current status of transition-state theory

Current Status of Transition-State Theory

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

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

Stephen J. KlippensteinDepartment 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 interfacewith 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 andtests of its accuracy via comparisons with both experimental and other theoretical dynamical approximations.

1. Introduction

Transition-state theory (TST) has a long history and a brightfuture. The status of the theory was reviewed in this journal in19831 on the occasion of a special issue dedicated to HenryEyring. The present status report will emphasize importantdevelopments since around that time. The reader is referred toa historical account of the origin of the theory2 and to severalbooks,3-10 pedagogical articles,11-13 handbook chapters,14-16 andreviews17,18 for background.The organization of this chapter is as follows. Section 2

reviews recent developments in the transition-state theory ofsimple barrier reactions in the gas phase, the original andprototypical type of system on which the transition-state storyhas unfolded. Section 3 considers reactions without an intrinsicbarrier, i.e., reactions that have no barrier in the exoergicdirection and whose barrier equals the endoergicity in the otherdirection. The most common examples are radical-radical andion-molecule associations that do not involve curve crossingand their reverse simple bond dissociations. Section 4 considersthe theory and application of TST for reactions in condensedphases and addresses the current “hot topic” of “environmentaleffects” (i.e., solvent effects and phonon assistance) on reactingspecies.Throughout this article, we use “transition-state theory” as a

general name for any theory based in whole or in part on thefundamental assumption of transition-state theory or somequantum mechanical generalization of this assumption. Clas-sically, the fundamental assumption19 is that there exists ahypersurface (or surface, for brevity) in phase space with twoproperties: (1) it divides space into a reactant region and aproduct region, and (2) trajectories passing through this “divid-ing surface” in the products direction originated at reactantsand will not reach the surface again before being thermalizedor captured in a product state. Part 2 of the fundamentalassumption is often called the no-recrossing assumption or thedynamical bottleneck assumption. In addition to the funda-mental assumption, transition-state theory invariably makes two

further assumptions:19 (II) The reactants are equilibrated in acanonical (fixed-temperature) or microcanonical (fixed-total-energy) ensemble; in the latter case one sometimes also takesaccount of conservation of total angular momentum. (III) Thereaction is electronically adiabatic (i.e., the Born-Oppenheimerseparation of electronic motion from internuclear motions isvalid) in the vicinity of the dynamical bottleneck. Within thiscontext we can identify several versions of transition-state theoryand related theories. For example, conventional transition-statetheory5 is distinguished by placing the dividing surface at thesaddle point and equating the net rate coefficient to the one-way flux coefficient.11 Variational transition-state theory(VTST)20-23 is distinguished by varying the definition of thedividing surface to minimize the one-way flux coefficient.RRKM theory24-33 is a name for transition-state theory appliedto a microcanonical ensemble of unimolecular reactions, andthe theory also included a thermal average incorporatingcollisional effects in its original form. Using the RRKM namefor transition-state theory places emphasis on the reactantequilibrium assumption of Rice and Ramsperger34,35 andKassel,36-38 which is consistent with assumption II (above) oftransition-state theory. (RRKM theory is a “quantum mechan-ical transition-state reformulation of RRK theory”.28) In theion dissociation literature, this theory is often termed the quasi-equilibrium theory (QET).26 The transition-state theory ofunimolecular reactions was developed by Marcus and Rice,24,25

Eyring and co-workers,26,27 Magee,39 Rabinovitch and co-workers,40 Bunker and co-workers,41-44 and a host of laterresearchers (for further references, see monographs3,4,7-10,29-33

on kinetics and unimolecular rate theory).Transition-state theory is directed to the calculation of the

one-way rate constant at equilibrium. It is usually assumed ininterpreting experimental data that the phenomenologicallydefined and measured rate constants under ordinary laboratoryconditions with reactants at translational and internal equilibriummay be interpreted as being reasonably independent of the extentof chemical disequilibrium. Then the observed one-way rateconstants should be well approximated by the one-way rateconstants corresponding to chemical as well as translationalX Abstract published inAdVance ACS Abstracts,June 15, 1996.

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

S0022-3654(95)03748-8 CCC: $12.00 © 1996 American Chemical Society

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internal equilibrium. Once we impose the condition of reactantequilibrium, the condition of transition-state equilibrium is notan additional assumption; classically, it is a consequence ofLiouville’s theorem. In other words, a system with an equi-librium distribution in one part of phase space evolves into asystem with an equilibrium distribution in other parts of phasespace. However, the quasiequilibrium assumption of transition-state theory is that all forward crossing trajectories thatoriginated as reactants and that will proceed to products withoutrecrossing the dividing surface also constitute a population thatis in equilibrium with reactants. Although all phase points onthe dividing surface are in equilibrium, it is not necessarily truethat this particular subset of all phase points is also inequilibrium. Since the dynamical and quasiequilibrium deriva-tions of transition-state theory are equivalent, the quasiequilib-rium assumption breaks down whenever any recrossing occurs.In classical mechanics, TST provides an upper bound12,13,20-22

to the rate constant if reactant equilibration replenishes reactantstates fast enough (and this leads to the variational approachby which the transition-state location is varied to minimize thecalculated rate, as mentioned above). For bimolecular reactionsin the gas phase, deviations from local equilibrium in the reactantstates are usually considered small,45-50 whereas for uni-molecular reactions in the gas phase one is almost always inthe “falloff” regime, where it is essential to consider competitionbetween energy transfer repopulating reactive states and reactivedepletion of those states.31 The enormous literature of the falloffproblem is beyond our scope here, but we note that inclusionof falloff effects is essential for using theory to predict the fateof reaction intermediates in complex mechanisms. For example,the dissociation rate of CH2ClO in N2 gas at 1 atm and 600 Kis an order of magnitude less than the infinite-pressure TSTvalue.51 There is another reason why reactant equilibrium statesmight be out of equilibrium with transition states in unimoleculargas-phase reactions, namely that the reactant phase space mighthave internal bottlenecks or be nonergodic and in particularmetrically decomposable, which again would violate the fun-damental assumptions of TST.4,41,42,44,52-56

Similar nonequilibrium issues arise for reactions in solution.In order for transition-state theory to be valid, the couplingbetween the solvent and the reacting solute molecules must besufficiently strong to maintain an equilibrium population ofreactants. If the coupling is too small, the rate of reaction islimited by the flow of energy into the reaction coordinate fromthe environment. This “energy diffusion’” regime in liquids issimilar to the low-pressure falloff region of unimolecular ratetheory. This region of low-to-moderate coupling has been thesubject of much research dating back to the seminal work ofKramers.57 Recent efforts have sought to obtain a unified theoryof the low- and high-friction theories. Since energy diffusionis not the emphasis of this review, we just provide a fewreferences.58-70 We note that in some cases weak couplingbetween solute and solvent can lead to seemingly anomalousbehavior when analyzed in terms of effective one-dimensionalmodels such as Kramer’s theory. When the coupling betweenthe solute reaction coordinate and the environment is modeledin terms of just the solute-solvent coupling, the theory predictsthat the system is in the energy diffusion regime. However,strong coupling between the reaction coordinate and internalmodes of the solute can lead to much faster dissipation of energyand fast equilibration (i.e., the system is not in the energydiffusion regime). This can be included in the theory if theinternal modes of the solute are included in modeling thecoupling between the reaction coordinate and environment. Thisimplies that energy diffusion should be rate limiting only forthose systems with weak coupling between the solute and

solvent and for which the solute molecule is sufficiently small(few body) so that the solute itself cannot provide a heat bathfor equilibrium.Because TST makes the equilibrium assumption, it can be

cast in a quasithermodynamic form. For example, it can beshown that VTST for a canonical ensemble is equivalent tominimizing the free energy of activation.71,72

A unifying element in several approaches to TST is theadiabatic theory of reactions. In this theory vibrations androtations (as well as electronic motionssassumption III above)are considered adiabatic as the system proceeds along thereaction coordinate. That the adiabatic assumption is relatedto the transition-state theory may be at first surprising sincethe adiabatic assumption involves global dynamics, but con-nections were pointed out by many workers,73-82 and it wasshown about 15 years ago that the adiabatic theory of reactionsis identical to microcanonical variational transition-state theoryas far as overall (i.e., non-state-specified) rate constants areconcerned.83,84 Perhaps more important for current thinking,though, is that local vibrational adiabaticity assumptions areuseful for classifying variational and supernumerary transitionstates (see section 2) and making extensions of TST for state-selected dynamics.One can distinguish the various transition-state theories by

the way in which quantal effects are incorporated, an area ofconsiderable current interest. The choice of coordinates forrepresenting the transition state dividing surface or the reactioncoordinate is another important distinguishing feature amongthe various transition-state theories. Other classification ele-ments include the recognition of multiple dynamical bottlenecksthat reflect flux through one another73,85-89 or other approxima-tions for estimating how much flux recrosses the assumeddynamical bottleneck. As one begins to include such transmis-sion coefficients, generalized transition-state theory begins toapproach accurate dynamics; the precise border where a theorystops being a generalized transition-state theory and becomessomething else, i.e., between what one calls a “statisticaltransition-state theory” and what one calls a “dynamical theory,”is certainly a matter of taste and semantics that cannot beresolved here. It is worthwhile to note, though, that a betterpair of names for what is usually meant by this distinction wouldbe “local dynamics” and “global dynamics” theories since theessenceof transition-state theory is that it is based on the localdynamics at the dynamical bottleneck.Moving into the condensed phase, one finds yet other

classifying elements: How is solute separated from solvent? Is“friction” included by letting solvent enter into the reactantcoordinate, or is it “added on”? Does one assume linearresponse of the solvent to solute motions? And so forth.Recognizing these generalizations, extensions, and distinc-

tions, we are left with a host of closely related transition-statetheories. Their usefulness for analyzing, correlating, predicting,and understanding a wide variety of rate processes and dynami-cal behaviors is beyond a doubt. What, though, is the currentstatus of the theory?

2. Simple Barrier Reactions

In 1983, one might have written: Transition-state theory isbasically a classical theory because the fundamental assumptionof transition-state theory for either bimolecular or unimolecularreactions (that one can define a phase space surface dividingreactants from products such that trajectories passing throughthis surface do not pass through it a second time on the timescale between thermalizing collisions with third bodies) isintrinsically classical. To apply transition-state theory in aquantal world, we combine classical assumptions from transi-

12772 J. Phys. Chem., Vol. 100, No. 31, 1996 Truhlar et al.

tion-state theory with quantal or semiclassical prescriptions forquantizing bound degrees of freedom and for including tunnelingand nonclassical reflection of the reaction coordinate motion.Transition-state theory is relevant to the total flux from reactantsto products; and only for modes with a well-understood adiabaticor diabatic correlation between reactants and a transition stateor between a transition state and products does transition-statetheory provide state-selective information. The goal of extend-ing the theory is to find suitable ways to merge classical andquantal concepts and to exploit adiabatic connections whenpossible.Today one might agree with most of this but look at it

differently and say: Transition states are quantum mechanicalmetastable states with intrinsic energies, widths, transmissioncoefficients, and partial widths. Their widths (in energy space)are due to (or control, depending on the point of view) theirlifetime and the extent of tunneling. Their partial widths (inthe language of resonance scattering theory90,91) give informationabout state-selective processes, and their transmission coef-ficients give a measure of their contribution to total reactivity.To some extent the properties of transition states can beunderstood in classical terms, but classical approximations thatdo not correspond closely enough to the true quantal nature ofthe transition states will often predict qualitatively incorrectresults. Our goal is to develop efficient and accurate ways tocalculate the properties of the quantized transition states orappropriate classical analogs when degrees of freedom areclassical enough.In contrast to the above, a 1984 review92 of the roles played

by metastable states in chemistry did not even mention barrierstates. This shifting perspective indicates a new appreciationfor the presence of quantized transition-state features in accuratequantal reaction probabilities. A shifting paradigm raises newquestions (and eyebrows!), and many of these questions are notanswered yet. Thus, it is an exciting and challenging time.The difficulty of incorporating quantum effects into transition-

state theory essentially results from the classical nature of theconcept of one-way flux through a surface. The direction ofmotion that takes the system through the surface (i.e., thecoordinate orthogonal to the surface) will be called the reactioncoordinates. Specifying a one-way flux through a surfacerequires simultaneous specification of the precise value ofsandof the sign of its conjugate momentumps. But when s isspecified precisely, we are forbidden by the uncertainty principlefrom knowing anything aboutps and in particular from knowingits sign. The uncertainty principle applies to any pair ofnoncommuting quantum mechanical variables,93 and all attemptsto translate the local one-way flux concept that is intrinsic totransition-state theory into quantal language result in noncom-muting variables94 and thus are uncertain.Although transition-state theory as originally formulated is

fundamentally a time-independent theory of stationary-statereaction processes, considerable insight can be gained byformulating it in time-dependent language. Both languagesplayed an important role in the 1930s with Eyring’s focus onthe static picture of time-independent quasiequilibrium betweenreactants and transition states95 and Wigner’s focus19 on thefundamental no-recrossing dynamical assumption discussed inthe Introduction. The quasiequilibrium language has dominatedtextbooks, but for at least 25 years advances in TST methodol-ogy have been dominated by the dynamical picture. The time-dependent approach to reaction rates was cast as a flux-fluxtime correlation functionCf(t) ) ⟨F(t)0) F(t)⟩, whereF is thenet flux through a dividing surface at timet, by Yamamoto,96

who also explained the relation of this formulation to transition-

state theory. The idea was further explicated in classicalmechanical terms by various workers.71,97,98

Miller et al.99,100proposed a quantum mechanical method tocalculate Yamamoto’s time-correlation function. This leads toaccurate reaction rates if fully converged, but it also affordsthe possibility of a short-time approximation that might beconsidered a quantal generalization of the transition-stateapproximation. A typical time dependence for the quantal flux-flux correlation function is shown in Figure 1. The accuraterate is obtained by integratingCf(t) from t ) 0 to t ) ∞; theproposed transition-state approximation100,101 is to stop theintegral att ) t0 wheret0 denotes the first time at whichCf(t)) 0. We will call this short-time quantum TST. Hansen andAnderson102 have suggested another approximation to thequantum mechanical flux-flux correlation function in whichthe flux-flux correlation function for a parabolic barrier is fittedso that it reproduces the actual correlation function and itssecond derivative att ) 0. Clearly, at timest ) t0 + ε, whereε is small, recrossing is dominating over flux moving in theinitial direction of the wave packet, but due to the uncertaintyprinciple, wave packets necessarily have a spread of momentaand coordinates, and the front or fastest edge of the wave packetmay have already recrossed the dividing surface beforet ) t0,whereas other parts of the wave packet have not yet revealedto what extent they will react or reflect. Thus, the short-timeapproximation is not a direct analog of the classical no-recrossing assumption. Furthermore, once one has the apparatusto calculate⟨F(t)0) F(t)⟩ from t to t0, it is often possible tocontinue on to convergence.103-107 Day and Truhlar107proposeda variational version of the short-time quantum TST (V-ST-QTST) in which they varied the location of the dividing surfaceto minimize the calculated rate. Comparing the results toaccurate quantum dynamics for the O+ HD, H + OD, and D+ OH reactions, they found typical differences of less than 10%with, however, an error of a factor of 1.8 for D+ OH at 200K. Such calculations are very powerful, but since theyintrinsically involve global dynamics, they do not provide thesame conceptual picture or applicability to complex systems astransition-state theory. Park and Light103and Seideman, Miller,and Manthe108-112have provided other formulations of the exactrate constant based on the flux concept, and thus theseformulations are related to transition-state theory with globaldynamics extensions.Another perspective on the time-correlation function approach

is offered in the work of Voth et al.113 This leads to anexpression for the rate constant in which the dynamical effectsare delineated from the energetic factors. In separate work, Vothet al.114proposed a Feynman path integral115,116(PI) formulationof quantum mechanical transition-state theory (QTST) based

Figure 1. Typical example of a flux autocorrelation function througha transition state as a function of time.

Current Status of Transition-State Theory J. Phys. Chem., Vol. 100, No. 31, 199612773

on earlier work of Gillan.117-119 Path-integral quantum transi-tion-state theory (PI-QTST) has recently been reviewed byVoth,120,121 including a review of applications and extensionsof the method. A similar approach has been presented byStuchebrukhov.122 These path-integral-based methods offer aconvenient way to include quantum mechanical effects for allmodes of the system on an equal footing and also provide ameans for including anharmonic effects of the potential energysurface. To date, the application of the method to gas-phasereactions has been limited to the collinear H+ H2 reaction,114

and the advantages of the method have been more fully exploitedfor treating condensed-phase systems (see section 4).The concept of a quantum state provides an alternative (and

perhaps more natural) way to incorporate the inevitable broad-ening associated with the uncertainty principle. Thus, whereasa classical equilibrium state of a bound oscillator hasx ) xeandpx ) 0, adding zero-point motion prevents violation of theuncertainty principle. This same concept can be used fortransition-state levels. (We will say levels in this contextbecause “transition-state states” involves using the classical andquantal meanings of “state” too close to each other.) Transition-state levels, however, are unbound states. Because unboundstates form a continuous spectrum, it has not been clear how torelate the discrete levels (“states”) in the quantized transition-state partition function (“sum over states”) to a continuum ofunbound quantal states. Recently, though, it has been pointedout123-127 that transition-state levels are associated with complex-energy poles of the scattering matrix (S matrix); such poles arevariously called metastable states, resonances, decaying states,quasibound states, or Siegert states. (Poles of the S matrix withreal energies are the ordinary bound states of a system.) Thewhole theory of metastable quantal states (quantum mechanicalresonance theory, a branch of quantum mechanical scatteringtheory) immediately becomes available when the S matrix poleidentification is made, and this provides a powerful tool foranalysis and interpretation of transition-state phenomena with,however, one rather serious limitationsnamely that transition-state levels are overlapping, broad resonances rather thanisolated, narrow resonances (INRs), whereas many of thetheoretical results90,91of resonance theory apply quantitativelyonly to INRs. Nevertheless, resonance theory has a lot to offerfor conceptualizing transition states, and it does provideimportant quantitative guidance; e.g., it allows us to assignlifetimes to transition states, and it provides much needed insightinto the role of initial rotational excitation enhancing orinhibiting reactivity.126

The discussion of transition-state levels as quantized discretestates whose effects on reactivity can be seen individually wasinitiated a few years ago by the analysis of accurate quantumscattering calculations on H+ H2, O+ H2, and X+ HX whereX is a halogen.123,125,126,128-131 (The interpretation using theconcepts of resonance theory arose as one aspect of thesestudies.) Individual quantized-transition-state energy levels havenow been seen and discussed for several reactions, namely H+ H2,123,125,130D + H2,132 O + H2,128,129,131F + H2,132-134 Cl+ H2,132,135H + XH,125 He + H2

+,136,137Ne + H2+,138-140 H

+ O2,141-144 Li + HF,145 and O+ HCl.146 The subject hasbeen reviewed very recently.132,147 A highlight of this kind ofanalysis is the ability to sort out “state-to-state-to-state” reactionprobabilities, i.e., from a specified level of the reactants to aspecified level of the transition state to a specified level of theproducts.123,126,131 This provides new insight into the originsof the effective threshold energies for reactions of individualvibrational-rotational states and how these depend on thequantum numbers.124,127,132

Sadeghi and Skodje148 have found the quantized transition-

state resonances associated with the reaction barriers for the H+ H2 and D + H2 reactions by using spectral quantizationtechniques. The resonance wave functions clearly demonstratedthe localization of probability densities on the variationaltransition state, and their procedures allow one to distinguishvarious kinds of transition-state resonances.149-151 Their analy-sis of the time-dependent quantum mechanics of barrierresonances leads to an understanding of the line shape in termsof the sequence of poles responsible for the barrier resonances.152

Experimental observation of quantized transition-state struc-ture is impeded by the difficulty of isolating contributions fromindividual total angular momentaJ since the structure is likelyto be harder to resolve or even unresolvable if spectra corre-sponding to variousJ are added together. Nevertheless,quantized structure associated with transition states or otherstates with significant amplitude in the transition-state regiondoes appear to have been observed in anion photodetachmentspectra,153-156 photofragmentation spectra,157-165 and photo-isomerization experiments.166 Even when discrete levels of thetransition state are not observed, anion photodetachment spec-troscopy provides a powerful probe of the transition-stateregions.167

In the most definitive of the experimental observations, Kimet al.160 and Green et al.158 point out that, for the photodisso-ciation of triplet ketene, observing thej′ ) 2 product state ofCO gives a signal with extra structure in the first several hundredcm-1 above threshold, which they interpret as an enhanced roleof flux through the CCO bend excited states of the transitionstate. This is fully consistent with theoretical analysis131whichshows that state-selected reaction probabilities often provide aless clustered view of certain levels of the quantized transitionstate. Kim et al. and Green et al. interpreted the spectrum asshowing a 250 cm-1 spacing for the CCO bend, in excellentagreement with ab initio electronic structure calculations168 thatpredict 252 or 282 cm-1 for this spacing, depending on the levelof theory. Electronic structure theory also predicts levels at150-154, 318-366, and 472-557 cm-1 that were not observed.In order to express the usual rate constantk(T) in terms of

quantized transition states, we start with the contributions ofindividual total energiesE and total angular momentaJ. First,we note that the canonical-ensemble rate constantk(T) for atemperatureT is83

whereâ ) (kBT)-1, kB is Boltzmann’s constant,FR(E) is thedensity of reactant states (per unit energy and volume forbimolecular reactions, per unit energy for unimolecular reac-tions), QR(T) is the partition function of reactants (per unitvolume for bimolecular reactions and unitless for unimolecularreactions), andk(E) is the rate constant in a microcanonicalensemble with energyE. The latter is given by

whereFR(E,J) is the density of states of a given total angularmomentum, andk(E,J) is the rate constant for the fixed-Jmicrocanonical ensemble. Note that

and

k(T) )∫dE e-âEFR(E) k(E)

QR(T)(1)

k(E) )

∑J

(2J+ 1)FR(E,J) k(E,J)

FR(E)(2)

FR(E) ) ∑J

(2J+ 1)FR(E,J) (3)


thus

Finally, for a bimolecular reaction, theE andJ resolved rateconstant is given by

where h is Planck’s constant, andN(E,J) is the cumulativereaction probability defined as86,130,169

In eq 7,Pifj(E,J) is a fully state-resolved quantal transitionprobability from statei to statej at total energyE and totalangular momentumJ, and the sums run over all energeticallyaccessible statesi of the reactantsR and all energeticallyaccessible statesj of the productsP. Finally putting (6) into(1) yields

Equations 1-8 are exact.78,170

In TST,N(E,J) becomes the number of energy states of thetransition state with total angular momentumJ and energy belowE, and the numerator of eq 8 becomes the canonical partitionfunction. Chatfield et al.130 interpreted the results of accuratequantal dynamics results using the relation

where PR(E,J) is the probability of reaction at energyEassociated with a quantized transition stateR of total angularmomentumJ with a particular valueMJ of the component oftotal angular momentum along an arbitrary space-fixed axis.This relation may be understood most easily by comparing itto earlier expressions based on adiabatic78,84,171or separable169

theory or later work based on resonance theory.172,173 In thelatter approach, theR and J quantum numbers specify atransition-state resonance. Because the resonances are notisolated, the transmission coefficients depend quantitatively onthe effects of additional poles of the S matrix, farther away inenergy or farther off the real axis. (This complexity is anintrinsic feature of interpreting the dynamics with overlapping,broad resonances.) Putting (9) into (8) yields

Note that eq 10 multiplies by (2J + 1) rather than summingoverMJ. Note that, as a function of total energyE, PR(E,J)typically rises from 0 to some finite valueκR (e1) in the vicinityof the resonance energyER for stateR; κR is the level-specifictransmission coefficient for transition-state levelR.

Equation 10 takes a form that allows a connection to be madebetween the resonance-state approach to QTST and path-integralQTST. It is convenient to define an aggregate quantum numberτ ) (R, J,MJ). First define a Boltzmann-weighted transmissionprobability for quantized transition stateτ by

[For a narrow resonance,Pτ is a step function,κτθ(ε), whereθis a Heaviside function, andγτ ) κτ]. SincePτ is independentof MJ, we can write eq 10 as

Equations 11 and 12 are identical to the resonance-stateversion of quantum TST (RS-QTST) proposed previously.172

In that theory the transmission coefficientγτ is evaluated byassuming thatPτ increases from 0 toκτ in the vicinity oftransition-state levelEτ with an energy dependence determinedby the properties of the metastablebarrier state, and we makethe transition-state assumption thatκτ equals unity. If, instead,we consider a unimolecular reactant and write the complexenergies of thereactantresonances by

and assume thatΓτ is small, then eqs 10 and 12 withγτ replacedby Γτ/kBT take the form of the rate constant from the imaginaryfree energy (ImF) method174-177

whereQ is the partition function for the entire system. Inprinciple, the partition functionQ is real and should divergefor a dissociative system, but complex values are obtained byanalytical continuation. For a unimolecular reactionQR andQboth refer to the whole system, and eq 14 becomes

where FR denotes the reactant free energy. Makarov andTopaler177 have demonstrated that the centroid-constraint rela-tionship in the PI-QTST method can be derived from the ImFmethod. The ImF formulation can be used for describingescape from a metastable potential, and as shown above, it hasa formal resemblance to the RS-QTST, which can be used todescribe either unimolecular or bimolecular reactions. However,the analogy is only formal since the ImF formulation is basedon the properties of the metastable states178-180of a unimolecularreactant that decays into a continuum, while RS-QTST is basedon the properties of the metastable states123,172 at the barriertop independent of the nature of the reactant. It would beinteresting to make further connections between these view-points.It is instructive to compare eq 10 to the most widely employed

version of variational transition-state theory (VTST), namelycanonical variational theory (CVT) with multidimensionalground-state transmission coefficientsκ(T).15,84,181 In this theory(to be denoted VTST/MT for VTST with multidimensionaltunneling), one defines a reaction path and a coordinatesdenoting progress along this path. Then we consider a sequenceof trial transition states withs fixed at various points along thepath. Sinces is fixed, these transition states are systems withdimensionality one less than the full dimensionality of the

QR(T) )∫dE e-âEFR(E) (4)

k(T) )

∫dE∑J

e-âE(2J+ 1)FR(E,J) k(E,J)

∫dE∑J

e-âE(2J+ 1)FR(E,J)(5)

k(E,J) )N(E,J)

hFR(E,J)(6)

N(E,J) ) ∑i∈R

∑j∈PPifj(E,J) (7)

k(T) )

∫dE e-âE∑J

(2J+ 1)N(E,J)

hQR(T)(8)

N(E,J) ) ∑RPR(E,J) (9)

k(T) )

∫dE e-âE∑J

(2J+ 1)∑RPR(E,J)

hQR(T)(10)

γτ )∫dε e-âεPτ(Eτ+ε,J)/kBT (11)

k(T) )kBT

hQR(T)∑

τ

exp(-âEτ)γτ (12)

Eres(τ) ) Eτ - iΓτ/2 (13)

k(T) )2

hQR(T)∑

τ

exp(-âEτ) Im Eres(τ)

2kBT/hQR Im Q (14)

k(T) ≈ 2 Im FR/h (15)


system. For each value ofs, we calculate the energy levelsER(s,J) of the trial transition state, and we find the location thatminimizes the transition-state canonical partition function; wecall the s value at the location of this “variational transitionstate” s*(T). In the absence of quantization, this procedurewould be equivalent to finding the dividing surface withminimum one-way flux coefficient. The rate constant calculatedfrom the partition functions ats*(T) is called kCVT(T); itcorresponds to the overbarrier contribution to the rate. We alsocalculate an approximate probability of reactionPG(E) associatedwith the ground state of the variational transition state to includethe effects of tunneling at energies below the effective barrierheight and nonclassical reflection (diffraction by the barrier top)at energies above the effective barrier. In the most reliableversion of such theories, the tunneling calculation is multidi-mensional, and the transmission coefficient may be denoted MT(multidimensional tunneling) or MTG (multidimensional tun-neling based on the ground state). Finally, the composite rateexpression is

where

E*R(J) is shorthand notation forER[s*(T),J], andE*G is shorthand

for E*0(J)0). Combining (16)-(18) yields

Comparing (19) to (10) shows that variational transition-statetheory with a ground-state transmission coefficient is equivalentto assuming that

i.e., the transmission probability for each successive transitionstate is just shifted from that of the ground state by the excitationenergy evaluated at the variational transition state.For gas-phase bimolecular reactions, accurate quantum rate

constants can be obtained by solving the Schro¨dinger equationby scattering theory, e.g., by expanding the scattering wavefunction in a basis set and converging the calculation withrespect to the basis set and all numerical parameters. At thetime of the previous status report,1 VTST/MT theory had beentested against accurate quantal results for 33 cases (30 reactionsin a collinear world and 3 in a three-dimensional world). Forthe 33 cases, conventional transition-state theory was accurate

within a factor of 2 in only 8 cases and within a factor of 5 inonly 23 cases, whereas VTST/MT theory was accurate within54% in 27 cases and within a factor ranging from 0.32 to 1.81in all cases. Since then, VTST/MT has been tested even morewidely against accurate quantal results23,135,182-191 with manymore tests in three dimensions.135,182,185,187-191 These testsconfirm the previous conclusions, and the most recent tests, forD + H2

191 and Cl+ H2135 in 3D, show very high accuracy

over a wide range of temperature.For D + H2 VTST/MT calculations192 of k(T) agreed with

accurate quantal calculations191performed on the same potentialenergy surface eight years later within 17% or better for thewhole temperature range of 200-1500 K. Agreement withexperiment,193 which also tests the potential energy surface,194

is 32% or better over this temperature range.The Cl + H2 reaction provides another test of VTST/MT

methods; for this reaction the average absolute deviation ofVTST/MT calculations from accurate quantum dynamics is only10% over the 200-1000 K range.135 The Cl + H2 reactionalso provides135 a very clear example of a supernumerarytransition state131 of the first kind. Such a transition state maybe interpreted as a secondary maximum on a vibrationallyadiabatic potential curve for some level, sayâ. If a system ina lower level, sayR, passes the location of the highest maximum(the variational transition state) of the curve for levelâ, it maythen undergo a vibrationally nonadiabaticR f â transition andreflect from the supernumerary transition state of levelâ. Thus,such nonconventional transition states provide a detailedexplanation of why the transmission coefficients are less thanunity in some cases.The O+ H2 reaction has also provided a prototype test case

for VTST/MT. VTST/MT calculations have been tested suc-cessfully against accurate quantal dynamics,185,187,189and theyhave been very successful at interpreting experimental kineticisotope effects.195,196 O+ H2(V)1)f OH+ H and OH(V)0,1)+ H2(V)0,1) f H2O + H, where V denotes a vibrationalquantum number, have been used to test the extension of VTSTconcepts to predict state-selected rates for high-frequency modeexcitation.187,197-210 The first analysis of quantized transition-state structure in a cumulative reaction probability based onaccurate quantal calculations128 was reported for the O+ H2

system by Bowman,129who noted the existence of structure dueto a bend excited transition state. Most recently, this systemprovided the most fertile ground to date for analyzing thedetailed state-to-state dynamics of a chemical reaction in termsof variational and supernumerary transition states observed inaccurate quantum dynamics calculations.131

A critical element in the success of VTST/MT theory is theaccuracy of the methods used to calculatePG(E). In particular,the most accurate calculations are based on multidimensionalsemiclassical methods for estimating the low-E tunneling tailsthat are critical for the thermal rate constants because of thee-âE factor in eq 19. To be satisfactory, multidimensionaltunneling methods must account both for zero-point variationsalong the reaction path and for corner cutting on the concaveside of the reaction path.170,186,201-210 The current status is thatconvenient and accurate multidimensional tunneling methodsare available for both small and large curvature of the reactionpath, namely the centrifugal-dominant small-curvature semiclas-sical adiabatic ground-state (CD-SCSAG) method211,212in theformer case, and the large-curvature ground-state method,version-3 (LCG3)211,213-215 in the latter. These methods differin the extent to which corner-cutting tunneling occurs, asdiscussed elsewhere.186,216 For atom-diatom reactions, a least-action method208 has been used to optimize the choice oftunneling paths more finely between these limits whereas for

kVTST/MT(T) = κMTG(T) kCVT(T) (16)

κMTG(T) )

∫dE e-âEPG(E)

∫E*G dE e-âE) (kBT)

-1eâE *0∫dE e-âEPG(E)

(17)

kCVT(T) )

kBT∑J∑

R(2J+ 1)e-âE *R (J)

hQR(T)(18)

kVTST/MT(T) )

∫dE e-âE∑J

(2J+ 1)∑RPG(E) e-â[E *R (J)-E*

G]

hQR(T)

)

∫dE e-âE∑J

(2J+ 1)∑RPG[E- E*R(J) + E*

G]

hQR(T)(19)

PR(E,J) ) PG[E- {E*R(J) - E*0(J)0)}] (20)


polyatomic reactions it has been considered sufficiently accurateto just optimize the tunneling path by choosing between theCD-SCSAG or LCG3 algorithms whichever predicts the greateramount of tunneling.215,217 Small-curvature tunneling calcula-tions, like VTST itself, require a knowledge of the reaction pathand the energies and frequencies along it; however, thisinformation may be required over a longer section of the paththan is required for VTST, especially at low temperature, wheretunneling through the lower, wider part of the barrier maycontribute significantly to the rate. Large-curvature tunnelingrequires, in addition, some information about the potential in awider region (termed the reaction swath) on the concave sideof the reaction path.184,209,210,218-220

The extension of validated multidimensional tunneling ap-proximations to arbitrary polyatomic systems will allow aconsiderable range of applications in the future. Some recentexamples based on electronic structure calculations for bi-molecular reactions are the reaction of CF3 with CH4,215 thereactions of OH with H2,200CH4,221CD4,222C2H6,223and NH3,224

and the reactions of H with H2O,225 NH3,226,227and CH4.228 Inthe CD4 case the predicted CH4/CD4 KIE at 416 K was 4.5, inpoor agreement with the only available measured value of 11.However, a new measurement yielded 4, in excellent agreementwith the prediction. For the first six examples mentioned above,the transmission coefficients at 300 K are calculated to be 67,5.9, 8.7, 7.6, 3.2, and 5.6, respectively. For the unimolecular[1,5] sigmatropic rearrangement ofcis-1,3-pentadiene (aninternal hydrogen transfer), the transmission coefficient for Htransfer is 6.5 at 470 K, leading to a KIE for H vs D transfer of4.9,212 in excellent agreement with the experimental value of5.2. The role of tunneling in this system had been verycontroversial prior to this full 36-dimensional VTST/MTcalculation.Miller, Hernandez, and co-workers229-232 have considered a

transition-state-like approximation to the rate constant by usingsemiclassical theory in action-angle variables. A difficulty inapplying this theory is that good global action-angle variableswill seldom be available, if they even exist. The theory hasbeen implemented using second-order perturbation theory. Thedisadvantages of this approach are that for tunneling, second-order perturbation theory is not very accurate for representingcorner-cutting effects,207,233and for overbarrier processes it isnot very accurate for representing large deviations of thevariational transition-state location from the saddle point.172 If,however, one has a situation where the second-order theory isadequate, the theory has the advantage that it is a convenientway234 to include mode-mode coupling. Recently,235a methodof treating anharmonicity in modes perpendicular to the reactioncoordinate has been developed for cases where they may exhibitwide-amplitude motion; this method may provide improvementsin cases where anharmonicity in the perpendicular modes ismore important than their coupling to the reaction coordinateand than reaction-coordinate anharmonicity (the latter twofeatures being responsible for variational displacements of thetransition state from the saddle point).Conventional (traditional) TST was concerned entirely with

the properties of the saddle point (the highest internal energypoint on the minimum-energy path), and indeed that myopicview of the potential energy surface was and is a strength ofthe theory because of the resulting low demand for structuraland energetic information about the system. Modern generalizedtransition-state theories are still relatively modest in their needsfor such input, but they typically require information along aconsiderable segment of the one-dimensional reaction path ratherthan at just two points. The local quadratic approximation alongthis path plays a central role in the theories as does the concept

of generalized normal modes, where the generalization refersto defining such modes in a subspace orthogonal to the reactioncoordinate at a nonstationary point (a point where the gradientof the potential does not vanish, as it does at a minimum or asaddle point).Even at the harmonic level, there are open research questions

about coordinate systems and vibrational energy levels ofgeneralized transition states. At stationary points (i.e., potentialminima and saddle points), which are the only points at whichinformation is required for conventional transition-state theorywithout tunneling, vibrational frequencies at the harmonic levelare independent of coordinate system. However, for nonsta-tionary points, even for a given choice of reaction path, harmonicfrequencies depend on the coordinate system; in particular, theydepend on the definition of the reaction coordinate for pointsoff the reaction path.236-238 Thus, it becomes very importantto choose a physically appropriate coordinate system. Mostreaction-path calculations to date use rectilinear coordinates, i.e.,coordinates writeable as linear combinations of Cartesians,whereas curvilinear coordinates (such as bond stretches, valencebends, and dihedral angles) are more physical. One manifesta-tion of the inadequacy of rectilinear coordinates is the frequentappearance of imaginary frequencies for bound modes inreaction-path calculations. Recently, a general formalism forcalculating reaction-path frequencies in curvilinear coordinateshas been presented and shown to eliminate (at least in the casesconsidered) the problem with unphysical imaginary frequen-cies.239,240 Even when the frequencies are not imaginary, theymay be inaccurate when computed with rectilinear coordinates;this may, for example, lead to an overestimate of the tunneling.It has been known for a long time that anharmonicity and

vibration-rotation coupling can have an important effect onTST calculations, even for tight transition states;182,231,233,241,242

however, progress in devising practical general methods forincluding anharmonicity of the generalized normal modes hasbeen slow. Three promising approaches include second-orderperturbation theory,231,243-248 especially in curvilinear internalcoordinates182,218,242,249,250where interaction terms are muchsmaller than in rectilinear coordinates, a classical configurationalintegral method,251 and Monte Carlo methods for quantummechanical path integrals.246 A convenient approximation forone-dimensional hindered rotations has been presented.215,252

Anharmonicity is even more important for loose transition states,and further discussion of anharmonic partition functions andnumbers and densities of states is provided in section 3.A promising avenue for future development is the unified

dynamical theory, in which recrossing corrections to VTST/MT calculations are evaluated from trajectories beginning at aquantized variational transition state.15,253-255 To the extent thatshort-time dynamics in the vicinity of a localized dynamicalbottleneck determines the rate, this includes quantum effectsand recrossing in a very effective way.If a particular vibrational mode is adiabatic (i.e., conserves

its quantum number) from reactants to the dynamical bottleneck,or from the dynamical bottleneck to products, TST can beextended to predict state-selected rate constants for thatmode.123,131,182,192,187,193,197-199,256-261 In some cases one canpredict state-selected rates due to state-specific tunnelingprocesses.184

Transition-state theory has traditionally been not only theprimary tool for interpreting kinetic isotope effects (KIEs) butpractically theonly tool.262 The standard interpretation of KIEsinvolves using conventional TST to infer detailed informationabout transition-state structure. The modern perspective differsfrom the conventional one in two ways:222,264-267 (i) VTSTindicates that the geometry (and hence the force field) can be


quite different for isotopic versions of the same reaction.264 (ii)Tunneling effects are often very significant even when the KIEdoes not exceed the maximum allowed by conventional modelsthat neglect tunneling. Recent progress in developing reliablemultidimensional methods for including tunneling contributionsin TST has assisted in identifying189,195,209,212,215the role oftunneling much more definitively than in the past where anOccam’s razor approach often underestimated its role. Par-ticularly noteworthy in recent work is a more sophisticatedtreatment of secondary kinetic isotope effects; factor analysesof partition function ratios and tunneling calculations, both basedon full-dimension transition-state models and electronic structurecalculations, have provided a better appreciation of the contribu-tions to the KIEs of each kind of motion at the transitionstate.217,268-278 These studies have tested the traditional viewthat secondary KIEs mainly reflect the position of the transitionstate on the loose-tight continuum of transition-state structure;a quick summary of the conclusions is that this factor isimportant, but high-frequency modes and tunneling are no lesssignificant in the general case. Another general conclusion fromthe recent work is that real-world KIEs are much morecomplicated than the enticingly simplesand even more en-ticingly successfulspopular models would have led us tobelieve. Nevertheless, TST interpretations of experimental KIEsremain one of the most powerful methods for testing modelsof dynamical bottlenecks and for inferring transition-statestructures and properties.Another kind of successful application279 of VTST/MT to

organic chemistry is provided by calculations on hydrogen 1,2-shifts in singlet carbenes. Heres-dependent zero-point effectson the effective barrier for tunneling were critical in calculatinga qualitatively correct entropy of activation.Transition-state theory is most directly applicable to reactions

which proceed on a single electronically adiabatic (i.e., Born-Oppenheimer) potential energy surface, although in some cases,for example, O(3P) + H2 f OH + H, it has been applied toreactions proceeding on multiple surfaces.196 A situation thatoccurs commonly is where the degeneracy of the single surfaceon which reaction occurs is smaller than the degeneracy of thereactants. This occurs, for example, in the Br reaction withH2. The conventional treatment in the case where state splittingof occupied reactant states is small compared tokBT is to assumethat the ratio of reactant to transition-state electronic degeneracydetermines the average fraction of collisions that proceed onthe reactive surface.280 More generally, if all reactive surfacesare similar, one can use the ratio of the transition-state electronicpartition function, counting only reactive surfaces, to the productof the reactants’ electronic partition functions, counting allsurfaces. A recent development is that advances in quantumscattering theory have allowed this kind of assumption to betested, and Schatz281 has presented such a test for the reactionCl(2P3/2,1/2) + H*Cl f HCl + *Cl, where *Cl is a tagged Cl.Schatz concludes that the approximation is good to better than20%. If reaction proceeds on more than one surface, and thesurfaces have significantly different properties, a reasonableassumption in the context of TST is to add the rates for thedifferent surfaces.190

Reactions with a barrier preceded by a well provide a morecomplicated scenario than simple barrier reactions. Ion-molecule reactions like the Cla

- + CH3Clb, Cl- + CH3Br, andF-(H2O)+ CH3Cl SN2 reactions fall into this more complicatedclass of reaction. Trajectory calculations on the former tworeactions indicate that multiple crossings of the barrier regionmay occur, leading to a breakdown of the fundamental assump-tion of TST.282,283 It is not clear to what extent these recrossingeffects might be due to the classical nature of the calculations,

the way in which initial phase space was sampled, the choiceof dividing surface, or the nature of the assumed potential energyfunctionsor to real recrossing effects in the actual ion-moleculereactions. The authors note that several aspects of the dynamicalbehavior are explained as being due to weak coupling betweenthe relative translational motions of reactants (or products) andthe reaction coordinate (asymmetric stretch) at the transitionstate.284 This raises the question of whether additional degreesof freedom (e.g., C2H5 instead of CH3 or the presence of solventor microhydration) would significantly increase this coupling.Interestingly, ab initio TST calculations285 of the CH3/CD3 andH2O/D2O KIEs for the microhydrated F-(H2O) + CH3Clreaction are in good agreement with experiment. There are tworecent studies of the KIE for Cl- + CH3Br, and they reachdifferent conclusions about agreement with experiment.285,286

However, combining the largest-basis-set harmonic KIE (0.95)from the latter study285 with the effect of anharmonicity (0.64/0.79) from the former286 yields a KIE of 0.77, in reasonableagreement with experiment (0.80-0.88) when one considersthe difficulty of the experiments, of converging the electronicstructure calculations, and of estimating the anharmonic effect.Gas-phase SN2 reactions illustrate some of the major unknownsin the current status of TST.282-288

Transition-state theory has had many names over the years,two of which are activated complex theory (ACT) and absolutereaction rate theory (ARRT). Nowadays, TST and ACT areconsidered to be exact synonyms, and ARRT has fallen out offavor. The disappearance of the ARRT moniker seems reason-able since it is much less descriptive than the other two namesof the type of physical approximation involved, but actuallythat is probably not the main reason for the ARRT name tohave fallen out of favor. When TST was first called ARRT,there was tremendous enthusiasm for combining it withelectronic structure theory to predict absolute reaction ratesentirely from theory. However, by the end of the 1930s it wascertainly clear that the goal of predicting chemically accuratepotential energy surfaces or even barrier heights was a muchmore difficult challenge than originally anticipated. The chal-lenge survives today, largely unmet, but nowsin the 1990ssitseems that a bit of cautious optimism may be in order, at leastfor few-body reactions. Two reasons for this may be advanced.First is the realization by a wider group of practitioners thatvery complete basis sets cannot be avoided when reliable barrierheights are desired, e.g., the use of a single set of d functionson C, N, O, or F, which was once considered a good basis set,is now recognized as at best semiquantitiative and but a smallfirst step toward completeness. Second is the development ofpractical size-consistent treatments of electron correlation thatinclude the dominant effects of double, triple, and higherexcitations. This array of techniques includes Møller-Plessetfourth-order (MP4) perturbation theory289sas used in theGaussian-2 (G2) semiempirical calculations,290 coupled clustertheory with single and double excitations and perturbativeinclusion of unlinked triples [CCSD(T)],291 and quadraticconfiguration interaction with singles and doubles and pertur-bative inclusion of unlinked triples [QCISD(T)].292 Note thatboth CCSD(T) and QCISD(T) include the dominant effects ofquadruples, and CCSD(T) has the extremely important potentialto make up for a poor reference set of orbitals. A non-size-consistent alternative that is competitive for very small systemsis multireference configuration interaction (MRCI) with singleand double excitations from a full-valence complete-active-spaceself-consistent-field (CASSCF) reference state,267,293-297but thismethod appears to have dimmer prospects for extendibility tolarger systems. Complete-active-space second-order perturba-tion theory (CASPT2) may play an important role in filling that


niche.298-300 Empirical correction and extrapolation schemesfor the high-level results, such as the bond-addivitiy-correctedMP4 (BAC-MP4),226,301-303 scaling external correlation(SEC),194,267,304,305and scaling all correlation (SAC)222,223,227,306-311

methods, are also very useful. For simple enough systems theuse of such ab initio calculations has replaced older techniquesof estimating entropies and energies of activation by empiricalgroup contributions,312,313but the empirical procedures still playan important role for larger species and for the correlation ofexperimental data.TST practitioners also have considerable interest in density

functional theory, which has had some notable successes fortransition states314-320 but which still appears to be basicallyunreliable for barrier heights.321 Density functional theory (likemany other levels of electronic structure theory) appears to bemore accurate for transition-state and reaction-path geometriesthan for absolute energies.322,323

Many other issues in the reliability of electronic structurecalculations of various types for transition states are also at theforefront of TST research, for example, the reliability ofunrestricted Hartree-Fock reference functions in comparisonto restricted open-shell Hartree-Fock (or in comparison toexperiment).A very important aspect of using the above theories con-

structively is the ability to calculate analytical gradients324 andeven analytical hessians. The former capability is essential tomaking geometry optimizations feasible, where “optimization”refers to the process of finding the zero-gradient structure of areagent or a saddle point. Geometry optimization of transitionstates is particularly important for comparative evaluation ofpossible reaction paths in complex systems, e.g., those wherecomplexed water molecules participate in the reaction.325,326Atpresent, geometry optimization is possible with some but notall of the methods mentioned above, and we can anticipate thatfurther advances in this area will have dramatic impacts on TSTapplications.The reader should keep in mind that the overall level and

reliability of electronic structure calculations depends not onlyon the level of treatment of electron correlation, as justdiscussed, but also on the completeness of the one-electron basisset. Thus, an encouraging development is the use of systematicbasis set studies to explore the convergence of transition-statebarrier heights with respect to the one-electron basis.327

Another computational issue of great importance for VTSTcalculations is the efficient calculation of the reaction path itself.Many algorithms have been advanced for this.15,328-341

A critical area of current research is designing new and moreefficient ways to interface electronic structure calculations withdynamics. The goal of such work is to find ways to calculatethe reaction-path and swath information needed for VTST andtunneling calculations from a minimum of electronic structureinformation. Two promising approaches are interpolated VTST(IVTST)343 and VTST with interpolated corrections (VTST-IC, also called dual-level dynamics).344 In IVTST, one carriesout high-level electronic structure calculations for reactants,products, the saddle point, and perhaps one or two additionalpoints near the saddle point; then all required reaction-pathinformation is interpolated. (Interpolation can also be carriedout by power series345or Shepard interpolation346,347methods.)In the IVTST-IC method, the high-level input is similar, butinterpolation is aided by the presence of an approximate potentialenergy function or additional lower-level electronic structurecalculations that are carried out for a longer segment of thereaction path and in the swath region. The goal of dual-levelvariational transition-state theory is not just to use high-levelelectronic structure calculations to correct the energies along a

reaction path calculated at a lower level and to correct thevibrational frequencies, but also to provide improved data foroptimized multidimensional tunneling calculations includingreaction-path curvature and tunneling paths that deviate fromthe minimum-energy path by more than can be predicted by aquadratic expansion about this path and/or more than the radiusof curvature. Recent successful examples of these interpolatoryapproaches are provided by the calculations, mentioned above,on OH+ CH4,309 CD4,222 and C2H6

223 (IVTST) and on OH+NH3

224 (VTST-IC).IVTST and IVTST-IC are examples of direct dynamics,

which is the calculation of dynamical attributes from electronicstructure calculations without the intermediacy of fitting apotential energy function. These methods involve interpolation,but straight direct dynamics has also been used for VTST cal-culations.200,212,215,224-228,269-271,279,303,310,314-316,322,323,344,348-352

In straight direct dynamics, one carries out an electronic structurecalculation directly every time that the algorithm requires anenergy, gradient, or hessian.Recent examples of combining high-level electronic structure

theory with transition-state theory for practical applications tolarger systems are provided by the work of Page and co-workerson the reactions CH3Of CH2O+ H353 (a dissociation reactionwith a barrier and a tight transition state), H+ HNO,354 O +NH2,355 and H+ NH2

296 and the reactions of H, OH, and NH2with N2H2.356 In each case the reaction path was calculated bythe CASSCF method, and single-point calculations with mul-tireference configuration interaction were used to improve theenergetics along these paths. These data were used as inputfor variational transition-state theory calculations and estimatesof tunneling corrections based on the zero-curvature vibra-tionally adiabatic approximation for the transmission coefficient.Reaction-path methods and the interface of electronic structure

theory with chemical kinetics using these methods for variationaltransition-state theory and semiclassical tunneling calculationshave recently been reviewed.220,357-361

As TST is being applied in recent years to complicatedreactions, a question that comes up is the treatment of competingpathways for a single set of reactants.362-364 In general, if bothreactions are slow, and their pathways have no part in common,TST can handle this just as it handles relative rates of differentreactants. When the competitive pathways share a commonintermediate or when one or both of the reactions are fast, onemust make additional assumptions. The simplest assumption,used in statistical phase space theory,365,366is that the scatteringmatrix is a random unitary matrix within a subset of the channelsspecified by some state-specific extension of TST,367,368e.g.,those channels with accessible orbiting transition states.

3. Reactions without an Intrinsic Potential Barrier

3.1. Theory. In the VTST treatment of unimolecularreactions33,369 the rate constant at energyE and total angularmomentumJ is given by

whereN*(E,J) is the number of energetically available quantumstates at the variational transition state, andFR(E,J) representsthe reactant density of states. In conventional TST,N*(E,J) isreplaced byN‡(E,J), which is evaluated at the saddle point, ifthere is one. The detailed aspects of applying VTST tounimolecular reactions with well-defined saddle point areessentially identical to those described in section 2 for thecorresponding bimolecular reactions. However, the implemen-tation of VTST for unimolecular dissociations and their reverse

k(E,J) )N*(E,J)

hFR(E,J)(21)


associations in cases where the association process is barrierlessraises new issues as do other processes without an intrinsicbarrier or with very flat potentials near the dynamical bottle-necks.For such “barrierless” reactions, it is especially important to

provide a proper treatment of the variation in location of thetransition state, due to its possible wide variation from inter-fragment separationsR as large as tens of angstroms at lowenergies, down to 2-3 Å at higher energies. (The extent ofvariation depends of course on the reaction, being larger, forexample, for a typical ion-molecule reaction than for a typicalradical-radical reaction. For example, for the CH2 + H fCH3 reaction, CASSCF-MRCI calculations indicate that thecanonical variational transition state moves from a C-H distanceof 3.7 Å at 159 K to 2.7 Å at 2850 K.370) The separation ofmodes into the vibrations of the fragments, termed the conservedmodes, and the rotational and orbital motions of the fragments,termed the transitional modes, has provided the basis for muchof the theoretical work in this area. This separation of modesis particularly meaningful at large subsystem separations, wherethe reaction coordinate is well described by the distance betweenthe centers of mass of the two fragments, and the transitionstate is “loose”; i.e., the interacting fragments have almost freerotation.Phase space theory (PST)365,366,371-377 is a version of TST

that focuses on the energetics of the separated fragments at acompletely loose transition state (i.e., rotations of the fragmentsare completely unhindered), and it often provides an accuratetreatment when the transition state is at large separations, as inion-molecule reactions. The simplest algorithms are based onlocating the transition state at the centrifugal barriers forspherically symmetricR-n potentials. This is sometimes calledthe orbiting transition-state model;378 for n ) 4 it is the modelof Langevin379 and Gioumousis and Stevenson,380 and forn )6 it is the model of Gorin.381,382 These models can beinvalidated in some cases by asymmetries in the long-rangepotential, arising, for example, from ion-dipole and other long-range dipole interactions. (Variational transition-state theory383

and the adiabatic channel model384,385have provided an accurateunderstanding of such effects.) Within PST,N*(E,J) is givenby the total number of asymptotic rovibronic states whoseeffective centrifugal barriers are below the available energy, andthe total angular momentum is explicitly conserved via theconsideration of the vector sums of the angular momentum ofeach of the fragments and of the orbital motion. An alternativeversion of PST, as reviewed by Peslherbe et al.,378 is to locatethe transition state atR) ∞, with properties identical to thoseof the completely separated collision partners or products. Anonstatistical phase space theory, called the intermediatecoupling probability matrix approach, includes effects of weakenergy transfer in associative or dissociative half-collisions bypostulating a finite but less than 100% probability of forminga complex in the former case or of coupling to a final state inthe latter.386,387 In this theory these probabilities depend on therate of energy transfer and the half-collision duration for a non-energy-mixing half-collision; the latter in turn depends on theasymptotic relative translational energy and the orbital angularmomentum. Recognition of the intermediate-coupling natureof the system has a significant effect on the temperaturedependence of association rate constants.387

In the following, we will focus on the description of thedeviations from PST-type capture rate constants due to short-range interactions, which are critical for association of neutralmolecules. At shorter separations where some of the rotationaldegrees of freedom have become internal rotations, librations,or bending vibrations, the intermolecular motions become

increasingly strongly coupled, resulting in the breakdown ofvarious aspects of the PST assumptions. The development ofaccurate and efficient procedures for treating these breakdownshas received considerable attention in recent years. Thestatistical adiabatic channel model (SACM)79,80,388-395providedan early and widely applied approach for treating such devia-tions. Though SACM is called an adiabatic theory,74-80,83,84itis based on a reference assumption of diabatic modes coupledwith a statistical distribution of energy among the modes.Diagonalizing the Hamiltonian to convert the diabatic energiesto adiabatic ones would not have a large effect on the sumsover states under typical non-state-selected conditions, though,so the adiabatic-diabatic distinction is not generally important.N*(E,J) is then approximated in terms of the number of channelswhose diabatic effective barriers are below the available energy.The primary difference of SACM from PST is the implicitimplementation of asymmetries in the intermolecular potentialand also the inclusion ofR-dependent variations in the fragmentrovibronic energies.Although VTST for a dissociation reaction may be based on

a very different reference assumption of a complete randomiza-tion of all motions of reactants, its estimates ofN*(E,J) areidentical to the SACM estimates, when implementing the sameestimates for the energetics and using equivalent symmetryfactors.83,84,396 However, the difference of adiabatic/diabaticassumptions from statistical ones does result in major differencesin estimates for the product state distributions,396which are reallybeyond the realm of ordinary TST, as mentioned in section 2.Comparisons164,397-399 with experiment, which are still morelimited than one would like, suggest that a combination ofadiabatic/diabatic assumptions for the conserved vibrationalmodes with a statistical assumption for the transitional modesis most appropriate,400although for larger molecules than thoseconsidered to date, one might expect some low-frequencyconserved modes to couple significantly with the transitionalmodes.The applications of SACM have generally focused on the

interpolation of energy levels from reactants to products,although some of the more recent studies have explicitlyconsidered the energetics of long-range potentials.392,394,395,401-403

Similar interpolations of the energetics have also been incor-porated within a VTST-based method.404,405 However, theexplicit consideration of the fragment-fragment potentialenergies of interaction, as in many of the recent VTST studies,provides more meaningful tests of the validity of the underlyingTST assumptions and also provides a direct means for makinga priori predictions.The implementations of VTST for barrierless reactions are

based on one of two alternative procedures.406,407 The firstalternative is based on the determination of a minimum-energyreaction path (RP) and the corresponding normal-mode harmonicfrequencies and moments of inertia along the reactionpath.378,405,408-410 The transition state partition function is thengenerally determined on the basis of rigid rotor harmonicoscillator (RRHO) type assumptions for the overall complexemploying classical expressions for the rotational motions andquantum expressions for the harmonic vibrations. The ques-tionable validity of harmonic oscillator assumptions for theintermolecular bending motions has also led to the use ofhindered rotor expressions. Unfortunately, the rigid rotorassumptions are of equal uncertainty and are difficult to removewithin the RP approach.The second alternative is based on an assumed decoupling

of the “conserved” and “transitional” modes with the quantityN*(E,J) evaluated via the convolution of a classical-phase-space-integral-based evaluation of the transitional mode contribution


with a direct quantum sum for the conserved mode contri-bution.411-413 In the original version of this approach, thereaction coordinate is taken as the separation between the centersof mass of the two reacting fragments. The classical treatmentof the transitional mode contributions is entirely satisfactorydue to their low-frequency nature as confirmed in a variety ofstudies.414-416 (Note that tunneling is usually expected to beof negligible importance for barrierless reactions due to thetypically large masses and widths for the effective centrifugalbarriers, in which case quantum corrections for the reactioncoordinate are also unimportant.) The ability to directly andaccurately incorporate quantum mechanical effects for the high-frequency modes provides one of the key advantages of TSTmethods over classical-trajectory-based methods. In some cases,this phase-space-integral-based VTST (PSI-VTST) gives similarresults to the RP approach.378

An important feature of the PSI-VTST approach is itsclassically accurate treatment of the interfragment couplings,low-frequency-mode anharmonicities, and low-frequency-modevibration-rotation couplings while conservingJ. Recentadvances in the methodology, involving analytic integrationsover the momentum portions of the phase space integrals, haveyielded algorithms of sufficient efficiency to be widelyapplicable.414,417-420 Furthermore, simplified expressions pro-viding extremely efficient approximate estimates have also beenpresented.421 Unfortunately, the implicit assumption of a center-of-mass separation distance reaction coordinate breaks downat shorter separation distances, particularly for those reactionswhere at least one of the atoms involved in the reactive bond iswell separated from the fragment center of mass.396

A recently developed approach has its basis in the PSImethodology but explicitly considers the variation in the formof the reaction coordinate.422-424 In particular, a variablereaction coordinate (VRC) is defined in terms of the distancebetween two variably located fixed points, with one fixed pointin each of the two fragments. An optimization is then carriedout not only of the value of the reaction coordinate, as in themost popular version of VTST for bimolecular reactions, butalso of the definition of the reaction coordinate in terms of thelocation of the two fixed points. This VRC-TST approachprovides a better representation of the reaction coordinate atclose separations, as in the reaction-path VTST method, whileretaining a classically accurate PSI-based treatment of thetransitional mode contributions. Analytic integrations over themomentum components of the integrals again yield an approachwhich is of sufficient efficiency to be widely applicable,425,426

while simplifying approximations provide an even more efficientprocedure.427

The precise location predicted for the transition state in VTSTis a key physical quantity since this location broadly determinesthe reaction rate. A primary importance of this location is, forexample, in directing the focus of quantum chemical estimatesof the interaction energies. Furthermore, the ability to predictthis location [and correspondinglyN*(E,J)] on the basis ofrelatively limited potential energy surface information is anotherkey advantage of VTST over more dynamical methods such asclassical trajectory simulations. (Another advantage is theability to quantize high-frequency modes, which can be crucialfor predicting accurate threshold energies and thermal rates.)Of course, this transition-state location depends not only on themethods employed in the state counting but also on the detailsof the potential energy surface employed. The implementationof high-level quantum chemical data in the formulation of suchpotentials, as in a number of recent VTST studies, is of theutmost importance due to the difficulty of obtaining informationabout the potential in the strong interaction region from

spectroscopic probes. Unfortunately, such data are moredifficult to obtain than for the corresponding equilibriumsituation due to the general occurrence of near degeneracies inthe electronic states in the transition-state region. One interest-ing byproduct of the recent developments in methodology forcounting states was an indication of just how rapidly the phasespace integrals converge with number of sampling points. Thisrapid convergence suggests the feasibility of bypassing theanalytic representation of the interfragment interaction potentialvia a direct ab initio quantum chemical evaluation of theinteraction energy for each phase space point sampled in theintegrationssimilar to the direct dynamics methods for bi-molecular reactions discussed in section 2. A first demonstrationof the feasibility and validity of this direct sampling approachhas recently been provided for the CH2CO dissociation.352,428

The energy and angular momentum resolved density of statesFR(E,J) for the reactant also plays a key role in the determinationof the rate constant. Evaluations ofFR(E,J) based on RRHOexpressions for the energetics provide a good first approximationto the density of states. The Beyer-Swinehart algorithm,429,430which is unfortunately limited in application to separableexpressions for the energy levels, provides the standard proce-dure for such evaluations. Improvements beyond the RRHOlevel are becoming more and more important as both theexperimental results and the other aspects of the theoreticalmethodology become increasingly accurate. The direct sum-mation over the nonseparable energy levels has been found tobe sufficiently efficient for the evaluation of the density of statesfor molecules as large as CH2CO,352,397,431 and simplifiedtreatments432,433are also useful. Alternatively, a method basedon the random sampling of the quantum numbers should beparticularly effective for larger molecules.434,435 However, inmany instances accurate expressions for the underlying energylevels are not available, in which case alternative proceduresbased on the integration of classical phase space integralsprovide useful means for estimating the anharmoniceffects.284,378,436-441 Again, the rapid convergence propertiesof recently developed phase space integration methods suggestthe possibility of a direct sampling of the potential in place ofits analytic representation. Adiabatic switching has been usedto compute the anharmonic density of states for Al3, and values2.5-2.9 times larger than the harmonic result have beenobtained.378 However, the anharmonic density for this systemdetermined by the phase space integration method is verysensitive to the assumed phase space boundary of the “reac-tant”.378 Peslherbe et al.284 have studied the effect of anhar-monicity on the RRKM rate constant for the unimoleculardissociation of Cl-‚‚‚CH3Cl. They find that anharmonicityincreases the reactant density of states by a factor of 2-3. Thiswould decrease the rate constant to the extent it is not canceledby transition-state anharmonicity.An alternative Monte Carlo random sampling based meth-

odology directly couples the evaluation of the reactant densityof states with the transition state number of states.442-449 Inthis method the rate constant is evaluated as the average velocitythrough the dividing surface for a random sampling over allavailable phase space. Unfortunately, the need to numericallyevaluate the delta function in the reaction coordinate makes thisapproach somewhat inefficient. Furthermore, the present imple-mentation of this methodology is restricted to a completelyclassical description for even the conserved vibrational modes.Another important advance in treating barrierless reactions

has been a coupling of the basic VTST methodology with otheraspects of the reaction kinetics beyond simply evaluatingk(E,J)and/or the high-pressure thermal rate constantk∞(T). Forexample, the VTST methodology has been combined with both


the standard RRKM formalism and the master equation approachin order to obtain a description of the pressure dependence ofthe reaction kinetics.450-453 Also, the coupling of the VRC-TST methodology with quantum chemical estimates of theradiative relaxation rate provides a novel route to the estimationof complex dissociation energies via comparison with experi-mental measurements of the zero-pressure radiative associationrate.454,455 The VTST calculations ofN*(E,J) have also providedmeaningful predictions for the product-state distributions viathe hybrid assumption of vibrational adiabaticity and rotationalmixing.164,397-400

The relation between transition-state theory and accuratequantum dynamics has also been pursued for unimolecularreactions and the related reverse recombination pro-cesses.178-180,456-470 For example, Bowman has presented adescription of the canonical rate constants in terms of thermalaverages over the trace of the Smith collision lifetime matrixand has also illustrated the relation to VTST.179 Bowman andWagner and co-workers have derived and applied an isolatedresonance version of TST theory to the H+ CO recombination/dissociation process.459-461,463 Miller and co-workers derivedand applied a random matrix based formulation of TST whichfurther explored the relation between scattering resonances andTST.180,465-467 Most recently, Miller has provided a formulationof recombination rate constants in terms of flux correlationfunctions.470 This formulation demonstrates how TST becomesexact for recombination processes in the high-pressure limit andfurthermore makes it clear how to evaluate deviations from theTST limit quantum mechanically.Berblinger and Schlier471 tested classical RRKM theory with

numerical phase space integration (i.e., no harmonic approxima-tion) against classical trajectory calculations for the reactionHD2

+ f D+ + HD and H+ + D2. They considered totalenergies 0.5-1.5 eV above the energetic threshold and totalangular momenta 0-50 p. Their results illustrate that there isan interesting theoretical subtlety in discussing unimoleculardecay due to the rapid dissociation, prior to equilibration, ofsome trajectories from an initially defined reactant ensemble.472

After that occurs, decay may be more statistical. In fact,statistical theories like TST do not apply to the direct, short-time component at all. In practical applications one cannotalways separate these effects, but for DH2

+ this was possible.The direct trajectories caused TST to underestimate the uni-molecular decay rate by up to 40%. Removing these trajecto-ries, one finds the encouraging result that TST overestimatesthe rate constant by only about 3% due to recrossing.In certain instances, such as the dissociation of van der Waals

molecules, there are substantial failings of the common formula-tions of TST. These failures are primarily the result ofbottlenecks to the redistribution of energy42within the molecularcomplex. Considerable progress in understanding these failureshas been made via analyses of the phase space structure of thesereactions.473-491 In fact, one recognizes two (or more) separatebottlenecks (or transition states), and the kinetics can beadequately modeled in terms of the statistical rates for crossingeach of them. In related work, Dumont has developed a“generalized flux renewal model” which in addition to the slowintramolecular energy flow also considers the effect of directcomponents to the unimolecular decay process.491 Bohigas etal.492and Tang et al.489have discussed the relation of this workto a quantum mechanical picture.A related concern regards the extent of randomization and

sharing of the rotational energy with vibrational modes. Theuncertainty in treating the rotational energy arises from the factthat the “quantum number”K, denoting the component of theangular momentum along a body-fixed axisz, is not conserved

when one includes vibration-rotation coupling. Within ap-plications of TST, the motion in the coordinate associated withK has generally been treated as either active, i.e., available tobe shared with the vibrational degrees of freedom, or inac-tive.81,407 An indication of the variation in the predictionsobtained from these limiting cases was presented by Hase andco-workers for the Cl+ C2H2 reaction.82 Gray and Davis havealso presented a classical trajectory study of the extent ofconservation ofK over a picosecond time scale for formaldehydeat moderate energies.493 General reviews of the role of angularmomentum in unimolecular reactions have been presented.407,494

3.2. Applications. The first application of the reaction-pathVTST methodology to barrierless reactions was to the O+ OHf HO2 f H + O2 reaction.405 Deviations as large as a factorof 5 were observed between quasiclassical trajectory and VTSTresults, and the importance of accurate estimates of theinteraction energies in the inner transition-state region from 2.5to 5.5 Å was noted for the first time. Furthermore, this studyprovided the first a priori indication of the presence of two well-separated transition states. Various other aspects of this reactionsystem have also been the subject of recent detailed VTSTcalculations.144,497-499 Broadening the scope beyond VTST,Duchovic and Pettigrew498 have recently compiled 100 refer-ences for the reverse H+ O2 f OH + H reaction, includingover 40 theoretical studies. The quantitative validity of VTSTfor the dissociation of HO2 has been demonstrated via directcomparison with averaged quantum scattering estimates.144

Meanwhile, VRC-TST results for the reverse bimolecularreaction are found to overestimate both quantum scatteringtheory and classical trajectory simulations by a factor of 2.497

Similar comparisons for the He and Ne+ H2+ reactions also

indicated a factor of 2 overestimate by VRC-TST calculationswhen an appropriate symmetry correction factor is included.137,139

(Note that, as described in a previous paper,497 a symmetrycorrection factor of 2 was neglected in these studies which leadsto the currently stated discrepancy between the TST andscattering theory results.) Such deviations appear to be the resultof the direct redissociation of a substantial fraction of theincoming trajectories as a result of an incomplete coupling inthe complex. This incomplete coupling is in turn related to thegenerally short lifetime (∼0.1 ps) and low density of states forthe HO2 complex. While numerous comparisons betweentheory and experiment for this reaction system have also beenpresented, such comparisons are unfortunately clouded byuncertainties in the potential energy surface and also in thecontribution from excited electronic states.The first detailed applications of the PSI-VTST approach were

to the recombinations of CH3 with CH3499-501and with H.502-505

For both reactions satisfactory agreement with the experimen-tally determined canonical rate constants was obtained whileemploying ab initio based potentials. Furthermore, the VTSTpredictions for the H+ CH3 recombination were in goodagreement with subsequent trajectory simulations.410 However,for the latter reaction the D isotope effect of 1.4 predicted byVTST does not agree with the experimentally observed valueof 2.5. Overall, the detailed VTST studies for these tworecombinations have provided an important testing ground forboth simplified VTST models421,506-512 and also for some ofthe inherent assumptions.409,414,505 Interestingly, earlier har-monic oscillator and hindered rotor based implementations ofthe reaction-path VTST method409 differ very little from thePSI-VTST results for the H atom recombination and also forthe association reactions of Li+ with H2O and (CH3)2O.513-515

Detailed comparisons for these reactions suggest that explicitvariation of the transition-state location at theE- andJ-resolvedlevel typically leads to an improvement by about 20% over its


consideration at only theE-resolved level. The methyl recom-bination reactions have also provided an important testingground for studies of the pressure dependence of the reactionkinetics450,451with subsequent applications made to the CH+H2

217 and CH3+ + CH3CN reaction kinetics.452 The detailedunderstanding of the CH3 plus H association has played animportant role in the development of models for the associationof alkyl radicals with H atoms, which in turn is of keyimportance in the understanding of the kinetics of diamondformation.516

The NCNO reaction is one of the first reactions for whichwide-ranging energy-resolved dissociation rate constants weredetermined experimentally.517 Such data provide a morestringent test for the theoretical predictions due in part to theabsence of any need to consider the collisional energy transferprocess. An initial model-potential-based PSI-VTST applicationindicated the occurrence of a transition from a long-rangetransition state to an inner transition state as the energy risesabove 100 cm-1, with only the ground singlet electronic statecontributing at the shorter separations.397 Subsequent compari-sons of related VRC-TST calculations with local trajectorypropagations suggest the quantitative validity of VTST, par-ticularly when a unified statistical treatment73,85-89 of the twotransition states is employed.518 Related indications of at leastthe qualitative validity of the unified statistical treatment werefound in a PSI-VTST study of the CH2CO product-statedistributions399 and also in a VRC-TST study of the Li+ HFreaction.145 The NCNO reaction also provided the first test ofthe VRC-TST approach with the optimization of the reactioncoordinate yielding a reduction in the rate constant by a factorof about 2-3.422-424 Similar estimates of the effect of optimiz-ing the form of the reaction coordinate have been obtained fora variety of related reactions, including the reaction of NC withO2

519 and the dissociations of NO2,164 CH2CO,352,416,428andC6H6

+.520 These studies further suggest that the bond lengthof the reacting bond generally provides a good first approxima-tion to the reaction coordinate in the inner transition-state regioncorresponding to atom-atom separations of about 2-3 Å. Foreach of these reactions the VRC-TST predicted rate constantshave been found to be in reasonable agreement with experi-mental predictions.The experimental studies of the singlet dissociation of CH2-

CO provide a detailed and wide ranging set of data forbarrierless reaction dynamics.158,521-526 In the latest VRC-TSTapplication, bothN*(E,J) and FR(E,J) were obtained whileemploying potentials based on high-level quantum chemicalinvestigations, including direct evaluations of the potential forboth bond length and center-of-mass separation distance reactioncoordinates.352,428 Furthermore, the indirect kinetic couplingbetween the conserved and transitional modes, as modulatedby the reaction coordinate, was explicitly treated. The resultingnonempirical estimates were found to quantitatively (i.e., within35% or better) reproduce the experimentally observed energydependence for the dissociation rate constant. In addition, PST,a RRHO-based implementation of VTST, and even the center-of-mass-separation-distance-based PSI implementation of VTSTare each in error by factors of 2 or greater.For ionic reactions, the stronger long-range interactions

generally lead to a better validity of the PST-type estimates.However, for large enough molecules and/or weak enoughattractions the short-range repulsions will still lead to a reductionin the flux. The recent VTST applications are beginning toaddress the point at which one needs to consider such short-range repulsions. For the dissociation of C6H6

+ into C6H5+ +

H, model potential based VRC-TST calculations indicate areduction by a factor of about 6 as compared to PST based

estimates.521 In contrast, sample applications to the Cl- +CH3Cl282,527-530 and Li+ + H2O or (CH3)2O513-515 reactionsprovide no indication of any reduction in the reactive flux dueto short-range interactions, and VTST appears to provide anaccurate description of the initial association process. However,quasiclassical calculations indicate that the overall reactionkinetics for the Cl- + CH3Cl reaction deviates from TSTexpectations as a result of the redissociation of the initiallyformed complex prior to a randomization of the energy,282,527-530

perhaps related once again to a short lifetime for the complex.The dissociation of C6H5Br+ into C6H5

+ + Br provides aninteresting intermediate example where the importance of short-range interactions was shown to depend on the particularparametrization of the potential employed.531 The interestingquestion of the occurrence of two separate transition states forionic proton transfer reactions has also been studied on the basisof the PSI-VTST methodology while employing model poten-tials.532,533

Unimolecular reactions provide some examples of very flatpotential energy surfaces where variational transition-state theoryis invaluable even for determining which structures are theintermediates and which are the activated complexes. Examplesare the tetramethylene534,535and trimethylene348 rearrangements.The latter species occurs as an intermediate in the cis-transisomerization of cyclopropane.In many cases, accurate ab initio data for the transition-state

region is unavailable. For such cases, simplified Gorin modeltype representations of the potential may prove useful asillustrated in a series of PSI-VTST studies of the reactions ofCH3 with CH3 or OH and the dissociation of C(CH3)4 andneopentane.510-512 The temperature dependence of the rateconstant for the reactions of HCO+ NO2

363 and of OH withHO2

536have been similarly predicted. A number of comparisonsbetween the Monte Carlo VTST methodology mentionedabove442-449 and trajectory simulations, also employing empiri-cal potentials for the transition-state region, have been presented.While good agreement was found for the dissociation of SiH2,447

the comparisons for the dissociations of Si2H6447,537 and

C2H4F2538,539 were not very favorable. The large deviationobserved in the Si2H6 study is somewhat surprising in light ofthe excellent agreement observed between VTST and experimentfor the closely related C2H6 dissociation. One point worthnoting is that the requirement of short propagation times (i.e.,10 ps or less) within the trajectory simulations means that thecomparisons must be made for higher excess energies than aregenerally considered in thermal studies. Furthermore, theempirical potential employed in these studies for the dynamicallyimportant inner transition-state region appears to be substantiallymore attractive than the values obtained in ab initio calculationsfor related reactions. For example, the CC bond strength inC2H4F2 is estimated to be 0.67 and 0.13 eV atRCC ) 3 and 4Å, respectively, whereas ab initio calculations501predict valuesof 0.34 and 0.03 eV at the same separations for the similar C2H6

dissociation. A detailed examination of the extent of the failurefor these reactions for more realistic representations of thepotential energy surface in the transition-state region will bean important issue for future studies.A detailed picture of the factors affecting both neutral and

ionic barrierless associations and their reverse dissociations isgradually emerging from these studies. Of particular note, forneutral reactions, is the general occurrence of a dominant innertransition state at separations of 2-3 Å between the atomsinvolved in the reactive bond. Overall, these applicationssuggest that the more sophisticated versions of VTST cangenerally be expected to describe the reactive flux within a givenpotential energy surface to within about a factor of 2. Further-


more, when the complex lifetime is on the order of a nanosecondor longer, even better agreement might be expected. Aninteresting procedure for making improved predictions for thereaction rate, and whose usefulness needs to be explored ingreater detail, involves a coupling of quantum mechanicallybased VTST estimates with classical trajectory based estimatesof nonstatistical effects.As for reactions with tight transition states, the greatest current

uncertainty in VTST estimates for the reaction rates of systemswith loose transition states involves the uncertainty in theunderlying potential energy surfaces. Thus, an important aspectof future work will involve the continued development ofaccurate descriptions of the potential energy surfaces, both forsample reactions and ultimately for larger classes of reactions.The availability of high-level quantum chemical data for the2-3 Å region of separations will be of the utmost importancein these developments. Related quantum chemical data willbe of use in providing more accurate descriptions of the reactantdensity of states.Many association reactions involve radicals with low-lying

excited states and/or for which the electronic state at infinitysplits into several electronic states upon interaction with theassociation partner. Thus, another large uncertainty in the apriori prediction of the rate constants regards the contributionfrom excited electronic states. At short separations, where thetransition state generally lies at higher energies, the electronicstates are often widely spaced so that an important contributionis expected from only the ground electronic state. However, atlarger separations electronic degeneracies often arise, and theproper description is then uncertain, depending on the strengthsof the couplings among the electronic states. Importantly, anaccurate a priori description of the temperature dependence (nearroom temperature and lower) of the radical-radical associationprocess will generally require an accurate description of theswitching of the transition state from large separations to shorterseparations and thereby the changing contribution of the excitedelectronic states.

4. Reactions in the Condensed Phase

Transition-state theory has been widely used for the calcula-tion and analysis of rate constants for chemical reactions in avariety of condensed-phase systems such as liquids, solids, andgas-solid interfaces. The use of TST in its simplest form forreactions in condensed phases dates back to the work of Evansand Polanyi539 for liquids andWert and Zener540and Vineyard541

for solids. Although there is a long history of applying TST tocondensed-phase systems, the accurate prediction of rateconstants in condensed phases still presents a major challengebecause of the complexity of including the extended nature ofthe system in the rate constant calculation as well as thedifficulty of accurately evaluating the interaction energies forthe extended systems. The condensed medium can profoundlyinfluence the reaction dynamics, for example, by inducingrecrossings of the transition-state dividing surface that lead toa breakdown of the fundamental assumption of TST. Acomputational issue that is still not fully solved is the inclusionof multidimensional quantum mechanical effects when they areimportant in these dissipative systems. There has been greatprogress in recent years in addressing these challenges. In thissection we review the advances made in the theoreticaldevelopment and applications of the theory to reactions in liquidsand to molecular processes in solids and on surfaces.4.1. Reactions in Liquids. At the most basic level, TST

for reactions in solution is based on the equilibrium solvationapproach. This involves evaluating the free energy of activationin solution, e.g., by adding the difference in free energy of

solvation between the saddle point and reactants to the gas-phase free energy of activation.542 At the next level ofsophistication one should include nonequilibrium effects of thedynamics of the solvent; the analytical theory of Kramers57,543

and the transmission coefficient expression of Grote and Hynes(G-H)544 have been particularly well studied. Severalreviews1,16,70,546-556of solution-phase reactions are available thatinclude discussions of TST and of these models. Within thepast decade, the state of the theory has advanced considerablyto include complex aspects not present in the earlier work. Theapplications of the newer theories are just beginning to appear.In this section we will review recent work with the discussionorganized as follows. First we consider classical mechanicalmodels, then we review quantum mechanical generalizationsto include important bound-state-quantization and tunnelingeffects, and finally we consider recent applications, with anemphasis on attempts to treat to realistic systems.4.1.1. Classical Theory and Application. A formal derivation

of classical TST for reaction in liquids was presented byChandler.71,555,557 The new element in the liquid phase is thatcollisions of solvent molecules with the reacting solute mol-ecules can lead to recrossings of the dividing surface that donot occur in the gas phase and, therefore, to a breakdown ofthe fundamental dynamical assumption of TST.Since, as mentioned in the Introduction, reactant activation

and equilibration are not the subject of this review, we proceedto consider the case where coupling energy into the reactantsis not rate limiting. One approach to approximating theinfluence of extended condensed-phase systems on the solutereaction dynamics is by separating static and dynamic effectsof the condensed phase (solvent); such effects are often calledequilibrium and nonequilibrium effects, respectively, and wewill sometimes follow this convention even though theseoverworked terms sometimes lead to confusion. The conceptof nonequilibrium solvation is old; modern work dates back atleast to the models of proton transfer developed by Kurz andKurz,558 and the concepts have been refined in more recentwork.1,545,546,548,555,559-561

Equilibrium solvation provides a good starting point fortreating the reaction energetics of the solute. The surroundingcondensed-phase molecules change (dress) the effective forcefield of the reacting molecules. The resulting mean-fieldpotential for the reacting molecule(s) is obtained from anequilibrium ensemble average over configurations of the othermolecules in the condensed phase. Since this mean-fieldpotential is obtained from an equilibrium ensemble average forfixed configurations of the reacting molecules, the equilibriumsolvation assumption implies that the solvating moleculesinstantaneously equilibrate to each new configuration of thereacting molecules. In the thermodynamic formulation ofTST,5,16,71,539the effect of the condensed phase on the reactionenergetics is included by the free energy of solvation that isobtained from equilibrium ensemble averages (a mean-fieldapproach) and is therefore an equilibrium solvent effect. Thisis the most common approach used for including solvationeffects in rate constants.542 Over the past several years therehas been increased interest in developing methods for calculatingthe equilibrium free energy of solvation as required to includeequilibrium solvation effects in TST. In one approach the freeenergy of solvation of a rigid solute complex is computed frommodel solute-solvent and solvent-solvent potentials usingclassical ensemble averaging and/or statistical perturbationtheory and added onto the gas-phase potential energy profileobtained from electronic structure calculations.562-570 (Thesolute and solvent molecules may be treated as either electroni-cally inert or polarizable when solvent is explicit, but in most


applications so far at least the solvent molecules are assumednonpolarizable.) Other approaches are based on representingthe solvent bath as a three-dimensional continuum;571-605 in suchmodels the electrostatic effects are treated by solving the Poissonequation for a dielectric medium or by an equivalent algorithmfor putting in the same physics, and nonelectrostatic effects areeither ignored or modeled based on the solvent-exposed atomicsurface areas. Most of this work treats the case where thesolvent polarization is assumed to be in the linear responseregime, although in some cases nonlinear effects in the firstsolvation shell (e.g., dielectric saturation) are included byempirical atomic surface tensions.580,581,604,605

Equilibrium solvation neglects any dynamical influence thecondensed phase may have on the reaction dynamics resultingfrom fluctuations of the solvent around equilibrium. Nonequi-librium or dynamic solvent effects can be separated into localand collective effects. Local effects involve only a limitednumber of solvent molecules such as can be included in a clustermodel. Transition-state theory can be applied to solution-phasereactions by separating the system into a cluster model thatcontains the part of the system undergoing reaction and thesolvent that is treated in an approximate manner. The clustermodel can include a finite collection of solvent molecules aswell as the reactants or solute molecules. The effects ofmicrosolvation217,269,270,273,285,303,343,560,578,598,606-615on reactiondynamics in small clusters have been studied using TST andVTST/MT in several papers; a review is available.616 TheVTST calculations have included only one or two solventmolecules; in principle, this approach can be extended to largerclusters, but very quickly (as the number of floppy degrees offreedom increases) the potential surface starts to exhibit multiplelow-energy pathways that are more appropriately treated bycondensed-phase methods such as those discussed in thefollowing paragraphs.When solvent molecules are treated explicit-

ly,285,560,578,585,586,598-600,612,614their motion can be included inthe reaction coordinate, and this can influence the geometry ofthe solute and the reaction energetics along the reaction pathand thereby the transition-state structure. Transition-stategeometries in solution can also be optimized by continuumsolvation methods.579,586,589,595-599,602,603 Both explicit-solventand continuum-solvent treatments show that the structure of thesolute at the liquid-phase transition state may be quite differentfrom the gas-phase transition-state structure or even from anypoint along the gas-phase reaction path.599

The approach of including one or a few solvent moleculesexplicitly can account for nonequilibrium solvation effects bothof the solvent caging type, in which the solvent molecules mustmove out of the way of the reacting molecules, and of the typewhere solvent molecules participate in the reaction as either areactant or a catalyst.303 Collective effects (including long-rangeelectric polarization of the solvent dielectric) involve cooperativemotions of the molecules in the condensed phase and are moredifficult to include by explicit few-body methods because ofthe large size of the system required. The larger number ofequilibrium geometries and transition states in liquid-phasereactions and even in medium-size clusters also motivates atransition from few-body gas-phase methods to many-bodycondensed-phase methods. Carter et al.617 have formulated anefficient molecular dynamics method for evaluating the classicalTST rate constant in terms of a constrained-reaction-coordinate-dynamics ensemble,618 which is applicable to any generaldefinition of the reaction coordinate. Harris and Stillinger619

discuss the application of VTST to the full coupled system ofthe solute and solvent and conclude that identifying the optimaldividing surface in the full space of the system may be

impractical. They suggest an approach based on an inherent-structure formalism in which the total rate constant is obtainedfrom a sum over TST rate constants for saddle points separatingreactant basins from product basins (the inherent structures inthe liquid).Another approach, complementary to the explicit-solvent and

continuum models, is to approximate the collective effects usingmodels of reduced dimensionality. Inclusion of collectiveeffects of nonequilibrium solvation in a TST framework findsits origin in the seminal work of Kramers.57,543 In the Kramers’theory, the reaction is treated in a highly simplified manner:the reacting solute is treated as a single reaction coordinate,and the rest of the system is treated as a bath in terms of aLangevin equation of motion. Takeyama620used this model toderive a transmission coefficient that accounts for the leadingeffect of friction when all friction components vanish exceptthe reaction-coordinate one. Grote and Hynes,544 retaining theone-dimensional reaction coordinate, obtained a more generalresult for the transmission coefficient via a more realistictreatment of the bath using a generalized Langevin equation621-627

(GLE). The GLE recognizes that friction does not act instan-taneously (as assumed in the original Langevin equation,sometimes called the assumption of ohmic friction), and thusits effect may be reduced for narrow barriers that can be crossedrapidly. The GLE for motion along a one-dimensional reactioncoordinate can be recast into Hamilton’s equation of motionfor a system in which the reaction coordinate is linearly coupledto a bath of harmonic oscillators;621,622,626,627we will call thisthe GLE model. Notice that in the GLE model, as in the originalLangevin model, all potentials involving the bath are quadraticor bilinear, and thus all forces on the solvent are linear.(Therefore, this model is a special case of treating the solventby linear response theory.) TST with the Grote-Hynestransmission coefficient (TST/G-H) is equivalent to classicalVTST for the GLE model in which the potential along thereaction coordinate is a parabolic barrier.553,628-635 For thispurely quadratic potential, VTST gives the exact result.The G-H expression may be interpreted as giving the effect

of friction (or microscopic viscosity) on the barrier traversalrate. Since the G-H equation can be derived by mixing bathmodes into the reaction coordinate,628-631,636 we see that“friction” and “nonequilibrium solvation” provide two differentways of looking at the same physical effect. Furthermore, sincefriction causes recrossing of a transition state that is definedwithout solvent participation in the reaction coordinate, we seethat for such a reaction coordinate nonequilibrium solvation isan example of the general phenomenon that was mentioned insection 1 by which recrossing can lead to a breakdown of thequasiequilibrium assumption of TST. The breakdown can becorrected by including solvent friction or by letting the solventparticipate in the reaction coordinate. The regime where thecoupling of solute to solvent is strong enough that energydiffusion and reactant equilibration need not be considered (butfriction and nonequilibrium solvation might be important) issometimes called the spatial diffusion regime, and it is thesubject of the rest of this section.Nonlinearities in the forces along the reaction coordinate can

cause the optimum dividing surface to be different from theapproximate one obtained in the parabolic barrier approximation.Variational TST has been widely used to treat this problem ofa nonquadratic reaction-coordinate potential that is linearlycoupled to a harmonic bath (i.e., the GLE model). Canonicaland microcanonical VTST theories have been applied to theGLE model to elucidate such effects.637-644 Microcanonicaland canonical VTST provide a significant improvement overTST/G-H for intermediate friction, where TST/G-H under-


estimates the effect of friction due to its neglect of nonlinearpotential forces.641,644 Berezhkovskii et al.645 have used varia-tional TST with planar dividing surfaces to develop an improvedapproximation for the case of nonlinear potential forces; ananalytical expression for the rate constant has been obtainedfor the quartic double-well potential and compared with accuratenumerical results.644 Frishman et al.646 have further extendedthis method to allow for bent planar dividing surfaces that areneeded for improved descriptions of some asymmetric potentials.Classical theories for reactions in solution that are based on

TST typically have as a goal either the minimization of thesolvent-induced recrossing by optimizing the dividing surfaceor estimating the recrossing of a given dividing surface. Pollakand Talkner647 have developed a statistical theory based uponthe unified statistical model73,85-89 that relates the averagenumber of recrossings of the dividing surface to the reactiveflux.89 Another method for including recrossing effects is toexplicitly calculate the reaction dynamics near the dividingsurface. Pollak and Talkner648 have developed a “dynamicalVTST” in which approximate dynamical corrections are in-cluded for an optimized dividing surface that is a function ofthe reaction coordinate and bath coordinates but goes throughthe saddle point. The dynamical corrections are approximatedby a perturbation approach similar to those used61 to describethe energy-diffusion regime.Kramers’ theory was extended to include spatially dependent

friction by Carmeli and Nitzan using an approach based on theFokker-Planck equation.649 Since then there have been manystudies of spatially dependent friction, especially friction thatis dependent upon the location along a reaction coordinate.650-656

Straus et al.651-655have considered a spatially dependent frictionthat is modeled by a Hamiltonian in which the coupling betweenthe reaction coordinate and the harmonic bath is a function ofthe reaction coordinate. They compared approximate VTSTresults with accurate results from simulations and found thatthe VTST expression was quite accurate. Voth653has presentedan “effective Grote-Hynes” method that provides a simpleprocedure to include spatially dependent friction. Haynes etal.656have applied the VTST approach of Berezhkovskii et al.645

to a spatially dependent and time-correlated friction model. Asin the previous study, the method was applied to a quarticdouble-well potential. The method is in excellent agreementwith exact simulations. This approach has been further extendedto allow for curved dividing surfaces.657

Pollak has extended the variational transition-state theoryapproach to treat condensed-phase reactions in which the“system” and bath forces are both of a general nonlinearform.658,659 Two orthogonal collective modes, an effectivereaction coordinate and a collective solvent mode, are definedthat are linear combinations of the bath and system coordinates,and canonical VTST is applied to the two-degrees-of-freedomproblem.637 This approach reduces to the TST/G-H ap-proximation for the case of a harmonic bath linearly coupledto a parabolic barrier.The Kramers and G-H models are for a one-dimensional

“system” (solute) coupled to a multidimensional bath. Multi-dimensional effects were discussed within the context of theseone-dimensional theories by Nitzan and Schuss.660 Berezhk-ovskii et al.645,661have developed a VTST with planar dividingsurfaces for systems with two degrees of freedom coupled to aGLE model. Multidimensional effects of the solute motion,such as the effects of reaction path curvature, have also beenexplored;635 this work showed that at low-to-intermediatecoupling between the solute and the bath the reaction-pathcurvature can induce recrossing of the transition-state dividing

surface which is not well described by the simple harmonicmodel underlying the Kramers and G-H expressions.Nonequilibrium solvation in charge transfer systems was

studied by van der Zwan and Hynes.559,628,629 In these studies,the nonequilibrium polarization effect was treated by a GLEand the rate constant was calculated by the TST/G-H method.Lee and Hynes662,663extended the treatment to include anhar-monic effects by defining an explicit solvation coordinate andan effective Hamiltonian for studying the reaction dynamics.An expression for the rate constant was obtained for the two-dimensional “solution reaction-path Hamiltonian” using varia-tional TST and applied to a model SN2 reaction. The work ofLee and Hynes stimulated further work in this area to developalternative definitions of the solvent polarization coordinatesand calculate rate constants for multidimensional systems withmethods based on TST.664-669 Some calculations667 indicatethe possibility of very large nonequilibrium effects, even as largeas a factor of 6, although we expect that nonequilibrium effectsare typically smaller in classical systems. van der Zwan andHynes670 have examined nonequilibrium solvation effects on amodel dipolar isomerization reaction in an electrolytic solutionusing TST/G-H.4.1.2. Quantum Mechanical Theory. Quantization and

tunneling effects are neglected in the classical approaches. Themajor contribution to a classical reaction rate at temperatureTtypically comes from energies aboutkBT above the barrierheight, whereas accurate quantum mechanical reaction prob-abilities are typically very small at such total energies becausezero-point energy requirements in modes transverse to thereaction coordinate are much greater thankBT. Quantitativestudies that do not enforce quantization conditions on transversemodes, at least approximately, have little relevance to thephysical world unless there is a fortuitous cancellation ofreactant and transition-state quantization effects. Classicalmethods also assume that reactive trajectories must surmountbarriers, whereas quantum mechanically the system can tunnelthrough barriers, and this can be the dominant mode of reaction,especially for systems where hydrogenic motions participate inthe reaction coordinate.The standard approach for including quantum mechanical

effects in TST for condensed-phase reactions is the same as forthe gas phasesan approximate, ad hoc procedure in which theclassical partition functions are replaced by quantum mechanicalones, and a factor is included to correct for quantum mechanicalmotion along a reaction coordinate (e.g., quantum mechanicaltunneling for energies below the classical barrier and nonclas-sical reflection above the barrier). One way to implement thisscheme would be (i) to quantize some of the solute modes buttreat the low-frequency modes of the environment classically(e.g., the solvation effect on the free energy of activation isincluded classically, but bound modes of the solute that changeappreciably in going from the reactants to saddle point arequantized) and (ii) to treat tunneling by a transmission coef-ficient. In recent years, significant progress has been made indeveloping consistent approaches for including quantum me-chanical effects in this way.A quantum mechanical analog of the classical Kramers and

G-H theories was first derived by Wolynes.671 Dakhnovskiiand Ovchinnikov630 and Pollak672,673 showed that applyingquantized TST with a parabolic tunneling correction factor tothe GLE model underlying these approximationssthat of areaction coordinate linearly coupled to harmonic oscillatorsrepresenting the bathsreproduced the Wolynes expression. Inthis quantum version of the model, tunneling is approximatedby the result for the one-dimensional parabolic barrier (whichdiverges at low temperature and that fails to capture the physics


of multidimensional tunneling at other temperatures674). Therehas been much interest in developing methods for going beyondthis simple approach to include tunneling effects, and reviewscovering selected aspects of the tunneling problem are avail-able.16,70,176,675,676

In this section we will focus on two approximate quantummechanical methods based on TST for treating reactions insolution. The first is quantized variational transition-state theorywith semiclassical corrections for quantum mechanical effectson reaction coordinate motion.15,16,84 Approaches to applyingthis method to reactions in solution have recently beendescribed561,677and are discussed below. The second methodis the path integral formulation of QTST114,117-122 discussedbriefly in section 2. This method has recently been reviewedby Voth120,121 including applications and extensions of themethod.Both the VTST/MT and PI-QTST approaches have been

applied to the GLE model of a reaction in solution.678 For thecase of a parabolic barrier the VTST/MT approach reduces tothe TST result already shown to reproduce Wolynes’ expression.The path integral method has also been show to reproduce thisexact result.114 These two approaches have been appliednumerically to the problem of an Eckart barrier linearly coupledto a harmonic bath, and although they employ differentapproximations, the results from the two methods agree wellwith each other and with accurate benchmark results.678 Infurther work along these lines,679 it was found that accuratetreatment of anharmonic effects is important in the VTSTcalculations for treating reactions in solution where low-frequency modes of the bath enhance the effects of anharmo-nicity. Furthermore, it was found that, for the model solution-phase reaction, VTST/MT calculations are often limited moreby the treatment of anharmonicity than by errors inherent inthe approximations to the reaction dynamics.Garrett and Schenter561 have described an approach for

applying VTST to activated chemical reactions in liquids. Inthis approach the total system is separated into a cluster modelthat is treated explicitly and a bath that is treated with a reduced-dimensionality model. Within an equilibrium solvation ap-proach, the effective potential for the cluster is the potential ofmean force as a function of the coordinates of the cluster. Whennonequilibrium solvation is included within a GLE approach,the effective potential becomes the sum of the potential of meanforce and a set of harmonic modes that are coupled to the clustercoordinates. The issue of how quantum mechanical tunnelingis implemented in this VTST approach has been discussed byTruhlar et al.677 A new solute-bath separation is presentedbased on tunneling through a canonically averaged mean-shapedpotential which can be evaluated from the bath contribution tothe potential of mean force. It is worth noting that the oppositepoint of view has been taken by Pollak,673 who advocates asudden approximation for the tunneling; this approach has beenmore fully developed by Levine et al.680

The PI-QTST approach has also been used to provide aquantum mechanical generalization of the G-H model.681,682

In this variational approach, a Gibbs-Bogoliubov-Feynmaninequality116 is used to derive an effective multidimensionalparabolic model that can be solved analytically. The quantummechanical generalization of the G-H recrossing factor takesthe same form as the classical factor, but with the classical valueof the imaginary barrier frequency replaced by an effectivequantum mechanical frequency. In a similar approach, Vothand O’Gorman683 obtained a simple analytical theory in whichan effective one-dimensional parabolic potential is used toeffectively include the nonquadratic nature of the potentialbarrier and the influence of linear dissipation. Haynes and

Voth654have examined the effect of spatially dependent frictionwithin the path-integral QTST method. At low temperatures,where quantum mechanical effects are important, the nonlineardissipation was seen to give large (order of magnitude)corrections to the results from a linear dissipation.Messina et al.684 have suggested a generalization of the PI-

QTST that allows for general dividing surfaces in phase space.Although this expression does not provide a rigorous upperbound to the quantum mechanical rate constant, in the samespirit as the quantum VTST approach, the optimum dividingsurface is found variationally to minimize the rate constant. Thisapproach has been applied to the model problem of an Eckartpotential coupled to a bath of oscillators,684 and a procedurefor optimizing planar dividing surfaces in the path-integralformalism (that is analogous to the classical variational methodof Berezhkovskii et al.645) has also been presented.685 Pollakhas also discussed variationally determining the optimumdividing surface within PI-QTST.686 This maximum free energyapproach is formulated for the GLE model and for the generalcase of nonlinear coupling to the bath. The variational PI-QTSTgreatly improves low-temperature tunneling corrections forasymmetric barriers.687

Schenter et al.688 have shown how dynamical corrections,which are based on classical trajectories on an effective potentialthat includes quantum mechanical effects, can be included inthe PI-QTST formalism. This work has similarities to theunified dynamical theory15,252-254discussed in section 2. Alongthe same lines, Sagnella et al.689have developed a semiclassicalTST, which is based on the semiclassical formulation ofChapman et al.,690 that estimates dynamical recrossings fromclassical trajectories initiated at the transition-state dividingsurface from a semiclassical phase-space distribution. The useof an effective potential that includes quantum mechanicaleffects is closely related to the approach used by Valone etal.691,692to study H diffusion on Cu surfaces (called EQP-TSTin section 4.2.2).The original path-integral-based rate theory by Gillan for-

mulated the problem in terms of the reversible work for movingthe centroid of the quantum mechanical paths from the reactantregion to the saddle point. This idea was extended to areversible-work formulation that involves moving a generalizedtransition-state dividing surface from the reactant region to ageneralized transition state in the interaction region.693,694 Whentranslations and rotations of the dividing surface are properlytaken into account, this method is rigorously equivalent to themethod of Voth et al.114 The reversible-work formulation hasthe advantage over the earlier approach114 that it requiresevaluation of averages of forces rather than the centroid-densityconstrained partition function that is more difficult computa-tionally.The theory developed by Marcus695-698 has played a central

role in describing electron transfer reactions in polar solvents.In this approach the solvent reorganization is the rate-limitingprocess for the electron transfer, and a solvent reorganizationenergy often plays the role of the reaction coordinate. In theelectronically adiabatic limit, this approach is equivalent to TST.For example, adiabatic electron transfer, in which the heavyparticle motion of the solvent limits the rate of reaction, hasbeen studied using a quantum mechanical TST approach forthe GLE model.699 Smith and Hynes700 have formulated a rateexpression for electron transfer in or near the electronicallyadiabatic regime based on the G-H approximation.4.1.3. Applications to Reactions in Solution. Over the past

decade, most theoretical studies of reaction in solution havestressed the importance of nonequilibrium solvation effects.However, in the analysis of real experimental data, it is often


hard to deconvolute equilibrium from nonequilibrium solvationeffects. For example, changing physical conditions such as thesolvent density to affect the friction will often also change thepotential of mean force of the system.560,633,701-703 An under-standing of the importance of nonequilibrium solvation effectsfor a given system often requires first the knowledge ofequilibrium solvation effects. Ladanyi and Hynes704have usedVTST to study equilibrium solvent effects on H atom transferreactions and simple geometric isomerizations in model com-pressed rare gas solvents. Compared to gas-phase rate constants,in solution large enhancements were seen for the H atomtransfers with much smaller enhancements for the isomeriza-tions.Garrett and Schenter703have argued that, for systems in which

the free energy of solvation is independent of solute mass,equilibrium solvation will not affect kinetic isotope effects(KIEs). Therefore, for these systems KIEs can be used to isolatenonequilibrium solvation effects. VTST with semiclassicaltunneling corrections was used to study a model of the reactionof H isotopes, including muonium (Mu), with benzene inaqueous solution. Anomalous Mu KIEs that were observedexperimentally could not be explained with an equilibriumsolvation model. Including nonequilibrium solvation showedsubstantial suppression of the Mu rate constant compared tothe H and D isotopes, in agreement with the experimentalfindings.The majority of applications of TST methods to reactions in

solution are based upon the simple Langevin equation inKramers’ theory or the GLE model in the G-H model. Wewill focus primarily on applications of TST or VTST tomultidimensional systems, and we will mention importantbenchmark tests of the Kramers and G-H transmission coef-ficients for realistic systems. Reference to recent work in whichthese theories were used to analyze experimental data includeSaltiel, Sun, and co-workers,705,706who explored the validityof Kramers’ model for isomerization reactions, Cho et al.,707

who review such analyses for three types of friction (mechanical,internal, dielectric), Schroeder and Troe,65who use pressure andtemperature as independent variables in a single solvent to testKramers’ theory, Sumi and Asano,708,709 who discuss thedifficulty of using the theory to understand isomerizationreactions of DBNA in an alkane solvent, and Anderton andKauffman,710who used Kramers’ expression to analyze dielectric-dependent activation energies for isomerizations.SN1 and SN2 Charge Transfer Reactions. Bimolecular

nucleophilic substitution (SN2) reactions in polar solvents haveprovided fairly extensive tests of the G-H model. Extensivework on the Cl- + CH3Cl reaction provided benchmark testsagainst classical trajectory simulations of realistic solute modelsin molecular solvents.634,711,712 In this work, the G-H modelwas found to be very accurate, whereas the Kramers model gaveresults that were much too low. Nonequilibrium effects reducedthe rate constants by about a factor of 0.5. Similar results werealso observed in comparisons of G-H model with trajectorysimulations for the process of ion pair interconversion betweencontact ion pairs and solvent-separated ion pairs in polarsolvents.713 Aguilar et al. explored the effect of solvent laggingbehind solute as the system proceeds along the reaction path(nonequilibrium solvation) for the F- + CH3F reaction; largeeffects on the effective barrier to reaction were predicted.714

Various formulations658,659,662,667of VTST have been appliedto SN2 reactions to examine the nature and effect of solute-solvent coupling.663,667,715,716

Transition-state theory has also been used to treat SN1 ionicdissociations in polar solvents. Zichi and Hynes717have appliedTST/G-H to a model for activated ionic dissociation with

substantial charge rearrangement in polar solvents. They foundthat Kramers’ theory predicted frictional transmission coef-ficients of 0.2-0.6 whereas the more realistic TST/G-H modelyielded 0.7-0.97, much closer to unity, by taking account ofthe fact that the reaction time scale is too fast for the full frictionto develop. For a narrow range of parameters leading to solutecaging by the solvent polarization field,κ values were calculatedto be as small as 0.1. Keirstead et al.718 have studied a modelSN1 reaction in water and also found that the G-H transmissioncoefficient leads to excellent agreement with trajectory simula-tions. Kim and Hynes719,720used a nonlinear response treat-ment574,575with the solvent reaction path (minimum-energy pathin the multidimensional space of solvent and solute coordinates)to develop an ionic dissociation model that was then used incalculations on nonequilibrium solvation effects based upon themethod of van der Zwan and Hynes.559,628,629 Mathis et al.668

extended these studies to analyze ionization oftert-butyl halides.Mathis and Hynes721 have studied anomalous behavior inionization of alkyl iodides using a similar approach to developthe reaction model and then either harmonic TST (e.g., themethod of van der Zwan and Hynes) or VTST where the barrieris not treated as parabolic. They have also studied the alkyliodide ionization using a formulation of TST in which thereaction coordinate is assumed to be the solvent coordinate.722

Although the above studies focus on dynamical issues relatedto the validity of transition-state theory and the additionalapproximations in the G-H model, one should keep in mindthat the dynamical, frictional, and nonequilibrium aspects ofthe problem often change the equilibrium TST prediction byless than a factor of 2, whereas the precise value of the solutebarrier height and equilibrium free energy of solvation571-605

of the transition state relative to reactants may have a muchlarger effect; e.g., errors of 1-2 kcal/mol lead to rate constanterrors of factors of 5.5 and 30 at 295 K. Thus, just asemphasized above for gas-phase reactions, the interface ofdynamics with electronic structure theory (or other methods ofapproximating solute potential energy surfaces and free energiesof solvation) assumes a critical role in predicting rate constants.Chandresekhar et al.723,724 calculated the free energy of

solvation of the Cl- + CH3Cl reaction at various points alongthe reaction path by classical mechanical simulations involvinghundreds of water molecules. They estimated a loss inequilibrium solvation energy of 23 kcal/mol on proceeding fromreactants to the transition state, which led to good agreementwith experiment. Later these studies were extended to OH- +H2CO.725 Continuum solvation models571-605 and molecularorbital-molecular mechanics563,567-569methods go beyond thesecalculations in allowing the solute charge distribution to polarizein solution. The techniques are maturing, and this kind ofcalculation should become even more useful in the future.Electron and Proton Transfer. TST can be applied to electron

transfer reactions, but such applications often involve additionalassumptions to handle the two-electronic-state aspects and theissues of solvent-induced charge localization. In the limit ofstrong electronic coupling of the initial and final valence states,electron transfer becomes a single-surface electronically adia-batic reaction and standard TST methods become applicable.Straus et al.726,727have studied adiabatic heterogeneous electrontransfer with both classical and quantum mechanical techniques.In the classical study, they compared the TST/G-H theory withtrajectory simulations and found that, as in the SN1 and SN2reactions, it accurately reproduces the trajectory results, and therecrossing factors are about a factor of 0.6. In a similar study,Rose and Benjamin728came to the same conclusions and foundrecrossing factors in the range 0.5-0.8. In contrast, PI-QTSTcalculations on the same system indicate that the quantum


mechanical effects are much larger than the classical recrossingeffects.726,727 Zichi et al.729 and Smith et al.730 also comparedTST/G-H and trajectory calculations for electron transferreactions.The model of slow solvent reorganization controlling charge

transfer reactions appears in both electron transfer and protontransfer reactions. The use of an effective solvent coordinateas the reaction coordinate for these processes has been ques-tioned recently. Path-integral QTST has been applied to thisproblem such that the proton tunneling coordinate and solventactivation were treated on equal footing.731 An application toa realistic model of proton transfer in a polar fluid, includingelectronic polarization, has been presented by Lobaugh andVoth.732,733 They conclude that solvent electronic polarizationcannot be neglected and must be included quantummechanicallyfor quantitative accuracy of the proton tunneling rates. Azzouzand Borgis734 have applied TST approaches to study anasymmetrical proton transfer model in liquid chloromethane.They compare results from a curve-crossing, transition-state rateconstant with those of PI-QTST and with conventional quantizedTST with a Bell tunneling correction factor. The agreementbetween the curve-crossing TST and PI-QTST results is fairlygood, ranging from differences of about 25% to just over a factorof 2 for different systems. They conclude that conventionalTST with parabolic tunneling is inadequate for these types ofsystems. It is noted that Warshel and Chu735 and Hwang andWarshel736 have also used PI-QTST for proton transfer, butbased on a different reaction coordinate. Similarly, Hwang etal.737 and Kong and Warshel738 have applied PI-QTST with anenergy-gap reaction coordinate to hydride transfer in enzymesand solution.Staib et al.739have carried out classical trajectory calculations

for proton transfer in a hydrogen-bonded acid-base complexin methyl chloride. Transition-state theory within an equilibriumsolvation model was compared with the trajectory results andalso with TST/G-H. The full dynamical nonequilibriumsolvation effect was calculated to be a factor of∼0.8, and theG-H transmission coefficient theory reproduced this value.Casamassina and Huskey used experimental KIEs to conclude

that motions of solvent hydrogens do not participate in thereaction coordinate for proton transfers from carbon acids (i.e.,acids in which the proton is bonded to carbon) to methoxide inmethanol or hydroxide in water.740

An important lesson learned in gas-phase dynamics is thattunneling probabilities are very sensitive to the quantitativeaspects of the barrier and the reaction path curvature.207,741,742

We should keep this in mind in assessing the reliability ofsimulations in the condensed phase.Reactions of Uncharged Species. Solvation effects can be

important for reactions of neutrals as well as ions, and theClaisen rearrangement, H2CdCH-O-CH2-CdCH2 f OdCH-CH2-CH2-CHdCH2, has served as a prototype for testingmethods.579,589,743-746 Gao calculated the potential of mean forcefor isomerization of dimethylformamide in water.747 Solventeffects on the ring opening of cyclopropanones were studied infour solvents using statistical perturbation theory, and theresulting shifts in free energies of activation were in goodagreement with experiment.748 A critical issue in predictingreactivities and solvent effects even for neutral molecules isthe set of values of atomic partial charges at the transitionstate.589

Enzyme Catalysis. Enzymatic reactions provide a specialcase. Nearly 50 years ago it was hypothesized that enzymesact by binding to and stabilizing transition states.749 Withinthis picture, knowledge of transition-state structures and chargedistributions is crucial to designing transition-state inhibitors

that bind tightly to active sites. Over the past several yearsthere have been efforts to model transition states from heavy-atom kinetic isotope effects;750-757 a review is available.758

Warshel and co-workers have used transition-state concepts todiscuss enzyme reactions in several studies,759-761and Warshelet al.762have argued that reactions of substrates at enzyme activesites do not proceed by displacing solvent molecules to createa gas-phase environment but that enzymes are designed tosolvate ionic transition states and act much like water in thisrespect. Free energy profiles have been computed for a fewenzymatic reactions.763

4.2. Molecular Processes in Solid-State Systems.Thetreatment of molecular processes in solid-state systems issomewhat simpler than the treatment of reactions in liquidsbecause the relative rigidity of solid systems often allowssimplifying approximations such as treating the solid as rigidor treating the phonon modes within a harmonic approximation.As a result, there have been many more applications of TST tosolid-state systems than to liquid-phase ones. These includedesorption/adsorption, diffusion, reactions, and surface recon-struction. The majority of these applications deal with diffusionand desorption/adsorption processes in which no chemical bondsare broken or made. The treatment of heterogeneous reactionsis more complicated primarily because of the increased com-plexity in describing the potential energy surface.4.2.1. Desorption from Surfaces. The recent body of work

from Pitt et al.764-766 provides an excellent discussion of theapplicability of TST to thermal surface adsorption in the absenceof an intrinsic barrier as well as a review of the relevantliterature. They argue that the variationally optimized dividingsurface need not necessarily be located at infinite separationfrom the surface as had been previously suggested in theliterature. They have developed a classical microcanonicalVTST approach that is valid for clean surfaces and for surfaceswith partial coverage. The method has been applied to a modelof Xe desorption from Pt.766

Doren and Tully767 have used classical TST and the unifiedstatistical model73,85-89 to study precursor-mediated adsorptionand desorption of molecules on surfaces. They find thatvariational optimization of the dividing surface (inherent in theUS model) can be very important, leading to order of magnitudechanges in the Arrhenius prefactor.Nagai768-770 has used transition-state theory based upon a

lattice gas model and the grand canonical ensemble to obtain asimple rate expression that depends upon lateral interactionsbetween adsorbates. For systems without a saddle point, thedividing surface is placed far away from the crystal where thepotential energy attains its maximum value and becomes flat.The validity of the model for the lateral interactions and thecoverage dependence has been questioned771,772 and de-fended.773,774

In related work, Anton775 attempted to include adsorbatecoverage dependence in classical TST and tested the methodfor desorption reactions. Pitt et al.765 argue that this derivationcannot be correct for barrierless adsorption. In a more empiricalapproach, the desorption of CO from metals has been modeledusing harmonic RRKM theory in which the vibrational frequen-cies of the reactants and transition state were taken fromexperimental data.776

Zhdanov has used a phenomenological lattice-gas TST modelto look at the effect of coverage dependence on the generalizedtransition-state partition function777 and to study the effect ofsurface reconstruction caused by adsorption on the desorptionrate.777,778

4.2.2. Diffusion on and in Solids. Classical Theories. Dolland Voter779 reviewed theories of diffusion for solid-state


systems, including methods based on TST. For systems withsufficiently large barriers, and strong enough adsorbate-substrate coupling, so that diffusion can be viewed as asuccession of uncorrelated hops between binding sites on thesurface, the diffusion constant can be related to the rate constantsfor jumps out of the binding sites. In these cases, TST can beapplied to calculate the unimolecular rate constants for thejumps, and these can be used to calculate diffusion coefficients.In its simplest form, the classical TST approximation to thesingle-hop rate constants is obtained from an effective one-dimensional diffusion model obtained by moving the diffusingatom along a 1D reaction coordinate and letting all the othercoordinates in the system relax adiabatically.780 In this formula-tion the rate constant takes the form of an attempt frequencyobtained from the one-dimensional model and a Boltzmannfactor in the difference in energy between the saddle point andreactants for the one-dimensional model. Vineyard541proposedthat the attempt frequency be given from a full classicalharmonic TST prescription which yields an expression that isthe ratio of the product of frequencies at the reactants to theproduct of saddle-point frequencies. For this pseudounimo-lecular reaction, the saddle point has one less bound frequencythan reactants. Voter and Doll have developed a Monte Carloprocedure for accurate numerical evaluation of the multidimen-sional expression of the classical TST hopping rate constantwithout resorting to a harmonic analysis.780 This approach hasbeen extended to include dynamical corrections based onclassical trajectories.781 Guo and Thompson782 compared asimple version of TST to full molecular dynamics for diffusionof C and H atoms in Au matrices and found agreement withina factor of 2.Although not explicitly recognized as such, a formulation of

classical TST was presented in which the transition-statedividing surface is curved.783,784 The curved dividing surfacegoes through the saddle point and is tangent to the conventionalTST dividing surface at the saddle point. It is defined by aquadratic expansion of the dividing surface (not the potential)about the saddle point and includes anharmonic effects. Ap-proximate dynamical corrections (short-term memory effects)have also been included in this formulation.785 These methodshave been applied to defect diffusion in solids (e.g., vacancydiffusion in metals). In a similar vein, the more conventionalclassical TST approach to diffusion in solids541 was used tostudy kink diffusion in a model system.786

Wahnstro¨m787 discussed the influence of dissipation onsurface diffusion and reviewed Kramers’ theory in the contextof surface diffusion and its extension to treat diffusion in aperiodic potential. Recent applications of Kramers’ theory andits extension have been made to treating hopping rates fordiffusion in periodic potentials.788-790

Quantum Mechanical Theories. For lower temperatures anddiffusion of light masses such as hydrogen, quantum mechanicaleffects are often important. It is interesting that the applicationof quantized conventional TST to the diffusion problem wasfirst proposed by Wert and Zener540 before the classical TSTtheory of Vineyard.541 It is only within the past decade thatmodern TST-based theories that include quantization of boundmodes and tunneling effects have been applied to diffusion. BothVTST/MT and PI-QTST methods have been used. In addition,Valone et al.691,692have proposed a method in which classicaltransition-state theory is applied to an Gaussian-averagedeffective potential energy surface that approximately includedquantum mechanical effects. We will refer to this version ofquantum mechanical TST as effective quantum potential TST(EQP-TST). Wahnstro¨m et al.791,792 have suggested an ap-proximation to the quantal flux-flux correlation functions in

which the rate constant is approximated from the correlationfunction up to the point that it first goes through its first zero.Since classical TST can be viewed as the short-time limit ofthe classical flux-flux correlation function,793 the authors havetermed this approximate method a quantum mechanical TST.As noted in section 2, this approximate QTST was first describedby Tromp and Miller.100,101 In this review we refer to thisversion of TST as short-time QTST.Self-Diffusion of Metal Atoms on Metal Surfaces. Monte

Carlo TST has been applied to self-diffusion on several metalsurfaces and compared with classical trajectory simulations ofthe mean-squared displacement of the adatom. Effects ofcorrelated hops and recrossing were studied, with dynamicaleffects accounting for changes from the TST activation energyof up to 6.5 kcal/mol.780 Dynamical corrections to classicalTST have been calculated for Rh diffusion on Rh(100)781 andfor adatom diffusion on the (111) face of a face-centered cubic(fcc) system model by a Lennard-Jones potential.794 For theRh system the TST results differed from the accurate, dynami-cally corrected ones by at most 6% in the temperature range200-1000 K. Differences were larger for the model fcc (111)system. At low temperatures TST exhibited small overestimates(less than about 20%) of the accurate diffusion constants becauseof recrossing of the transition-state dividing surface. As thetemperature increases, larger underestimates (greater than afactor of 2) of the diffusion constant were observed because ofthe importance of multiple hops in the dynamical simulationsthat are not included in the TST results.Atomic Diffusion in and on Solids. Zhang et al.795 compared

classical trajectory diffusion constants for hydrogen on the (100)face of Ni with classical TST calculations. The accuracy ofTST for surface diffusion is limited by the neglect of hops tononadjacent sites, or multiple hops, and by recrossings of thedividing surface. This study showed that multiple jumps canbe important, increasing the diffusion constant by as much as afactor of 3 over that assuming only single jumps. Recrossingfactors were found to be less important causing decreases ofthe diffusion constant by only about 25%.Engberg et al.796have examined the validity of classical TST

for the diffusion of H in Pd at 800 K and showed that thedistribution of transition-state configurations (i.e., the probabilityof finding a H atom at a transition state) determined fromclassical trajectory simulations is well reproduced by the TSTapproximation in terms of the Boltzmann factor of the potentialof mean force. They emphasize that a diffusive jump event“should be treated as a fluctuation in a many-body system atthermal equilibrium”. They concluded that the coupled H-Pdfluctuations are adequately treated within the TST approxima-tion.The diffusion of H atoms on Si(111) with partial hydrogen

coverage has been studied by Raff and Thompson and co-workers797,798using a classical Monte Carlo canonical VTSTmethod,799,800 which is closely related to the microcanonicalmethod442-449employed by this group for gas-phase studies anddiscussed in section 3.1. These methods were also applied tothe diffusion of Si atoms on the reconstructed Si(111)-(7× 7)surface.801,802

The diffusion of oxygen atoms in Ar and Xe matrices hasalso been studied with this Monte Carlo approach to classicalvariational transition-state theory.803 An underestimate of theexperimental diffusion constants by several orders of magnitudelead the authors to suggest that the experimentally observeddiffusion is not for a perfect crystal, but must occur primarilyalong defect pathways in the lattice.Blochl et al.804have studied proton diffusion in silicon using

classical TST. The lattice is allowed to adjust adiabatically to


the diffusing proton, reducing the problem to one of a singleparticle in a three-dimensional potential.Simplified TST has been used to study Si adatom diffusion

on Si surfaces to model the dynamics of surface rearrange-ment.805 This approach was also used to study diffusion of Hand CH3 on diamond surfaces.806

Jaquet and Miller807 have compared accurate quantum me-chanical diffusion constants with TST for a model of H on Win which the H atom is treated as two-dimensional and is coupledto a phonon bath treated by harmonic oscillators. Theirharmonic TST results were for a dividing surface that was afunction of only solute coordinates and neglected quantummechanical tunneling. Therefore, their correction factor to TSTincluded both quantum mechanical tunneling effects and aGrote-Hynes-type correction for phonon-induced classicalrecrossing.The H on Cu(100) system is especially important for

understanding the current status of TST because several differentversions of TST have been applied for the rigid-surface case,and the EQP-TST, VTST/MT, and PI-QTST approaches havebeen applied with movable metal atoms. The various studiesall use the same potential energy function, which is not veryaccurate, so the system should probably be regarded as “modelCu”, but we still learn about dynamics.For H on rigid Cu, Valone et al.691,692 applied EQP-TST,

Lauderdale and Truhlar808,809applied VTST/MT [in particularVTST with the original SCSAG method, which is identical toCD-SCSAG for surface diffusion on a rigid fcc (100) surface],and Sun and Voth810 applied PI-QTST.114 Accurate multidi-mensional quantum mechanical results and short-time QTSTcalculations for H on rigid Cu were presented by Haug et al.,792

and the former can be used to test the transition-state theories,which prove to be quite accurate at low temperatures, with errorsof 32%, 7%, 9%, and 24% at 200 K and 9%, 25%, 30%, and20% at 300 K for the short-time QTST, EQP-TST, VTST/MT,and PI-QTST results, respectively. At higher temperaturesrecrossings and multiple hops become more important, and theerrors in the VTST methods grow to factors greater than 2 fortemperatures above 400 K.Short-time QTST has also been applied to reduced-dimen-

sional models of H diffusion on the rigid Cu (100) surface andcompared with the accurate quantummechanical results.792,811,812

In addition, this approximate version of QTST was comparedwith the VTST/SCSAG calculations of Lauderdale andTruhlar808,809and the EQP-TST method of Valone et al.,691 aswell as the accurate quantum mechanical results. It is interestingthat the errors in short-time QTST are typically larger than thosefor VTST/MT and EQP-TST.The calculations with a nonrigid lattice are based on the

embedded cluster method of Lauderdale and Truhlar.813 Sunand Voth810 applied PI-QTST to diffusion of H on nonrigidCu(100). Allowing the substrate to move suppressed the rateconstant slightly at the lowest temperature (e.g., by about 40%at 100 K) and increased it by 2-20% at temperatures from 120to 300 K. Including the effect of electron-hole pairs by adissipative Langevin-like model decreased the rate constantsby factors of about 40% at 100 K to 1% at 300 K.Wonchoba and Truhlar814,815reported VTST/MT calculations

using the CD-SCSAG tunneling method for H diffusion on Cu;the difference between rate constants calculated with movingand fixed lattices increases from a factor of 3.7 at 300 K tofactors of 24-27 at 80-120 K. The comparison of the PI-QTST and VTST/MT calculations showed that the two quitedifferent methods predict similar effects of including quantumtunneling for 30 moving Cu atoms (93 degrees of freedom) atT g 120 K. The VTST/MT calculations included up to 56

moving Cu atoms (171 degrees of freedom) to achieveconvergence. Comparisons were also made to EQP-TSTcalculations816 with 36 moving Cu atoms, and reasonableagreement was obtained forT g 200 K. We know of no othercase where alternative quantum TST methods are validated bysuch comparisons with so many degrees of freedom.Wonchoba et al.817 also applied the VTST/MT method to H

on Ni(100) and found smaller effects of lattice motion than inthe Cu model system. This study explained the previouslyconfusing phenomenon of a low-temperature transition temper-ature at which the Arrhenius plot shows a dramatic change inslope. This was previously interpreted by experimentalists asa transition between overbarrier and tunneling dynamics, butWonchoba et al.817 showed it is really a transition betweenmultistate tunneling and ground-state tunneling, which isconsistent with earlier VTST/MT calculations of the state-dependent tunneling probabilities by Rice et al.818 Wonchobaet al. obtained an analytic approximation that fits the fullcalculations well.Mak and George819applied conventional TST with quantized

vibrations for H diffusion on Ru(001). The surface was treatedas rigid. The calculated Arrhenius preexponential factor washigher than the experimental value by about a factor of 4.Haug and Metiu820have studied H diffusion on Ni(100). The

motion of surface atoms were treated within a mean-fieldapproximation. For this model, quantum mechanical resultswere compared with the short-time QTST method.791,792

The diffusion of H isotopes on a rigid Ru(0001) surface hasalso been studied using VTST with SCSAG tunneling821 and apotential energy function based on ab initio pseudopotentialcalculations. Kinetic isotope effects were in good agreementwith experiment.The first applications of PI-QTST to surface diffusion were

for H and D diffusion on niobium using a reversible workformulation, although it seems that the calculation was formoving the centroid along a reaction path, rather than formoving a dividing surface that constrains the centroid.822 Path-integral QTST was used to study H and D diffusion on thePd(111) surface and diffusion into subsurface sites below thefirst layer of surface atoms.823 Quantum mechanical effects forsurface diffusion are modest and tend to increase the diffusionconstants compared to purely classical results. The subsurfacetransitions are more constricted and show an unusual quantummechanical behavior. The quantum mechanical rate constantsfor transition to the subsurface are significantly lower than theclassical ones.Perry et al.824 have applied a Monte Carlo approach to

classical variational transition-state theory to the diffusion ofH atoms in xenon matrices at 12-80 K. Tunneling contribu-tions to the diffusion coefficient are estimated by Boltzmannaverage of the tunneling probabilities through the one-dimensional potential along the minimum-energy path.Molecular Diffusion on Surfaces. Lakhlifi and Girardet825

have applied a TST-like approach (termed the transit timeconcept826) to the diffusion of Xe and small molecules on crystalsurfaces for temperatures from 20 to 100 K. The rate constantfor a diffusive jump takes the form of a harmonic TSTexpression for an approximate Hamiltonian representing a rigidadsorbate molecule on the surface. The bound degrees offreedom are treated quantum mechanically, and tunneling isneglected. The reaction coordinate is described in terms ofmotion in the two coordinates parallel to the plane of the surface.The vibrational modes are taken to be the motion of the moleculeperpendicular to the surface plane, the molecular rotation, andthree-dimensional vibration of each substrate atom. Thisapproach seems to neglect the bound mode in the planeperpendicular to the reaction coordinate.


Dobbs and Doren827 have compared classical TST estimateswith classical trajectory simulations of the diffusion constantfor CO diffusion on Ni(111). The TST estimates are based onapproximating the diffusion constant by single hops betweenadjacent sites, whereas the trajectory simulations are obtainedfrom the long-time limit of the mean-squared displacement. Forthis system multiple hops are important, and the TST resultsunderestimate the accurate values by factors of 100 to 20 overthe temperature range from 175 to 1000 K. Pai and Doren828

have studied the diffusion of a model rigid linear triatomic ona metal surface. Numerically accurate classical TST diffusioncoefficients are compared with exact classical diffusion coef-ficients for three models in which the mass distribution withinthe triatomic is different for each model, but the potentialremains the same. The change in the distribution of massesalters the frequency of the bending motion of the triatomicrelative to the surface. Classical TST yields the same diffusioncoefficient for each of these models, whereas marked changesare observed in the exact diffusion coefficients. Furthermore,classical TST underestimates the diffusion constant for all threemodels. The authors find that as the bending frequencydecreases the dissipation of energy in the motion along thesurface is more rapid, and multiple hops become less important.Calhoun and Doren829 have used the PI-QTST method to

study a two-dimensional model of CO diffusion on Ni(111).Comparisons with purely classical results indicated that quantummechanical effects on the rate constant became important fortemperatures below about 100 K. Large enhancements at 50K, larger than both the classical and the extrapolated Arrheniusfit to the high-temperature quantum mechanical results, wereattributed to quantum mechanical tunneling.Diffusion in Zeolites and Polymers. June et al.830 have used

classical, Monte Carlo TST to calculate hop rates betweendifferent sites for two Lennard-Jones spheres representing xenonand SF6 at infinite dilution in the zeolite silicalite. Then Poissonstatistics were assumed to calculate the diffusivity. Comparisonof the TST results with trajectory simulations of the diffusivityfor Xe diffusion at 150 and 200 K gave good agreement. Snurret al.831have used a similar process to obtain diffusion constantsfor benzene in the zeolite silicalite. In this case, rate constantsfor hops between adjacent sites were approximated usingclassical canonical TST. Diffusion constants were computedby a Monte Carlo simulation of the master equation describingthe time evolution of populations at different adsorption sitesin the silicalite structure. Schro¨der and Sauer have also studiedthe diffusion of benzene in silicalite by calculating enthalpiesand entropies in the rigid rotor-harmonic oscillator approxima-tion along reference paths parallel to the crystallographic axes.832

A TST model was developed and applied to the diffusion oflight gases such as He and H2 in rigid matrices of densepolymers such as rubbery polyisobutylene and glassy bisphenolA polycarbonate.833

Baker834,835has applied approximate formulas for diffusionin melts based on classical, conventional TST to interdiffusionin complex aluminosilicate melts.4.2.3. Surface Reactions. To date, most applications of TST

to surface reactions have been to relatively simple reactiveprocesses. Most of this work has focused on dissociativechemisorption of H2 on metals. VTST with SCSAG tunnelingwas applied to H2 and D2 dissociative chemisorption on threerigid surfaces, Ni (100), (110), and (111).836,837 The effects ofsurface defects (steps) and adsorbed H atoms on the chemi-sorption kinetics were studied using VTST/MT. The reversiblework formulation of PI-QTST was applied to the dissociativechemisorption of H2 on Cu.838,839 Quantum mechanical effectswere shown to be very important for both the Ni and Cu cases.

Reaction of gas-phase molecules with adsorbates can proceedby two mechanisms: direct collision of gas-phase moleculeswith the adsorbates (Eley-Rideal mechanism) and a trappingmediated process in which the gas-phase molecule first adsorbsand then reacts with another adsorbate (Langmuir-Hinshelwoodmechanism). Weinberg840 has presented a TST analysis of therates of these two reaction processes on surfaces. From thisanalysis he concludes that under most conditions the rate ofthe trapping mediated process dominates that for the directreaction process.The interaction of H atom with surfaces of solids like silicon

and carbon can be viewed as a covalent bonding interaction.Thus, the desorption or adsorption of H atom on these surfacesis viewed more like an association or unimolecular dissociationreaction. For silicon an important question is whether hydrogencomes off the surface as atoms or molecules. Classical TSTrate constants for H atom desorption and H-H recombinationand desorption from Si(111) have been calculated using a MonteCarlo approach.799 H atom desorption was found to benegligible, in agreement with experiment. These calculationswere extended to classical VTST calculations of the rate constantfor the H-H recombination and desorption from Si(111).800

Canonical VTST calculations with quantum mechanicalpartition functions have been carried out for the association ofH atoms with the (111) surface of diamond.516,841,842 VTSTrate constants for this process have been compared with thoseof the association reaction of H atoms with alkyl radicals usingempirical potentials.516 The two systems show similar behaviorwith the exception of the rotational motion of the alkyl radical.VTST calculations for the H atom association with the (111)diamond surface have been performed for a potential energysurface that was developed based on accurate ab initio calcula-tions.841 Rate constants obtained from quasiclassical trajectoriesfor the same potential agree well with the VTST rate con-stants.842

Conventional microcanonical transition-state theory has beenused to study activated dissociative adsorption of CH4 onPt(111).843

5. Concluding Remarks

Looking forward from the vantage point of 1983, one mighthave predicted that the basic formalism of transition-state theorywas well established, and the future would consist of variousquantitative refinements, especially taking advantage of theanticipated advances in electronic structure theory. Theseadvances have indeed occurred, but along with further concep-tual refinements in the dynamics. Probably the chief noteworthyadvance of the past 10 years is that the relationship of transition-state theory to accurate quantum dynamics has been greatlyclarified. Especially rewarding developments include advancesin exploiting the flux-flux correlation function formulation ofreaction rate theory, the discovery and analysis of quantizedtransition-state structure in microcanonical ensemble rate con-stants, the extension of well-validated multidimensional tun-neling approximations to polyatomic systems, the developmentof the path integral approach to TST, techniques for consideringalternative dividing surfaces, and the development of MonteCarlo sampling techniques. Simultaneously, new frameworkshave been proposed for treating solvent effects on complexsystems.On the applications side it is clear that we stand on several

thresholds. As far as electronic structure methodology forapplications, direct dynamics or statistics methods are poisedto have a major impact, so the difficulties of the creation ofanalytic potential energy surfaces will not so readily impedeprogress. Furthermore, electronic structure theory itself is


making especially noteworthy progress in several directionssthreeof which are quantitative accuracy for systems bigger than H3,linear-scaling algorithms to allow the treatment of very largesystems, and practical methods for including solvent effects inquantum mechanical calculations of solute structure, energetics,and reactivity. As far as dynamics methodology, realisticmultidimensional tunneling calculations have become practicalfor large systems, and algorithms for using more appropriatedefinitions of the reaction coordinate have been developed bothfor barrier reactions and for barrierless associations. For thelatter reactions the classically accurate treatment of the low-frequency mode anharmonicities and vibration-rotation cou-plings are also now feasible.In the future we can expect further progress on the quantum

dynamical aspects. Such progress, combined with paralleladvances in direct dynamics, can be expected to blur thedistinctions between transition-state theory and quantum scat-tering theory on one hand and between structure and dynamicson the other. The future does not seem nearly as predictableas it might have 10 years ago!

Acknowledgment. The authors are grateful to Joel Bowman,Laura Coitino, Chris Cramer, Bill Hase, Steven Mielke, GregMills, Kiet Nguyen, Greg Schenter, Rex Skodje, Don Thomp-son, Thanh Truong, Suzi Tucker, and Greg Voth for helpfuldiscussions or comments on the manuscript. This work wassupported in part by the Division of Chemical Sciences, Officeof Basic Energy Sciences, U.S. Department of Energy, throughGrant DE-FG02-86ER13579 (to D.G.T.) and through ContractDE-AC06-76RLO 1830 with Battelle Memorial Institute whichoperates the Pacific Northwest National Laboratory (PNNL) andby the National Science Foundation through Grant CHE-9423927 (to D.G.T.) and Grant CHE-9423725 (to S.J.K.).

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current status of transition-state theory

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