Reactive Intermediate Chemistry

952 POTENTIAL ENERGY SURFACES AND REACTION DYNAMICS

of Figure 21.8 (b). It shows the time dependence of the methyl group dissociation from the acetone radical cation. One sees evidence of oscillatory behavior in which the first methyl to be lost is the newly formed one. A short while later there is a preference for loss of the existing methyl, and then the graph shows a damped oscillatory behavior between the two. The integral of the branching-fraction curve does not converge on zero at long times (i.e., there is an asymmetry in the final product ratio that favors loss of the newly formed methyl). This result is consistent with the experimental observations.

The simulation suggests that the geometries of the [1,3] migration TS and the methyl-dissociation asymptote (there is not a maximum in PE along the dissociation pathway) are crucial to understanding this behavior. The methyl dissociation occurs preferentially from a geometry in which one OCC angle is small (near 90 ) and the other large (>130 ). This results because the product acylium ion has an OCC angle of 180 . The [1,3] migration TS has a similar asymmetry in the OCC bond angles, caused by the need to get the oxygen and methylene carbon sufficiently close to transfer a hydrogen atom. Thus the migration TS starts out closer in geometry to one of the dissociation exits than to the other. The simulation reveals that some trajectories simply leave the [1,3] migration TS and take the nearer dissociation exit without ever accessing the acetone radical ion minimum. These trajectories constitute the first peak in Figure 21.8 (b). They necessarily lose the newly formed methyl. Those trajectories that fail to take the nearer exit do enter the acetone radical ion minimum, and they pick up roughly 40 kcal/mol of kinetic energy as they do so. This kinetic energy appears largely in a vibrational motion consisting of an in-plane bend of the carbonyl oxygen, first toward one methyl and then toward the other. Importantly, this vibration is excited with a particular phase; as trajectories leave the [1,3] migration TS, the OCC angle to the newly formed methyl is increasing and the other is decreasing. This means that the existing methyl becomes the first to experience a geometry of the acetone radical cation appropriate for dissociation. This behavior is responsible for the large downward peak in Figure 21.8 (b). After that the carbonyl oxygen swings backward and forward for several hundred femtoseconds, encouraging one methyl and then the other to dissociate. The dissociation peaks are damped both because of IVR, which is spreading the kinetic energy into other vibrational modes, and because of loss of population due to dissociation.

This case is clearly one in which nonstatistical dynamics (indicated both by the time-dependent branching ratio and the lack of correspondence between the static symmetry of the intermediate and its branching ratio) plays an important role, despite the depth of the PE minimum in which the acetone radical cation resides.

3.2.3.3. The Rearrangement of 1,2,6-Heptatriene. The experimental facts7 and basic mechanistic idea8 behind this reaction were outlined in section 1.2. The molecular dynamics study9 began with a single CASSCF(8,8)/6-31G(d) trajectory started from TS1 (see Fig. 21.2) with no kinetic energy (not even ZPE) in any of the real-frequency normal modes. The purpose of such an unphysical trajectory calculation is to see what is the steepest descent path down from the transition state

and which modes of the product or intermediate receive most of the kinetic energy picked up along the way. In this case, the trajectory led to a local minimum conformation of the biradical intermediate. Its vibrational motions turned out to be very interesting. As soon as it was formed, the biradical started to exhibit a large-ampli- tude stretching motion of the C4 C5 bond (see Scheme 21.3). No other bond in the molecule showed such behavior. It was of interest because scission of this very bond would be necessary to complete the conversion of the intermediate to the product. The mechanism by which kinetic energy could be ‘‘channeled’’ into this motion was revealed when a second trajectory was run. It was identical to the first save for the fact that H10 (see Scheme 21.3) was replaced by a deuterium. Now the amplitude of the C4 C5 stretch induced in the biradical was much reduced. The picture that arises from these observations is as follows. In TS1, the C3 H10 bond is almost normal to the C1 C2 C3 plane, reflecting the allenic origin of this moiety. However, in the biradical, the C3 H10 bond is in the C1 C2 C3 plane, as required for the intermediate to enjoy allylic stabilization. Thus, an important motion taking one from TS1 to the biradical is the bending of the C3 H10 bond into the C1 C2 C3 plane. Consequently, the out-of-plane bend of C3 H10 receives a large fraction of the kinetic energy released in the formation of the intermediate. It then turns out that the C3 H10 bend is in near resonance with the C4 C5 stretch, which leads to very efficient energy transfer between the two. The evidence for this picture comes from the deuterium substitution, which significantly influences the frequency of the bend but hardly changes the frequency of the stretch, and therefore ‘‘detunes’’ the resonance. The significance of the excitation of the C3 H10 bend and C4 C5 stretch is that the combination of these two geometrical changes is precisely what is required to take the biradical over the second transition state and on to the product. And, indeed, AM1–SRP simulations reveal a significant number of trajectories that enter the biradical minimum but then exit to product in <500 fs. This reaction is much faster than would be expected on the basis of a statistical model kinetic analysis.

The trajectory calculations reveal another important piece of information about the reaction. When CASSCF(8,8)/6-31G(d) trajectories are started from TS1 with ZPE in the vibrational modes, it is found that they can either go to the biradical or they can cross the second transition state directly, without ever accessing the biradical minimum. The difference between the two sets is only the relative phases of the vibrational motions. This result is also seen in the AM1–SRP simulation. Thus, while the chemistry is very different, the general picture that emerges for the 1,2,6- heptatriene rearrangement is strikingly similar to that found for the acetone radical cation dissociation. In both cases, a significant fraction of the trajectories leaving the rate-limiting TS fail to follow the steepest descent path, and thereby miss the subsequent intermediate minimum altogether. Those that do enter the minimum experience nonstatistical dynamical effects that have an important influence on how the intermediate proceeds on to the products.

3.2.3.4. Thermal Deazetization of 2,3-Diazabicyclo[2.2.1]hept-2-ene.33 In the previous two examples, the reactions involved intermediates residing in potential

954 POTENTIAL ENERGY SURFACES AND REACTION DYNAMICS

Figure 21.9. (a) Experimental data on the thermal deazetization of DBH-d2. (b) Schematic enthalpy profile for the reaction. The cyclopentane-1,3-dyl intermediate is calculated to have a C2-symmetry equilibrium structure.

energy wells that were 10–20-kcal/mol deep. The nonstatistical dynamics observed for such intermediates, while chemically significant, amounted to no more than a 60% deviation in branching ratio from that expected on the basis of the statistical approximation (ignoring contributions from trajectories that bypassed the local intermediate minimum). In the thermal deazetization of 2,3-diazabicyclo[2.2.1]- hept-2-ene-exo,exo-d2, (DBH-d2, see Fig. 21.9) the deviation from the statistical prediction is nearly 500%. The reason is that the PE profile for the reaction looks like that in Figure 21.9(b): The intermediate has a barrier-to-product formation of only 1 kcal/mol, but is formed with excess kinetic energy of roughly 14 kcal/mol. Reactions of this kind can be expected to show large effects from nonstatistical dynamics, because the PE surface in the region of the intermediate is relatively flat, and so it is possible to take non-IRC paths to the products with an energy penalty that is much smaller than the excess energy available to the system. This means that the dynamics for formation of the intermediate do not need to be perfectly coupled to those for passage of the intermediate over its product-forming transition state, because direct trajectories can take slightly higher energy routes to the products. In the case of DBH-d2, CASSCF(6,6) calculations show a Cs symmetry transition structure for loss of N2. The resulting cyclopentane-1,3-diyl has a C2 symmetry equilibrium geometry. However, a CASSCF trajectory run from the rate-determining TS maintains Cs symmetry throughout—it does not ever access the minimum energy structure of the biradical. Nor does it pass through the C1 symmetry transition structure for ring closure. Instead, it takes a higher barrier ( 3 kcal/ mol) Cs symmetry path, because this is the direction that the dynamics of N2 loss have imposed on it. Most important is the fact that the trajectory shows formation of the inverted product bicyclo[2.1.0]pentane-exo,exo-2,3-d2. This product is in fact favored experimentally (ratio 4.7 0.9:1 exo/endo in the gas phase). The calculations suggest that its preferential formation is due to the Newtonian mechanics of N2 loss from the reactant. As the N2 departs in one direction, the carbons to which it

was attached move in the opposite direction to conserve momentum. This phenomenon drives an out-of-plane bending motion in the cyclopentane-1,3-diyl with sufficient energy to carry it over the barrier (although not via the transition state) to formation of the inverted product.

3.3. Experimental Tests for Effects Due to Nonstatistical Dynamics

The most direct evidence for nonstatistical behavior comes from state-to-state laserinduced reactions conducted in molecular beams. In such studies, the tracking of energy flow during chemical reactions is sometimes possible, and in those cases the validity of the statistical approximation can be assessed directly.34 However, for reactions conducted under conditions used by most organic chemists, the unambiguous detection of nonstatistical dynamics is more challenging. Thus far only two techniques have been developed, and both really need the support of high-level electronic-structure theory to make a convincing case.

The first technique is relatively simple: It is the measurement of the temperature

dependence of product ratios, particularly in the case where products are related to each other as optical isomers or isotopic label isomers.6b,35 For reactions of this

kind the thermodynamic product ratio can usually be deduced by symmetry, provided any equilibrium isotope effects are small. If the observed product ratio differs significantly from the thermodynamic value, one can be confident that the products are formed under kinetic control. In a mechanistic interpretation that relied on the validity of the statistical approximation, the product ratio would then be ascribable to competitive reaction paths with barriers of different heights to formation of the products. Generally, one would expect these barriers to have different heights on the potential energy hypersurface, since the supposed competitive mechanisms will almost always have transition states of different geometry, and, barring coincidence, different geometry usually means different energy. If indeed there are PE barriers of different heights to formation of the products, then any of the statistical kinetic models will predict that there should be a temperature dependence to the product ratio. Often that turns out to be the case, and then one can be satisfied with the conventional mechanistic description. However, in several instances studies have found temperature-independent kinetic product ratios.6b,35 These experiments present a problem for the standard mechanistic description, because in each case such a result requires that the barriers to product formation be identical on the PE surface but different on the standard free energy surface. In other words, the transition states would have to have coincidentally identical heats of formation but very different entropies. There could be odd circumstances where such a thing might happen occasionally, but the repeated observation of this behavior in reactions of structurally diverse molecules stretches the limits of credulity for the ‘‘coincidence’’ explanation. Actually, models based on nonstatistical dynamics also fail to predict absolute rigorous temperature independence to product ratios. However, in those cases where the question has been studied, the energy or temperature dependence is found to be very weak, and well within the error of a typical experiment.5b,36

956 POTENTIAL ENERGY SURFACES AND REACTION DYNAMICS

Somewhat more direct evidence for nonstatistical behavior has come from studies in supercritical fluids.33 In these experiments, the idea is to try to intercept by collision any molecules on direct, nonstatistical trajectories to the products. The expectation is that collisions should promote IVR, and that increasing collision frequency should therefore lead to increasing conformance of the product ratio to that predicted from a statistical kinetic model. The reason that supercritical fluids are useful in such studies is that their high compressibility means that the density of the medium, and hence the collision frequency, is controllable by changing the pressure. This finding is also true for gases, of course, but in the gas phase it is difficult to attain the collision frequency necessary to intercept an intermediate whose lifetime might be only 100 fs. At least in one case, quantitative fitting of the pressure dependence of the product ratio has permitted an experimental estimate of the lifetime of the nonstatistical population component for a reactive intermediate.33 This number can be compared directly with results from MD simulation.

4. CONCLUSION AND OUTLOOK

Two principal points serve to encapsulate the discussion in this chapter. The first is that it is necessary to think about the PE hypersurface for a chemical reaction in a higher dimensionality projection than has typically been employed by mechanistic organic chemists to date. Even the three-dimensional projections on which most of the discussions in this chapter have been based are undoubtedly concealing important topological features that a still higher dimensional view would reveal, but for now the features summarized in Section 1.1 provide more than enough material for research. The second point is that kinetic models employing the statistical approximation may not have served us as well as we had thought. Increasing application of MD simulation, made possible by rapidly increasing computer power, is continuing to show the shortcomings of the TST and RRKM models, and hence of the mechanistic descriptions that were based on them.

There are (at least) two major opportunities for research by those interested in this topic. On the computational side, there is definite room for improvement in simulation methods. Right now none of the simulation approaches has the ‘‘user friendliness’’ that has brought electronic-structure calculation into the realm of routine applicability by nonspecialists. Nor has the field seen the development of the qualitative or semiquantitative models that did so much to make the results of molecular orbital calculations useful to organic chemists. On the experimental side, it will be obvious to the reader that the techniques for detecting the effects of nonstatistical dynamics are still very rudimentary and indirect. There is clearly room for creative scientists to come up with techniques whose results can give us more direct insight into these issues.

Looking a little further out, it is already apparent that the dynamical models outlined in this chapter are themselves inadequate. There is growing evidence that some fundamentally quantum mechanical phenomena, such as tunneling, can have important roles to play in everyday reactions.37 These phenomena are not

REFERENCES 957

describable by models based on classical mechanics. Progress is being made on the incorporation of quantum mechanical components into MD simulations, although the full quantum mechanical description of dynamics remains far from computationally feasible for systems of more than a few degrees of freedom.38 Similarly, nonadiabatic dynamic simulations, in which trajectories (or, better, wave packets) are propagated simultaneously on more than one PES, are becoming increasingly sophisticated and are starting to prove their value in analyzing photochemical (and some kinds of thermal) reactions.39

Even for purely adiabatic reactions, the inadequacies of classical MD simulations are well known. The inability to keep zero-point energy in all of the oscillators of a molecule leads to unphysical behavior of classical trajectories after more than about a picosecond of their time evolution.40 It also means that some important physical organic phenomena, such as isotope effects, which are easily explained in a TST model, cannot be reproduced with classical molecular dynamics. So it is clear that there is much room for improvement of both the computational and experimental methods currently employed by those of us interested in reaction dynamics of organic molecules. Perhaps some of the readers of this book will be provide some of the solutions to these problems.

SUGGESTED READING

L.Sun, K. Song, and W. L. Hase, ‘‘A SN2 Reaction that Avoids its Deep Potential Energy Minimum,’’ Science, 2002, 296, 875.

C.Doubleday, ‘‘Mechanism of the Vinylcyclopropane–Cyclopentene Rearrangement Studied by Quasiclassical Direct Dynamics,’’ J. Phys. Chem. A 2001, 105, 6333.

B.K. Carpenter, ‘‘Dynamic Behavior of Organic Reactive Intermediates,’’ Angew. Chem., Int. Ed. Engl. 1998, 37, 3341.

R.D. Levine, ‘‘Molecular Reaction Dynamics Looks Toward the Next Century: Understanding Complex Systems,’’ Pure Appl. Chem. 1997, 69, 83.

R. Q. Topper, ‘‘Visualizing Molecular Phase Space: Nonstatistical Effects in Reaction Dynamics,’’ Rev. Comput. Chem. 1997, 10, 101.

R.D. Levine and R. B. Bernstein, Molecular Reaction Dynamics and Chemical Reactivity, Oxford University Press, New York, 1987.

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