Reactive Intermediate Chemistry

942 POTENTIAL ENERGY SURFACES AND REACTION DYNAMICS

k 1½M& k2, in which case kuni ¼ k2ðk1=k 1). This is condition called the highpressure limit. The trick then is to note that k1=k 1 defines the equilibrium ratio [A*]/[A], which can be determined from the Boltzmann distribution. Thus, if one calculates k2(E) for a range of energies above E0 (the higher the value of E and the larger the number of steps between E0 and E, the more accurate the calculation, but the longer it takes) from the RRKM Eq. (2), multiplies each of these microcanonical rate constants by the Boltzmann factor expð E=kBTÞ, and then adds up the results, the result is the RRKM value for kuni. Importantly, provided the corrections for rotational effects are treated properly, the RRKM expression for kuni in the highpressure limit is the same as the transition state theory result, which is discussed next.

2.4. Transition State Theory21

Transition state theory, as worked out by Evans and Polanyi, and by Eyring, uses the techniques of equilibrium statistical mechanics to derive an expression for kuni. It depends explicitly (and not surprisingly) on the transition state hypothesis and implicitly on the statistical approximation. The original papers on TST do not discuss any of the versions of the statistical approximation listed in Section 2.2.3, but that the theory does nonetheless make that approximation is clear given that the TST result can be derived as the high-pressure-limit version of RRKM theory where the invocation of the statistical approximation is unambiguous. Alternatively, one can recognize that the assumed maintenance of a canonical ensemble of reactant molecules at a well-defined temperature throughout the reaction only makes sense if the repopulation of all energetically accessible parts of the reactant phase space occurs more rapidly than the passage through the transition state.

One version of the TST result is given by Eq. 4:

where Q and Qz are the partition functions of the reactant and transition state, respectively. Alternatively, one can recognize that the ratio of partition functions and the exponential term together define the hypothetical equilibrium constant between reactant and transition state. This transformation leads to the familiar thermodynamic formulation of TST (Eq. 5) that most organic chemists will encounter:

where Gz0 is defined as the standard free energy of activation (which is itself decomposable into enthalpy and entropy components in the usual way).

It is perhaps worth pointing out that several texts claim that the TST result is exact provided that quantum mechanical effects such as tunneling are negligible. However, for the purposes of the discussion in Section 3, one needs to be clear

what ‘‘exact’’ means. An expression for a rate constant can only be exact if the rate constant itself is well defined. That will be true only if the reactant population decays exponentially with time (the unimolecular rate constant then being the proportionality constant in the exponent), and that in turn will be generally true only if the statistical approximation is correct. Thus the TST result is exact if quantum mechanical effects are negligible, and if the statistical approximation is correct.

2.5. Variational Transition State Theory22

We have seen that TST depends on the nonrecrossing of the transition state plane for the accuracy of its predicted rate constant. To the extent that trajectories do recross the transition state, the real rate constant will be reduced. One could consequently seek to find the optimum location for the transition state plane by looking for the position along the reaction coordinate that minimizes the computed rate constant. This idea is behind variational transition state theory (VTST). It comes in two varieties that differ in how the idea is implemented. In canonical VTST, one admits the possibility that the ideal transition state may not be located at the maximum in the potential energy profile, but wherever the ideal position is, one assumes that it is fixed and independent of the energies of individual molecules. Thus the calculation is to find the transition state position along the reaction coordinate, qz, that makes dkuniðTÞ=dqz ¼ 0. This condition will always locate the transition state at a maximum on the standard free energy surface rather than a maximum on the PE surface. Obviously, this search for the optimum transition state is more work than simply assuming that it occurs at the PE maximum as conventional TST does. For reactions with large barriers, the improvement in kuni by using the VTST approach may not be worth the extra effort because the optimum transition state is likely to be very close to the conventional one. However, for reactions with small or zero conventional barriers, the VTST approach is essential.

The most sophisticated and computationally demanding of the variational models is microcanonical VTST. In this approach one allows the optimum location of the transition state to be energy dependent. So for each k(E) one finds the position of the transition state that makes dk(E)/dqz ¼ 0. Then one Boltzmann weights each of these microcanonical rate constants and sums the result to find kuni. There is general agreement that this is the most reliable of the statistical kinetic models, but it is also the one that is most computationally intensive. It is most frequently necessary for calculations on reactions with small barriers occurring at very high temperatures, for example, in combustion reactions.

3. NONSTATISTICAL DYNAMICS

3.1. Introduction: Elements of Molecular Dynamics23

The basic idea behind molecular dynamics (MD) simulation was presented in Section 2.2.1. From a specified set of initial conditions one wishes to solve Newton’s

944 POTENTIAL ENERGY SURFACES AND REACTION DYNAMICS

second law of motion or, equivalently, Hamilton’s equations, in order to predict how the system under study will evolve in time. This task has three prerequisites:

(a) some way of selecting the initial conditions; (b) some way of calculating the potential energy of the system, as well its first derivatives with respect to nuclear positions, at all steps of the trajectory; (c) some way of numerically integrating the equations of motion. It will be convenient to discuss these requirements in reverse order.

The reason that one needs a numerical integration algorithm is easy to understand. The argument will be framed in terms of Newton’s equations, but a similar analysis applies to the Hamiltonian formulation. In the first step of a MD simulation, Newton’s second law allows one to calculate the accelerations experienced by each atom of a system, given their masses and the forces acting on them. What one then needs to do is to use the computed accelerations, plus the velocities and positions specified in the initial conditions to predict some new set of coordinates for the atoms. However, since velocities are first-time derivatives and accelerations are second-time derivatives of position, this task is going to require some sort of integration. If the forces (and hence accelerations) remained constant, or varied in some simple way with time one might be able to do the integration analytically. Unfortunately, for all systems of chemical interest the forces acting on the atoms change in very complicated ways as the system moves around on the PES. It is for that reason that one needs some numerical integration scheme. Inevitably, the trajectory will be calculated as a sequence of small time steps. How small depends on the rate of change of the forces with time, and the sophistication of the integration algorithm, but for systems of interest to the organic chemist time steps will typically be 0.1 to 1 fs (i.e., 10 16–10 15 s). Whether the steps have been small enough is easily checked by making sure that total energy conservation is obeyed (i.e., the sum of the potential and kinetic energies must be constant). It is also prudent to use the more rigorous (but costly) check of time reversal. Since the system being simulated is purely classical, the equations of motion are deterministic and so at any point of a trajectory one should be able to reverse the arrow of time (by changing the signs of the velocity or momentum components on the atoms) and follow the trajectory backward to its starting point.

For the systems discussed in this chapter, a typical duration of a complete trajectory might be of the order of a picosecond (10 12 s), for other kinds of problems it could be several orders of magnitude longer. Even a 1-ps trajectory is likely to require 103–104 steps, and at each step one must calculate the PE of the system plus the 3N first derivatives with respect to nuclear coordinates. In principle, there are two general ways of approaching this task. One is to have calculated ahead of time a complete PE hypersurface for the system, and to have expressed it as some sort of algebraic function of the nuclear coordinates. This task turns out to be almost impossible for systems of a size interesting to organic chemists. The alternative is to use so-called direct dynamics in which some sort of computation of the PE and its derivatives is done on the fly as the trajectory is evolving. The problem with this approach is easily seen. Suppose one used a sophisticated post-Hartree– Fock electronic structure calculation, as one might want to do in order to get an

accurate estimate of the energy. With current computational resources and algorithms, such a calculation might take on the order of 1 min for a medium-sized organic molecule. That means a 1-ps trajectory might take as long as a week of CPU time, at the end of which you would know how one single molecule had reacted. A typical simulation requires several thousand complete trajectories with different initial conditions in order to get a sample of how the whole system will behave.

An obvious solution to this problem would be to compromise on the quality of method used to estimate the PE at each point. Indeed, if one goes as far as using molecular mechanics (MM) for the purpose, it becomes possible to run MD simulations on full-size proteins.24 The drawback is that MM methods are not capable of describing the making and breaking of covalent bonds, and so such techniques have not been useful for the kinds of reactions discussed in this chapter. There have been efforts to hybridize molecular mechanics and quantum mechanics (QM) so that the reactive centers in a molecule can be treated by the QM part and the nonreactive ones by the MM part.25 Some considerable successes have been reported with these techniques, but they are not yet easy to use for the nonspecialist because the division of molecules into reactive and nonreactive components, and then the correct blending of the MM and QM descriptions of these two parts, requires careful thought and a fair amount of experience. One might expect that the recent great success of density functional theory (DFT) for electronic-structure calculations would carry over into MD simulation, and indeed there has been available for some time a direct dynamics technique due to Car and Parrinello that relies on DFT methods. The Car–Parrinello approach has also enjoyed some considerable success, but has not been widely used in reactions involving covalent bonding changes, perhaps because DFT with a plane-wave basis set is not well suited to describing such processes.26

Probably the most widely used method for carrying out MD simulations on reactions of the kind considered here is the AM1–SRP technique introduced by Truhlar and co-workers.27 In this approach, one carries out high-level ab initio calculations on the stationary points for a reaction of interest and then reparametrizes a semiempirical molecular orbital model, such as AM1, to fit the results. The resulting model is capable of the very fast calculation of PE and its derivatives required for direct dynamics, but has parameters that are specific for the reaction of interest, and have to be readjusted for a new reaction. The SRP part of the acronym stands for specific reaction parameters. Even this technique is not without its problems. The fitting procedure requires that one try to reproduce not only the relative potential energies of the key stationary points but also their molecular geometries and vibrational frequencies (which give information about the local curvature of the PE hypersurface). Unfortunately, energy, geometry, and frequency are measured in different units, and so there is no algorithmic way to specify an optimum fit to them all simultaneously. The reparametrization requires some subjective assessments of the relative weights to be given to various properties of the stationary points, and so it could well be that two people, given the same set of ab initio results, would end up with two different ‘‘best fit’’ AM1 parameters. Despite these

946 POTENTIAL ENERGY SURFACES AND REACTION DYNAMICS

difficulties, the AM1–SRP approach seems to be the most generally applicable one at the moment, at least for reactions involving covalent bonding changes in medium-sized organic molecules.

The selection of initial conditions for a trajectory needs to be done carefully. A single trajectory, even if completely and accurately representative of reality, would still tell one only about the behavior of a single molecule (assuming the reaction is unimolecular). The behavior of that molecule would almost certainly depend heavily on the initial conditions (its geometry and the velocity or momentum components assigned to its atoms) that were selected to start the calculation. In order to assess how a collection of molecules will behave it is therefore necessary to take some sort of statistical sample of the possible initial conditions, and to run trajectory calculations for each of them. What that sample looks like depends on a number of things. First, one needs to decide what the population of starting molecules would look like in the experiment to be simulated. If the reaction is thermal, then a canonical ensemble might be expected; if it is a state-to-state laser study, then a microcanonical ensemble might be a better representation. Next, one needs to decide where on the PE hypersurface to begin the simulation. This choice may seem trivial—why would it be anywhere but in the reactant region? The answer is that, for a thermal, unimolecular reaction, this is rarely feasible. As discussed in Section 2.3, all current kinetic theories rely on some version of the Lindemann– Hinshelwood model for thermal reactions. In this picture, the promotion of a molecule to an energy high enough for reaction is a very rare event if the barrier is much bigger than kBT. Even when a molecule makes it to this elevated position, there is, in the high-pressure limit, a much higher chance of its being deactivated by the next collision than there is of sufficient vibrational energy happening to accumulate in the reaction coordinate to permit passage through the transition state region. This reason is why thermal reactions take minutes or hours rather than picoseconds. The upshot is that attempted simulation of a thermal reaction of a typical organic molecule, starting from the reactant region is not computationally feasible, by 12–14 orders of magnitude! Consequently, all of the MD simulations described in this chapter were initiated in the vicinity of the transition state for formation of the reactive intermediate of interest. Typically, the assumption is made that molecules in this region form a microcanonical or canonical ensemble. To the extent that assumption is wrong, it would presumably tend to bias the results in favor of a statistical model for the overall kinetics, and so any finding of nonstatistical behavior probably cannot be explained away on that basis. Finally, one has to decide how to apportion kinetic energy to the molecules in one’s sample. There are many schemes for doing this, each with their own virtues and liabilities. Most common for the kinds of study described here is so-called quasiclassical normal-mode sampling.28 In this procedure, kinetic energy is supplied in quantized fashion to the vibrational normal modes of the molecule, with randomization of the vibrational phases. This last stipulation means that different trajectories will start from different molecular structures in the vicinity of the stationary point; if all of the vibrational oscillators were at the minima of their potential curves, the relative phase angles between them could be only 0 or 180 . In order to start with phase angles other than these two

values, oscillators have to be ‘‘caught’’ at some position other than that corresponding to the minimum in PE. This requirment implies that the molecule itself will start with a distorted geometry. Usually, zero-point energy (0.5 hn) is supplied to each normal mode. In addition, some of the modes may be excited to levels above v ¼ 0, in a way that preserves the energetic requirements for the ensemble as a whole. Quantized initial conditions are used, despite the fact that the subsequent trajectory calculations employ only the equations of classical mechanics, because quantization ensures a proper distribution of energy among modes of different frequency. For reactions run under typical thermal conditions, the low-frequency modes should have average energies near the classical limit of kBT (above the zero-point level), whereas high-frequency modes will generally have average energies below this limit.

3.2. The Dynamics of Reactive Intermediates

3.2.1. Influence of the PE Profile. Although we saw at the beginning of this chapter that PE profiles should be treated with extreme caution when one is analyzing mechanistic possibilities, they are nevertheless useful devices for classifying the kinds dynamic of behavior one might expect for reactive intermediates.

For reactions whose PE profile is like that in of Figure 21.7 (a), there is neither experimental nor theoretical reason to doubt that the statistical kinetic models will do a good job of describing both absolute and, in the case where there are alternative rate-determining exits from the intermediate minimum, relative rates of product formation. However, the same cannot be said for any of the other three profiles in Figure 7. In Figure 21.7 (b) the intermediate is formed with kinetic energy (largely in the form of molecular vibrations) that far exceeds kBT, and hence also far exceeds the energy needed to traverse the barrier to products. Such reactions have a high probability of exhibiting large nonstatistical dynamical effects.

Figure 21.7 (c) is a representation of the kind of PE profile found for many reactions involving the opening of a strained hydrocarbon ring. As described in Section 1.2, the species residing on the energy plateau is a singlet biradical. In this kind of reaction (of which eight examples have been studied at the time of writing) there is not an intermediate with a single well-defined equilibrium geometry. Rather, the biradical is highly conformationally flexible and usually capable of forming a number of stereoisomeric products with little or no barrier. The statistical kinetic models are ill equipped to deal with such situations. When trajectory calculations are run, one typically finds that all stereoisomeric products can be formed, but not with equal probability. A combination of the entrance dynamics (as defined roughly by the direction of the reaction-coordinate eigenvector at the transition state for formation of the biradical) and the presence of very small barriers ( kBT) appear to determine the favored products. Thus far it has proven difficult to predict how these factors will combine, and hence which product(s) will be favored before doing the molecular dynamics simulation. It would be a significant contribution to this kind of chemistry if qualitative or semiquantitative rules

948 POTENTIAL ENERGY SURFACES AND REACTION DYNAMICS

Figure 21.7. Schematic PE profiles. All except the one in panel (a) represent reactions that are potentially susceptible to nonstatistical dynamical effects. See text for further discussion.

for making such predictions could be developed, and that is an area of active research.

Figure 21.7 (d) represents the kind of profile for which it has typically been thought that the statistical models will work. However, in two examples cited below, this turns out not to be the case. Even when the intermediate sits in a relatively deep potential energy minimum, the kinetic energy acquired during its formation can be an important factor controlling its fate. Of particular importance is the fact that this kinetic energy will not be statistically deposited into all vibrational modes, but rather will selectively excite certain modes of the intermediate. Which ones can be identified by running a single trajectory from the rate-determining TS toward the intermediate, with just enough kinetic energy in the reaction coordinate (say 0.1 kcal/mol) to get things moving and with no kinetic energy (even zero-point motion) in any other mode. Such a trajectory will almost never pass exactly through the minimum-energy geometry of the intermediate, because that would require the highly unlikely event that its vibrational modes be excited with relative phase angles of exactly 0 or 180 . Instead, the trajectory is likely to follow a path that explores various regions of the PE minimum corresponding to the intermediate. If the statistical approximation were correct, this motion would be chaotic (i.e., there would be no discernible periodicity). However, in reality one typically finds quite clear periodicity for at least a picosecond or so. Whether this has consequences for the subsequent chemistry of the intermediate is a question that is still

under investigation, but in two cases that have been studied in detail, it seems that it does. They are described below.

3.2.2. The Subtle Role of Symmetry. An appeal to symmetry seems like one of the least controversial and most rigorous bases for making an argument about the behavior of a physical system. And indeed it is. However, it is possible for arguments that appear to be based solely on symmetry actually to depend on some additional ancillary assumptions that may be less obviously valid. An important example for the present discussion concerns prediction of product ratios from reactive intermediates.

Almost every textbook on introductory organic chemistry will tell its readers that a reaction occurring via an achiral intermediate in an achiral medium can lead only to achiral and/or racemic products, even if the reactant was chiral and optically pure. The argument seems unassailable: If there were a chiral product available from an achiral intermediate, then the transition structures leading to the two enantiomers of the product would themselves have to be related as enantiomers. Since enantiomers have identical thermochemical properties in an achiral medium, the rate constants for conversion of the intermediate to the enantiomeric products would have to be identical by symmetry. Hence, the product would have to be racemic. However, there is a covert assumption in the argument. We have treated the intermediate as if it were the reactant for the product-forming step of the process. By so doing, we have implicitly ignored the history of formation of the intermediate. This decision is equivalent to making the statistical approximation, because we are saying that the dynamics of formation of the intermediate are not relevant to its subsequent chemistry. In reality, there are several cases that show such an assumption is not generally valid. In effect, the symmetry of the phase space in the vicinity of an intermediate can be lower than the symmetry of the PE surface; the momentum components describing the dynamics of the system can carry chiral information even if the static minimum-energy geometry of the intermediate appears to be achiral. A similar argument applies to the formation of isotopic isomers of products formed from a labeled intermediate, under circumstances where kinetic isotope effects are negligible.

The analysis summarized in Section 3.2.1 suggests that the reactions of an intermediate can depend on how it was made, because its formation from the preceding TS imprints it with dynamic information that can influence its subsequent selection of exit channels to the products. This claim flies in the face of some distinguished physical organic history. Some of the classic experiments for probing the existence of a reactive intermediate have relied on the product ratios from the intermediate both reflecting its static symmetry and being independent of its mode of generation. And, in fact, such behavior has been demonstrated in some celebrated cases.29 However, recognize that in almost all of these cases the product-forming step was a bimolecular trapping event. Even if the PE barrier is very small, a bimolecular reaction will usually be relatively slow because it can occur only when the trapping reagent happens to be near to and correctly oriented to the intermediate. There are few cases in which independent generation of a reactive intermediate from

950 POTENTIAL ENERGY SURFACES AND REACTION DYNAMICS

different sources has led to the same ratio of products when the product-forming step is unimolecular.30

3.2.3. Examples. There are by now several reactions for which the best available levels of ab initio electronic structure theory find a plateau on the PE hypersurface, in the vicinity of a singlet-state biradical.6 Several of these have been studied by MD simulation and/or experiment, and in each case the conclusion is that application of statistical kinetic models will give a misleading description of how the reaction really occurs. There is not room to describe each of these studies, and so just one is chosen as a representative, and described in Section 3.2.3.1.

The demonstration that reactions with more traditional PE profiles, similar to those in Figure 21.7 (b) and (d), may also exhibit significant nonstatistical dynamics has broadened the range of processes for which this effect may need to be considered. For that reason, three examples are described below.

3.2.3.1.The Vinylcyclopropane Rearrangement. In Section 1.2, we saw that the vinylcyclopropane rearrangement is a reaction for which the standard classification of concerted versus stepwise mechanism seems inadequate. It turns out that, in large measure, this is because the statistical kinetic models are also inadequate for describing the reaction. Electronic structure calculations at various levels of the-

ory agree that the reaction occurs with the involvement of a biradical that sits on a plateau on the potential-energy hypersurface.5 From this plateau region more-or-

less isoenergetic paths lead to the four stereochemically distinct products (see Scheme 21.1). Doubleday, et al.5a carried out extensive quasiclassical trajectory stu-

dies on this reaction using an AM1–SRP model (see Section 3.1) fit to the highest level ab initio results—those from a multireference configuration interaction calculation. Their simulations matched the experimental outcome very well: the com-

puted product ratios were 42:30:10:18 (si/sr/ar/ai), whereas the experimental ratios3c were 40:23:13:24. Importantly, the simulations revealed that the reaction

was dominated by nonstatistical dynamics. One symptom of the nonstatistical behavior was a highly time-dependent product ratio. Thus, trajectories that completed the passage to products in <200 fs gave a ratio of 53:43:0:4, whereas those taking >600 fs gave 20:22:30:28. Clearly, the reaction becomes more stereorandom the longer the trajectories last. This picture is just what one would expect if IVR were occurring on a time scale comparable to that for product formation. The simulations also revealed that the product ratio could be strongly influenced by the distribution of kinetic energy among the vibrational modes of the TS from which the trajectories were initiated. This behavior is also clearly nonstatistical.

There is every reason to believe that the results found for the vinylcyclopropane rearrangement are typical of those to be expected for reactions involving biradicals (and presumably other reactive intermediates) on energetic plateaus. Such reactions simply cannot be understood within the context of any statistical kinetic model.

3.2.3.2.Generation and Dissociation of the Acetone Radical Cation. It has been known for some time, and verified by a number of research groups, that the