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## Reaction rate theory: what it was, where is it today, and where is it going

Venue: | CHAOS |

Citations: | 15 - 0 self |

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@ARTICLE{Pollak_reactionrate,

author = {Eli Pollak and Peter Talkner},

title = {Reaction rate theory: what it was, where is it today, and where is it going},

journal = {CHAOS},

year = {}

}

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### Abstract

A brief history is presented, outlining the development of rate theory during the past century. Starting from Arrhenius ͓Z. Phys. Chem. 4, 226 ͑1889͔͒, we follow especially the formulation of transition state theory by Wigner ͓Z. Phys. Chem. Abt. B 19, 203 ͑1932͔͒ and Eyring ͓J. Chem. Phys. 3, 107 ͑1935͔͒. Transition state theory ͑TST͒ made it possible to obtain quick estimates for reaction rates for a broad variety of processes even during the days when sophisticated computers were not available. Arrhenius' suggestion that a transition state exists which is intermediate between reactants and products was central to the development of rate theory. Although Wigner gave an abstract definition of the transition state as a surface of minimal unidirectional flux, it took almost half of a century until the transition state was precisely defined by Pechukas ͓Dynamics of Molecular Collisions B, edited by W. H. Miller ͑Plenum, New York, 1976͔͒, but even this only in the realm of classical mechanics. Eyring, considered by many to be the father of TST, never resolved the question as to the definition of the activation energy for which Arrhenius became famous. In 1978, Chandler ͓J. Chem. Phys. 68, 2959 ͑1978͔͒ finally showed that especially when considering condensed phases, the activation energy is a free energy, it is the barrier height in the potential of mean force felt by the reacting system. Parallel to the development of rate theory in the chemistry community, Kramers published in 1940 ͓Physica ͑Amsterdam͒ 7, 284 ͑1940͔͒ a seminal paper on the relation between Einstein's theory of Brownian motion ͓Einstein, Ann. Phys. 17, 549 ͑1905͔͒ and rate theory. Kramers' paper provided a solution for the effect of friction on reaction rates but left us also with some challenges. He could not derive a uniform expression for the rate, valid for all values of the friction coefficient, known as the Kramers turnover problem. He also did not establish 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 thermodynamic TST. Both of these questions were resolved in the 1980s when Pollak ͓J. Chem. Phys. 85, 865 ͑1986͔͒ showed that Kramers' expression in the moderate to strong friction regime could be derived from TST, provided that the bath, which is the source of the friction, is handled at the same level as the system which is observed. This then led to the Mel'nikov-Pollak-Grabert-Hänggi ͓Mel'nikov and Meshkov, J. Chem. Phys. 85, 1018 ͑1986͒; Pollak, Grabert, and Hänggi, ibid. 91, 4073 ͑1989͔͒ solution of the turnover problem posed by Kramers. Although classical rate theory reached a high level of maturity, its quantum analog leaves the theorist with serious challenges to this very day. As noted by Wigner ͓Trans. Faraday Soc. 34, 29 ͑1938͔͒, TST is an inherently classical theory. A definite quantum TST has not been formulated to date although some very useful approximate quantum rate theories have been invented. The successes and challenges facing quantum rate theory are outlined. An open problem which is being investigated intensively is rate theory away from equilibrium. TST is no longer valid and cannot even serve as a conceptual guide for understanding the critical factors which determine rates away from equilibrium. The nonequilibrium quantum theory is even less well developed than the classical, and suffers from the fact that even 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. ͓DOI: 10.1063/1.1858782͔ Rate theory provides the relevant information on the long-time behavior of systems with different metastable states and therefore is important for the understanding of many different physical, chemical, biological, and technical processes. Einstein indirectly contributed to this theory by clarifying the nature of Brownian motion as being caused by the thermal agitation of surrounding molecules on an immersed small particle. Kramers later pointed out that the thermally activated escape from a metastable state is nothing else but the Brownian motion of a fictitious particle along a reaction coordinate leading from an initial to a final locally stable state. In order to overcome the energetic barrier separating the two states, the particle has to "borrow" energy from its surroundings, an extremely rare event if as is usually the case the activation energy is much larger than the thermal energy. So, many attempts will take place until the particle has overcome the barrier separating the two states. During these many unsuccessful events, the particle completely loses any memory on how it had come to its initial state. Due to this loss of memory, the waiting time in the initial well will be random with an exponential distribution whose average coincides with the inverse of the decay rate. In this article the historical development of rate theory is outlined, the concepts are discussed, and present day generalizations to quantum nonequilibrium systems are presented. Our review ends with a brief summary of some challenges facing rate theory which remain open even a century after Einstein's seminal paper. I. RATE THEORY IN THE FIRST HALF OF THE 20TH CENTURY: ARRHENIUS, WIGNER, EYRING AND KRAMERS A. Arrhenius and activated molecules The father of reaction rate theory is Arrhenius who in his famous 1889 paper 1 investigated the temperature dependence of the rates of inversion of sugar in the presence of acids. As noted by Hänggi et al., 2 Arrhenius himself cites van't Hoff 3 as the person who suggested an e −A/T temperature dependence of reaction rates. However, Arrhenius is the father of rate theory since he postulated that this relationship indicates the existence of an "activated sugar" whose concentration is proportional to the total concentration of sugar, but is exponentially temperature dependent. But perhaps there is another reason why Arrhenius is so highly respected by the physical chemistry community. In 1911 he traveled to the United States and gave there a series of lectures, summarized in his book Theories 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 extensive science. Physics on the other hand works with rather few substances, such as mercury, water, alcohol, glass, air but analyzes the experimental results very thoroughly; it is an intensive science. Physical chemistry is the child of these two sciences; it has inherited the extensive character from chemistry…it has its profound quantitative character from the science of physics." He ends his Introduction by noting that, "The theoretical side of physical chemistry is and will probably remain the dominant one." It is this theoretical side which lies at the heart of this review. We will try briefly to follow the history of the development of rate theory in chemistry and physics, its impact on present day science, and its prospect for the 21st century-is there still anything that can be added to it? B. Wigner and Eyring: The transition state method Arrhenius' idea of an activated intermediate was amplified upon by a number of authors during the next 35 years. Christiansen and Kramers 5 were able to provide a rationale for the Arrhenius form based on the kinetic theory of gases. They realized already in 1923 that the activation energy could be understood by assuming that a minimum amount of energy is needed before reaction could occur. The probability for attaining this energy is given by the canonical distribution and thus one obtains the Arrhenius factor. They then provided a heuristic estimate for the magnitude of the prefactor. In 1935, Eyring published a paper titled "The Activated Complex in Chemical Reactions." 6 By this time, it was well established that reaction rates ͑k͒ should be written in the form where is a prefactor with the dimensions of 1 / s for unimolecular reactions and 1 / ͑s ·cm 3 ͒ 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 down a formula for the rate which allowed one to estimate the prefactor in the rate expression. While most previous works dealt with the relative rates of reactions, in which the prefactor would be eliminated, Eyring gave a heuristic derivation of an expression for the prefactor based on the assumption of an equilibrium between the activated complex and reactants. To obtain the time constant, he postulated, that at the saddle point, any quantum state perpendicular to the reaction coordinate reacts with the same universal time constant k B T /2ប. The rate is then given by the product of this universal time constant with the ratio of the partition function of the activated complex ͑which has one degree of freedom less than the reactants͒ to the partition function of the reactants. Eyring's formulation, in terms of partition functions allowed a heuristic quantum mechanical formulation for the rate constant and indeed in his 1935 paper he used quantum mechanical partition functions. 6 Noteworthy also is the work of Farkas 7 and Szilard ͑who was credited by Farkas, but a specific citation was not given͒ who realized and implemented the flux over population definition of the rate constant. This was then picked up by Pelzer and Wigner in their 1932 paper 8 in which they estimated the rate of conversion of parahydrogen into normal hydrogen. In this very early paper one may find all the elements of much more sophisticated work which abounded in the second half of the 20th century. They adapt the Eyring-Polanyi representation 9 of the ground Born-Oppenheimer potential energy surface for the motion of the nuclei, even showing a fictitious trajectory leading from reactants to products. They estimate the effect of electronically nonadiabatic interactions and show that they are negligible. To compute the reaction rate, they use a thermal equilibrium distribution in the vicinity of the saddle point of the potential energy surface and estimate the unidirectional classical flux in the direction from reactants to products. Already here, they note that they ignore the possibility of recrossings of the saddle point, pointing out that their probability at room temperature would be rather small. To get the rate they use the flux over population method after harmonically expanding the potential energy surface about the saddle point. The Pelzer and Wigner paper 8 is the very first use of transition state theory to estimate reaction rates. It is however written in a rather specific form, as applied to the hydrogen ex- change reaction. Eyring's later paper of 1935 6 provides general formulas which were then applied to many different activated reactions. In 1932 Wigner 10 made an additional seminal contribution to rate theory by providing an estimate for the tunneling contribution to the thermal flux of particles crossing a barrier. Using the same parabolic barrier expansion as in the work with Pelzer and his recently formulated quantum distribution function in phase space, he derives a series in ប 2 for the thermal tunneling corrections. His derivation is heuristic, he uses quantum mechanics for the thermal density of particles but treats the motion across the barrier as classical. The competition between Wigner and Eyring seems to have reached a head at a Faraday discussion, the papers of which were published in 1938. Wigner 11 and Eyring 12 had back-to-back papers; Wigner's was titled "The Transition State Method," Eyring used "The Theory of Aboslute Reaction Rates." Wigner notes here that the transition state method is inherently a classical theory, since the uncertainty principle forbids the simultaneous determination of a dividing surface and the sign of the momentum across the surface. Eyring ignores this, and uses his heuristic thermodynamic formulation within a quantum mechanical context, applying the theory to the reaction of NO with O 2 . One of the most interesting aspects of this Faraday meeting was the emerging ambiguity with respect to the definition of the "activated complex." In Wigner's approach it is defined through the dividing surface. However, when using equilibrium thermodynamics as a basis for the theory, ambiguities arise. Evans 13 in his paper titled "Thermodynamical Treatment of the Transition State" defines it as "The least probable configuration along the reaction path" but does not really define this probability. Guggenheim and Weiss in their paper "The Application of Equilibrium Theory to Reaction Kinetics" 14 are very forthcoming: "We are not always quite sure whether the expression the activated complex refers to AЈ ͑an energetic molecule͒ or A * ͑a reacting molecule͒ or to something intermediate between the two." This ambiguity continues for three more decades. In a symposium held in Sheffield, in April 1962 a lively discussion takes place between G. Porter and Eyring. 15 Porter asks: "May we begin by making sure that we know what we are talking about? In the papers presented at this meeting…potential-energy and free energy maxima are used rather indiscriminately to define the transition state…. Would Professor Eyring give us a rigorous definition to the transition state?" Eyring's answer is "the concept of an activated complex…provides us with the same theoretical tools for discussing reaction kinetics that have been so successfully used in discussing the equilibrium constants. The current literature is an eloquent testimonial to the fecundity of the concept." It is noteworthy, that this kind of discussion did not arise through Wigner's work. Wigner's approach was based on classical mechanics. Early on, Wigner realized that classical transition state theory provides an upper bound for the classical canonical net flux going from reactants to products. In his 1937 paper 16 he used this property to derive an upper bound to the rate of association reactions, using a dividing surface in energy rather than in configuration space. The upper bound leads to the variational property, which says that the best dividing surface is that which minimizes the unidirectional flux from reactants to products. With this definition, there is no ambiguity. The classical formulas for estimating this unidirectional flux are identical to the classical limit of the formulas used by Eyring and the chemistry community, this is not an accident, we remember that Pelzer and Wigner 8 had already given the foundations for the transition state theory ͑TST͒ method in their 1932 paper and that Eyring made sure that his formulation would correctly reduce to theirs. Wigner's interest in the TST method waned after 1938. In a personal meeting with him in the early 1980s, he showed no further interest in the issue. C. Kramers-Brownian motion in a field of force Kramers, whose earlier paper on rate theory was with Christiansen 5 as mentioned above, published a seminal paper of his own in 1940 titled "Brownian Motion in a Field of Force and the Diffusion Model of Chemical Reactions," 17 which may be thought of as the direct descendent of Einstein's famous paper of 1905. Although he does not provide a citation to Einstein's work 18 he does note, "A theory of Brownian motion on the Einstein pattern can be set up…" under appropriate conditions. Using the Langevin equation as a model in which a particle is moving under the influence of a field of force and a frictional force characterized by a damping coefficient ␥ he derives a Fokker-Planck equation in phase space ͑known also as the Kramers equation͒ for the motion. Kramers then proceeds to use the flux over population method to derive the rate of passage over a barrier in three limits-weak damping, intermediate, and strong damping. He derives the famous prefactor for the rate which is valid for the intermediate to strong damping regimes and independently a different prefactor, obtained by deriving a diffusion equation in energy in the underdamped limit, where the reaction rate is limited by the rate of flow of energy from the surrounding to the particle. Kramers notes that the intermediate friction range is the one in which one obtains the transition state limit for the rate. His paper also presented a challenge-deriving a uniform expression for the rate valid for all values of the friction, known as the turnover problem. Kramers realized that in the intermediate friction range his expression for the rate is identical to the ones derived by Pelzer and Wigner 8 and Eyring 6 and cites them accordingly. He considers his derivation as support for the transition state method, noting that "the transition state method gives results which are correct, say within ten percent in a rather wide range of ͑friction coefficient͒ values." However, Kramers does not state nor does he derive any formal equivalence between his results and transition state theory. Moreover, he refers to the ambiguity of choice of transition state, noting that for small friction it should be a state of definite energy while for larger friction it is characterized by the spatial coordinate. His work which was used by the physics community for the next four decades or so, evolved independently of the TST approach to rate theory used by the chemistry commu- nity. Kramers' equation was put to use for solving a variety of problems, and was even generalized to include quantum effects such as tunneling by Caldeira and Leggett in 1983. 19 Kramers himself considered his model as having no room for quantum mechanical effects such as tunneling. Kramers' rate theory was generalized by a number of groups mainly in the 1970s and 1980s to systems with more than one degree of freedom In passing we note that any inverse mean first passage time which results from a Fokker-Planck equation can formally be represented as a flux over population expression. 2,33 Recently the same result was obtained for mean first passage times of any time-homogeneous process. Keck had an additional contribution. The chemistry community became interested not only in thermal reaction rates but also in energy dependent microcanonical rates. For this purpose, he formulated a statistical theory of reaction rates 46 which could be considered as an adaptation to molecular dynamics of the statistical theories of nuclear reaction rates 47 developed earlier in the physics community. Rate theory and especially TST became an object of quantitative studies with the introduction of computers. In the early 1960's people started using them to solve numerically the classical motion of atoms and molecules evolving on a single Born-Oppenheimer potential energy surface. They were able to compute numerically exact reaction rates and compare them with theory. One of the interesting results was that microcanonical TST gave energy dependent reaction probabilities that were greater than unity. This difficulty was also resolved through use of VTST. The variational dividing surface which truly minimizes the reactive flux will never lead to reaction probabilites that are greater than unity. Only when one uses a "bad" dividing surface for which there are many recrossings does one encounter the problem. When accounting for recrossings, it was shown 50 that the unidirectional flux across any dividing surface is identical to the average number of recrossings of the dividing surface. This average number can of course be greater than unity. When it is less than unity, it gives a nontrivial upper bound on the reaction rate. B. What is the activated complex? In pioneering work, De Vogelaere and Boudart realized that periodic orbits play a special role in collinear atom diatom reactions. 55 For reactive systems with two degrees of freedom, one typically finds that for a small energy range above the saddle point energy, there exists only one pods between reactants and products. In this energy region, there are no recrossings and TST is exact. 56 When more than one pods exists, the flux through the pods leads to both upper and lower bounds to the microcanonical reaction probability. The identity of the activation energy in the Arrhenius expression was resolved in 1978 by Chandler. C. Unification of the TS method with Kramers' Brownian motion theory and solution of the Kramers turnover problem Originally, Kramers' prefactor for the rate was considered as a dynamic friction induced correction to TST. Only in 1986 was it demonstrated that Kramers' result for the moderate-to-strong damping regime may be derived from classical variational transition state theory. In the underdamped energy diffusion limited regime, a variational TST approach shows that the dividing surface is now in energy space, 65 similar to the dividing surface used by Wigner in his treatment of three-body dissociation. 16 However, it is not possible to construct a simple surface which would totally eliminate the recrossings of this energy dividing surface. The observation that in dissipative systems one should use collective mode reaction coordinates then led to a flurry of activity, culminating by a two step solution to the old Kramers' turnover problem. Mel'nikov and Meshkov showed how one could derive a uniform expression for the rate which covered the underdamped-to-moderate damping limit. 66 Pollak, Grabert, and Hänggi then used Mel'nikov and Meshkov's formalism but adapted it to the collective mode to derive an expression which is valid for all values of the damping as well as memory friction. 67 D. Quantum TST? Another question which remains open even today is that of a quantum mechanical analog to TST. One can follow Eyring's approach 6 and replace classical partition functions with quantum partition functions, make a separable approximation about the saddle point, and use it to treat the unstable mode quantum mechanically thus introducing tunneling corrections. This "engineering" approach was used rather successfully by Truhlar et al., A serious attempt to formulate a quantum mechanical TST comes in 1974 with the paper of McLafferty and Pechukas. 78,79 The great advantage of the centroid approach is that it can be computed for many dimensional systems, however, the lack of a derivation from first principles leaves one with an uneasy feeling. At this point, the search for a rigorous expansion, in which the centroid based formulas would be the first term in a series is open. Just as the Wigner ansatz for the tunneling correction was a wonderful guess, so is the centroid approach. But one should keep in mind that even to date, it is more of an ansatz than a theory. The centroid approach served though as an impetus for other approximate thermodynamic rate expressions. Pollak and Liao 80 followed an earlier idea of Voth et al. A different thermodynamic approach was suggested by Hansen and Andersen 84 who noted that the first few initial time derivatives of the flux flux correlation function involve only thermodynamic averages. They used various extrapolation formulas to bridge the gap between the initial time and final time and thus derive a thermodynamic rate theory. Here too though, the extrapolation is fraught with danger, and it is difficult to obtain systematic corrections and convergence toward an exact result. 85 In the early 1970s Miller 86 showed that semiclassically the tunneling rate is determined by a periodic trajectory moving on the upside down potential energy surface, with period ប / k B T. The physics community, rediscovered this same object E. Reactive flux method One of the central difficulties in computing reaction rates for large systems is the rare sampling of events that lead to reaction. Especially if the barrier height is large compared to k B T the probability of finding an initial condition which will lead to barrier crossing becomes small and the computational cost high. To overcome this problem, Chandler 58 used Onsager's regression hypothesis 93 to derive a reactive flux formula whose great advantage was that it provided a technique in which one could sample the phase space in the vicinity of the barrier to reaction instead of reactants, thus leading to an enormous saving in computational effort. 97 The reactive flux method also served as the basis for the application of variational TST to condensed phases, 103 F. Numerically exact quantum methods The short paper by McLafferty and Pechukas 70 also provided the formulation of a precise quantum mechanical expression for the reactive flux as with j͑x͒ being the standard quantum mechanical current operator. An important construct here is the projection operator P + which projects onto the scattering wave functions with the appropriate boundary conditions. At the same time Miller was interested in a semiclassical version of reaction rate theory and transition state theory. He too wrote down a quantum expression for the reactive flux which was identical to that of Pechukas and McLafferty, and then proceeded to estimate it semiclassically. 104 It took almost another decade until in 1983 Miller, Schwartz, and Tromp 105 formulated the exact quantum reaction rate in terms of flux correlation functions, which turned out to be generalizations of Yamamoto's expressions for reaction rates 106 which were based on linear response theory. McLafferty and Pechukas 70 and Miller and coworkers 105 used the known scattering theory expressions for the rate and rewrote them as a reactive flux divided by the density of reactants. It cannot be overstressed that the scattering theory based expression is correct. As discussed below, it has also been adapted to reactions in condensed phases where the boundary conditions differ from scattering boundary conditions. The formal proof of the validity of the flux-flux correlation function formalism for the quantum ͑and even classical mechanics͒ rate of reaction in condensed phases remains open to date, see also below. Topaler and Makri, 107 in their ground breaking quantum numerical computations, used the reactive flux methodology to compute the numerically exact quantum rate in a symmetric double well potential coupled bilinearly to a harmonic bath using what they called the quasiadiabatic path integral method. Their computations then served as benchmarks for many other approximate quantum theories. At the end of the day, we do not know whether the formal rate expression used by Topaler and Makri is exact. The reduced barrier height used in their computations was ⌬V / k B T = 5, where we already know classically that the reactive flux over population method starts to deviate from the lowest nonzero eigenvalue of the Kramers equation. Quantum rate theory in condensed phases remains an open problem. Enormous progress has been made toward solution of the scattering dynamics of molecule-molecule collisions. Reviewing the molecular scattering theory literature is beyond the scope of the present review, suffice it to note that the present day state of the art is a full quantum mechanical solution for systems with up to seven degrees of freedom. 108 The limitation is that even the most sophisticated method relies on using grids in configuration space and their dimensionality grows exponentially with the number of degrees of freedom. It is therefore virtually impossible even with present day computers to go with these methods beyond at most ten dimensions. The evident strategy to overcome the dimensionality problem is to resort to path integral methods. These are prohibitive for real time because of the sign problem, the integrand in the real time path integral is too oscillatory. However when considering imaginary time path integrals, the exponent is real and negative and so there is no serious convergence problem. Imaginary time path integrals have been computed for systems with more than 100 dimensions. 109 The natural extension of rate theory was then to attempt to devise a quantum mechanical framework which would rely only on imaginary time data. One natural choice is to use the inverse Laplace transform to get the real time results. 110-112 The inverse Laplace transform is though ill behaved, the noise in the Monte Carlo estimates is typically too large to allow more than short time inversion and this is just not sufficient, even when using sophisticated numerical methods such as maximum entropy inversion. 112 G. Rates in nonequilibrium systems Thermal equilibrium is of eminent importance for the understanding of many processes in physics and chemistry but, in many other cases, as for example in living matter, fluxes of energy and matter prevent a system from approaching a thermal equilibrium state. Time and space dependent structures may then persist in the asymptotic long-time behavior of such a nonequilibrium system. In general, the presence of fluxes in nonequilibrium systems breaks the microscopic reversibility and consequently these systems do not obey the principle of detailed balance. For autonomous nonequilibrium systems, in principle, both the flux over population method and the mean first passage time approach can be used to determine the rate. For each method, the asymptotic probability distribution is required, but most often is not known. In particular cases, perturbational treatments about known stationary distributions are feasible. In the presence of time dependent driving, noise induced transitions give rise to important effects which are strictly absent in equilibrium systems. These effects comprise stochastic resonance, 135 III. FUTURE: OPEN PROBLEMS A. Classical rate theory at equilibrium Pechukas 52 gave a clear identification for the activated complex in terms of a hypersurface of classical bound states embedded in the continuum. For closed systems with two degrees of freedom, the fixed energy surface is well characterized as a periodic orbit dividing surface. However for systems with more than two degrees of freedom, the nature of this surface remains elusive. It can no longer be considered as a dividing surface in configuration space, one must consider it as a dividing surface in phase space. We also noted that for canonical systems, the activated complex is identifiable as a pods with infinite period moving on an effective temperature dependent potential energy surface. 55 Although Miller gave a general formal solution to the problem 104 the actual solution for systems with more than two degrees of freedom remains an open problem. B. Classical rate theory away from equilibrium As detailed in this paper, equilibrium rate theory is by and large well understood. However, as also outlined above, classical rate theory for systems outside of equilibrium is poorly understood. There is not any clear characterization of the structures that determine the flow such as the pods in the equilibrium case. The variational minimum principle of Wigner no longer exists and except for numerical simulations, we do not have any good theory of reaction rates away from equilibrium except in a few particular limiting cases. The key information that in most cases is missing is the asymptotic probability distribution in the state space of the considered system. Once this distribution is known one further has to identify the relevant transition regions corresponding to the saddle points in the free energy landscape in equilibrium problems, and has to determine the local dynamics in these regions. In principle, this then would allow one to estimate the rate by means of the flux over population expression, or by a conveniently defined mean first passage time. A further complication may arise from the fact that the topology of the energy landscape may be very complicated as, for example, for glass-forming liquids, Another question that only recently has been posed is related to the influence of a possibly non-Gaussian, algebraically correlated, random force on the escape dynamics from a metastable state. Apart from numerical simulations, fractional Fokker-Planck equations might prove to be a convenient starting point for such investigations. 141-143 However, even the best of available results runs into problems when the barrier crossing dynamics is not rapid on the molecular time scales. It is very difficult to obtain long real time quantum mechanical data. Various authors have been suggesting methods for overcoming this problem. We note especially the recent multiconfiguration time-dependent Hartree method 144 which claims to give numerically exact results for dissipative systems with up to 1000 bath modes. A different approach is based on the semiclassical initial value series representation of the exact propagator. 145 Instead of using the path integral, one uses a semiclassical propagator, which is a leading order term in a series expansion. Since one knows all the terms in the expansion, one can now compute them one by one to obtain the exact propagator. Experience with some model systems shows that the series converges rapidly 146 so that it can be used to generate numerically exact real time quantum data using a classical trajectory based Monte Carlo algorithm. When considering dissipative systems, it is very appealing to attempt and derive reduced equations of motion for the system. A recent review of these may be found in Ref. 147. In the weak damping regime, these lead to Redfield theory 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 between systems that are driven out of equilibrium by external time dependent fields and systems with different reservoirs maintaining currents of energy and matter in the system. Very little is known about this latter case in general, see also the article by Hänggi and Ingold 61 in this Focus Issue. The brute force numerical approach, which allows one to get insight into the classical problem is not available. In the presence of time-dependent forcing, the Zwanzig-Caldeira-Leggett harmonic oscillator bath model E. Concluding remarks This paper attempted to provide some insight into the development of rate theory. It is though limited, due to the strict length limitations set by the editors of this Focus Issue and the probably subjective point of view of the authors. We have brought almost no formulas, instead pointed out to the interested reader what we believe are the "important" references. The paper is not comprehensive; for example, we have not discussed the rates of electron transfer reactions, where much remains for future work, especially when considering molecular electronics or electron transfer in biological systems. We have not provided a detailed history of the development of quantum scattering theory. Here too, progress has been made in inventing algorithms which allow extension of the computational horizon to increasingly larger systems, though as already noted, the largest to date is with seven degrees of freedom. The theory of surface diffusion, surface reactions, and catalysis which is closely related to rate theory has also been left out. We also have not considered the coherent control of rate processes, 160 a topic of intensive activity and interest, whose details are becoming clearer as people devise better quantum methods for dealing with multidimensional systems. Another "hot topic" having to do with classical-quantum correspondence and the influence of classical chaos on quantum dynamics has also been set aside. Our central purpose was to point out some of the important milestones in the development of rate theory and to encourage a new generation to continue the study of rate theory. Due justice to the many people who have contributed significantly to the theory and from whose knowledge we have all gained would probably be only possible if one would write up a comprehensive book on rate theory. ACKNOWLEDGMENTS