Transition state theories

Transition-state theory is one of the earliest attempts to explain chemical reaction rates from first principles. It was initially developed by Eyring [124] and Evans and Polayni [122,123], The conventional transition-state theory (CTST) discussed here provides a relatively straightforward method to estimate reaction rate constants, particularly the preexponential factor in an Arrhenius expression. This theory is sometimes also known as activated complex theory. More advanced versions of transition-state theory have also been developed over the years [401], 

Transition-state theory is based on the assumption of chemical equilibrium between the reactants and an activated complex, which will only be true in the limit of high pressure. At high pressure there are many collisions available to equilibrate the populations of reactants and the reactive intermediate species, namely, the activated complex. When this assumption is true, CTST uses rigorous statistical thermodynamic expressions derived in Chapter 8 to calculate the rate expression. This theory thus has the correct limiting high-pressure behavior. However, it cannot account for the complex pressure dependence of unimolecular and bimolecular (chemical activation) reactions discussed in Sections 10.4 and 10.5. 

For this simple derivation of CTST assume that in order for chemical species A and B to react to form products D and E, a barrier to reaction must be surmounted. The system 

The reaction pathway is shown schematically in Fig. 10.4. The assumptions implicit in transition-state theory are discussed next. 

The CTST reaction scheme could have been written with any number of reactant or product species. In addition the theory says nothing about the mechanism of excitation to form C or the de-excitation of the activated complex back to the reactant species. The theory simply assumes that an equilibrium exists between C and the reactants. More rigorous derivations also treat the equilibrium between C and the product molecules. 

Transition state theory assumes an equilibrium energy distribution among all possible quantum states at all points along the reaction coordinate. The probability of finding a molecule in a given quantum state is proportional to which is a Boltzmann 

From the TST expression (12.2) it is clear that if the free energy of the reactant and TS can be calculated, the reaction rate follows trivially. Similarly, the equilibrium constant for a reaction can be calculated from the free energy difference between the reactant(s) and product(s). 

Transition state theory describes the changes of geometrical configuration which occur when suitably activated molecules, having the required critical energy, react. It gives a detailed account of the absolute rate of reaction. The physical steps of energy transfer in activation and deactivation are outside the remit of the theory. 

TRANSITION STATE THEORY Conventional transition state theory 

One of the fundamental assumptions of TST is that there exists a divide in the PES that separates the reactant and product regions. This divide contains the transition state, which is defined as the maximum value on the minimum energy path of the PES that connects reactant(s) and product(s). Any trajectory passing through the divide from the reactant side is assumed to form products eventudly this is often referred to the as the non-recrossing rule. Consider the generalized elementary gas phase abstraction reaction 

Assuming equilibrium between reactants and the activated complexes, the rate for Equation 3 may also be represented by 

Solving for k/T) using Equations (4) and (5), and assuming for simplicity an ideal system, gives 

If all species are treated as ideal gases and the quasi-equilibrium constant is represented in terms of molecular partition functions qi, we obtain (McQuarrie 1973) 

In transition state theory a transitory geometry is formed by the reactant(s), (A) and (B, C), as they proceed to products. First the molecules A and BC react to form an intermediate called an activated complex, (ABC)+ 

This complex can either decompose back to give the initial reactants 

We can follow the pathway of this reaction by plotting the energy of interaction of molecules A and BC as a function of distance between the species A and B (rj) and B and Ci (r2) molecule 

As the molecules come together, the interaetion energy increases to some maximum value at the transition state. As the reaction proceeds past the transition state the energy of interaction between molecules A and BC decreases as speeies A and B move close together and species B and C move farther apart. Figwe G-3 shows this concept schematically. 

The rate of reaction is the rate at which the complex (ABC) crosses the energy barrier that is, the rate of formation by A is 

The transition state theory describes reactants combining to form unstable intermediates called activated complexes, which rapidly 

The overall rate of reaction depends on the rate of decomposition of AB to product P. 

The collision theory considers the rate to be governed by the number of energetic collisions between the reactants. The transition state theory considers the reaction rate to be governed by the rate of the decomposition of intermediate. The formation rate of the intermediate is assumed to be rapid because it is present in equilibrium concentrations. 

Atoms and free radicals are highly reactive intermediates in the reaction mechanism and therefore play active roles. They are highly reactive because of their incomplete electron shells and are often able to react with stable molecules at ordinary temperatures. They produce new atoms and radicals that result in other reactions. As a consequence of their high reactivity, atoms and free radicals are present in reaction systems only at very low concentrations. They are often involved in reactions known as chain reactions. The reaction mechanisms involving the conversion of reactants to products can be a sequence of elementary steps. The intermediate steps disappear and only stable product molecules remain once these sequences are completed. These types of reactions are referred to as open sequence reactions because an active center is not reproduced in any other step of the sequence. There are no closed reaction cycles where a product of one elementary reaction is fed back to react with another species. Reversible reactions of the type A+B C + Dare known as open sequence mechanisms. The chain reactions are classified as a closed sequence in which an active center is reproduced so that a cyclic reaction pattern is set up. In chain reaction mechanisms, one of the reaction intermediates is regenerated during one step of the reaction. This is then fed back to an earlier stage to react with other species so that a closed loop or 

A chain reaction consists of three main steps  

The transition state theory requires the conception of an activated complex named by H. Eyring (1935) or transition state by M.G. Evans and M. Polanyi (1935). According to the transition state theory, the rate of a reaction is defined as 

The factor ( Tt/iksTf is the partition function for the translation in the reaction path. So, combining Equations 5.3 and 5.8 in Equation 5.7 yields 

we have to define the rate constant in terms of the transition state thermodynamic properties  

At this point, it might be useful to remember the timescales of the pro- timescales cesses during a catalytic reaction. The electronic processes of the potential energy surface of the reaction have characteristic times of 10s, while the vibrational motions of the atoms are in the order of 10 s. The timescales of the bond formation and breaking of the catalytic processes are reported to be in the order of 10 to 10 s (van Santen and Neurock, 

Under these circumstances, it is fair to assume that all vibrational motions are equilibrated with the exception of those on the reaction coordinate (this condition is satisfied when Sk T (Kramers, 1940). 

The transition state theory, also called the theory of the activated complex, was known as the theory of absolute reaction rates in its early days. According to Henry Eyring (American chemist, 1901-1981) and Michael Polanyi (1891-1976) (see section 1.2), many reactions proceed via a pre-equilibrium mechanism. For instance, a bimolecular reaction 

Nowadays, there is a tendency to define the transition state more broadly. The definition of the transition state (TS) used recently by John C. Polanyi and Ahmed H. ZewaiP is the full family of configurations through which the reacting particles evolve en route from reactants to products. They believe that this definition of the TS (usual symbol, displayed at the crest of the energy barrier to reaction) is likely to prove most enduring. 

The assumption of a pre-equilibrium mechanism implies that the rate constant of the overall second order reaction (eq. 1.4.1a) should be much lower 

the task of the theory is to calculate the rate constant k of the unimolecular decay of (AB ) and the activation equilibrium constant K.  

One of the vibrational modes of the activated complex leads to the A B bond cleavage (in the above H2/HD example asymmetric stretching of H- H -D complex). The frequency of this vibration is closely related to the (AB)t decay frequency  

The concepts we introduce here form the basis of a theory that explains the rates of 

Although collision theory is a useful starting point for the discussion of reactions in the atmosphere, it has little relevance to the reactions that interest biologists the most those taking place in the aqueous environment of a cell. A more sophisticated theory, transition state theory (or activated complex theory), builds on collision theory but is applicable to a wider range of reaction environments and introduces a more sophisticated interpretation of the empirical Arrhenius parameters A and E.  

According to the transition-state theory, this elementary reaction is described by the following model  

Cj represents the transition state or activated complex, denoted by the subscript J. 

It is assumed that reaction (8) is at quasi-equilibrium and that reaction (9) has the following rate  

C is the concentration of the activated complexes, calculated using the quasiequilibrium hypothesis  

Equation (11) also assumes that the reaction is ideal, i.e. that the fugacity coefficients are equal to 1. The equilibrium constant K contains a partition function corresponding to the vibrational frequency vt. Transition-state theory assumes that the vibration has a very low frequency, so that the corresponding partition function can be factorized in the following form  

Frying absolute rate theory or transition state theory allows a convenient dissection of the Arrhenius expression, k = A exp(—Fact/ T), into thermodynamic quantities since the theory assumes a near equilibrium between the reactant and transition state. In transition state theory, the first-order rate constant (at the high pressure 

We shall mention the ideas of transition state theory, since the kinetic relationships derived in Section 3.1.5 will be interpreted according to this theory. 

FIGURE 3.3. Plot of potential energy along the reaction coordinate (transition state theory). 

By analogy with the thermodynamics of stable species, K is expressed in terms of a free energy of activation G, which is the difference between the free energy of the reactants and that of the activated complex, in standard states. Hence we can write  

We shall use rate processes, to derive quantitative reiationships for the rate of electrode processes, which are examples of heterogeneous reactions. 

Kinetic Isotope Effects Continued Variational Transition State Theory and Tunneling 

Abstract Some of the successes and several of the inadequacies of transition state theory (TST) as applied to kinetic isotope effects are briefly discussed. Corrections for quantum mechanical tunneling are introduced. The bulk of the chapter, however, deals with the more sophisticated approach known as variational transition state theory (VTST). 

1 Introduction Transition State Theory, Variational Transition State Theory, and Tunneling 

To begin we are reminded that the basic theory of kinetic isotope effects (see Chapter 4) is based on the transition state model of reaction kinetics developed in the 1930s by Polanyi, Eyring and others. In spite of its many successes, however, modern theoretical approaches have shown that simple TST is inadequate for the proper description of reaction kinetics and KIE s. In this chapter we describe a more sophisticated approach known as variational transition state theory (VTST). Before continuing it should be pointed out that it is customary in publications in this area to use an assortment of alphabetical symbols (e.g. TST and VTST) as a short hand tool of notation for various theoretical methodologies. 

In conventional transition state theory (TST) (see Chapter 4) the first approximation for the thermal rate constant k is given  

It is a fundamental supposition of TST that one can define a region of the potential surface, identified with a small length 3 of the reaction coordinate around the maximum of V(q), as corresponding to a transition structure or activated complex (Fig. 11). The reaction is then thought of as proceeding from reactants to products via this transition structure 

In order to derive a rate-coefficient expression, reactants and activated complexes are assumed to be in equilibrium 

Equation (3.48) describes the rate of forward reaction at chemical equilibrium between reactants and products. It is assumed in TST that Eq. (3.48) also describes the rate in the forward direction even if the system is not at equilibrium, i.e., when net forward reaction occurs. One condition for this is that the reactants are in internal thermodynamic equilibrium, so that their states are populated according to the Boltzmann distribution law. If this is true, Eq. (3.48) is unaffected by the extent to which the reverse reaction is occurring (Smith, 1980). 

We are thus able to calculate a rate coefficient if we have a means to evaluate the concentration of activated complexes ABC and their decomposition frequency. Assuming that [ABC is given by an equilibrium 

This rate-coefficient expression can be evaluated from molecular properties by using statistical mechanics formulas. To do this one writes the equilibrium constant K T) in terms of partition functions per unit volume Q 

Free energy of activation Energy required to raise the energy of the reactant to the energy of the transition state Transition state Highest-energy arrangement of atoms that occurs between the reactants and product 

Transition state theory tells us that when a molecule of substrate has enough energy to jump the barrier, its structure is intermediate between that of the substrate and that of the product. Some bonds are stretched, partially broken, partially formed, and so forth. The arrangement of atoms that has the highest energy between the substrate and product is called the transition state. Transition state theory assumes that the transition state doesn t exist for more than the time required for one bond vibration (about 10 s)—so the transition state really doesn t exist, but we can talk about it as if it did. The AG s of activation are always positive. The more positive, the slower. 

A FREE-ENERGY REACTION COORDINATE DIAGRAM shows the free energy of the substrate, product, and transition state of a chemical reaction. It tells you how favorable the overall reaction is (AGeq) and how fast (AG ). 

In this chapter we explain the algorithms used to implement VTST, especially CVT, and multidimensional tunneling approximations in the POLYRATE computer program. We also include some discussion of the fundamental theory underlying VTST and these algorithms. Readers who want a more complete treatment of theoretical aspects are referred to another review. 

VARIATIONAL TRANSITION STATE THEORY FOR GAS-PHASE REACTIONS 

Transition state theory (TST), also known as conventional TST, goes back to the papers of Eyring and Evans and Polanyi in 1935. Eor a general gas-phase reaction of the type 

In a world where nuclear motion is strictly classical, we need not consider (4), and the TST classical rate constant, for Eq. [1] is given by 

We can establish a connection between Eq. [2] and thermodynamics by starting with the relation between the free energy of reaction, AGj at temperature T, and the equilibrium constant K, which is given by 

The two important theories of chemical kinetics, the collision theory and the transition-state theory, both depend on essentially the same assumption. Since this is concerned with the existence of a kind of equilibrium in the system, even during the course of its reaction, it is appropriate to give a short outline of one of these theories in the present volume. 

We commence with a discussion of the potential energy surface. If n is the number of atomic nuclei which are involved in each elementary reaction, their positions relative to a frame of reference could be 

The surface can be visualized at all easily only in the simplest instances. As an example consider the reaction 

as Z comes near to XY the potential energy of the trio rises on account of the repulsive forces between the electronic envelopes. Y gradually stretches away from X until it is somewhere about midway between X and Z. The system X YZ is then on the top of the pass or col and is said to be in the transition state. Finally, the atom X breaks away, leaving the new molecule YZ and the co-ordinates of the system simultaneously move down into the YZ vaUey. Plotted in terms of distance along the reaction path, or reiadtion co-ordinate the potential energy would thus appear somewhat as shown in Fig. 47. 

In the general case, as mentioned above, the potential energy surface cannot be so easily visualized, but the transition state can be thought of as being the configuration at the top of the lowest pass between the reactants and the products. The term activated complex is also used as referring to a set of nuclei which has this configuration 

The first step in the above reaction is the formation of a transient complex, called the activated complex or transition state  

Transition state theory (also known as activated-complex theory) assumes that the transition state is much more likely to decay back to the original reactants than proceed to the stable products if this is the case, then first two reactions can be assumed to be in equilibrium. The reactive process can then be represented as 

The rate of reaction is that at which ABC passes to products (as a result of translational or vibrational motions along the reaction coordinate). 

We will not go through the full derivation of the transition state theory, for which there are many excellent references [e.g., Laidler (1987), Pilling and Seakins (1995)]. The result is that the rate coefficient is expressed as 

In many cases the preexponential factor can be considered to be independent of temperature, and the rate coefficient is written as 

The activated complex of the transition state is at the highest energy along the reaction coordinate, i.e., its energy is higher than that of the reactants by the activation energy. In the transition state theory, it is further assumed that the 

Since the activated complex is highly unstable, it may not survive the vibration of the complex. Hence, the rate coefficient k can be thought to be proportional to the fundamental frequency v i.e., we may write 

Based on thermodynamic considerations, the equilibrium constant can be written as 

In the context of the transition state theory, the activation energy for the reaction can be approximately identified as the enthalpy necessary to form the activated complex  

This understanding can be used to infer the pressure dependence of k because enthalpy depends on pressure. It can be shown that 

The rate of the reaction is given by the classical rate of crossing the dividing surface. The crossing rate is expressed as the product of the transition state concentration, [A ], the frequency of crossing, v, and a transmission coefficient, k 

In this equation the transition states are considered to be formed only when their total energy is equal to or greater than Eq. The transmission coefficient provides a correction factor for quantum effects. It may be necessary to correct the classical rate coefficient for quantum mechanical tunneling through the barrier for reactants with total energy less than To, or for non-adiabatic reactions. The transition state concentration, [A ], refers to only those transition states that are moving toward product formation, and does not include those that may come from the 

If the Born-Oppenheimer approximation is valid for both the reactants and the transition state, the equilibrium constant, is related to molecular partition functions, = eV n Qh as shown in almost any text on statistical thermodynamics. The rate coefficient can be expressed 

Here Q is the zero point energy-corrected molecular partition function per unit volume for each species, k-Q the Boltzmann s constant, T the absolute temperature, h the Planck constant, and Eo=V] + Y eq, where sq is the zero point energy of the transition state minus the sum of the zero point energies of the reactants. Details for the calculation of 

A collision theory of even gas phase reactions is not totally satisfactory, and the problems with the steric factor that we described earfier make this approach more empirical and qualitative than we would like. Transition state theory, developed largely by Henry Eyring, takes a somewhat different approach. We have already considered the potential energy surfaces that provide a graphical energy model for chemical reactions. Transition state theory (or activated complex theory) refers to the details of how reactions become products. For a reaction fike 

The essential feature of transition state theory is that there is a concentration of the species at the saddle point, the transition state or activated complex, that is in equihbrium with reactants and products. The Boltzmann Distribution Law governs the concentration of that transition state, and the rate of reaction is proportional to its concentration. Since the concentration 

The concentration of the transition state is not the only factor involved, since the frequency of its dissociation into products comes into play because the rate at which it decomposes must also be considered. Therefore, the rate can be expressed as 

Rate = (Transition state concentration) x (Decomposition frequency) 

In order for the transition state to separate into products, one bond (the one being broken) must acquire sufficient vibrational energy to separate. When it does separate, one of the 3N — 6 vibrational degrees of freedom (for a Hnear molecule it is 3N — 5) is lost and is transformed into translational degrees of freedom of the products. Central to the idea of transition state theory is the assumption that the transition state species is in equilibrium with the reactants. Thus, 

Building on the Lindemarm Theory described above, Henry Eyring, and independently also M.G. Evans and Michael Polanyi, developed around 1935 a theory for the rate of a reaction that is still used, namely the transition state theory. 

In qualitative terms, the reaction proceeds via an activated complex, the transition state, located at the top of the energy barrier between reactants and products. Reacting molecules are activated to the transition state by collisions with surrounding molecules. Crossing the barrier is only possible in the forward direction. The reaction event is described by a single parameter, called the reaction coordinate, which is usually a vibration. The reaction can thus be visualized as a journey over a potential energy surface (a mountain landscape) where the transition state lies at the saddle point (the col of a mountain pass). 

We express the equilibrium constant in terms of the partition functions of both the reactant and the transition state, and we take the partition function of the reaction coordinate separately  

Because the frequency of a weakly bonded vibrating system is relatively small, i.e. kBT hu we may approximate its partition function by the classical limit k T/hv, and arrive at the rate expression in transition state theory  

For many years the Tafel equation was viewed as an empirical equation. A theoretical interpretation was proposed only after Eyring, Polanyi and Horiuti developed the transition-state theory for chemical kinetics, in the early 1930s. Since the Tafel equation is one of the most important fundamental equations of electrode kinetics, we shall derive it first for a single-step process and then extend the treatment for multiple consecutive steps. Before we do that, however, we shall review very briefly the derivation of the equations of the transition-state theory of chemical kinetics. 

Consider the simple isotope-exchange reaction in which a hydrogen atom reacts with a deuterium molecule, D2, to form a molecule of HD and a free deuterium atom  

One could imagine this reaction occurring by D2 first splitting into two atoms, followed by one of the atoms combining with the hydrogen atom. This is, however, 

2) The December 2005 issue of the journal Corrosion was dedicated to the 100 anniversary of the publication of tiie Tafel Equation. It contains mostly review articles, which can be useful for a deeper understanding of electrode kinetics. 

We shall now write the reaction in more general form as 

In a dynamical simulation the calculation of the dynamical factor (eq. (2.19)) is the most demanding task. Therefore approximate treatments of the dynamical factor can greatly simplify the problem. Transition state theory (TST) is an extremly important approximation in this context. It assumes that the value of the dynamical factor is determined by the initial velocity (or momentum) of the trajectory  

Trajectories with initial momentum directed towards products are considered to be reactive trajectories with initial momentum directed towards reactants are assumed to be nonreactive. Thus, recrossings of the dividing surface are completely neglected. 

Inserting eq. (2.20) into eq. (2.18), one obtains the thermal rate constant in TST-approximation  

As discussed above, recrossing always results in a decrease of the (classically) calculated rate constant. Therefore the TST rate constant is a strict upper bound of the classical rate  

The information content of AV and AS is similar. Values of AV are usually more precise than those of AS, although they require specialized apparatus for their measurement. If ions are being formed in the activation step, AV may be -20 cm3 mol-1. This effect reflects the electrostriction of the solvent. If the transition state features bond breaking, as in an SnI reaction, AV 10 cm3 mol-1. Conversely, AV -10 cm3 mol-1 is characteristic of bond making. 

Quite a contrasting effect was found for the solvolysis of sterically hindered palla-dium(II) complexes of the ligand Et2NCH2CH2NHCH2CH2NEt2 (= Et4dien).16 The reactions, with X = halide or pseudohalide, are 

These reactions have negative values of both AV and AS, as presented in Table 7-3, consistent with an associative mechanism. That being so, the activation process is dominated by the formation of the Pd-H20 bond, not by dissociation of Pd-X. 

The most widely accepted treatment of reaction rates is transition state theory (TST), devised by Henry Eyring.17 It has also been known as absolute rate theory and activated complex theory, but these terms are now less widely used. 

A graphical representation of the potential energy surface or reaction coordinate. The transition state occurs at the saddle point. ( Adapted from Ref. 18.) 

Fortunately, the reaction rates of many important processes can be obtained without a full molecular dynamics simulation. Most reaction rate theories for elementary processes build upon the ideas introduced in the so-called transition state theory [88-90]. We shall focus on this theory here, particularly because it (and its harmonic approximation, HTST) has been shown to yield reliable results for elementary processes at surfaces. 

If the transition state is assumed to have zero thickness, it is of a dimensionality of one lower than the configuration space of the system. If entropy effects were to be neglected, the natural choice would be to let the transition state follow the 

Under these assumptions transition state theory states that the rate constant of an elementary reaction (the rate assuming that one has the reactant in the initial state) is given by  

Where A, Ax, and Av are normalization constants. Since the potential and kinetic energies are additive, the probability of finding the system in the configuration interval ([3c, 3c + dx], [v, v + dv]) turns out to be separable into probabilities of position and velocity. We can now determine the probability of finding the system in the infinitesimal vicinity of thickness, Sx, around the transition state to be  

This velocity can be evaluated directly from the Boltzmann distribution  

In the preceding sections, the rate coefficient, k, was considered as a parameter to be derived from experimental data. From the 1930s onwards, theoretical work was undertaken to model k and to calculate it from first principles. Today, progress in theoretical chemistry and computational power allows the calculation of k with a satisfactory accuracy, at least for elementary steps in homogeneous media. An elementary step does not have any detectable intermediate between the reactants and products. It corresponds to the least change in structure at the molecular level. There are various approaches for the modeling of k, like the collision theory and the transition state theory (TST), presently the favored one. 

The rate will now be related to the change in energy of A as it passes the potential energy barrier towards the products. This passage is linked to the vibrational contribution to Eabx- 

The partition functions express the distribution of the energy states of an entity, be it the molecule. 4 or 5 or the activated complex A. They are referred to the selected zero-point energy-level. The exponential function in (1.7.1-5) takes care of the adaptation to the actual reaction temperature. represents the difference between the selected zero-point molar energy levels of the activated complex and the reactants. In practical terms Eo is the activation energy required by the reactants at 0°K. 

The probability Pi that a molecule is in the i-th quantum state with 

ABLE 2.4 Rate Constants for Indigenous Phosphorus Release from a Thiokol Silt Loam Soil  

Transition-state theory or reaction-rate theory was extensively developed by H. Eyring and collaborators (Glasstone et al., 1941 Frost and Pearson, 1961). 

For a given reaction in accordance with the absolute rate theory 

Application of Chemical Kinetics to Soil Chemical Reactions 

In order to derive an equation for the reaction rate, let t denote the average time for an activated complex to decay to products. Then, from the definition of co, 

The assumption of equilibrium between the complex and reactants implies that 

If the velocity distribution in the reaction coordinate is assumed to be Maxwellian (because of the hypothesis of thermodynamic equilibrium for the complex), then a simple integration shows that v = where 

From the preceding paragraph, the reader will note that many assumptions are involved in transition-state theory. Alternative derivations exhibit differing hypotheses. In a quicker but perhaps less intuitive derivation, translation in the reaction coordinate is treated formally as the low-frequency limit of a vibrational mode. Expansion of the vibrational partition function given in Section A.2.3 then yields Q = Q (k T/hv), which is substituted into equation (A-24), to be used directly in equation (66), thereby producing equation (69) when v = 1/t. The decay time thus is identified as the reciprocal of the small frequency of vibration in the direction of the reaction coordinate. 

Comparison between transition-state theory and collisional theory 

Potential energy profiles for the elementary reaction A + B endothermic reaction and (b) an exothermic reaction. 

Consider again the reaction given by Equation (2.2.2). At thermodynamic equilibrium, Equation (2.2.3) is satisfied. In the equilibrated system, there must also exist an equilibrium concentration of transition states Cjs- The rates of the forward and the reverse reactions are equal, implying that there is an equivalent number of species traversing the activation barrier from either the reactant side or the product side. Thus, 

Using this formulation for Cts in Equation (2.3.2) allows the reaction rate to be of the form  

A fundamental assumption of transition-state theory is that is a universal frequency and does not depend upon the nature of the reaction being considered. It can be proven that (see, for example, M. Boudart, Kinetics of Chemical Processes, Butterworth-Heinemann, 1991, pp. 41-45)  

This is the general equation of transition-state theory in its thermodynamic form. 

Consider a chemical reaction of the type A + B — C + D. The rate of reaction may be written as 

Transition State Theory (TST) assumes that a reaction proceeds from one energy minimum to another via an intermediate maximum. The Transition State is the configuration which divides the reactant and product parts of the surface ti.e. a molecule which has reached the transition state will continue on to product), while the geometrical configuration of the energy maximum is called the Transition Structure. Within standard TST the transition state and transition structure are identical, but this is not necessarily The case for more refined models. Nevertheless, the two terms are often used 

If krate IS kuowu, the concentration of the various species can be calculated at any given time from the initial concentrations. At the microscopic level, the rate constant is a 

Introduction to Computational Chemistry, Second Edition. Frank Jensen. 2007 John Wiley Sons, Ltd 

The Gibbs free energy is given in terms of the enthalpy and entropy, G = H - TS, and the enthalpy and entropy for a macroscopic ensemble of particles may be calculated from properties of a relatively few molecules by means of statistical mechanics, as discussed in Section 13.4. 

Chemical reactions involve the making and breaking of chemical bonds. The energy associated with a chemical bond is a form of potential energy. Reactions are accompanied by changes in potential energy. Consider the following hypothetical, one-step reaction at a certain temperature. 

Unless otheiwise noted, all content on this page is Cengage Leaining. 

0 A reaction that releases energy (exothermic). An example of an exothermic gas-phase reaction is 

The reaction of potassium metal with water is spontaneous and has a very low activation energy (fa). The small f, means that the reaction will be very fast. 

0 A reaction that absorbs energy (endothermic). An exampieof an endothermic gas-phase reaction is 

FIGURE lA Successful (a) and unsuccessful (b) transfer of a hydrogen atom from HI to Cl. 

From the saddle point, the energy increases in all the principal directions except along the direction that leads to reaction (forward) or (backward) to reform the starting materials. The course of a simple reaction may be represented as motion along the reaction coordinate, which is a combination of atomic coordinates leading from the initial configuration (reactants) through the transition state to the final 

FIGURE 1.5 A portion of a potential-energy surface E(x,y), showing a saddle point. 

FIGURE 1.6 (a) The reaction pathway of least energy and (b) the profile along the pathway, 

FIGURE 1.7 A possible, more realistic reaction profile for a ligand-exchange reaction, showing reactants (a), precursor (b), and successor (d) complexes, the transition state (t), the possibility of the formation of a reactive intermediate (c), and products (e). Redrawn after 

FIGURE 20.22 The oscillating nature of the Belousov-Zhabotinsky reaction can be illustrated by the varying concentrations of some of the species involved. Many of these concentrations are measured electrochemically. Source Reprinted with permission from the Journal of the American Chemical Society, Vol. 94, No. 25, p. 8651. 

FIGURE 20.23 Under certain conditions, oscillating reactions can produce different colors over time, leading to some fascinating visual displays. 

Not all kinetics is phenomenological. In recent years, there have been advances in understanding the kinetics of reactions from a theoretical perspective. In this section, we will review the basics of some theoretical kinetics. 

Collision theory is a simple description of reacting molecules that treats them as hard spheres. Some of the basic concepts of collision theory were considered at the end of section 20.6. Although this model does predict some numerical reaction parameters having about the right order of magnitude, its description of molecules as hard spheres and use of steric factors as fudge factors ignores the complex nature of even simple molecular reactions. A more realistic approach is necessary. 

A key point in transition-state theory is to calculate a theoretical rate constant k for the bimolecular elementary process. In terms of transition-state theory, the bimolecular elementary process given as 

It may look easier than it typicaUy is to determine a good TS on the PES. Since the TS has zero thickness along the reaction path, it is an object of dimensionaUty one 

FIGURE 4.1 Separation of the configuration space used in transition state theory into three regions the reactant region, the product region, and the transition state, which is a dividing surface that separates the reactant and product regions. 

The PES defines a unique correspondence between a potential and the nuclear positions of the system. When developing a rate theory based on the existence of a PES, we are therefore implicitly invoking the Bom-Oppenheimer approximation. This approximation assumes that motion of the electrons is instantaneous compared with the motion of the nuclei, such that for whatever motion, we shall never move on an electronically excited state not corresponding to the ground-state potential energy. This is often a reasonable assumption, because the mass of any nucleus is on the order of 2 X 10 -5 X 1(F heavier than an electron. Since the forces exchanged between the nuclei and electrons are of similar size, the electrons can be assumed to be in their ground state in the electrostatic potential set up by the environment (typically the nuclei). 

We shall in addition assume that the rate of quantum tunneling through the potential barriers is negligible compared with the rate obtained from the classical treatment. This is an assumption that generally always breaks down when the temperature becomes low enough. However, the typical crossover temperature below which the rate becomes dominated by quantum tunneling is very low (dependent on the barrier thickness and the masses of the tunneling particles). 

Finally, we define the rate constant for an elementary process as the rate of the process under the assumption that the system starts out by being in the reactant region. The rate constant is thus equal to the reaction rate, if the system is in the reactant configuration. This concept will be used to derive microkinetics in Chapter 5. 

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As LEED studies have shown, the stmcture of a chemisorbed phase can change with 6. In terms of transition state theory, we can write A = (I/tq) and a common observation is that while E may change with a phase change, AS will tend to change also, and similarly. The result, again known as a compensation effect, is that the product remains relatively constant... 

Flere, we shall concentrate on basic approaches which lie at the foundations of the most widely used models. Simplified collision theories for bimolecular reactions are frequently used for the interpretation of experimental gas-phase kinetic data. The general transition state theory of elementary reactions fomis the starting point of many more elaborate versions of quasi-equilibrium theories of chemical reaction kinetics [27, M, 37 and 38]. 

The quasi-equilibrium assumption in the above canonical fonn of the transition state theory usually gives an upper bound to the real rate constant. This is sometimes corrected for by multiplying (A3.4.98) and (A3.4.99) with a transmission coefifiwient 0 < k < 1. 

A3.4.7 STATISTICAL THEORIES BEYOND CANONICAL TRANSITION STATE THEORY... 

Finally, the generalization of the partition function q m transition state theory (equation (A3.4.96)) is given by... 

These equations lead to fomis for the thermal rate constants that are perfectly similar to transition state theory, although the computations of the partition functions are different in detail. As described in figrne A3.4.7 various levels of the theory can be derived by successive approximations in this general state-selected fomr of the transition state theory in the framework of the statistical adiabatic chaimel model. We refer to the literature cited in the diagram for details. 

It may be iisefiil to mention here one currently widely applied approximation for barrierless reactions, which is now frequently called microcanonical and canonical variational transition state theory (equivalent to the minimum density of states and maximum free energy transition state theory in figure A3,4,7. This type of theory can be understood by considering the partition fiinctions Q r ) as fiinctions of r similar to equation (A3,4.108) but with F (r ) instead of V Obviously 2(r J > Q so that the best possible choice for a... 

Truhiar D G, Garrett B C and Kiippenstein S J 1996 Current status of transition-state theory J. Phys. Chem. 100 12 771-800... 

Bennett C H 1977 Molecular dynamics and transition state theory the simulation of infrequent events Algorithms for Chemical Computation (ACS Symposium Series No 46) ed R E Christofferson (Washington, DC American Chemical Society)... 

Poliak E 1993 Variational transition state theory for dissipative systems Acf/Vafed Barrier Crossinged G Fleming and P Hanggi (New Jersey World Scientific) p 5... 

Gershinsky G and Poliak E 1995 Variational transition state theory application to a symmetric exchange in water J. Chem. Phys. 103 8501... 

Poliak E 1987 Transition state theory for photoisomerization rates of f/ a/rs-stilbene in the gas and liquid phases J. Chem. Phys. 86 3944... 

Poliak E, Tucker S C and Berne B J 1990 Variational transition state theory for reaction rates in dissipative systems Phys. Rev. Lett. 65 1399... 

Poliak E 1990 Variational transition state theory for activated rate processes J. Chem. Phys. 93 1116 Poliak E 1991 Variational transition state theory for reactions in condensed phases J. Phys. Chem. 95 533 Frishman A and Poliak E 1992 Canonical variational transition state theory for dissipative systems application to generalized Langevin equations J. Chem. Phys. 96 8877... 

Voth G A, Chandler D and Miller W H 1989 Rigorous formulation of quantum transition state theory and its dynamical corrections J. Chem. Phys. 91 7749... 

Voth G A 1990 Analytic expression for the transmission coefficient in quantum mechanical transition state theory Chem. Phys. Lett. 170 289... 

Miller W H 1975 Semiclassical limit of quantum mechanical transition state theory for nonseparable systems J. Chem. Phys. 62 1899... 

Makarov D E and Topaler M 1995 Quantum transition-state theory below the crossover temperature Phys. Rev. E 52 178... 

Stuchebrukhov A A 1991 Green s functions in quantum transition state theory J. Chem. Phys. 95 4258... 

Shao J, Liao J-L and Poliak E 1998 Quantum transition state theory—perturbation expansion J. Chem. Phys. 108 9711 Liao J-L and Poliak E 1999 A test of quantum transition state theory for a system with two degrees of freedom J. 

The first two of these we can readily approach with the knowledge gained from the studies of trappmg and sticking of rare-gas atoms, but the long timescales involved in the third process may perhaps more usefiilly be addressed by kinetics and transition state theory [35]. 

This is connnonly known as the transition state theory approximation to the rate constant. Note that all one needs to do to evaluate (A3.11.187) is to detennine the partition function of the reagents and transition state, which is a problem in statistical mechanics rather than dynamics. This makes transition state theory a very usefiil approach for many applications. However, what is left out are two potentially important effects, tiiimelling and barrier recrossing, bodi of which lead to CRTs that differ from the sum of step frmctions assumed in (A3.11.1831. 

Miller W H 1974 Quantum mechanical transition state theory and a new semiclassical model for reaction rate constants J. Chem. Phys. 61 1823-34... 

In deriving the RRKM rate constant in section A3.12.3.1. it is assumed that the rate at which reactant molecules cross the transition state, in the direction of products, is the same rate at which the reactants fonn products. Thus, if any of the trajectories which cross the transition state in the product direction return to the reactant phase space, i.e. recross the transition state, the actual unimolecular rate constant will be smaller than that predicted by RRKM theory. This one-way crossing of the transition state, witii no recrossmg, is a fiindamental assumption of transition state theory [21]. Because it is incorporated in RRKM theory, this theory is also known as microcanonical transition state theory. 

Miller W H 1976 Importance of nonseparability in quantum mechanical transition-state theory Acc. Chem. Res. 9 306-12... 

Hu X and Hase W L 1989 Properties of canonical variational transition state theory for association reactions without potential energy barriers J. Rhys. Chem. 93 6029-38... 

Song K and Chesnavich W J 1989 Multiple transition states in chemical reactions variational transition state theory studies of the HO2 and HeH2 systems J. Chem. Rhys. 91 4664-78... 

Miller W H, Hernandez R, Moore C B and Polik W F A 1990 Transition state theory-based statistical distribution of unimolecular decay rates with application to unimolecular decomposition of formaldehyde J. Chem. Phys. 93 5657-66... 

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