Reaction rate theory: What it was, where is it today, and where is it going?

Reaction rate theory: What it was, where is it today, and where is it going?Eli Pollak and Peter Talkner Citation: Chaos 15, 026116 (2005); doi: 10.1063/1.1858782 View online: http://dx.doi.org/10.1063/1.1858782 View Table of Contents: http://chaos.aip.org/resource/1/CHAOEH/v15/i2 Published by the American Institute of Physics. Related ArticlesRate coefficients and kinetic isotope effects of the X + CH4 → CH3 + HX (X = H, D, Mu) reactions from ringpolymer molecular dynamics J. Chem. Phys. 138, 094307 (2013) Multiple population-period transient spectroscopy (MUPPETS) in excitonic systems J. Chem. Phys. 138, 034201 (2013) Study on gas phase collisional deactivation of O2(a1Δg) by alkanes and alkenes J. Chem. Phys. 138, 024320 (2013) Thresholds for igniting exothermic reactions in Al/Ni multilayers using pulses of electrical, mechanical, andthermal energy J. Appl. Phys. 113, 014901 (2013) Reaction dynamics with the multi-layer multi-configurational time-dependent Hartree approach: H + CH4 → H2 +CH3 rate constants for different potentials J. Chem. Phys. 137, 244106 (2012) Additional information on ChaosJournal Homepage: http://chaos.aip.org/ Journal Information: http://chaos.aip.org/about/about_the_journal Top downloads: http://chaos.aip.org/features/most_downloaded Information for Authors: http://chaos.aip.org/authors

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Reaction rate theory: What it was, where is it today, and where is it going?Eli PollakChemical Physics Department, Weizmann Institute of Science, 76100 Rehovot, Israel

Peter TalknerInstitut für Physik, Universität Ausgburg, D-86135 Augsburg, Germany

sReceived 16 December 2004; accepted 23 December 2004; published online 17 June 2005d

A brief history is presented, outlining the development of rate theory during the past century.Starting from ArrheniusfZ. Phys. Chem.4, 226 s1889dg, we follow especially the formulation oftransition state theory by WignerfZ. Phys. Chem. Abt. B19, 203 s1932dg and EyringfJ. Chem.Phys. 3, 107 s1935dg. Transition state theorysTSTd made it possible to obtain quick estimates forreaction rates for a broad variety of processes even during the days when sophisticated computerswere not available. Arrhenius’ suggestion that a transition state exists which is intermediate betweenreactants and products was central to the development of rate theory. Although Wigner gave anabstract definition of the transition state as a surface of minimal unidirectional flux, it took almosthalf of a century until the transition state was precisely defined by PechukasfDynamics of Molecu-lar Collisions B, edited by W. H. MillersPlenum, New York, 1976dg, but even this only in the realmof classical mechanics. Eyring, considered by many to be the father of TST, never resolved thequestion as to the definition of the activation energy for which Arrhenius became famous. In 1978,ChandlerfJ. Chem. Phys.68, 2959s1978dg finally showed that especially when considering con-densed phases, the activation energy is a free energy, it is the barrier height in the potential of meanforce felt by the reacting system. Parallel to the development of rate theory in the chemistrycommunity, Kramers published in 1940fPhysicasAmsterdamd 7, 284 s1940dg a seminal paper onthe relation between Einstein’s theory of Brownian motionfEinstein, Ann. Phys.17, 549 s1905dgand rate theory. Kramers’ paper provided a solution for the effect of friction on reaction rates butleft us also with some challenges. He could not derive a uniform expression for the rate, valid forall values of the friction coefficient, known as the Kramers turnover problem. He also did notestablish the connection between his approach and the TST developed by the chemistry community.For many years, Kramers’ theory was considered as providing a dynamic correction to the thermo-dynamic TST. Both of these questions were resolved in the 1980s when PollakfJ. Chem. Phys.85,865 s1986dg showed that Kramers’ expression in the moderate to strong friction regime could bederived from TST, provided that the bath, which is the source of the friction, is handled at the samelevel as the system which is observed. This then led to the Mel’nikov–Pollak–Grabert–HänggifMel’nikov and Meshkov, J. Chem. Phys.85, 1018s1986d; Pollak, Grabert, and Hänggi,ibid. 91,4073 s1989dg solution of the turnover problem posed by Kramers. Although classical rate theoryreached a high level of maturity, its quantum analog leaves the theorist with serious challenges tothis very day. As noted by WignerfTrans. Faraday Soc.34, 29 s1938dg, TST is an inherentlyclassical theory. A definite quantum TST has not been formulated to date although some very usefulapproximate quantum rate theories have been invented. The successes and challenges facing quan-tum rate theory are outlined. An open problem which is being investigated intensively is rate theoryaway from equilibrium. TST is no longer valid and cannot even serve as a conceptual guide forunderstanding the critical factors which determine rates away from equilibrium. The nonequilib-rium quantum theory is even less well developed than the classical, and suffers from the fact thateven today, we do not know how to solve the real time quantum dynamics for systems with “many”degrees of freedom. ©2005 American Institute of Physics. fDOI: 10.1063/1.1858782g

Rate theory provides the relevant information on thelong-time behavior of systems with different metastablestates and therefore is important for the understanding ofmany different physical, chemical, biological, and techni-cal processes. Einstein indirectly contributed to thistheory by clarifying the nature of Brownian motion asbeing caused by the thermal agitation of surroundingmolecules on an immersed small particle. Kramers later

pointed out that the thermally activated escape from ametastable state is nothing else but the Brownian motionof a fictitious particle along a reaction coordinate leadingfrom an initial to a final locally stable state. In order toovercome the energetic barrier separating the two states,the particle has to “borrow” energy from its surround-ings, an extremely rare event if as is usually the case theactivation energy is much larger than the thermal energy.

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So, many attempts will take place until the particle hasovercome the barrier separating the two states. Duringthese many unsuccessful events, the particle completelyloses any memory on how it had come to its initial state.Due to this loss of memory, the waiting time in the initialwell will be random with an exponential distributionwhose average coincides with the inverse of the decayrate. In this article the historical development of ratetheory is outlined, the concepts are discussed, and presentday generalizations to quantum nonequilibrium systemsare presented. Our review ends with a brief summary ofsome challenges facing rate theory which remain openeven a century after Einstein’s seminal paper.

I. RATE THEORY IN THE FIRST HALF OF THE 20THCENTURY: ARRHENIUS, WIGNER, EYRINGAND KRAMERS

A. Arrhenius and activated molecules

The father of reaction rate theory is Arrhenius who in hisfamous 1889 paper1 investigated the temperature dependenceof the rates of inversion of sugar in the presence of acids. Asnoted by Hänggiet al.,2 Arrhenius himself cites van’t Hoff3

as the person who suggested ane−A/T temperature depen-dence of reaction rates. However, Arrhenius is the father ofrate theory since he postulated that this relationship indicatesthe existence of an “activated sugar” whose concentration isproportional to the total concentration of sugar, but is expo-nentially temperature dependent.

But perhaps there is another reason why Arrhenius is sohighly respected by the physical chemistry community. In1911 he traveled to the United States and gave there a seriesof lectures, summarized in his bookTheories of Solutions.4

In the Introduction he made the following observations:“Chemistry works with an enormous number of substances,but cares only for some few of their properties; it is an ex-tensive science. Physics on the other hand works with ratherfew substances, such as mercury, water, alcohol, glass, airbut analyzes the experimental results very thoroughly; it isan intensive science. Physical chemistry is the child of thesetwo sciences; it has inherited the extensive character fromchemistry…it has its profound quantitative character fromthe science of physics.” He ends his Introduction by notingthat, “The theoretical side of physical chemistry is and willprobably remain the dominant one.”

It is this theoretical side which lies at the heart of thisreview. We will try briefly to follow the history of the devel-opment of rate theory in chemistry and physics, its impact onpresent day science, and its prospect for the 21st century—isthere still anything that can be added to it?

B. Wigner and Eyring: The transition state method

Arrhenius’ idea of an activated intermediate was ampli-fied upon by a number of authors during the next 35 years.Christiansen and Kramers5 were able to provide a rationalefor the Arrhenius form based on the kinetic theory of gases.They realized already in 1923 that the activation energycould be understood by assuming that a minimum amount ofenergy is needed before reaction could occur. The probability

for attaining this energy is given by the canonical distribu-tion and thus one obtains the Arrhenius factor. They thenprovided a heuristic estimate for the magnitude of theprefactor.

In 1935, Eyring published a paper titled “The ActivatedComplex in Chemical Reactions.”6 By this time, it was wellestablished that reaction ratesskd should be written in theform

k = ne−E/kBT, s1.1d

wheren is a prefactor with the dimensions of 1/s for unimo-lecular reactions and 1/ss ·cm3d for bimolecular reactions.Eyring proposed a method by which one could calculate the“absolute reaction rate.” Eyring, though a devout Mormon,probably did not really mean “absolute” in the divine sense,rather his claim to fame at this point was that he wrote downa formula for the rate which allowed one to estimate theprefactor in the rate expression. While most previous worksdealt with the relative rates of reactions, in which theprefactor would be eliminated, Eyring gave a heuristic deri-vation of an expression for the prefactor based on the as-sumption of an equilibrium between the activated complexand reactants. To obtain the time constant, he postulated, thatat the saddle point, any quantum state perpendicular to thereaction coordinate reacts with the same universal time con-stantkBT/2p". The rate is then given by the product of thisuniversal time constant with the ratio of the partition func-tion of the activated complexswhich has one degree of free-dom less than the reactantsd to the partition function of thereactants. Eyring’s formulation, in terms of partition func-tions allowed a heuristic quantum mechanical formulationfor the rate constant and indeed in his 1935 paper he usedquantum mechanical partition functions.6

Noteworthy also is the work of Farkas7 and Szilardswhowas credited by Farkas, but a specific citation was not givendwho realized and implemented the flux over population defi-nition of the rate constant. This was then picked up by Pelzerand Wigner in their 1932 paper8 in which they estimated therate of conversion of parahydrogen into normal hydrogen. Inthis very early paper one may find all the elements of muchmore sophisticated work which abounded in the second halfof the 20th century. They adapt the Eyring–Polanyirepresentation9 of the ground Born–Oppenheimer potentialenergy surface for the motion of the nuclei, even showing afictitious trajectory leading from reactants to products. Theyestimate the effect of electronically nonadiabatic interactionsand show that they are negligible.

To compute the reaction rate, they use a thermal equilib-rium distribution in the vicinity of the saddle point of thepotential energy surface and estimate the unidirectional clas-sical flux in the direction from reactants to products. Alreadyhere, they note that they ignore the possibility of recrossingsof the saddle point, pointing out that their probability at roomtemperature would be rather small. To get the rate they usethe flux over population method after harmonically expand-ing the potential energy surface about the saddle point. ThePelzer and Wigner paper8 is the very first use of transitionstate theory to estimate reaction rates. It is however writtenin a rather specific form, as applied to the hydrogen ex-

026116-2 E. Pollak and P. Talkner Chaos 15, 026116 ~2005!


change reaction. Eyring’s later paper of 19356 provides gen-eral formulas which were then applied to many different ac-tivated reactions.

In 1932 Wigner10 made an additional seminal contribu-tion to rate theory by providing an estimate for the tunnelingcontribution to the thermal flux of particles crossing a bar-rier. Using the same parabolic barrier expansion as in thework with Pelzer and his recently formulated quantum dis-tribution function in phase space, he derives a series in"2 forthe thermal tunneling corrections. His derivation is heuristic,he uses quantum mechanics for the thermal density of par-ticles but treats the motion across the barrier as classical.

The competition between Wigner and Eyring seems tohave reached a head at a Faraday discussion, the papers ofwhich were published in 1938. Wigner11 and Eyring12 hadback-to-back papers; Wigner’s was titled “The TransitionState Method,” Eyring used “The Theory of Aboslute Reac-tion Rates.” Wigner notes here that the transition statemethod is inherently a classical theory, since the uncertaintyprinciple forbids the simultaneous determination of a divid-ing surface and the sign of the momentum across the surface.Eyring ignores this, and uses his heuristic thermodynamicformulation within a quantum mechanical context, applyingthe theory to the reaction of NO with O2.

One of the most interesting aspects of this Faraday meet-ing was the emerging ambiguity with respect to the definitionof the “activated complex.” In Wigner’s approach it is de-fined through the dividing surface. However, when usingequilibrium thermodynamics as a basis for the theory, ambi-guities arise. Evans13 in his paper titled “ThermodynamicalTreatment of the Transition State” defines it as “The leastprobable configuration along the reaction path” but does notreally define this probability. Guggenheim and Weiss in theirpaper “The Application of Equilibrium Theory to ReactionKinetics”14 are very forthcoming: “We are not always quitesure whether the expression the activated complex refers toA8 san energetic moleculed or A* sa reacting moleculed or tosomething intermediate between the two.”

This ambiguity continues for three more decades. In asymposium held in Sheffield, in April 1962 a lively discus-sion takes place between G. Porter and Eyring.15 Porter asks:“May we begin by making sure that we know what we aretalking about? In the papers presented at thismeeting…potential-energy and free energy maxima are usedrather indiscriminately to define the transition state…. WouldProfessor Eyring give us a rigorous definition to the transi-tion state?” Eyring’s answer is “the concept of an activatedcomplex…provides us with the same theoretical tools fordiscussing reaction kinetics that have been so successfullyused in discussing the equilibrium constants. The current lit-erature is an eloquent testimonial to the fecundity of theconcept.”

It is noteworthy, that this kind of discussion did not arisethrough Wigner’s work. Wigner’s approach was based onclassical mechanics. Early on, Wigner realized that classicaltransition state theory provides an upper bound for the clas-sical canonical net flux going from reactants to products. Inhis 1937 paper16 he used this property to derive an upperbound to the rate of association reactions, using a dividing

surface in energy rather than in configuration space. The up-per bound leads to the variational property, which says thatthe best dividing surface is that which minimizes the unidi-rectional flux from reactants to products. With this definition,there is no ambiguity. The classical formulas for estimatingthis unidirectional flux are identical to the classical limit ofthe formulas used by Eyring and the chemistry community,this is not an accident, we remember that Pelzer and Wigner8

had already given the foundations for the transition statetheory sTSTd method in their 1932 paper and that Eyringmade sure that his formulation would correctly reduce totheirs. Wigner’s interest in the TST method waned after1938. In a personal meeting with him in the early 1980s, heshowed no further interest in the issue.

C. Kramers–Brownian motion in a field of force

Kramers, whose earlier paper on rate theory was withChristiansen5 as mentioned above, published a seminal paperof his own in 1940 titled “Brownian Motion in a Field ofForce and the Diffusion Model of Chemical Reactions,”17

which may be thought of as the direct descendent of Ein-stein’s famous paper of 1905. Although he does not providea citation to Einstein’s work18 he does note, “A theory ofBrownian motion on the Einstein pattern can be set up…”under appropriate conditions. Using the Langevin equationas a model in which a particle is moving under the influenceof a field of force and a frictional force characterized by adamping coefficientg he derives a Fokker–Planck equationin phase spacesknown also as the Kramers equationd for themotion.

Kramers then proceeds to use the flux over populationmethod to derive the rate of passage over a barrier in threelimits—weak damping, intermediate, and strong damping.He derives the famous prefactor for the rate which is validfor the intermediate to strong damping regimes and indepen-dently a different prefactor, obtained by deriving a diffusionequation in energy in the underdamped limit, where the re-action rate is limited by the rate of flow of energy from thesurrounding to the particle. Kramers notes that the interme-diate friction range is the one in which one obtains the tran-sition state limit for the rate. His paper also presented achallenge—deriving a uniform expression for the rate validfor all values of the friction, known as the turnover problem.

Kramers realized that in the intermediate friction rangehis expression for the rate is identical to the ones derived byPelzer and Wigner8 and Eyring6 and cites them accordingly.He considers his derivation as support for the transition statemethod, noting that “the transition state method gives resultswhich are correct, say within ten percent in a rather widerange ofh sfriction coefficientd values.” However, Kramersdoes not state nor does he derive any formal equivalencebetween his results and transition state theory. Moreover, herefers to the ambiguity of choice of transition state, notingthat for small friction it should be a state of definite energywhile for larger friction it is characterized by the spatial co-ordinate.

His work which was used by the physics community forthe next four decades or so, evolved independently of theTST approach to rate theory used by the chemistry commu-

026116-3 Reaction rate theory Chaos 15, 026116 ~2005!


nity. Kramers’ equation was put to use for solving a varietyof problems, and was even generalized to include quantumeffects such as tunneling by Caldeira and Leggett in 1983.19

Kramers himself considered his model as having no room forquantum mechanical effects such as tunneling.

Kramers’ rate theory was generalized by a number ofgroups mainly in the 1970s and 1980s to systems with morethan one degree of freedom20,21and to systems with memoryfriction in the moderate to strong friction regime22–24 and inthe weak friction regime.25 These authors determined the rateby a flux over population expression where the flux is con-sidered as the number of particles per unit time crossing aconveniently chosen hypersurface in the phase space of thesystem that divides between reactants and products. Thepopulation is by definition the density of reactants. The fluxand the population are calculated for a stationary distributionin which those particles that have escaped from the state ofreactants through the separating hypersurface are replen-ished. The validity of this method is based on the assumptionthat the reinjected particles thermalize before they escape thenext time. This requires that the reactants are separated fromthe products by a barrier that is substantially higher than thethermal energy. The necessary minimal barrier height for theflux over population method to hold strongly depends on theconsidered system. In the presence of a slow degree of free-dom which is different from the reaction coordinate the truerates may be strongly suppressed compared to the classicalKramers–Langer–Grote–Hynes expression.26–29

Another approach that basically gives the same rate ex-pressions as the flux over population method relies on thedetermination of the mean time that the reactants need totransform into the activated complex, or more precisely, onthe mean first passage time of trajectories leading from thereactant state to the separating hypersurface. Elaborateasymptotic methods have been developed30–32 for the calcu-lation of the mean first passage time.

In passing we note that any inverse mean first passagetime which results from a Fokker–Planck equation can for-mally be represented as a flux over population expression.2,33

Recently the same result was obtained for mean first passagetimes of any time-homogeneous process.34 Yet there are con-ceptual differences between the two methods. For the meanfirst passage time, trajectories arriving at the consideredseparating surface are immediately stopped. Consequentlyany recrossings of this surface are strictly excluded from thecorresponding formal rate. This is quite in contrast to theKramers flux over population expression which allows forrecrossings of the barrier. Absorption only takes place be-yond the barrier from where it is most likely that reactionwill be completed before any eventual back reaction takesplace. In simple cases, the effect of recrossings can be incor-porated in the rate resulting from a mean first passage timeby a numerical factor which is 1/2 if the separating surfacecoincides with the so-called stochastic separatrix.35–37 Inthese works it was also shown that the stochastic separatrixcoincides with the usual, deterministic separatrix in the limitof infinitely high barriers. On the other hand, the recrossingsbecome immaterial for mean first passage times of a separat-

ing surface that is sufficiently close to the final product state.In general, however, there is no easy way to infer the truerate from a mean first passage time.38–41

II. RATE THEORY IN THE SECOND HALF OF THE20TH CENTURY

A. Variational transition state theory

The next milestone in the development of rate theory iswhat is termed variational transition state theorysVTSTd.Keck42 used the ideas of Marcelin,43 Wigner,10,11,16 andHoriuti44 and in his definitive review of 196745 presented thesystematic application of the variational property to activatedchemical reactions.

In VTST one varies a surface that divides between reac-tants and products so as to minimize the unidirectional fluxthrough the surface. Considering that the rate is given by theratio of the reactive flux and the population of reactants onehas that “such a calculation gives an upper limit to the truereaction rate since passage at least once through the trialsurface is a necessary condition for reaction.” Keck specifi-cally shows that VTST reduces to Eyring’s TST under appro-priate conditions. For Keck, as for Wigner and Horiuti, thereis no ambiguity in the definition of the activated complex, itis the solution of the variational minimization problem. If theactivation energy is large as compared withkBT then natu-rally this surface will be in the vicinity of the barrier and willbe very close to the surface perpendicular to the unstablemode at the saddle point.

Keck had an additional contribution. The chemistrycommunity became interested not only in thermal reactionrates but also in energy dependent microcanonical rates. Forthis purpose, he formulated a statistical theory of reactionrates46 which could be considered as an adaptation to mo-lecular dynamics of the statistical theories of nuclear reactionrates47 developed earlier in the physics community.

Rate theory and especially TST became an object ofquantitative studies with the introduction of computers. Inthe early 1960’s people started using them to solve numeri-cally the classical motion of atoms and molecules evolvingon a single Born–Oppenheimer potential energy surface.They were able to compute numerically exact reaction ratesand compare them with theory. One of the interesting resultswas that microcanonical TST gave energy dependent reac-tion probabilities that were greater than unity.48 This problemwas actually realized already back in the Faraday discussionby Hinshelwood.49 It usually does not arise in practice inactivated reactions for which the barrier height is muchlarger thankBT, but it is striking when considering microca-nonical reaction rates.

This difficulty was also resolved through use of VTST.The variational dividing surface which truly minimizes thereactive flux will never lead to reaction probabilites that aregreater than unity. Only when one uses a “bad” dividingsurface for which there are many recrossings does one en-counter the problem. When accounting for recrossings, it wasshown50 that the unidirectional flux across any dividing sur-face is identical to the average number of recrossings of the



dividing surface. This average number can of course begreater than unity. When it is less than unity, it gives a non-trivial upper bound on the reaction rate.

B. What is the activated complex?

In pioneering work, De Vogelaere and Boudart realizedthat periodic orbits play a special role in collinear atom dia-tom reactions.51 In 1976, Pechukas52 resolved the mystery bynoting that classically, the solution to microcanonical varia-tional TST is a classical bound state embedded in the con-tinuum. For a system withN degrees of freedom, it is a phasespace manifold with dimensionality 23 sN−1d and so it istruly an intermediate entity between reactant and products.For a system with two degrees of freedom the energy depen-dent activated complex is a periodic orbit dividing surface,named a pods,53,54 moving between the two equipotentialenergy surfaces. One can also show that the optimal canoni-cal dividing surface in two degrees of freedom is a trajectoryembedded in the continuum, however it moves on an effec-tive temperature dependent surface.55

For reactive systems with two degrees of freedom, onetypically finds that for a small energy range above the saddlepoint energy, there exists only one pods between reactantsand products. In this energy region, there are no recrossingsand TST is exact.56 When more than one pods exists, the fluxthrough the pods leads to both upper and lower bounds to themicrocanonical reaction probability.57 The activated complexis by definition the pods with the minimal unidirectional fluxthrough it.

The identity of the activation energy in the Arrheniusexpression was resolved in 1978 by Chandler.58 He presenteda formal derivation of the classical transition state theoryexpression for the rate of reaction in liquids. Using the re-gression hypothesis, he identified the rate as the time rate ofchange of the reactants density autocorrelation functionwhich can be expressed as the ratio of the “reactive flux” tothe density of reactants. Then he noted that classical TSTgives an upper bound to the reactive flux and used the uni-directional flux of TST to obtain the TST expression for therate. His innocuous looking one-dimensional TST result isobtained by hiding the condensed phase in a one-dimensional potential of mean force. The activation energyin his expression is now a free energy, answering the 50-year-old question as to whether it is the energy or the freeenergy.

C. Unification of the TS method with Kramers’Brownian motion theory and solution of the Kramersturnover problem

Originally, Kramers’ prefactor for the rate was consid-ered as a dynamic friction induced correction to TST. Only in1986 was it demonstrated that Kramers’ result for themoderate-to-strong damping regime may be derived fromclassical variational transition state theory.59 Until then, ev-eryone used the “simple” dividing surface perpendicular tothe system coordinateq. The identity between variationalTST and Kramers’ result was derived bysad using the well-known identity of the Langevin equation with a Hamiltonian

in which the system is bilinearly coupled to a harmonicbath,60 and thensbd allowing the dividing surface to be per-pendicular to a collective mode reaction coordinater whichis a linear combination of the system and harmonic bathmodes. The coefficients of the linear combination are deter-mined by minimizing the unidirectional flux. For a parabolicbarrier potential this exactly gives the Kramers result. In fact,the ratio of the “standard” TST result based on use of thedividing surface perpendicular to the systemq coordinateand Kramers’ expression is just a reflection of the large num-ber of times that trajectories recross the “standard” dividingsurface. For the parabolic barrier, the optimized collectivemoder is separable from all other modes, trajectories crossthe surface perpendicular to it only once and VTST is exact.

In the underdamped energy diffusion limited regime, avariational TST approach shows that the dividing surface isnow in energy space,65 similar to the dividing surface usedby Wigner in his treatment of three-body dissociation.16

However, it is not possible to construct a simple surfacewhich would totally eliminate the recrossings of this energydividing surface.

The observation that in dissipative systems one shoulduse collective mode reaction coordinates then led to a flurryof activity, culminating by a two step solution to the oldKramers’ turnover problem. Mel’nikov and Meshkovshowed how one could derive a uniform expression for therate which covered the underdamped-to-moderate dampinglimit.66 Pollak, Grabert, and Hänggi then used Mel’nikov andMeshkov’s formalism but adapted it to the collective mode toderive an expression which is valid for all values of thedamping as well as memory friction.67

D. Quantum TST?

Another question which remains open even today is thatof a quantum mechanical analog to TST. One can followEyring’s approach6 and replace classical partition functionswith quantum partition functions, make a separable approxi-mation about the saddle point, and use it to treat the unstablemode quantum mechanically thus introducing tunneling cor-rections. This “engineering” approach was used rather suc-

cessfully by Truhlaret al.,68,69 who devised strategies forcalculating quantum reaction rates in rather complex sys-tems. Although useful in practice, and typically accurate towithin an order of magnitude and often much less, this ap-proach remains unsatisfying due to itsad hocnature.

A serious attempt to formulate a quantum mechanicalTST comes in 1974 with the paper of McLafferty andPechukas.70 They demonstrated that although one could de-rive upper bounds of a TST form to the reaction rate thebounds were not very good even for a free particle or aparabolic barrier potential. The quantum upper bound prop-erty remained a topic of active research for the next twodecades71–74 however the quantum upper bounds to the rateare not as useful as in the classical theory. For deep tunnelingthey go as the amplitude of the tunneling, instead of theamplitude squared and for above barrier activation, they arenot accurate enough to justify the numerical effort needed incomputing them.



Classical canonical transition state theory relies on ther-modynamic averages, so that the search for a quantum tran-sition state theory is really a search for a rate theory whichrelies only on imaginary time data. An important milestonein this direction was the centroid based approximation for therate developed during the past 15 years.75–77

In this approach, the rate expression is approximated as aproduct of a centroid density containing all closed paths withcentroid at the transition state, multiplied by a thermal veloc-ity factor. The centroid expression for the rate is exact for aparabolic barrier. However, due to the intuitive derivation,centroids do not give upper bounds to the rate and so thelocation of the transition state in asymmetric systems canbecome quite a problem. The centroid TST has been im-proved upon by employing centroid dynamics.78,79

The great advantage of the centroid approach is that itcan be computed for many dimensional systems, however,the lack of a derivation from first principles leaves one withan uneasy feeling. At this point, the search for a rigorousexpansion, in which the centroid based formulas would bethe first term in a series is open. Just as the Wigner ansatz forthe tunneling correction was a wonderful guess, so is thecentroid approach. But one should keep in mind that even todate, it is more of an ansatz than a theory.

The centroid approach served though as an impetus forother approximate thermodynamic rate expressions. Pollakand Liao80 followed an earlier idea of Vothet al.81 and re-placed the exact projection operator appearing in the rateexpression with two choices, one being the parabolic barrierprojection operator in phase space, this approximation wastermed by them as quantum TST, the other using classicalmechanics to compute the projection operator.80,82 Thismixed quantum classical theory was rederived by Wang andcoworkers83 and termed the linearized approximation, sinceit also comes about from linearization of the action appearingin the path integral about the classical dynamics. The formaladvantage of these two methods over the centroid method isthat one can at least in principle write down quantum correc-tion terms. In practice, although demonstrated to be usefulfor a one-dimensional symmetric Eckart potential,82 no onehas attempted to compute these correction terms for reac-tions with two or more degrees of freedom.

A different thermodynamic approach was suggested byHansen and Andersen84 who noted that the first few initialtime derivatives of the flux flux correlation function involveonly thermodynamic averages. They used various extrapola-tion formulas to bridge the gap between the initial time andfinal time and thus derive a thermodynamic rate theory. Heretoo though, the extrapolation is fraught with danger, and it isdifficult to obtain systematic corrections and convergence to-ward an exact result.85

In the early 1970s Miller86 showed that semiclassicallythe tunneling rate is determined by a periodic trajectory mov-ing on the upside down potential energy surface, with period" /kBT. The physics community, rediscovered this sameobject87,88 and named it the instanton. In a seminal paper,Caldeira and Leggett derived the formulas for the instantonin the case of a particle moving under the influence of dissi-pation, introduced by coupling the system bilinearly to a har-

monic bath. In fact, for many years, starting with Kramers’paper of 1940, the physics community was interested in re-action rates in condensed matter and modeled the “bath” interms of the Langevin equation, which had its roots in Ein-stein’s paper of 1905. It was though only in 1983 that Cal-deira and Leggett applied the Feynman–Vernon influencefunctional approach89 to a quantum system bilinearlycoupled to a harmonic bath to derive an instanton based rateexpression for tunneling in the presence of dissipation. Theirtheory became the standard model of quantum friction,which continues to be a subject of intense study to this veryday due to its applicability as a model for a broad variety ofphysical phenomena. In particular, the influence of tempera-ture and friction on quantum rates has been studied exten-sively by means of this theory. For a review we refer toHänggiet al.2

Originally, the concept of the instanton came from semi-classics. Only lately Miller and coworkers have managed toformulate a quantum mechanical object which in the semi-classical limit reduces to the instanton.90 This quantum me-chanical instanton turns out to be an extremum of the offdiagonal matrix elements of the Boltzmann operatorkque−bHuq8l with respect to the positionsq andq8. They thenuse the quantum instanton coupled with the short timeLaplace inversion method of Plimak and Pollak91 to obtain anew rate expression. Although the tests on this quantum in-stanton theory were rather successful,92 this approach toosuffers from the fact that it is not a theory in the sense of aset of approximations that converge towards the correct an-swer.

E. Reactive flux method

One of the central difficulties in computing reaction ratesfor large systems is the rare sampling of events that lead toreaction. Especially if the barrier height is large compared tokBT the probability of finding an initial condition which willlead to barrier crossing becomes small and the computationalcost high. To overcome this problem, Chandler58 used On-

sager’s regression hypothesis93 to derive a reactive flux for-mula whose great advantage was that it provided a techniquein which one could sample the phase space in the vicinity ofthe barrier to reaction instead of reactants, thus leading to anenormous saving in computational effort.94 This approachlater led to more sophisticated methods for identifying thebarriers to reaction in multidimensional systems.95,96The re-active flux method which originally was restricted to pro-cesses with a well-defined velocity, was later also general-ized to jump and diffusion processes.97

The reactive flux method also served as the basis for theapplication of variational TST to condensed phases,29,55,98,99

as the reactive flux is bounded from above by the unidirec-tional flux. By varying the dividing surface one could im-prove upon the Kramers–Grote–Hynes estimate of the rate inthe spatial diffusion limit and obtain temperature dependentcorrections.100 The same corrections were also found in theframework of the Fokker–Planck description of thermallyactivated escape.101



One has to keep in mind though that any meaningful ratedescription is based on the assumption that the time scale onwhich the reaction takes place must be well separated fromother time scales that describe the relaxation to, say, productsonce the separating barrier was crossed, and vice versa toreactants from the other side of the barrier. The time-scaleseparation is determined by the height of the separating bar-rier as seen from the reactants side. Recently Drozdov andTucker102 determined the lowest nonvanishing eigenvalue ofthe Fokker–Planck equation and the reactive flux rate in anumerical study of an overdamped bistable Brownian oscil-lator. The lowest eigenvalue describes the asymptotic expo-nential approach to equilibrium and therefore can be identi-fied with the equilibrium rate constant. If the rate pictureholds, the equilibrium rate should agree with twice the reac-tive flux rate. For reduced barrier heightsDV/kBT smallerthan 5 the deviation exceeds 1% and grows rapidly for lowerbarriers. For too low barriers the time scales start to mergeand consequently the reactive flux rate no longer displays apronounced plateau as a function of time because the popu-lation of reactants no longer can be considered as constant. Ifthe correct time dependence of the reactant population isincluded the reactive flux rate converges to a plateauvalue.103

F. Numerically exact quantum methods

The short paper by McLafferty and Pechukas70 also pro-vided the formulation of a precise quantum mechanical ex-pression for the reactive flux as

Flux = Trf jsxdP+e−bHg s2.1d

with jsxd being the standard quantum mechanical current op-erator. An important construct here is the projection operatorP+ which projects onto the scattering wave functions withthe appropriate boundary conditions. At the same time Millerwas interested in a semiclassical version of reaction ratetheory and transition state theory. He too wrote down a quan-tum expression for the reactive flux which was identical tothat of Pechukas and McLafferty, and then proceeded to es-timate it semiclassically.104 It took almost another decadeuntil in 1983 Miller, Schwartz, and Tromp105 formulated theexact quantum reaction rate in terms of flux correlation func-tions, which turned out to be generalizations of Yamamoto’sexpressions for reaction rates106 which were based on linearresponse theory. McLafferty and Pechukas70 and Miller andcoworkers105 used the known scattering theory expressionsfor the rate and rewrote them as a reactive flux divided by thedensity of reactants. It cannot be overstressed that the scat-tering theory based expression is correct. As discussed be-low, it has also been adapted to reactions in condensedphases where the boundary conditions differ from scatteringboundary conditions. The formal proof of the validity of theflux–flux correlation function formalism for the quantumsand even classical mechanicsd rate of reaction in condensedphases remains open to date, see also below.

Topaler and Makri,107 in their ground breaking quantumnumerical computations, used the reactive flux methodologyto compute the numerically exact quantum rate in a symmet-ric double well potential coupled bilinearly to a harmonic

bath using what they called the quasiadiabatic path integralmethod. Their computations then served as benchmarks formany other approximate quantum theories. At the end of theday, we do not know whether the formal rate expression usedby Topaler and Makri is exact. The reduced barrier heightused in their computations wasDV/kBT=5, where we al-ready know classically that the reactive flux over populationmethod starts to deviate from the lowest nonzero eigenvalueof the Kramers equation. Quantum rate theory in condensedphases remains an open problem.

Enormous progress has been made toward solution ofthe scattering dynamics of molecule-molecule collisions. Re-viewing the molecular scattering theory literature is beyondthe scope of the present review, suffice it to note that thepresent day state of the art is a full quantum mechanicalsolution for systems with up to seven degrees of freedom.108

The limitation is that even the most sophisticated methodrelies on using grids in configuration space and their dimen-sionality grows exponentially with the number of degrees offreedom. It is therefore virtually impossible even withpresent day computers to go with these methods beyond atmost ten dimensions.

The evident strategy to overcome the dimensionalityproblem is to resort to path integral methods. These are pro-hibitive for real time because of the sign problem, the inte-grand in the real time path integral is too oscillatory. How-ever when considering imaginary time path integrals, theexponent is real and negative and so there is no serious con-vergence problem. Imaginary time path integrals have beencomputed for systems with more than 100 dimensions.109

The natural extension of rate theory was then to attempt todevise a quantum mechanical framework which would relyonly on imaginary time data. One natural choice is to use theinverse Laplace transform to get the real time results.110–112

The inverse Laplace transform is though ill behaved, thenoise in the Monte Carlo estimates is typically too large toallow more than short time inversion and this is just notsufficient, even when using sophisticated numerical methodssuch as maximum entropy inversion.112

G. Rates in nonequilibrium systems

Thermal equilibrium is of eminent importance for theunderstanding of many processes in physics and chemistrybut, in many other cases, as for example in living matter,fluxes of energy and matter prevent a system from approach-ing a thermal equilibrium state. Time and space dependentstructures may then persist in the asymptotic long-time be-havior of such a nonequilibrium system. In general, the pres-ence of fluxes in nonequilibrium systems breaks the micro-scopic reversibility and consequently these systems do notobey the principle of detailed balance.93,113Yet different lo-cally stable states may coexist. Ubiquitous thermal or possi-bly also external noise will generally induce transitions be-tween these states. The determination of the respectivetransition rates though is hampered by the fact that theasymptotic distribution of the system is not known in mostcases due to the lack of detailed balance. Often, numericalsimulations of the stochastic dynamics114,115 of the consid-ered system are the only possible approach.



Before discussing some of the available approximatemethods we note that a general distinction can be made be-tween systems that are driven out of equilibrium by station-ary fluxes of matter and energy and others on which time-dependent external forces act preventing the system fromreaching equilibrium. In the first case the system’s dynamicsis still autonomous and, in presence of noise, a stationaryprobability distribution will be approached asymptotically atlarge times. In the second case, asymptotically, the probabil-ity distribution will still depend on time. Accordingly, differ-ent methods are needed to determine transition rates in thesecases.

For autonomous nonequilibrium systems, in principle,both the flux over population method and the mean first pas-sage time approach can be used to determine the rate. Foreach method, the asymptotic probability distribution is re-quired, but most often is not known. In particular cases, per-turbational treatments about known stationary distributionsare feasible.116–119 For weak noise, a singular perturbationtheory30,120,121which is analogous to WKB theory in quan-tum mechanics, can be employed to first find the stationarydistribution and then the transition rates. Within this ap-proach, the stationary probability distribution contains an ex-ponentially leading part exph−Fsxd /ej wheree denotes thenoise strength,Fsxd is a generalized potential, andx a pointin the respective state space. The generalized potential is acontinuous function ofx but may show discontinuities of itsderivatives. These singularities must be compensated by apreexponential factor to assure that a smooth probability dis-tribution results. Decay rates of metastable states were deter-mined for special two-dimensional Fokker–Planck cases forwhich the generalized potential is sufficiently smooth122 andalso for simple one-dimensional noisy maps.123,124In the lat-ter case, as a consequence of the singularities of the gener-alized potential, the prefactor of the rate depends on thenoise strength in a nonanalytical way.

In the presence of time dependent driving, noise inducedtransitions give rise to important effects which are strictlyabsent in equilibrium systems. These effects comprise sto-chastic resonance,125,126 and resonance activation127,128 aswell as the rectification of noise in ratchets or Brownianmotors.129,130In general, the time-dependent forcing leads totime-dependent rates, considerably complicating any analyti-cal description. For a periodically driven bistable systemtime-dependent rates as well as the time averaged rates weredetermined in the limit of weak noise by means of the pathintegral representation of the transition probability from onelocally stable state to the moving separatrix.131 For the sametype of systems a different limit results if the noise strengthis small but fixed and the external driving is much slowerthan the intrawell relaxation. The time dependent rate is thengiven by the adiabatic expression for the frozen force. Forthis approximation to hold, no further restriction is to beimposed on the period of the driving force which may still belarger or smaller than the resulting transition time. This kindof kinetic description has been applied to describe a largevariety of phenomena132–134but only recently a strict deriva-tion from a Fokker–Planck model was provided.135

III. FUTURE: OPEN PROBLEMS

A. Classical rate theory at equilibrium

Pechukas52 gave a clear identification for the activatedcomplex in terms of a hypersurface of classical bound statesembedded in the continuum. For closed systems with twodegrees of freedom, the fixed energy surface is well charac-terized as a periodic orbit dividing surface. However for sys-tems with more than two degrees of freedom, the nature ofthis surface remains elusive. It can no longer be consideredas a dividing surface in configuration space, one must con-sider it as a dividing surface in phase space.136 The study ofvariational dividing surfaces in systems with more than twodegrees of freedom remains a topic of active research even todate137 as the classical dynamics of such nonlinearly coupledsystems is rather complicated.

We also noted that for canonical systems, the activatedcomplex is identifiable as a pods with infinite period movingon an effective temperature dependent potential energysurface.55 Although Miller gave a general formal solution tothe problem104 the actual solution for systems with more thantwo degrees of freedom remains an open problem.

B. Classical rate theory away from equilibrium

As detailed in this paper, equilibrium rate theory is byand large well understood. However, as also outlined above,classical rate theory for systems outside of equilibrium ispoorly understood. There is not any clear characterization ofthe structures that determine the flow such as the pods in theequilibrium case. The variational minimum principle ofWigner no longer exists and except for numerical simula-tions, we do not have any good theory of reaction rates awayfrom equilibrium except in a few particular limiting cases.The key information that in most cases is missing is theasymptotic probability distribution in the state space of theconsidered system. Once this distribution is known one fur-ther has to identify the relevant transition regions corre-sponding to the saddle points in the free energy landscape inequilibrium problems, and has to determine the local dynam-ics in these regions. In principle, this then would allow oneto estimate the rate by means of the flux over populationexpression, or by a conveniently defined mean first passagetime. A further complication may arise from the fact that thetopology of the energy landscape may be very complicatedas, for example, for glass-forming liquids,138 or for the mo-tion and folding of proteins.139 In such cases the resultingrelaxational dynamics is no longer governed by an exponen-tial decay law, but by some slower, possibly stretched expo-nential or even algebraic law.

Another question that only recently has been posed isrelated to the influence of a possibly non-Gaussian, algebra-ically correlated, random force on the escape dynamics froma metastable state. Apart from numerical simulations, frac-tional Fokker–Planck equations might prove to be a conve-nient starting point for such investigations.140



C. Quantum equilibrium rate theory

At this point, there are a few benchmark computations ofnumerically exact quantum rates for dissipative systems. Oneof the promising routes is the use of stochastic Schrödingerequations.141–143However, even the best of available resultsruns into problems when the barrier crossing dynamics is notrapid on the molecular time scales. It is very difficult toobtain long real time quantum mechanical data. Various au-thors have been suggesting methods for overcoming thisproblem. We note especially the recent multiconfigurationtime-dependent Hartree method144 which claims to give nu-merically exact results for dissipative systems with up to1000 bath modes.

A different approach is based on the semiclassical initialvalue series representation of the exact propagator.145 Insteadof using the path integral, one uses a semiclassical propaga-tor, which is a leading order term in a series expansion. Sinceone knows all the terms in the expansion, one can now com-pute them one by one to obtain the exact propagator. Expe-rience with some model systems shows that the series con-verges rapidly146 so that it can be used to generatenumerically exact real time quantum data using a classicaltrajectory based Monte Carlo algorithm.

When considering dissipative systems, it is very appeal-ing to attempt and derive reduced equations of motion for thesystem. A recent review of these may be found in Ref. 147.In the weak damping regime, these lead to Redfieldtheory148–150 and there have been attempts at solution forquantum reaction rates using these equations. In the strongfriction limit, Ankerhold151 has derived low temperature ex-tensions of the Caldeira–Leggett reduced equation ofmotion.19 An application of these reduced equations of mo-tion to quantum barrier crossing problems has recently beencarried out.152 The resulting rate coincides with the largefriction limit of Wolynes’ rate expression.153

D. Quantum rate theory away from equilibrium

This is perhaps the most open problem remaining today.Again, as for classical systems, one may distinguish betweensystems that are driven out of equilibrium by external timedependent fields and systems with different reservoirs main-taining currents of energy and matter in the system. Verylittle is known about this latter case in general, see also thearticle by Hänggi and Ingold61 in this Focus Issue. The bruteforce numerical approach, which allows one to get insightinto the classical problem is not available. In the presence oftime-dependent forcing, the Zwanzig–Caldeira–Leggett har-monic oscillator bath model19 and the related spin-boson

model154 can be extended to include the forcing.155 Withinthis framework particular aspects of quantum stochastic

resonance126,156and quantum ratchets129,157have been inves-tigated. But a comprehensive theoretical understanding isstill lacking. A serious general problem in the field of openquantum systems is that seemingly harmless and even plau-sible approximations may entail inconsistencies with generalstatistical mechanical laws158 and may introduce Maxwelldemons that survive even in thermal equilibrium.159

E. Concluding remarks

This paper attempted to provide some insight into thedevelopment of rate theory. It is though limited, due to thestrict length limitations set by the editors of this Focus Issueand the probably subjective point of view of the authors. Wehave brought almost no formulas, instead pointed out to theinterested reader what we believe are the “important” refer-ences. The paper is not comprehensive; for example, wehave not discussed the rates of electron transfer reactions,where much remains for future work, especially when con-sidering molecular electronics or electron transfer in biologi-cal systems. We have not provided a detailed history of thedevelopment of quantum scattering theory. Here too,progress has been made in inventing algorithms which allowextension of the computational horizon to increasingly largersystems, though as already noted, the largest to date is withseven degrees of freedom. The theory of surface diffusion,surface reactions, and catalysis which is closely related torate theory has also been left out. We also have not consid-ered the coherent control of rate processes,160 a topic of in-tensive activity and interest, whose details are becomingclearer as people devise better quantum methods for dealingwith multidimensional systems.

Another “hot topic” having to do with classical-quantumcorrespondence and the influence of classical chaos on quan-tum dynamics has also been set aside.161 Interesting phenom-ena such as quantum tunneling in multidimensional systemsand its relationship to the underlying real time classical dy-namics, or the effects of external fields on quantum tunnelingrates have also not been included in this review.

Our central purpose was to point out some of the impor-tant milestones in the development of rate theory and to en-courage a new generation to continue the study of ratetheory. Due justice to the many people who have contributedsignificantly to the theory and from whose knowledge wehave all gained would probably be only possible if onewould write up a comprehensive book on rate theory.

ACKNOWLEDGMENTS

This work has been supported by grants of the IsraelScience Foundation, the US–Israel Binational Science Foun-dation, and the Deutsche Forschungsgemeinschaft.

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Reaction rate theory: What it was, where is it today, and where is it going?