Reactive Intermediate Chemistry

932 POTENTIAL ENERGY SURFACES AND REACTION DYNAMICS

derivatives of the PE with respect to all geometrical coordinates are simultaneously zero. Two familiar classes of features on the PES meet this criterion: minima— corresponding to reactants, products, or intermediates—and transition states. These classes are distinguished by the partial second derivatives of the PE with respect to the geometrical coordinates. The second derivatives describe the direction of curvature of the hypersurface. For a minimum, all of the partial second derivatives are positive (or, more accurately, all of the eigenvalues of the Hessian or secondderivative matrix are positive) because the PES is heading upward in all directions. For a transition state, there is one (and only one) direction in which the PES is curved downward. Computing partial second derivatives is an important part of the calculation of the vibrational frequencies for a molecule or transition state, although the calculation is usually done in so-called mass weighted coordinates, which allow convenient separation of kinetic and potential energy terms in the equations of motion. The curvature of the PES at a transition state means that it will always have a single normal mode of imaginary frequency. The atomic displacements of this normal mode serve to define the reaction coordinate.

The geometrical properties of the PES in the vicinity of a transition state mean that the steepest descent path down from the transition state (also generally calculated in mass-weighted coordinates, and called the intrinsic reaction coordinate or IRC) will usually lead only to a single reactant in one direction and a single product (or intermediate) in the other. However, a transition state can sometimes be ‘‘shared’’ by more than one reactant and/or product. One of these cases arises when the PES possesses a so-called valley-ridge inflection point (VRI).10

It is easy to construct a simple mathematical function that has such a point. One begins by plotting out a simple cubic equation of the form shown in Eq. 1.

Then, at each of the turning points of the curve one constructs a parabola in the xz plane. The parabola at the higher (larger z value) of the turning points is chosen to curve upward while the one at the lower turning point is chosen to curve downward. Now suppose that the z direction represents the PE of a molecule, while the x and y directions represent two geometrical coordinates (q1 and q2). The resulting PE surface is shown in Figure 21.3.

Because of our construction choices, both of the stationary points on this surface are transition states. However, one of them (TS1) has its reaction coordinate along the y direction (q1), whereas the other (TS2) has it along the x direction (q2). Imagine now trying to follow the IRC down from TS1 toward TS2. At first, the IRC must be along the q1 direction because that is the reaction coordinate for TS1. However, at some point, as we begin to approach TS2 the reaction coordinate direction has to change from q1 to q2. The point where this occurs is the VRI point. It is located, in this simple case, by calculating the second derivative of the PE with respect to q2 at each point along the IRC. At TS1 this quantity must be positive, since q2 is the upward-curved direction. However, as we proceed along the IRC it must decrease in magnitude, on its way to becoming negative at TS2. The

Figure 21.3. A schematic PE surface possessing a VRI point (see color insert).

only way to get smoothly from a positive quantity to a negative one is to go through zero. The place where the second derivative goes to zero is the VRI.

The chemical significance of this rather abstract idea is that the IRC bifurcates at the VRI, which is shown by the blue line in Figure 21.3. Consequently, the reactant and product that are linked by TS2 are either both products for TS1 or both reactants for TS1, depending on which way the reaction proceeds. In principle, there could be a VRI on each side of TS1, meaning that two reactants and two products would be interconverted via a single transition state. There could even be additional VRIs on either side of TS2, which would increase the number of reactants and products that could share TS1.

The VRIs are chemically relevant features on a PE hypersurface even though they do not happen to be stationary points. They represent perplexing places for traditional kinetic models, such as TST, because these models have no way of predicting what fraction of molecules will choose one path or the other at the bifurcation. In other words, TST cannot tell you what the product ratio will be in a reaction that occurs via a VRI. Several examples of such reactions are now known.11

The VRIs are not the only place where bifurcations can occur. The mechanism of the 1,2,6-heptatriene rearrangement that is summarized in Figure 21.2 also depicts a

934 POTENTIAL ENERGY SURFACES AND REACTION DYNAMICS

bifurcation (the red arrow or the blue arrow) occurring at TS1. At first sight, this would seem to contravene the requirement that the PE hypersurface be downward curved in only one direction at a transition state. However, it really does not. It is true that the steepest descent path is uniquely defined at a transition state, but that fact does not stop one from taking a nonsteepest descent path that is still all downhill but not along the IRC. For the surface depicted in Figure 21.2, the IRC begins along the direction of the blue arrow. The red arrow is a nonsteepest descent path. Why would one ever follow anything but the IRC down from a transition state? That is also a question to which transition state theory can provide no insight because the answer depends on details of reaction dynamics to be described in Section 3.

1.4. Conical Intersections

So far our discussion has presumed that, for a given reaction, a single PES is sufficient to describe all of the interesting chemistry. For thermal reactions, this is usually (but not always) a reasonable approximation. For photochemistry it never is. By its very nature, photochemistry involves the generation of one or more electronic excited states, each of which has its own PES. How excited-state surfaces are related to each other and to the ground-state surface is an issue of crucial importance to understanding photochemistry. It is an interesting historical fact that the discussion of this topic in almost all of the organic photochemistry literature was based on erroneous understanding, until about the mid-1980s. In order to grasp the somewhat challenging concept of conical intersections, which turn out to be at the root of the problem, it is perhaps useful to repeat this error and then to discuss its origins and consequences. The photochemistry of butadiene serves as a convenient example.

The photochemical disrotatory closure of butadiene to cyclobutene has been described with a state-correlation diagram, like that shown in Figure 21.4.12 It is based on the familiar orbital-correlation diagram of Woodward and Hoffmann,4 from which the intended correlations indicated by the dashed lines can readily be deduced. The solid lines indicate that there is an avoided crossing, which is put in as a result of the quantum mechanical noncrossing rule. It says that two states of the same total symmetry cannot cross. Instead, as they approach each other in energy, they will mix and separate, as the solid lines indicate.

The description of the photochemical ring closure of butadiene that derives from this picture is as follows. Absorption of an ultraviolet (UV) photon by the butadiene generates the 1A state. The 1A state evolves along the correlation line to the 1A state of cyclobutene until it encounters an allowed crossing with the excited 1S state. It hops over to that state and rolls down to the local minimum on the upper surface. From there it drops down to the maximum on the ground-state surface, allowing it either to return to the reactant or proceed on to ground-state cyclobutene.

So what’s the error in this description? It turns out to be the invocation of the noncrossing rule. It has been known for many decades that PE hypersurfaces for states of the same total symmetry actually can cross.13 Quite why this information

Figure 21.4. A state-correlation diagram depicting the avoided-crossing representation of butadiene photochemistry.

did not make it from the chemical physics community to the organic chemistry community is not clear, but its consequences are significant.

For an N-atom molecule, the PES of each state has 3N 5 dimensions (3N 6 geometrical coordinates plus the energy coordinate). Two hypersurfaces for electronic states of the same symmetry are allowed to cross in 3N 7 dimensions. Potential energy hypersurfaces for electronic states of different spin and/or spatial symmetry can cross in 3N 6 dimensions. Note that diatomic molecules (N ¼ 2) represent a special case. For them, 3N 7 ¼ 1 and so in this case the noncrossing rule is rigorously obeyed. But for all molecules of three or more atoms, crossing between PE hypersurfaces for states of the same symmetry is permitted.13 The crossing is usually depicted in the two special coordinates (i.e., the difference between 3N 5 and 3N 7) along which the degeneracy of the two states is lifted. These coordinates are usually given the somewhat forbidding names of the nonadiabatic coupling vector and the gradient difference vector.14 When projected onto these two coordinates, the crossing between the two surfaces takes on the geometry of a pair of cones touching at a point—hence the name ‘‘conical intersection.’’ It is shown schematically in Figure 21.5.

The chemical significance of conical intersections is that they provide sites of unit efficiency for return from an excited electronic state to the ground state. It turns out that the probability of (nonradiative) hopping between two electronic states is inversely dependent on the energy gap between them. So the return from the excited-state minimum to the ground-state maximum in Figure 21.4 would be a

936 POTENTIAL ENERGY SURFACES AND REACTION DYNAMICS

Figure 21.5. A schematic representation of a conical intersection between two electronic states of a molecule. Coordinates q1 and q2 are the nonadiabatic coupling vector and the gradient difference vector, along which the degeneracy between the states is lifted (see color insert).

low-probability event. By contrast, since the energy gap between states at a conical intersection is zero, the probability of returning from the excited to the ground state is unity.

Since the crossing between electronic states is itself describable as a hypersurface, there is no single molecular geometry at which this can be said to occur. Instead, it is usually assumed that the most probable geometries at which return to the ground state will occur correspond to energy minima on the crossing hypersurface. Modern computer programs capable of doing ab initio electronic structure calculations allow one to search for such structures.

The significance of conical intersections for organic photochemistry is twofold. First, their molecular geometries are frequently quite unlike any of the minima or transition states on the ground-state surface.14 Consequently, the return to the ground state can occur at structures and energies that are essentially never accessed in any thermal reaction. Second, the initial direction of the trajectories across the ground-state surface is restricted to the plane defined by the nonadiabatic coupling vector and the gradient difference vector. Inspection of Figure 21.5 may suggest that this latter requirement hardly represents much of a restriction since it would appear to allow motion in any direction at all. However, it is important to remember that Figure 21.5 is a schematic projection of a hypersurface onto just the two unique coordinates. The restriction of the initial trajectories to this plane is actually quite severe.

In the case of butadiene photocyclization, detailed computational studies have revealed that some features of the avoided-crossing model were correct, but others were not.15 As the older mechanism had suggested, there is a crossing from the initially accessed excited state to a nominally doubly excited state that is dropping in energy along a disrotatory coordinate. However, this crossing occurs at a geometry of C1 symmetry, and is therefore necessarily a conical intersection. The

disrotatory motion evolves on the new excited-state surface until a conical intersection between it and the ground state is encountered. Examination of the nonadiabatic coupling and gradient difference vectors shows that the disrotatory ring closure to cyclobutene is completed on the ground-state surface.

2. STATISTICAL KINETIC MODELS

2.1. Introduction

Construction of at least a partial PE hypersurface for a reaction is obviously an important step in the elucidation of its mechanism. Of special interest in the context of this volume is investigation of the part of the PES in the vicinity of a putative intermediate. Typically, one would like to know whether there actually is an intermediate formed during a reaction of interest, and if so what its properties are. Modern ab initio electronic-structure calculations can be very helpful in providing some of that information, but so far there are few who would be willing to accept their results as substitutes for experimental facts. The principal historical challenge for the experimental study of reactive intermediates has been that many have lifetimes that are too short to permit direct detection. The development of lasers with pulse widths in the nanosecond to subpicosecond range has gone a long way to overcoming that problem, but laser-flash experiments are still largely limited to reactions that can be initiated photochemically. Even today, there exist few good ways to detect transient intermediates directly formed in thermal reactions, because the steady-state concentrations of such species are too low, especially given the fact that they have to be distinguished from structurally related reactants and products present in concentrations that are orders of magnitude higher.

The paucity of generally applicable direct methods for observing thermally generated reactive intermediates has led over the years to the development of a variety of techniques for indirectly detecting their presence and deducing their properties. Many of these techniques depend, explicitly or implicitly, on kinetic models, particularly TST. Since the later discussions in this chapter will question the general applicability of these models, at least as they have been typically employed by organic chemists, it seems appropriate to begin by reviewing their basic assumptions.

2.2. Concepts and Approximations

2.2.1. Phase Space.16 It will be useful here to anticipate a formulation that we will use in more detail in Section 3, namely, the solution of the classical equations of motion for the atoms of a molecule undergoing a chemical reaction. One starts with a molecule of defined geometry (say, in Cartesian coordinates) and with defined velocities for each of its atoms (expressible as components in the x, y, and z directions). The problem then is to solve Newton’s second law of motion, F ¼ mA, for each atom. The force, F, can be calculated as the first derivative of

938 POTENTIAL ENERGY SURFACES AND REACTION DYNAMICS

the PE with respect to the x, y, and z coordinates of each atom, the mass, m of each atom is known, and so Newton’s law allows one to deduce the acceleration, A, on each atom. Together with its known velocity, the acceleration permits the calculation of a new position for each atom (i.e., a new geometry for the molecule) after some time step dt. At the new geometry one starts the calculation all over again. When these time steps are strung together, they define the dynamics of the mole- cule—the way that its shape and PE change with time. The change in shape could be traced out as a path across the PE hypersurface, but this would not be a complete record of the calculation, because at each point we need to know not only the molecular geometry but also the velocity components on each atom. A complete specification of these quantities for a N-atom molecule thus requires 6N variables (or 6N 12 if we are not interested in the translational and rotational motion of the molecule). The time evolution of the coordinates and velocities can be described as the path followed by a point in a (6N 12) dimensional space, called the phase space of the system. The path for the molecule is called a trajectory. In order for the reader to make a connection between this very sketchy description of classical trajectory calculations and the much more detailed ones in the cited references, it is necessary to mention one technical issue. It turns out that Newton’s second law can be transformed from a set of second-order differential equations in positions and velocities to a set of first-order differential equations in positions and momenta. This reformulation, which is due to Hamilton, permits an easier numerical solution, and so is the basis for most descriptions of trajectory calculations that one finds in textbooks. Similarly, the concept of phase space is often cast in terms of the positions and momentum components of each atom rather than their positions and velocities, but the two formulations are equivalent.

For reasons described in Section 3, solving the equations of motion is usually time consuming, even with modern computers. In the 1930s, when many of the theories of chemical kinetics were being developed, electronic computers did not exist, and so there was no hope of taking this approach for systems of any complexity. Instead, theories were developed that sought to avoid the calculation of individual molecular trajectories, while still being able to describe how an ensemble of trajectories, for a large number of molecules simultaneously, would behave on average. For the later discussion, it will be useful to describe here two different kinds of ensembles that appear frequently in the discussion of kinetic models.17 In a microcanonical ensemble, every molecule has the same total energy, but a different position in phase space. In a canonical ensemble, the molecules have a range of energies corresponding to a Boltzmann distribution at a defined temperature, and again they have different positions in phase space.

2.2.2. The Transition State Hypothesis. The general idea that a transition state is located at a saddle point on the PES, as detailed in Section 1.3, is familiar to most organic chemists. However, the original concept of a transition state started out as something rather different. In the development of both transition state and RRKM theory, the transition state was defined as the location of a plane (actually a hyperplane) in phase space, perpendicular to the reaction coordinate.18

Figure 21.6. Schematic representation of the relative phase-space volumes available to reactant, transition state, and product. A plane located at the most constricted place has the highest probability of being crossed only once by a molecular trajectory, which is the location of the transition state.

The location was chosen so that all trajectories starting on the reactant side of the phase space would proceed on to the product side with unit probability once they had crossed this plane. In other words, no trajectory should recross the plane and return to the reactant side. The proposed existence of such a plane was called the transition state hypothesis. Its location is shown schematically in Figure 21.6. The peanut-shaped object in this figure is supposed to be a symbolic representation of the volume of phase space for a microcanonical ensemble of molecules in the reactant region, product region, and between the two. For passage to occur between reactant and product regions, the total energy of our ensemble must be at least slightly higher than the value of the maximum PE along the reaction coordinate. In the reactant and product regions, where the minimum PEs are low, our molecules will consequently have considerable amounts of excess energy. That allows them to have large velocities for their atoms and/or large numbers of structures that deviate significantly from the minimum-energy ones. In other words, the volume of phasespace accessible to the ensemble is large. However, in the vicinity of the PE maximum the amount of excess energy is greatly reduced, and the phase-space consequently becomes constricted. Hence, the peanut shape. If one now imagines the trajectories of our molecules, rattling more or less randomly around in the available phase space, it is apparent that the constriction in the vicinity of the potential energy maximum should force most of the trajectories to flow roughly parallel to the reaction coordinate. Consequently, this is the location for the plane that should minimize the chance of trajectory recrossing.

The recrossing question is important because, if none occurs, the number of trajectories traversing the plane per unit time defines the rate of product formation, which is one of the fundamental quantities one wants to get from any kinetic theory. It also turns out that a famous approximation of TST—the supposed thermal equilibrium between reactant and transition state molecules—arises as a direct consequence of the nonrecrossing hypothesis.18c

940 POTENTIAL ENERGY SURFACES AND REACTION DYNAMICS

2.2.3. The Statistical Approximation. This approximation gives the statistical kinetic models their name; it is the one that permits estimations of rate constants to be made without the need for any trajectory calculations, and it is also the one whose careless application is going to be most heavily criticized in the following pages of this chapter. There are a variety of equivalent statements of the statistical approximation.18 One is to say that, for a microcanonical ensemble of reactant molecules, all states of the same total energy are equally likely to be populated throughout the reaction. Another is to say that reactant–molecule trajectories are ergodic (i.e., they explore all of the available phase space in the reactant region before passage through the transition state). A third is to say that the lifetime (time prior to reaction) of a molecule in the reactant region is random, and therefore that the population in this region decays (by reaction) exponentially. In the end, though, what these statements all imply is that the rate of intramolecular vibrational energy redistribution (IVR) is much faster than the rate of passage through the transition state.

Some classic experiments19 that have contributed to the acceptance of the statistical approximation have led to the conclusion that the effective ‘‘rate constant’’ for IVR in a typical polyatomic molecule is of the order of 1012 s 1. In other words, if an ensemble of molecules could be prepared in a defined vibrational state, they would be roughly one-half way toward having a statistical distribution of their vibrational energy (in the absence of collisions) within about a picosecond. No serious disagreement with this general picture arises from the dynamical simulations and experiments cited here. Proponents of a statistical model go on to say, correctly, that if the lifetime of an intermediate is very much longer than a picosecond, then IVR will be largely complete before it reacts, and therefore the statistical approximation will be valid by definition. It is the final step in the argument that is brought into question by the recent studies. The assumption is made that an intermediate facing a PE (or, more accurately, standard free energy) barrier to product formation of more than a few kBT, where kB is the Boltzmann constant and T is the temperature, will have a lifetime significantly longer than a picosecond, and by the preceding arguments, will consequently have suffered near-complete IVR. The problem is that the argument is circular. By associating a lifetime (or equivalently a rate constant) with a barrier height one is accepting a statistical kinetic model whose validity the exercise was supposed to be testing. In molecular dynamics simulations, one often finds that there exists no well-defined rate constant for the reaction of a thermally generated reactive intermediate—its concentration is not describable by a single-exponential decay. In particular, there can be components of the population that react much faster than would have been expected from any statistical kinetic model because the dynamics of their formation place them on more-or-less direct trajectories over the barrier(s) to formation of one (or sometimes more) of the products. At this stage in the research, it is not clear whether one can ever say that a barrier to product formation of a given magnitude will be sufficient to ensure statistical behavior of any reactive intermediate.

If the statistical approximation were correct, one could estimate the rate constant for a microcanonical ensemble of reactant molecules by estimating the volume of

phase space available to the reactant and the transition state or, in a quantized version, the number of rotational and vibrational states available in each region. It is this latter, counting approach that turns the estimation of rate constants into a statistical calculation.

2.3. RRKM Theory18

The RRKM theory is the most widely used of the microcanonical, statistical kinetic models.20 It seeks to predict the rate constant with which a microcanonical ensemble of molecules, of energy E (which is greater than E0, the energy of the barrier to reaction) will be converted to products. The theory explicitly invokes both the transition state hypothesis and the statistical approximation described above. Its result is summarized in Eq. 2

where k(E) is the rate constant, s is a statistical factor that counts reaction path degeneracies arising from possibly different symmetries of the reactant and transition state, NzðE E0Þ is the number of vibrational states of the transition state between E0 and E, h is Planck’s constant, and r(E) is the density of vibrational states in the reactant at energy E. This expression needs some modification in order to include rotational states properly (and to ensure conservation of angular momentum), but that will not be discussed further here.18d

Of course, in a thermal reaction, molecules of the reactant do not all have the same energy, and so application of RRKM theory to the evaluation of the overall unimolecular rate constant, kuni, requires that one specify the distribution of energies. This distribution is usually derived from the Lindemann–Hinshelwood model, in which molecules A become activated to vibrationally and rotationally excited states A* by collision with some other molecules in the system, M. In this picture, collisions between M and A* are assumed to transfer energy in the other direction, that is, returning A* to A:

k2

A* Products

Application of the usual steady-state approximation to this mechanism reveals that

Under the conditions used for most organic reactions, it is reasonable to assume that the concentration of collision partners, [M], is sufficiently high that