687 exact result.114 These two approaches have been applied asymmetric barriers. numerically to the problem of an Eckart barrier linearly coupled Schenter et al.688 have shown how dynamical corrections, to a harmonic bath, and although they employ different which are based on classical trajectories on an effective potential approximations, the results from the two methods agree well that includes quantum mechanical effects, can be included in with each other and with accurate benchmark results.678 In the PI-QTST formalism. This work has similarities to the further work along these lines,679 it was found that accurate unified dynamical theory15,252-254 discussed in section 2. Along treatment of anharmonic effects is important in the VTST the same lines, Sagnella et al.689 have developed a semiclassical calculations for treating reactions in solution where low- TST, which is based on the semiclassical formulation of frequency modes of the bath enhance the effects of anharmo- Chapman et al.,690 that estimates dynamical recrossings from nicity. Furthermore, it was found that, for the model solution- classical trajectories initiated at the transition-state dividing phase reaction, VTST/MT calculations are often limited more surface from a semiclassical phase-space distribution. The use by the treatment of anharmonicity than by errors inherent in of an effective potential that includes quantum mechanical the approximations to the reaction dynamics. effects is closely related to the approach used by Valone et 691,692 Garrett and Schenter561 have described an approach for al. to study H diffusion on Cu surfaces (called EQP-TST applying VTST to activated chemical reactions in liquids. In in section 4.2.2). this approach the total system is separated into a cluster model The original path-integral-based rate theory by Gillan for- that is treated explicitly and a bath that is treated with a reduced- mulated the problem in terms of the reversible work for moving dimensionality model. Within an equilibrium solvation ap- the centroid of the quantum mechanical paths from the reactant proach, the effective potential for the cluster is the potential of region to the saddle point. This idea was extended to a mean force as a function of the coordinates of the cluster. When reversible-work formulation that involves moving a generalized nonequilibrium solvation is included within a GLE approach, transition-state dividing surface from the reactant region to a the effective potential becomes the sum of the potential of mean generalized transition state in the interaction region.693,694 When force and a set of harmonic modes that are coupled to the cluster translations and rotations of the dividing surface are properly coordinates. The issue of how quantum mechanical tunneling taken into account, this method is rigorously equivalent to the is implemented in this VTST approach has been discussed by method of Voth et al.114 The reversible-work formulation has Truhlar et al.677 A new solute-bath separation is presented the advantage over the earlier approach114 that it requires based on tunneling through a canonically averaged mean-shaped evaluation of averages of forces rather than the centroid-density potential which can be evaluated from the bath contribution to constrained partition function that is more difficult computa- the potential of mean force. It is worth noting that the opposite tionally. point of view has been taken by Pollak,673 who advocates a The theory developed by Marcus695-698 has played a central sudden approximation for the tunneling; this approach has been role in describing electron transfer reactions in polar solvents. more fully developed by Levine et al.680 In this approach the solvent reorganization is the rate-limiting The PI-QTST approach has also been used to provide a process for the electron transfer, and a solvent reorganization quantum mechanical generalization of the G-H model.681,682 energy often plays the role of the reaction coordinate. In the In this variational approach, a Gibbs-Bogoliubov-Feynman electronically adiabatic limit, this approach is equivalent to TST. inequality116 is used to derive an effective multidimensional For example, adiabatic electron transfer, in which the heavy parabolic model that can be solved analytically. The quantum particle motion of the solvent limits the rate of reaction, has mechanical generalization of the G-H recrossing factor takes been studied using a quantum mechanical TST approach for the same form as the classical factor, but with the classical value the GLE model.699 Smith and Hynes700 have formulated a rate of the imaginary barrier frequency replaced by an effective expression for electron transfer in or near the electronically quantum mechanical frequency. In a similar approach, Voth adiabatic regime based on the G-H approximation. and O’Gorman683 obtained a simple analytical theory in which 4.1.3. Applications to Reactions in Solution. Over the past an effective one-dimensional parabolic potential is used to decade, most theoretical studies of reaction in solution have effectively include the nonquadratic nature of the potential stressed the importance of nonequilibrium solvation effects. barrier and the influence of linear dissipation. Haynes and However, in the analysis of real experimental data, it is often 12788 J. Phys. Chem., Vol. 100, No. 31, 1996 Truhlar et al. hard to deconvolute equilibrium from nonequilibrium solvation substantial charge rearrangement in polar solvents. They found effects. For example, changing physical conditions such as the that Kramers’ theory predicted frictional transmission coef- solvent density to affect the friction will often also change the ficients of 0.2-0.6 whereas the more realistic TST/G-H model potential of mean force of the system.560,633,701-703 An under- yielded 0.7-0.97, much closer to unity, by taking account of standing of the importance of nonequilibrium solvation effects the fact that the reaction time scale is too fast for the full friction for a given system often requires first the knowledge of to develop. For a narrow range of parameters leading to solute equilibrium solvation effects. Ladanyi and Hynes704 have used caging by the solvent polarization field, κ values were calculated VTST to study equilibrium solvent effects on H atom transfer to be as small as 0.1. Keirstead et al.718 have studied a model reactions and simple geometric isomerizations in model com- SN1 reaction in water and also found that the G-H transmission pressed rare gas solvents. Compared to gas-phase rate constants, coefficient leads to excellent agreement with trajectory simula- in solution large enhancements were seen for the H atom tions. Kim and Hynes719,720 used a nonlinear response treat- transfers with much smaller enhancements for the isomeriza- ment574,575 with the solvent reaction path (minimum-energy path tions. in the multidimensional space of solvent and solute coordinates) Garrett and Schenter703 have argued that, for systems in which to develop an ionic dissociation model that was then used in the free energy of solvation is independent of solute mass, calculations on nonequilibrium solvation effects based upon the equilibrium solvation will not affect kinetic isotope effects method of van der Zwan and Hynes.559,628,629 Mathis et al.668 (KIEs). Therefore, for these systems KIEs can be used to isolate extended these studies to analyze ionization of tert-butyl halides. nonequilibrium solvation effects. VTST with semiclassical Mathis and Hynes721 have studied anomalous behavior in tunneling corrections was used to study a model of the reaction ionization of alkyl iodides using a similar approach to develop of H isotopes, including muonium (Mu), with benzene in the reaction model and then either harmonic TST (e.g., the aqueous solution. Anomalous Mu KIEs that were observed method of van der Zwan and Hynes) or VTST where the barrier experimentally could not be explained with an equilibrium is not treated as parabolic. They have also studied the alkyl solvation model. Including nonequilibrium solvation showed iodide ionization using a formulation of TST in which the substantial suppression of the Mu rate constant compared to reaction coordinate is assumed to be the solvent coordinate.722 the H and D isotopes, in agreement with the experimental Although the above studies focus on dynamical issues related findings. to the validity of transition-state theory and the additional The majority of applications of TST methods to reactions in approximations in the G-H model, one should keep in mind solution are based upon the simple Langevin equation in that the dynamical, frictional, and nonequilibrium aspects of Kramers’ theory or the GLE model in the G-H model. We the problem often change the equilibrium TST prediction by will focus primarily on applications of TST or VTST to less than a factor of 2, whereas the precise value of the solute multidimensional systems, and we will mention important barrier height and equilibrium free energy of solvation571-605 benchmark tests of the Kramers and G-H transmission coef- of the transition state relative to reactants may have a much ficients for realistic systems. Reference to recent work in which larger effect; e.g., errors of 1-2 kcal/mol lead to rate constant these theories were used to analyze experimental data include errors of factors of 5.5 and 30 at 295 K. Thus, just as Saltiel, Sun, and co-workers,705,706 who explored the validity emphasized above for gas-phase reactions, the interface of of Kramers’ model for isomerization reactions, Cho et al.,707 dynamics with electronic structure theory (or other methods of who review such analyses for three types of friction (mechanical, approximating solute potential energy surfaces and free energies internal, dielectric), Schroeder and Troe,65 who use pressure and of solvation) assumes a critical role in predicting rate constants. temperature as independent variables in a single solvent to test Chandresekhar et al.723,724 calculated the free energy of 708,709 - Kramers’ theory, Sumi and Asano, who discuss the solvation of the Cl + CH3Cl reaction at various points along difficulty of using the theory to understand isomerization the reaction path by classical mechanical simulations involving reactions of DBNA in an alkane solvent, and Anderton and hundreds of water molecules. They estimated a loss in Kauffman,710 who used Kramers’ expression to analyze dielectric- equilibrium solvation energy of 23 kcal/mol on proceeding from dependent activation energies for isomerizations. reactants to the transition state, which led to good agreement - SN1 and SN2 Charge Transfer Reactions. Bimolecular with experiment. Later these studies were extended to OH + 725 571-605 nucleophilic substitution (SN2) reactions in polar solvents have H2CO. Continuum solvation models and molecular provided fairly extensive tests of the G-H model. Extensive orbital-molecular mechanics563,567-569 methods go beyond these - work on the Cl + CH3Cl reaction provided benchmark tests calculations in allowing the solute charge distribution to polarize against classical trajectory simulations of realistic solute models in solution. The techniques are maturing, and this kind of in molecular solvents.634,711,712 In this work, the G-H model calculation should become even more useful in the future. was found to be very accurate, whereas the Kramers model gave Electron and Proton Transfer. TST can be applied to electron results that were much too low. Nonequilibrium effects reduced transfer reactions, but such applications often involve additional the rate constants by about a factor of 0.5. Similar results were assumptions to handle the two-electronic-state aspects and the also observed in comparisons of G-H model with trajectory issues of solvent-induced charge localization. In the limit of simulations for the process of ion pair interconversion between strong electronic coupling of the initial and final valence states, contact ion pairs and solvent-separated ion pairs in polar electron transfer becomes a single-surface electronically adia- solvents.713 Aguilar et al. explored the effect of solvent lagging batic reaction and standard TST methods become applicable. behind solute as the system proceeds along the reaction path Straus et al.726,727 have studied adiabatic heterogeneous electron - (nonequilibrium solvation) for the F + CH3F reaction; large transfer with both classical and quantum mechanical techniques. effects on the effective barrier to reaction were predicted.714 In the classical study, they compared the TST/G-H theory with 658,659,662,667 Various formulations of VTST have been applied trajectory simulations and found that, as in the SN1 and SN2 to SN2 reactions to examine the nature and effect of solute- reactions, it accurately reproduces the trajectory results, and the solvent coupling.663,667,715,716 recrossing factors are about a factor of 0.6. In a similar study, 728 Transition-state theory has also been used to treat SN1 ionic Rose and Benjamin came to the same conclusions and found dissociations in polar solvents. Zichi and Hynes717 have applied recrossing factors in the range 0.5-0.8. In contrast, PI-QTST TST/G-H to a model for activated ionic dissociation with calculations on the same system indicate that the quantum Current Status of Transition-State Theory J. Phys. Chem., Vol. 100, No. 31, 1996 12789 mechanical effects are much larger than the classical recrossing that bind tightly to active sites. Over the past several years effects.726,727 Zichi et al.729 and Smith et al.730 also compared there have been efforts to model transition states from heavy- TST/G-H and trajectory calculations for electron transfer atom kinetic isotope effects;750-757 a review is available.758 reactions. Warshel and co-workers have used transition-state concepts to The model of slow solvent reorganization controlling charge discuss enzyme reactions in several studies,759-761 and Warshel transfer reactions appears in both electron transfer and proton et al.762 have argued that reactions of substrates at enzyme active transfer reactions. The use of an effective solvent coordinate sites do not proceed by displacing solvent molecules to create as the reaction coordinate for these processes has been ques- a gas-phase environment but that enzymes are designed to tioned recently. Path-integral QTST has been applied to this solvate ionic transition states and act much like water in this problem such that the proton tunneling coordinate and solvent respect. Free energy profiles have been computed for a few activation were treated on equal footing.731 An application to enzymatic reactions.763 a realistic model of proton transfer in a polar fluid, including 4.2. Molecular Processes in Solid-State Systems. The electronic polarization, has been presented by Lobaugh and treatment of molecular processes in solid-state systems is Voth.732,733 They conclude that solvent electronic polarization somewhat simpler than the treatment of reactions in liquids cannot be neglected and must be included quantum mechanically because the relative rigidity of solid systems often allows for quantitative accuracy of the proton tunneling rates. Azzouz simplifying approximations such as treating the solid as rigid and Borgis734 have applied TST approaches to study an or treating the phonon modes within a harmonic approximation. asymmetrical proton transfer model in liquid chloromethane. As a result, there have been many more applications of TST to They compare results from a curve-crossing, transition-state rate solid-state systems than to liquid-phase ones. These include constant with those of PI-QTST and with conventional quantized desorption/adsorption, diffusion, reactions, and surface recon- TST with a Bell tunneling correction factor. The agreement struction. The majority of these applications deal with diffusion between the curve-crossing TST and PI-QTST results is fairly and desorption/adsorption processes in which no chemical bonds good, ranging from differences of about 25% to just over a factor are broken or made. The treatment of heterogeneous reactions of 2 for different systems. They conclude that conventional is more complicated primarily because of the increased com- TST with parabolic tunneling is inadequate for these types of plexity in describing the potential energy surface. systems. It is noted that Warshel and Chu735 and Hwang and 4.2.1. Desorption from Surfaces. The recent body of work Warshel736 have also used PI-QTST for proton transfer, but from Pitt et al.764-766 provides an excellent discussion of the based on a different reaction coordinate. Similarly, Hwang et applicability of TST to thermal surface adsorption in the absence al.737 and Kong and Warshel738 have applied PI-QTST with an of an intrinsic barrier as well as a review of the relevant energy-gap reaction coordinate to hydride transfer in enzymes literature. They argue that the variationally optimized dividing and solution. surface need not necessarily be located at infinite separation Staib et al.739 have carried out classical trajectory calculations from the surface as had been previously suggested in the for proton transfer in a hydrogen-bonded acid-base complex literature. They have developed a classical microcanonical in methyl chloride. Transition-state theory within an equilibrium VTST approach that is valid for clean surfaces and for surfaces solvation model was compared with the trajectory results and with partial coverage. The method has been applied to a model also with TST/G-H. The full dynamical nonequilibrium of Xe desorption from Pt.766 solvation effect was calculated to be a factor of 0.8, and the Doren and Tully767 have used classical TST and the unified ∼ G-H transmission coefficient theory reproduced this value. statistical model73,85-89 to study precursor-mediated adsorption Casamassina and Huskey used experimental KIEs to conclude and desorption of molecules on surfaces. They find that that motions of solvent hydrogens do not participate in the variational optimization of the dividing surface (inherent in the reaction coordinate for proton transfers from carbon acids (i.e., US model) can be very important, leading to order of magnitude acids in which the proton is bonded to carbon) to methoxide in changes in the Arrhenius prefactor. methanol or hydroxide in water.740 Nagai768-770 has used transition-state theory based upon a An important lesson learned in gas-phase dynamics is that lattice gas model and the grand canonical ensemble to obtain a tunneling probabilities are very sensitive to the quantitative simple rate expression that depends upon lateral interactions aspects of the barrier and the reaction path curvature.207,741,742 between adsorbates. For systems without a saddle point, the We should keep this in mind in assessing the reliability of dividing surface is placed far away from the crystal where the simulations in the condensed phase. potential energy attains its maximum value and becomes flat. Reactions of Uncharged Species. Solvation effects can be The validity of the model for the lateral interactions and the 771,772 important for reactions of neutrals as well as ions, and the coverage dependence has been questioned and de- fended.773,774 Claisen rearrangement, H2CdCH-O-CH2-CdCH2 f OdCH- 775 CH2-CH2-CHdCH2, has served as a prototype for testing In related work, Anton attempted to include adsorbate methods.579,589,743-746 Gao calculated the potential of mean force coverage dependence in classical TST and tested the method for isomerization of dimethylformamide in water.747 Solvent for desorption reactions. Pitt et al.765 argue that this derivation effects on the ring opening of cyclopropanones were studied in cannot be correct for barrierless adsorption. In a more empirical four solvents using statistical perturbation theory, and the approach, the desorption of CO from metals has been modeled resulting shifts in free energies of activation were in good using harmonic RRKM theory in which the vibrational frequen- agreement with experiment.748 A critical issue in predicting cies of the reactants and transition state were taken from reactivities and solvent effects even for neutral molecules is experimental data.776 the set of values of atomic partial charges at the transition Zhdanov has used a phenomenological lattice-gas TST model state.589 to look at the effect of coverage dependence on the generalized Enzyme Catalysis. Enzymatic reactions provide a special transition-state partition function777 and to study the effect of case. Nearly 50 years ago it was hypothesized that enzymes surface reconstruction caused by adsorption on the desorption act by binding to and stabilizing transition states.749 Within rate.777,778 this picture, knowledge of transition-state structures and charge 4.2.2. Diffusion on and in Solids. Classical Theories. Doll 779 distributions is crucial to designing transition-state inhibitors and Voter reviewed theories of diffusion for solid-state 12790 J. Phys. Chem., Vol. 100, No. 31, 1996 Truhlar et al. systems, including methods based on TST. For systems with which the rate constant is approximated from the correlation sufficiently large barriers, and strong enough adsorbate- function up to the point that it first goes through its first zero. substrate coupling, so that diffusion can be viewed as a Since classical TST can be viewed as the short-time limit of succession of uncorrelated hops between binding sites on the the classical flux-flux correlation function,793 the authors have surface, the diffusion constant can be related to the rate constants termed this approximate method a quantum mechanical TST. for jumps out of the binding sites. In these cases, TST can be As noted in section 2, this approximate QTST was first described applied to calculate the unimolecular rate constants for the by Tromp and Miller.100,101 In this review we refer to this jumps, and these can be used to calculate diffusion coefficients. version of TST as short-time QTST. In its simplest form, the classical TST approximation to the Self-Diffusion of Metal Atoms on Metal Surfaces. Monte single-hop rate constants is obtained from an effective one- Carlo TST has been applied to self-diffusion on several metal dimensional diffusion model obtained by moving the diffusing surfaces and compared with classical trajectory simulations of atom along a 1D reaction coordinate and letting all the other the mean-squared displacement of the adatom. Effects of coordinates in the system relax adiabatically.780 In this formula- correlated hops and recrossing were studied, with dynamical tion the rate constant takes the form of an attempt frequency effects accounting for changes from the TST activation energy obtained from the one-dimensional model and a Boltzmann of up to 6.5 kcal/mol.780 Dynamical corrections to classical factor in the difference in energy between the saddle point and TST have been calculated for Rh diffusion on Rh(100)781 and reactants for the one-dimensional model. Vineyard541 proposed for adatom diffusion on the (111) face of a face-centered cubic that the attempt frequency be given from a full classical (fcc) system model by a Lennard-Jones potential.794 For the harmonic TST prescription which yields an expression that is Rh system the TST results differed from the accurate, dynami- the ratio of the product of frequencies at the reactants to the cally corrected ones by at most 6% in the temperature range product of saddle-point frequencies. For this pseudounimo- 200-1000 K. Differences were larger for the model fcc (111) lecular reaction, the saddle point has one less bound frequency system. At low temperatures TST exhibited small overestimates than reactants. Voter and Doll have developed a Monte Carlo (less than about 20%) of the accurate diffusion constants because procedure for accurate numerical evaluation of the multidimen- of recrossing of the transition-state dividing surface. As the sional expression of the classical TST hopping rate constant temperature increases, larger underestimates (greater than a without resorting to a harmonic analysis.780 This approach has factor of 2) of the diffusion constant were observed because of been extended to include dynamical corrections based on the importance of multiple hops in the dynamical simulations classical trajectories.781 Guo and Thompson782 compared a that are not included in the TST results. simple version of TST to full molecular dynamics for diffusion Atomic Diffusion in and on Solids. Zhang et al.795 compared of C and H atoms in Au matrices and found agreement within classical trajectory diffusion constants for hydrogen on the (100) a factor of 2. face of Ni with classical TST calculations. The accuracy of Although not explicitly recognized as such, a formulation of TST for surface diffusion is limited by the neglect of hops to classical TST was presented in which the transition-state nonadjacent sites, or multiple hops, and by recrossings of the dividing surface is curved.783,784 The curved dividing surface dividing surface. This study showed that multiple jumps can goes through the saddle point and is tangent to the conventional be important, increasing the diffusion constant by as much as a TST dividing surface at the saddle point. It is defined by a factor of 3 over that assuming only single jumps. Recrossing quadratic expansion of the dividing surface (not the potential) factors were found to be less important causing decreases of about the saddle point and includes anharmonic effects. Ap- the diffusion constant by only about 25%. proximate dynamical corrections (short-term memory effects) Engberg et al.796 have examined the validity of classical TST have also been included in this formulation.785 These methods for the diffusion of H in Pd at 800 K and showed that the have been applied to defect diffusion in solids (e.g., vacancy distribution of transition-state configurations (i.e., the probability diffusion in metals). In a similar vein, the more conventional of finding a H atom at a transition state) determined from classical TST approach to diffusion in solids541 was used to classical trajectory simulations is well reproduced by the TST study kink diffusion in a model system.786 approximation in terms of the Boltzmann factor of the potential Wahnstro¨m787 discussed the influence of dissipation on of mean force. They emphasize that a diffusive jump event surface diffusion and reviewed Kramers’ theory in the context “should be treated as a fluctuation in a many-body system at of surface diffusion and its extension to treat diffusion in a thermal equilibrium”. They concluded that the coupled H-Pd periodic potential. Recent applications of Kramers’ theory and fluctuations are adequately treated within the TST approxima- its extension have been made to treating hopping rates for tion. diffusion in periodic potentials.788-790 The diffusion of H atoms on Si(111) with partial hydrogen Quantum Mechanical Theories. For lower temperatures and coverage has been studied by Raff and Thompson and co- diffusion of light masses such as hydrogen, quantum mechanical workers797,798 using a classical Monte Carlo canonical VTST 799,800 effects are often important. It is interesting that the application method, which is closely related to the microcanonical 442-449 of quantized conventional TST to the diffusion problem was method employed by this group for gas-phase studies and first proposed by Wert and Zener540 before the classical TST discussed in section 3.1. These methods were also applied to theory of Vineyard.541 It is only within the past decade that the diffusion of Si atoms on the reconstructed Si(111)-(7 7) 801,802 × modern TST-based theories that include quantization of bound surface. modes and tunneling effects have been applied to diffusion. Both The diffusion of oxygen atoms in Ar and Xe matrices has VTST/MT and PI-QTST methods have been used. In addition, also been studied with this Monte Carlo approach to classical Valone et al.691,692 have proposed a method in which classical variational transition-state theory.803 An underestimate of the transition-state theory is applied to an Gaussian-averaged experimental diffusion constants by several orders of magnitude effective potential energy surface that approximately included lead the authors to suggest that the experimentally observed quantum mechanical effects. We will refer to this version of diffusion is not for a perfect crystal, but must occur primarily quantum mechanical TST as effective quantum potential TST along defect pathways in the lattice. (EQP-TST). Wahnstro¨m et al.791,792 have suggested an ap- Blo¨chl et al.804 have studied proton diffusion in silicon using proximation to the quantal flux-flux correlation functions in classical TST. The lattice is allowed to adjust adiabatically to Current Status of Transition-State Theory J. Phys. Chem., Vol. 100, No. 31, 1996 12791 the diffusing proton, reducing the problem to one of a single moving Cu atoms (171 degrees of freedom) to achieve particle in a three-dimensional potential. convergence. Comparisons were also made to EQP-TST Simplified TST has been used to study Si adatom diffusion calculations816 with 36 moving Cu atoms, and reasonable on Si surfaces to model the dynamics of surface rearrange- agreement was obtained for T g 200 K. We know of no other ment.805 This approach was also used to study diffusion of H case where alternative quantum TST methods are validated by 806 and CH3 on diamond surfaces. such comparisons with so many degrees of freedom. 817 Jaquet and Miller807 have compared accurate quantum me- Wonchoba et al. also applied the VTST/MT method to H chanical diffusion constants with TST for a model of H on W on Ni(100) and found smaller effects of lattice motion than in in which the H atom is treated as two-dimensional and is coupled the Cu model system. This study explained the previously to a phonon bath treated by harmonic oscillators. Their confusing phenomenon of a low-temperature transition temper- harmonic TST results were for a dividing surface that was a ature at which the Arrhenius plot shows a dramatic change in function of only solute coordinates and neglected quantum slope. This was previously interpreted by experimentalists as mechanical tunneling. Therefore, their correction factor to TST a transition between overbarrier and tunneling dynamics, but 817 included both quantum mechanical tunneling effects and a Wonchoba et al. showed it is really a transition between Grote-Hynes-type correction for phonon-induced classical multistate tunneling and ground-state tunneling, which is recrossing. consistent with earlier VTST/MT calculations of the state- 818 The H on Cu(100) system is especially important for dependent tunneling probabilities by Rice et al. Wonchoba understanding the current status of TST because several different et al. obtained an analytic approximation that fits the full calculations well. versions of TST have been applied for the rigid-surface case, 819 and the EQP-TST, VTST/MT, and PI-QTST approaches have Mak and George applied conventional TST with quantized been applied with movable metal atoms. The various studies vibrations for H diffusion on Ru(001). The surface was treated all use the same potential energy function, which is not very as rigid. The calculated Arrhenius preexponential factor was higher than the experimental value by about a factor of 4. accurate, so the system should probably be regarded as “model 820 Cu”, but we still learn about dynamics. Haug and Metiu have studied H diffusion on Ni(100). The motion of surface atoms were treated within a mean-field For H on rigid Cu, Valone et al.691,692 applied EQP-TST, approximation. For this model, quantum mechanical results Lauderdale and Truhlar808,809 applied VTST/MT [in particular were compared with the short-time QTST method.791,792 VTST with the original SCSAG method, which is identical to The diffusion of H isotopes on a rigid Ru(0001) surface has CD-SCSAG for surface diffusion on a rigid fcc (100) surface], also been studied using VTST with SCSAG tunneling821 and a and Sun and Voth810 applied PI-QTST.114 Accurate multidi- potential energy function based on ab initio pseudopotential mensional quantum mechanical results and short-time QTST calculations. Kinetic isotope effects were in good agreement calculations for H on rigid Cu were presented by Haug et al.,792 with experiment. and the former can be used to test the transition-state theories, The first applications of PI-QTST to surface diffusion were which prove to be quite accurate at low temperatures, with errors for H and D diffusion on niobium using a reversible work of 32%, 7%, 9%, and 24% at 200 K and 9%, 25%, 30%, and formulation, although it seems that the calculation was for 20% at 300 K for the short-time QTST, EQP-TST, VTST/MT, moving the centroid along a reaction path, rather than for and PI-QTST results, respectively. At higher temperatures moving a dividing surface that constrains the centroid.822 Path- recrossings and multiple hops become more important, and the integral QTST was used to study H and D diffusion on the errors in the VTST methods grow to factors greater than 2 for Pd(111) surface and diffusion into subsurface sites below the temperatures above 400 K. first layer of surface atoms.823 Quantum mechanical effects for Short-time QTST has also been applied to reduced-dimen- surface diffusion are modest and tend to increase the diffusion sional models of H diffusion on the rigid Cu (100) surface and constants compared to purely classical results. The subsurface 792,811,812 compared with the accurate quantum mechanical results. transitions are more constricted and show an unusual quantum In addition, this approximate version of QTST was compared mechanical behavior. The quantum mechanical rate constants with the VTST/SCSAG calculations of Lauderdale and for transition to the subsurface are significantly lower than the 808,809 691 Truhlar and the EQP-TST method of Valone et al., as classical ones. well as the accurate quantum mechanical results. It is interesting Perry et al.824 have applied a Monte Carlo approach to that the errors in short-time QTST are typically larger than those classical variational transition-state theory to the diffusion of for VTST/MT and EQP-TST. H atoms in xenon matrices at 12-80 K. Tunneling contribu- The calculations with a nonrigid lattice are based on the tions to the diffusion coefficient are estimated by Boltzmann embedded cluster method of Lauderdale and Truhlar.813 Sun average of the tunneling probabilities through the one- and Voth810 applied PI-QTST to diffusion of H on nonrigid dimensional potential along the minimum-energy path. Cu(100). Allowing the substrate to move suppressed the rate Molecular Diffusion on Surfaces. Lakhlifi and Girardet825 constant slightly at the lowest temperature (e.g., by about 40% have applied a TST-like approach (termed the transit time at 100 K) and increased it by 2-20% at temperatures from 120 concept826) to the diffusion of Xe and small molecules on crystal to 300 K. Including the effect of electron-hole pairs by a surfaces for temperatures from 20 to 100 K. The rate constant dissipative Langevin-like model decreased the rate constants for a diffusive jump takes the form of a harmonic TST by factors of about 40% at 100 K to 1% at 300 K. expression for an approximate Hamiltonian representing a rigid Wonchoba and Truhlar814,815 reported VTST/MT calculations adsorbate molecule on the surface. The bound degrees of using the CD-SCSAG tunneling method for H diffusion on Cu; freedom are treated quantum mechanically, and tunneling is the difference between rate constants calculated with moving neglected. The reaction coordinate is described in terms of and fixed lattices increases from a factor of 3.7 at 300 K to motion in the two coordinates parallel to the plane of the surface. factors of 24-27 at 80-120 K. The comparison of the PI- The vibrational modes are taken to be the motion of the molecule QTST and VTST/MT calculations showed that the two quite perpendicular to the surface plane, the molecular rotation, and different methods predict similar effects of including quantum three-dimensional vibration of each substrate atom. This tunneling for 30 moving Cu atoms (93 degrees of freedom) at approach seems to neglect the bound mode in the plane T g 120 K. The VTST/MT calculations included up to 56 perpendicular to the reaction coordinate. 12792 J. Phys. Chem., Vol. 100, No. 31, 1996 Truhlar et al.

Dobbs and Doren827 have compared classical TST estimates Reaction of gas-phase molecules with adsorbates can proceed with classical trajectory simulations of the diffusion constant by two mechanisms: direct collision of gas-phase molecules for CO diffusion on Ni(111). The TST estimates are based on with the adsorbates (Eley-Rideal mechanism) and a trapping approximating the diffusion constant by single hops between mediated process in which the gas-phase molecule first adsorbs adjacent sites, whereas the trajectory simulations are obtained and then reacts with another adsorbate (Langmuir-Hinshelwood from the long-time limit of the mean-squared displacement. For mechanism). Weinberg840 has presented a TST analysis of the this system multiple hops are important, and the TST results rates of these two reaction processes on surfaces. From this underestimate the accurate values by factors of 100 to 20 over analysis he concludes that under most conditions the rate of the temperature range from 175 to 1000 K. Pai and Doren828 the trapping mediated process dominates that for the direct have studied the diffusion of a model rigid linear triatomic on reaction process. a metal surface. Numerically accurate classical TST diffusion The interaction of H atom with surfaces of solids like silicon coefficients are compared with exact classical diffusion coef- and carbon can be viewed as a covalent bonding interaction. ficients for three models in which the mass distribution within Thus, the desorption or adsorption of H atom on these surfaces the triatomic is different for each model, but the potential is viewed more like an association or unimolecular dissociation remains the same. The change in the distribution of masses reaction. For silicon an important question is whether hydrogen alters the frequency of the bending motion of the triatomic comes off the surface as atoms or molecules. Classical TST relative to the surface. Classical TST yields the same diffusion rate constants for H atom desorption and H-H recombination coefficient for each of these models, whereas marked changes and desorption from Si(111) have been calculated using a Monte are observed in the exact diffusion coefficients. Furthermore, Carlo approach.799 H atom desorption was found to be classical TST underestimates the diffusion constant for all three negligible, in agreement with experiment. These calculations models. The authors find that as the bending frequency were extended to classical VTST calculations of the rate constant decreases the dissipation of energy in the motion along the for the H-H recombination and desorption from Si(111).800 surface is more rapid, and multiple hops become less important. Canonical VTST calculations with quantum mechanical Calhoun and Doren829 have used the PI-QTST method to partition functions have been carried out for the association of study a two-dimensional model of CO diffusion on Ni(111). H atoms with the (111) surface of diamond.516,841,842 VTST Comparisons with purely classical results indicated that quantum rate constants for this process have been compared with those mechanical effects on the rate constant became important for of the association reaction of H atoms with alkyl radicals using temperatures below about 100 K. Large enhancements at 50 empirical potentials.516 The two systems show similar behavior K, larger than both the classical and the extrapolated Arrhenius with the exception of the rotational motion of the alkyl radical. fit to the high-temperature quantum mechanical results, were VTST calculations for the H atom association with the (111) attributed to quantum mechanical tunneling. diamond surface have been performed for a potential energy Diffusion in Zeolites and Polymers. June et al.830 have used surface that was developed based on accurate ab initio calcula- 841 classical, Monte Carlo TST to calculate hop rates between tions. Rate constants obtained from quasiclassical trajectories different sites for two Lennard-Jones spheres representing xenon for the same potential agree well with the VTST rate con- 842 and SF6 at infinite dilution in the zeolite silicalite. Then Poisson stants. statistics were assumed to calculate the diffusivity. Comparison Conventional microcanonical transition-state theory has been of the TST results with trajectory simulations of the diffusivity used to study activated dissociative adsorption of CH4 on for Xe diffusion at 150 and 200 K gave good agreement. Snurr Pt(111).843 et al.831 have used a similar process to obtain diffusion constants for benzene in the zeolite silicalite. In this case, rate constants 5. Concluding Remarks for hops between adjacent sites were approximated using Looking forward from the vantage point of 1983, one might classical canonical TST. Diffusion constants were computed have predicted that the basic formalism of transition-state theory by a Monte Carlo simulation of the master equation describing was well established, and the future would consist of various the time evolution of populations at different adsorption sites quantitative refinements, especially taking advantage of the in the silicalite structure. Schro¨der and Sauer have also studied anticipated advances in electronic structure theory. These the diffusion of benzene in silicalite by calculating enthalpies advances have indeed occurred, but along with further concep- and entropies in the rigid rotor-harmonic oscillator approxima- tual refinements in the dynamics. Probably the chief noteworthy tion along reference paths parallel to the crystallographic axes.832 advance of the past 10 years is that the relationship of transition- A TST model was developed and applied to the diffusion of state theory to accurate quantum dynamics has been greatly light gases such as He and H2 in rigid matrices of dense clarified. Especially rewarding developments include advances polymers such as rubbery polyisobutylene and glassy bisphenol in exploiting the flux-flux correlation function formulation of 833 A polycarbonate. reaction rate theory, the discovery and analysis of quantized Baker834,835 has applied approximate formulas for diffusion transition-state structure in microcanonical ensemble rate con- in melts based on classical, conventional TST to interdiffusion stants, the extension of well-validated multidimensional tun- in complex aluminosilicate melts. neling approximations to polyatomic systems, the development 4.2.3. Surface Reactions. To date, most applications of TST of the path integral approach to TST, techniques for considering to surface reactions have been to relatively simple reactive alternative dividing surfaces, and the development of Monte processes. Most of this work has focused on dissociative Carlo sampling techniques. Simultaneously, new frameworks chemisorption of H2 on metals. VTST with SCSAG tunneling have been proposed for treating solvent effects on complex was applied to H2 and D2 dissociative chemisorption on three systems. rigid surfaces, Ni (100), (110), and (111).836,837 The effects of On the applications side it is clear that we stand on several surface defects (steps) and adsorbed H atoms on the chemi- thresholds. As far as electronic structure methodology for sorption kinetics were studied using VTST/MT. The reversible applications, direct dynamics or statistics methods are poised work formulation of PI-QTST was applied to the dissociative to have a major impact, so the difficulties of the creation of 838,839 chemisorption of H2 on Cu. Quantum mechanical effects analytic potential energy surfaces will not so readily impede were shown to be very important for both the Ni and Cu cases. progress. Furthermore, electronic structure theory itself is Current Status of Transition-State Theory J. Phys. 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