Chapter 1: Introduction

The field of molecular reaction dynamics encompasses a broad range of research areas united by the common goal of gaining a truly fundamental understanding of chemical reactivity. A single chemical event may be thought of in terms of a collision in which the species involved change their chemical identity through the cleavage and formation of chemical bonds. When we look at many such collisions, we find that every chemical reaction bears its own unique fingerprint, embodied in the kinetic energy, angular distribution, and rotational and vibrational motion of the newly formed reaction products. These quantities reflect the forces acting during the reaction, particularly in the transition state region, and their measurement often provides unparalleled insight into the basic physics governing chemical reactivity.

The ‘basic physics’ we are interested in is in principle very simple. Any collision must satisfy certain physical laws, such as conservation of energy and momentum, with the consequence that the final result of a chemical reaction is determined completely by the forces and energetics involved in the reactive collision. However, despite this apparent simplicity, the intricacies of molecular structure and dynamics mean that gaining a detailed understanding of even a fairly simple reaction presents a challenging problem. The outcome of a reactive collision may be exquisitely sensitive not only to the chemical identity and structure of the reactants, but also to their relative orientation and velocity, their electronic, vibrational and rotational states, and so on. The many variables affecting reactive collisions give rise to a rich and varied spectrum of chemical reactivity. Gaining an appreciation of the various factors at play is the focus of this book.

As an example of the types of questions that a reaction dynamics study can answer, consider an S_N2 type reaction (e.g., F⁻+CH₃Cl→CH₃F+Cl⁻) and contemplate the type of scattering dynamics we might expect to see. If we could control the reactant orientation so that the F⁻ ion strikes either the Cl or the CH₃ end of the methyl chloride molecule (which is not beyond current experimental technology) we might expect to find that in the former case we would see very little reaction, while in the latter case we would see Cl⁻ products scattered forwards with respect to the initial direction of the F⁻ ion. These products might be expected to have fairly high velocities since they are likely to receive most of the energy released in the reaction as the C-Cl bond breaks. We might find that exciting the C-Cl stretch (say with a pulse of laser light) leads to a much higher reaction cross section by making the bond easier to break, or that increasing the velocity of the attacking F⁻ increases the range of angles over which reaction can occur (often called the ‘cone of acceptance’). This would occur if it is the velocity component along the breaking bond that is important for reaction, and would lead to a broader angular distribution for the scattered Cl⁻ product.

Molecular reaction dynamics is one area of chemistry in which the link between theory and experiment is particularly strong. The bridge between experiment and theory is the reaction potential energy surface (PES), a key concept in reaction dynamics, as we will see in Chapter 2. A PES describes the potential energy of the system under study as a function both of the nuclear coordinates of each atom involved in the reaction and of the electronic state of the system. The potential energy arises from the electrostatic interactions between the electrons and nuclei involved in the reaction in a given electronic state, and is generally a complicated multidimensional function. As explained in Chapter 3, from a theoretical point of view, once the PES is known, in principle the dynamics of the reaction may be understood completely, provided that the reaction does not involve more than one electronic state of the system. Reactions involving multiple electronic states, and therefore multiple PESs, are considered in Chapter 4.

A single point on the surface represents the electronic energy of the system at a set of fixed atomic positions. Some regions of the surface correspond to reactants, some to products, and others to the transition state structures linking the two (see Section 1.4.5). The gradient of the surface at a given point determines the forces acting on the atoms at that point, defining their subsequent motion. An analogy may be drawn between the dynamics of a reaction over a potential energy surface and the trajectory of a ball bearing or marble rolling over a curved surface (this analogy simply replaces the electrostatic potential relevant to chemistry with a gravitational potential). We explore the idea of nuclear motion over a potential energy ‘landscape’ using the simple example of a three-atom reaction proceeding over a reaction PES. This is shown in ‘cartoon’ form in Figure 1.1. Since each position on the PES represents the potential energy at a particular molecular geometry, then as shown schematically in the figure, if we place a marble on the surface and track its path as it rolls, it traces out the changing geometry of the reacting molecules as they collide and scatter, effectively allowing us to construct a ‘movie’ of the sequence of events played out in the reaction. The initial position of the marble on the surface and the amount of energy imparted to it affects its trajectory in the same way as the initial geometry, relative positions and energies of the reacting molecules define the way in which they react.

Using state-of-the-art theoretical techniques, some of which are explored in Chapter 2 of this book, potential energy surfaces for reactions involving a few atoms may now be calculated from first principles to a high degree of accuracy. This involves solving the Schrödinger for the electronic motion at many geometries and interpolating the result to produce a smooth continuous surface. The dynamics of the reaction may then be predicted by carrying out either quasi-classical trajectory calculations (similar in spirit to the ‘rolling the marble’ approach described above) or full quantum mechanical scattering calculations over the surface, as discussed primarily in Chapters 3 to 7.

The ultimate test of our understanding of a chemical system, and the accuracy of a calculated potential energy surface, arises when the predictions of theory are compared with the results of experiment. There are a number of experimental probes of the potential energy surface, each of which provide information on different features of the reaction dynamics. Product quantum state distributions, velocity distributions, angular distributions, and even preferred axes of rotation are all amenable to experimental measurement and may be compared with the predictions of theory. The techniques developed to perform these measurements are the subjects of Chapters 4 to 7 of this book. The topic of stereochemistry, discussed in Chapter 9, introduces the notion of performing a complete experiment, in which the vectorial nature of linear and angular momenta are both recognized and exploited. This allows the effect of reactant bond-axis or angular momentum orientation on chemical reactivity, and the disposal of angular momentum after molecular collisions, to be fully quantified.

The above discussion has been focussed on gas phase bimolecular reactions. However, similar ideas involving the motion of particles over potential energy surfaces can be used to interpret other processes, such as collisional energy transfer (Chapter 5) and molecular photodissociation (Chapter 8), i.e. the break up of molecules subsequent to absorption of light. Chapter 10 illustrates how scattering of molecules at solid surfaces can also be tackled in an atomistic fashion, similar to that employed to interpret processes occurring in the gas phase. The final two chapters of these Tutorials provide a forward look to two important emerging areas in reaction dynamics; namely the study and control of chemical processes using femtosecond laser pump-probe techniques (Chapter 11), and the behaviour of molecules at extremely low temperatures, where only a few molecular quantum states are accessible, and classical models of reactivity are likely to be of little use (Chapter 12).

Having briefly introduced the topics to be covered over the course of the book, the remainder of this introductory chapter aims to provide an overview of some of the concepts and principles that form the basis of molecular reaction dynamics, setting the stage for the more detailed treatments of different aspects of the field provided in later chapters. We begin by introducing the idea of a reaction cross section, the ‘single collision’ version of a rate constant, and explore some of its properties. We then look at some of the experimental techniques used in reaction dynamics measurements, before considering a number of factors involved in interpreting the data from such experiments.

The rate at which reactions occur is usually quantified by either a cross section or a rate constant, depending on the type of measurement being made.^* Rate constants are most commonly used for cases in which the reactants are in thermal equilibrium, and are then usually referred to as thermal rate constants. Rate constants will be employed particularly in the discussions of Chapters 7 and 12. Cross sections are more commonly encountered in situations in which the collision energy is well defined (see Section 1.3.3), or when the system is not at thermal equilibrium. Understanding cross sections, and their dependence on energy, quantum state, or the polarization of reactants or products, is a major goal of reaction dynamics studies. You should already be familiar with the concept of a rate constant, but for many readers the idea of a reaction cross section will require further explanation.

To gain an understanding of what a cross section represents, we start with a simple classical picture of a binary collision between two reactants A and B, of mass m₁ and m₂, respectively, as illustrated in Figure 1.2. We define the impact parameter, b, as the distance of closest approach in the absence of an interaction between the two species. Although the impact parameter is a difficult quantity to control experimentally, we will see throughout this book that it plays an important role in molecular collisions.

Let us assume that the two species approach each other with a fixed relative velocity, v_rel, where v_rel=v₂−v₁ and v₁ and v₂ are the velocities of the two reactants measured with respect to some reference point (e.g., in the laboratory). Classically, the impact parameter defines the magnitude of the orbital angular momentum, , of the two reactants

where μ=m₁m₂/(m₁+m₂) is the reduced mass of the two particles, and R is the relative position of particle B with respect to particle A. Quantum mechanically, the square of the orbital angular momentum is quantized, and the magnitude of the orbital angular momentum is defined by

where  is the orbital angular momentum quantum number. In the correspondence limit, in which  takes integer values much greater than zero, Eqs. (1.1) and (1.2) provide a useful link between the classical and quantum descriptions of angular momentum. As we shall see in Chapters 3, 5, and 12, in the specific case of a collision between two structureless particles, the orbital angular momentum is conserved, but in general for more complex systems it is not (see Section 1.4.2).

The collision depicted in Figure 1.2 immediately raises two important general issues about scattering processes. It is clear that the probability of a scattering event, whether it be reaction or merely an energy transfer process, is likely to vary with impact parameter or orbital angular momentum. Collisions at high impact parameters, i.e. those which involve a ‘glancing blow’ between the two particles, are inherently less likely to lead to reaction or energy transfer than ‘head on’ collisions, which correspond to low-impact-parameter encounters. As we shall see in Section 1.4.3, the underlying reason for the difference between low and high impact parameter collisions is that only the kinetic energy associated with the relative motion of the particles along their line-of-centres is available for overcoming the reaction barrier. For high-impact-parameter collisions, the large amount of centrifugal energy associated with the orbiting motion of the two reactants cannot easily be used, for example, to promote reaction. For inelastic collisions (those in which kinetic energy is not conserved – see Section 1.4.2) and reactions, the variation in the probability of the process of interest with impact parameter is know as the opacity function, and is generally written P(b). The equivalent quantum mechanical opacity function, P(), defines the variation in the probability with orbital angular momentum quantum number, .

The second important issue suggested by Figure 1.2 concerns the azimuthal angle, ϕ, associated with v_rel lying out of the plane of the page. It turns out that if the reactants are not oriented in space, or indeed are spherical particles, then the probability of the process in question becomes independent of ϕ, and we simply need to integrate over this angle when determining the reaction cross section. Figure 1.3 illustrates the averaging involved.

As discussed in more detail in Chapter 5, the cross section for the process of interest can therefore be written as a ‘dart-board’ average over the reaction probability^*

Note that the integral over the opacity function runs from b=0, for head on collisions, to some maximum value, b_max. Beyond this value the collision becomes too glancing a blow for inelastic scattering or reaction to take place.^† As a particularly simple example, if we set the reaction probability to unity below this cut-off impact parameter the cross section becomes

In spite of the rather unfamiliar form of Eq. (1.3), it reassuringly yields a cross section with the correct dimensions of area, as expected on the basis of simple collision theory^1,2  (see Figure 1.4 and Figure 1.5). The cross section can therefore be thought of as an effective target area within which the colliding particles must approach for the particular process of interest to occur. Different types of collisional process have very different cross sections, reflecting the different effective target areas, resulting from the relevant probabilities for the various possible outcomes of a collision and the range of impact parameter for each process. It will not come as a particular surprise that reactions taking place on PESs with high barriers tend to have small reaction cross sections, since the probability of reaction is small unless reactants of sufficient energy approach in some well-defined direction. On the other hand, reactions on attractive surfaces, e.g., those occurring between oppositely charged ions or between species possessing large dipole moments, tend to have large cross sections.

By transforming the integral in Eq. (1.3) to a sum over , and using the fact that the momentum associated with the relative motion of the colliding particles is p=μv_rel=kℏ, where k is the wavenumber, (k=2π/λ, i.e. 2π times the number of waves per meter), it can be shown that the corresponding quantum expression for the cross section can be written (see Chapter 12)

In quantum mechanics the cross section can be expressed as the weighted sum of reaction probabilities for each orbital angular momentum, with the (2+1) factor arising from the degeneracy of the m_ projections, reflecting the different orientations of the collision plane indicated in Figure 1.3.

The right hand panel of Figure 1.3 shows the products scattering at a particular angle relative to the initial approach direction of the atom. The angular distribution of the products turns out to be an important characteristic feature of a given collision process. In the simple process under consideration, namely the collision between two spherical particles, the figure indicates, and indeed it is the case, that there is a one-to-one correspondence between impact parameter and the angle at which the products depart. This angle is known as the scattering angle, or specifically in this case, the centre-of-mass (CM) scattering angle (we return to a discussion of the centre-of-mass in Section 1.3.3).

For more complex systems, there is no longer a direct one-to-one relationship between impact parameter and scattering angle, but nonetheless the angular distribution of the scattered products provides valuable clues about the mechanism of the collisional process under study. As will be evident from the discussion in the preceding subsection, provided that the reactants are initially randomly oriented, the angular distribution of the products is independent of azimuthal angle ϕ, i.e. we may write P(θ,ϕ)=P(θ)/2π.

The angular distribution does not quantify the number of products scattered into a particular direction per unit time; it simply reflects the probability that the products are scattered in a particular direction assuming that they are formed in the first place. To quantify the number of products scattered into a particular direction, or more particularly their flux, one needs to define a quantity known as the differential cross section. The differential cross section, , characterizes the effective target area of the colliding particles that leads to scattering into a particular angular range, dω. The angular range is defined in terms of the solid angle ω (with units of steradian), with dω=sin θ dθdϕ. The angular probability distribution, P(θ,ϕ), is then related to the differential cross section by the equation

where P(θ,ϕ)sin θ dθdϕ represents the probability of scattering into solid angles in the range θ→θ+dθ and ϕ→ϕ+dϕ. Integration of the differential cross section over all scattering angles yields the so called integral cross section discussed already in Section 1.2.2:

As we have noted already, the above discussion assumes the reactants to be randomly oriented in space. Chapter 9 discusses, amongst other things, the interesting issue of what happens to the reactivity when reactants are polarized.

Reactions that are dominated by long range attractive forces, which take place preferentially via glancing blow collisions at high impact parameters, tend to have large cross sections and show scattering into the forward direction. For these processes, little deflection of the atoms occurs during reaction, and the products fly away at small scattering angles in a similar direction to that of the reactants. Conversely, reactions that take place via the more repulsive short range region of the PES tend to require more head-on collisions. These have small impact parameters, and lead to backward scattering of the reaction products, i.e. scattering in the opposite direction to that in which the reactants were moving. These simple concepts are developed much more in subsequent chapters, particularly in Chapters 5 and 6.

We have thus far neglected the fact that atoms and molecules have internal degrees of freedom, associated with electronic, rotational and vibration motion. Atoms and molecules often react with cross sections that vary considerably with initial quantum state. For example, vibrational excitation of a bond that is broken in a reaction often enhances the reactivity more than excitation of a ‘spectator’ bond which is preserved during reaction. Furthermore, the quantum states of the products of inelastic and reactive scattering processes are often populated in highly specific ways. As we shall see in Chapters 6 and 7, some reactions are so selective in their energy disposal that they can lead, for example, to vibrational population inversions in the products. The dependence of the cross section for a collision on initial and final quantum states is a recurring theme throughout this book, and is a particular focus of Chapter 7. If we write the state-to-state cross section as σ_if, where i and f refer to the initial and final states, then the initial state specific cross sections can be obtained simply as a sum over the final states

To obtain the integral cross section from the initial-state-dependent cross sections one needs to know what the populations are in the initial states, P(i), since the integral cross section is a weighted sum over the initial state specific cross sections

In the case that the system is at thermal equilibrium, then the populations over the initial states will simply be characterized by the Boltzmann distribution

where k_B is Boltzmann's constant, q is the appropriate partition function (e.g., for the rotational or vibrational degrees of freedom of the reactants), and g_i and ε_i are the degeneracies and energies of level i.

As already noted, it is also common to quantify the rates of state-specific processes in terms of state specific rate constants. At well-defined relative velocity the relationship between the rate constants and the cross sections is simply given by (see Figure 1.4.)

In this expression we emphasize that the cross section also depends in general on the relative velocity. Extending simple collision theory, one can think of the state-to-state rate constant as an effective volume swept out by the reactants in specific quantum states and leading to products departing in specific final states.

We return finally in this subsection to the thermal rate constant. Thermal rate constants are a primary focus of Chapter 12, which investigates chemistry at very low temperatures. As we have observed, the term thermal rate constant implies that the system is at thermal equilibrium, with each state populated according to the Boltzmann distribution law. For a bimolecular reaction, the thermal rate constant can be obtained from the integral cross section by averaging over the Maxwell-Boltzmann distribution of velocities, f(v_rel)

where

It should be clear from the above that the thermal rate constant is a highly averaged quantity, and that to gain the most insight into the dynamics of a reaction it is helpful to avoid as much as possible the blurring effects of this averaging over quantum states and relative velocity. For this reason, in most of the remainder of this book we will be using cross sections to quantify the rates of processes of interest.

Most of the experiments to be described in these Tutorials are concerned with isolated collisions of molecules, either with other atoms or molecules, with photons, or with surfaces. We need to quantify what we mean by an ‘isolated collision’, and also to indicate the experimental methods that are most commonly used to study such events.

One of the most important considerations when carrying out an experiment aimed at probing the dynamics of a reactive collision is the need to isolate the event of interest amongst a perpetual background of other collisions. Collisions generally change the direction of motion, kinetic energy or internal state of a molecule, which presents a serious problem when these are precisely the properties we wish to measure. For this reason, it is vitally important to detect the products of the collision of interest before the information they hold is lost through secondary collisions. When this is achieved, the experiment is said to be carried out under single-collision conditions.

There are a number of ways in which we can satisfy the criterion of single-collision conditions. The two general approaches are often referred to in terms of spatial isolation and temporal isolation, though a given approach is often a combination of the two. Spatial isolation is usually employed in experiments that require the newly formed reaction products to travel some distance to a detector before their properties are measured. In this case we must ensure that the mean free path of the particles (i.e. the average distance travelled between collisions) is larger than the distance the particles must travel to the detector. This is achieved by maintaining a very low background pressure inside the experimental apparatus. An example of spatial isolation is found in crossed molecular beam experiments (see Section 1.3.2), in which the background pressure is often kept at less than 10⁻⁷ Torr. At this pressure, the mean free path may be several hundreds of metres, much longer than the distance the products must travel from the reaction centre to the detector. Temporal isolation is achieved when we ensure that the mean time between collisions is longer than the time required to make a measurement. This condition is also satisfied by the crossed molecular beam example given above, but the term is often used to describe experiments carried out at much higher pressures in which products are detected almost instantaneously after their formation, usually by some kind of spectroscopic technique. An example is a laser pump-probe experiment, in which the first laser initiates reaction, usually by breaking a chemical bond, and the second probes one or more products by a spectroscopic technique such as laser-induced fluorescence (LIF) or resonantly-enhanced multiphoton ionization (REMPI) (see Section 1.3.4 and Study Box 5.2). The mean free path and time between collisions in such an experiment may be relatively short, but the use of pulsed lasers allows the pump and probe laser pulses and pump-probe delay times to be even shorter. In many experiments these times are on the order of nanoseconds, allowing measurements to be made in gases at relatively high pressures (a few tenths of a Torr – see Problem 1). State-of-the-art pump-probe experiments using femtosecond lasers allow detection over such fast time scales that measurements are no longer restricted to the gas phase, and the dynamics of liquids and other condensed phases may be probed (see Chapter 11 and Study Box 7.4).

Product angular and velocity distributions, as introduced in Section 1.2.4, were first measured using the technique of crossed molecular beams. These experiments operate on a very simple principle: two molecular beams containing the reactants are crossed, usually at right angles, reaction occurs at the point of intersection, and products scattered at a particular angle are detected by a mass spectrometer equipped with an electron-impact ion source, which ionizes the neutral products prior to entry into the mass analyzer. The flight time from the crossing region to the detector yields the product velocity, and by stepping the detector through the possible scattering angles, the entire product velocity-angle distribution may be obtained. Such an experiment is shown schematically in Figure 1.6.

The experiment described above is the classic ‘universal’ crossed beam experiment, so called because the detection technique (electron impact ionization) works for any molecule and the experiment therefore has the capacity to measure the scattering distribution of virtually any chemical species, regardless of its spectroscopy (see Section 6.2). However, in modern crossed beam experiments, the moveable mass spectrometer is often replaced with a laser-based detection scheme, as discussed in Chapters 5 and 6. Such schemes provide product quantum state selectivity and improved sensitivity, together with the possibility of measuring product angular momentum polarization effects (see Chapters 5 and 9).

The collision energy in a crossed beam experiment is determined by the speeds of the molecules in the two beams and by the angle at which they cross. Using a simple model in which the thermal energy of the molecules inside the beam source is converted into translational kinetic energy of motion along the beam direction, the terminal velocity of a molecular beam is found to be^3,4 

where T₀ is the temperature inside the source, m is the mass of the molecules in the beam, and γ=C_p/C_V is the ratio of heat capacities at constant pressure and volume for a particular gas. In the case of gas mixtures, weighted averages of m and γ are used in Eq. (1.14), and a range of reactant speeds may therefore be obtained by ‘seeding’ the reactant in different inert carrier gases (He, H₂, N₂, Ne, Ar, Kr, and Xe are common choices). This provides a relatively straightforward means by which to vary the collision energy. Whether or not the beam crossing angle can be altered depends on the design of the instrument, but there are several crossed beam experiments in existence for which the collision energy can be controlled in this way (see, for example, refs. [^5,6 ]).

The scattering distribution of the products recorded in a reaction dynamics experiment is measured in some fairly arbitrary laboratory (LAB) frame, determined by the geometry of the experiment. Extracting the scattering distribution from the experimental data collected in a crossed beam experiment generally requires a LAB-frame to centre-of-mass (CM) frame transformation. Consider the crossed beam experiment shown in Figure 1.6. The fact that the beams cross at right angles in this example is not a necessary requirement for measuring the scattering distribution. We could just as easily have crossed the beams at a different angle, in which case conservation of momentum and energy would have given the measured scattering distribution (in the LAB-frame) an entirely different appearance. We could have carried out a different experiment entirely, in which case the measured scattering distribution would change again. However, no matter what experiment we choose to carry out, we are probing the same chemical process. The scattering distribution provides a ‘fingerprint’ for a chemical reaction in much the same way as a spectrum provides a ‘fingerprint’ for a molecule, so there must be some way in which we can compare the scattering distributions from different experiments and obtain chemically meaningful information. As it turns out, all we need to do is to transform the results into the centre-of-mass (CM) frame. In this frame, the collision is seen from the viewpoint of an observer travelling along with the centre-of-mass of the system, and the two reactants appear to undergo a collision at the position of their centre-of-mass. This transformation is illustrated for a crossed beam experiment in Figure 1.7.

The CM-frame is independent of experimental geometry, allowing results from different types of experiments to be compared, and also provides a much more intuitive picture of the collision dynamics. It is helpful at this stage to work through some of the details concerning the LAB to CM transformation.

Before the collision, the two particles have LAB-frame velocities v₁ and v₂, and kinetic energies and , such that the total kinetic energy is

Because the CM-frame is simply the frame in which we are travelling along with the centre-of-mass of the system, all we need to do to determine the velocities of the particles in the CM-frame is to subtract the velocity of the CM from v₁ and v₂. We will call the resulting CM-frame velocities u₁ and u₂ to differentiate them from the LAB velocities.

We can determine v_CM by using the fact that the total momentum may be written either as the momentum of the centre-of-mass, or as the sum of the momenta of the two individual particles:

where M is the total mass of the two particles. This rearranges to give

In the above, we have defined a momentum associated with the motion of the centre-of-mass. We can also define the kinetic energy associated with this motion.

In the absence of an external force (e.g., as might be applied by an external electric or magnetic field), the momentum of the centre-of-mass must be conserved. Hence, the velocity, momentum, and kinetic energy of the centre-of-mass are constant throughout the collision (this is true for any type of collision, whether elastic, inelastic, or reactive). Energy ‘tied up’ in the motion of the centre-of-mass is therefore not available for the collision. For a reactive collision, this energy does not help to overcome any activation barrier that may be present.

We have now defined the total kinetic energy and the kinetic energy associated with the motion of the centre-of-mass. The remaining kinetic energy is the energy associated with relative motion of the two particles. This energy is available for the collision, and consequently, it is often referred to as the collision energy, E_c, or sometimes the ‘CM-frame kinetic energy’.

Having defined all of the relevant parameters involving the reactants we now turn to the products. The product velocities may be determined by requiring that momentum and kinetic energy (or total energy in the case of an inelastic or reactive collision) are conserved during the collision. Usually it is most straightforward to carry out this calculation in the CM-frame, though the same results are obtained if it is carried out in the LAB-frame;

For elastic and inelastic scattering, the masses of the reactants and products are the same , whilst for reactive scattering the reactant and product masses will generally differ. Note also that in a treatment of inelastic or reactive scattering, the second of the above equations, requiring conservation of energy, would contain contributions from reactant and product internal energies and reaction endo- or exoergicities, as discussed in Section 1.4.2. At this point we have two equations in the two unknowns and (the final CM-frame velocities). Remember that in the CM-frame the total momentum is zero both before and after the collision; this simplifies the solution of these equations considerably, since the first equation becomes simply .

Once we have determined the CM-frame velocities of the collision products, we can find the equivalent LAB-frame velocities simply by adding on the velocity of the centre-of-mass (which, as we have already seen, stays constant throughout the collision):

Further details concerning the LAB to CM transformation can be found in Chapter 6 and Study Box 6.1, whilst details concerning the mathematics of changing variables of integration are given in Study Box 1.1.

The pump-probe experiments to be described here have their origins in the technique of flash photolysis, pioneered by Norrish and Porter in the 1950's and 60's.⁷  The basic idea is to excite a molecule with an intense pulse of light, and then monitor one or more of the different processes that the molecule undergoes subsequently. This is generally achieved using a spectroscopic technique, such as absorption or fluorescence. If the excited molecule undergoes dissociation to produce atoms or radicals, these transient species may be studied directly, or they may be allowed to react with other species. In the latter case, either the rate of these secondary reactions or the nature of the products formed might in turn be studied.

The advent of tuneable lasers (see Study Box 1.2) has opened the way to a whole range of new experimental techniques, many of which employ the basic principles of flash-photolysis experiments. Laser-based studies often rely on a two-laser ‘pump-probe’ approach, in which the first (‘pump’) laser initiates reaction and a short time later the second (‘probe’) laser detects the products using a spectroscopic technique such as LIF or REMPI (see Study Box 5.2). As an example, reactions of fast atoms are often initiated by UV photolysis of a diatomic precursor, with the speed of the atoms being determined by the bond dissociation energy and the laser wavelength. To study reactions of fast chlorine atoms, for example, Cl₂ is often dissociated at 308 or 355 nm, giving Cl atom speeds of 2040 or 1641 m s⁻¹, respectively.^*

Since we require single-collision conditions (see Section 1.3.1) for a reaction dynamics experiment, the maximum permissible delay between the pump and probe lasers is determined by the average time between collisions for the molecules in the gas sample. This is typically a few tens of nanoseconds for a gas-phase pump-probe experiment, depending on the pressure. However, much shorter delays are possible. Chemical reactions usually occur over the course of a few tens to a few hundreds of femtoseconds. The development of femtosecond laser sources has opened up an entirely new field of chemistry known as femtochemistry, in which pump-probe experiments on the femtosecond timescale are used to probe chemical reactions in a time-resolved manner. These experiments literally allow the course of the reaction, and the changes in energy level structure as the system evolves from reactants to products, to be probed in real time as the reactive collision occurs. As noted in Section 1.3.1, the extremely short time scales accessible with femtosecond lasers also make it possible to carry out pump-probe measurements on liquids and other condensed phases, as discussed in Chapter 11 and Study Box 7.1.

We have thus far introduced some of the basic machinery and language of molecular reaction dynamics, and given an indication of the ways in which dynamics experiments are generally performed. Next we might ask whether there are any simple guiding principles that can help with the interpretation of the results from such experiments. Fortunately there are, and they are principles which will be familiar to many students of physics and chemistry.

Perhaps the most important approximation in chemistry is the Born-Oppenheimer approximation, in which the motions of the electrons and nuclei are assumed separable. Because nuclei are much heavier than electrons, they may be treated as approximately stationary on the timescale of electronic motion. Similarly, the electrons respond essentially instantaneously to motions of the nuclei, and are said to adiabatically follow the nuclear motion.^* This allows the total wavefunction of the system Ψ(r, R) to be written in the product form

Here χ_n(R) is the nuclear wavefunction, describing the nuclear positions in terms of nuclear coordinates R, and ψ_e(r; R) is the electronic wavefunction, describing the positions of the electrons. The electronic wavefunction depends on the coordinates of the electrons, r, and also parametrically on the nuclear coordinates, R. As we will see in detail in Chapters 2 and 4, solution of the Schrödinger equation for the motion of the electrons

yields the total electronic energies, E(R) (the electronic energy levels), for a particular nuclear geometry. If calculations are carried out for many nuclear configurations, the R dependence of E(R) can be mapped out. For a particular electronic state, E(R) is effectively the potential energy that the nuclei experience at a given configuration, and is in fact the familiar potential energy surface of Section 1.1. It is usually given the symbol V(R). Note that each electronic state of the system will have a different PES. As we saw qualitatively in Section 1.1, the potential energy surface is of use because it can be used to determine the forces between particles at any given configuration. More quantitatively, the force on the system is simply the negative gradient of the potential. For example, in one dimension we have

whilst in general we may write

Once the PES is known then, classically, one has to solve Newton's equations subject to a given set of initial conditions to obtain the motion of the nuclei, or the ‘trajectory’, over the surface. Classical methods to determine the dynamics of nuclear motion are discussed further in Chapter 9. To solve the nuclear problem quantum mechanically is quite a complex task, requiring solution of the Schrödinger equation for the motion of the nuclei, and is the main topic for discussion in Chapter 3.

As outlined above, the Born-Oppenheimer approximation leads to the important concept of the electronic state, and of the potential energy surface which defines the variation in electronic energy of a particular electronic state with nuclear coordinates. However, there are occasions when the Born-Oppenheimer approximation fails, and it is important to understand the underlying reasons for such non-Born-Oppenheimer (or non-adiabatic) behaviour. This is the main topic of Chapter 4.

In Section 1.3.3 we touched on the important roles played by the conservation of energy and linear momentum. For inelastic and reactive collisions only the total energy is conserved in general; the partitioning between kinetic and internal energy before and after such a collision is illustrated in Figure 1.9. For a simple A+BC collision, the total energy, E, can be written

where E_c is the collision energy (see Section 1.3.3), E_r and E_v are the rotational and vibrational energies before collision, and , and are the translational, rotational and vibrational energies after collision. The energy available to the products, E_avl, is the sum of the product energies , and one often talks about the fraction of the available energy deposited into a particular degree of freedom. For example, the fraction of the energy released into vibration could be calculated from the equation .

Another important conserved quantity is angular momentum. We have already noted that in the absence of external forces the orbital angular momentum, , is conserved in the elastic scattering of structureless particles. More generally, account also needs to be taken of the internal rotational and/or electronic angular momentum of the colliding particles, and it is only the total angular momentum, J, which is conserved. For a simple A+BC→AB+C reaction, if we neglect electronic angular momentum of the atoms and molecules, which is generally only a small contribution, angular momentum conservation can be written

where j_i refer to the internal rotational angular momentum of species i, and  and ′ are the initial and final orbital angular momenta, respectively.

In many cases the constraints of angular momentum conservation are more restrictive than implied by Eq. (1.28). There are two reasons for this. Firstly, many experiments in reaction dynamics employ molecular beam expansion techniques, and the internal degrees of freedom of molecules entrained in such beams are often cooled down to temperatures as low as a few Kelvin.^* Under these conditions, the initial rotational angular momentum can often be neglected and we may write

The mass combination may also play an important role, something which is usually referred to as a kinematic effect. If we label heavy and light atoms as and , respectively, then for a light atom transfer reaction, i.e.

one might expect rather low levels of rotational excitation of the products, because of the low moment of inertia of the ‘’ AB molecule, and hence its wide rotational energy level spacing. Under these circumstances, together with the initial condition of low rotational excitation, angular momentum conservation reduces to

We see that in this case initial orbital angular momentum is channeled preferentially into final orbital angular momentum. Given the link between the orbital angular momentum and relative velocity (see Eq. (1.1)), it is perhaps not surprising that light atom transfer reactions also often display a propensity to conserve kinetic energy (such that ).

Another class of reaction, that involving a light attacking or departing atom, also displays characteristic angular momentum constraints. In the latter case we have

and the product orbital angular momentum is generally small compared with the AB rotational angular momentum, again due to the much lower moment of inertia involved. The resulting angular momentum constraint is

and initial orbital angular momentum is channeled preferentially into product rotational excitation. This has the practical consequence that a measurement of the rotational population distribution of the product AB molecule can be used to determine the opacity function of the reaction, P() or P(b).⁸ 

The role of kinematics in bimolecular chemical reactions is returned to again in several places in these Tutorials, but particularly in Study Box 7.1, which discusses the use of mass weighted coordinates for potential energy surfaces.

Angular momentum plays another important role in molecular collisions. Associated with the orbital angular momentum is a centrifugal kinetic energy, which at a separation R between the reactants is given classically by^*

If the orbital angular momentum is conserved during the collision, then one may think of the reaction as occurring on an effective potential

which can possess a centrifugal barrier, even in the absence of a barrier on the potential energy surface. This is illustrated in Figure 1.10.

Although, the orbital angular momentum is only rigorously conserved in the case of elastic scattering of atoms, the centrifugal barrier remains an important feature of all binary collisions, and will be returned to frequently in this book, particularly so in Chapter 12. Reactants must overcome the barrier in the effective potential if reaction is to occur. This centrifugal barrier may be overcome if the reactants have sufficient kinetic energy along the radial coordinate – known as the radial kinetic energy. In addition, quantum mechanically the reactants can tunnel through the barrier in the effective potential. Finally, as we have seen, for collisions between real atoms and molecules the orbital angular momentum may not be conserved, and can be exchanged with other sources of angular momentum (e.g., molecular rotation), such that only the total angular momentum is conserved. This angular momentum exchange provides another means by which the barrier in the effective potential may be overcome.

As a simple example of the relevance of the effective potential, consider the line-of-centres model of chemical reactions. Imagine two spherical reactants, A and B, undergoing a collision as in Section 1.2.1. Assume that reaction can only take place if the kinetic energy along the line-of-centres, the radial kinetic energy, is greater than zero at the location of the barrier, R₀, i.e.

where . In the line-of-centres model we may calculate the radial kinetic energy at any separation R simply as the initial collision energy minus the effective potential, i.e.

If the interaction potential, V(R), has a significant barrier (see Figure 1.10(a)), then it is reasonable to assume that the barrier in the effective potential will be located close to that in V(R), i.e. we need only consider the barrier in the effective potential at R₀, and we can therefore write (from Eqs. (1.34) and (1.35))

where E₀ is the barrier height on the potential curve, located at R₀. Substituting Eq. (1.38) into Eq. (1.37) and rearranging yields an equation for the maximum impact parameter at which reaction can occur

If we assume that reaction occurs with unit probability provided that the radial kinetic energy at the barrier is greater than or equal to zero, then Eq. (1.4) may be used to estimate the cross section as

This equation predicts a cross section that increases monotonically with collision energy, as shown in Figure 1.11, with a threshold at the energy of the barrier E₀, and an asymptotic cross section at high energy of . The variation in the cross section with collision energy is usually referred to as an excitation function. The physical reason for the rise in the cross section with collision energy is that higher energy collisions have more energy to surmount the centrifugal barrier in the effective potential, and reaction can hence take place over a wider range of impact parameters. The functional form derived from the simple line-of-centres model is qualitatively observed in many reactions with high potential energy barriers (see also Problem 3 of this chapter). As we will see in Chapter 12, the same treatment can be used to model reactions without barriers, which dominate chemistry at low temperatures.

Another important general factor in determining both the rate of a chemical reaction or energy transfer process, and the utilization and disposal of energy, is whether or not the reaction behaves statistically. At fixed energy, the term statistical is taken to mean that each quantum state of the system is equally accessible. Whether or not a reaction behaves statistically often comes down to a question of timescales, and in particular the timescale for the process of interest compared with that required to randomize the energy. The principle mechanism for the latter is intramolecular vibrational redistribution (IVR), a process which is discussed in detail in Chapter 7. IVR is responsible for the flow of internal energy around a molecule. If one excites a particular bond in a molecule, by direct absorption of light, for example, then that energy tends to leak out of the bond initially excited due to the coupling between the different vibrational modes of the molecule. If the timescale for reaction is long compared with that for IVR then the internal energy in the system becomes randomized (or ‘statistical’) over the course of the reaction, and there is usually a clear signature of that randomization in the reaction products, for example, in the population distributions over the product quantum states. On the other hand, if the reaction takes place on a timescale faster than IVR, then it is said to be under dynamical control, and, for example, the shape of the potential energy surface will dictate the disposal of energy in the reaction products, as opposed to the number of accessible product quantum states.

Statistical reactions are usually those which take place on potential energy surfaces with deep wells, such that the reaction can be thought of as occurring via an intermediate or long-lived collision complex. The timescale for such reactions can extend for many vibrational periods, and the intermediates may be long-lived even on the timescale of molecular rotation (i.e. picoseconds). In the latter case, the angular distribution of the reaction products can provide a useful signature of complex formation, as discussed in Chapter 6.

An important issue addressed in both Chapters 7 and 11 is whether or not it is possible to ‘beat’ IVR, and to perform state or bond selective chemistry, or even control the outcome of chemical reactions before IVR occurs. In some cases the answer is ‘yes’: IVR is insufficiently quick on the timescale of the reaction for complete randomization of energy to occur, and in such cases state and bond selective chemistry is to be expected. It is also possible for the experimenter to actively intervene by probing the reaction after a short delay time, before IVR is complete. As we will see in Chapter 11, it is now becoming possible to tailor short pulses of light to cause dissociation of a molecule in a particular way, and this can only be achieved if IVR occurs on a longer timescale than the excitation pulse employed.

Transition state theory is an example of a statistical theory of reaction rates, and is widely applied to both bimolecular and unimolecular reactions. Transition state theory is covered in many texts on kinetics and dynamics (see, for example, refs. [^2,9–11 ]), and it will not be a primary focus of this book. The concept of the transition state is nonetheless an important one, particularly for the discussions in Chapters 6 and 7, and it will be touched on repeatedly in the pages that follow.

The transition state of a chemical reaction is defined formally as a dividing surface separating reactants from products (see Figure 1.12). The potential energy often passes through a maximum along one degree of freedom (where the barrier along the reaction coordinate is located), and a minimum along all other directions. This feature corresponds to a saddle point on the PES, and is discussed further in Chapter 2. The location of the saddle point is usually close to where the transition state is placed. The transition state can be thought of qualitatively as a ‘bottleneck’ through which the reactants must pass in order for them to become products,^* and the shape of the PES in this region tends to determine the overall reaction rate. In transition state theory, one attempts to estimate the rate of reaction by simply considering the properties of the reactants and the transition state alone, without taking account of any other details of the PES. Transition state theory is often referred to as a ‘statistical rate theory’ because the reaction rate is calculated on the basis of a statistical estimate of the probability of the reactants reaching the transition state. Once the reactants reach the transition state, the reaction is assumed to proceed with unit efficiency.

For the purposes of reaction dynamics, it is often more common to think of a transition state region on the PES, rather than a formal dividing surface. This is the region where the reactants are close enough to interact with each other, and consequently where chemical bonds are made and broken. In the majority of this book we will be using the term ‘transition state’ in this rather looser sense. The reactants only sample the transition state region of the PES for a time on the order of a few tens to a few hundreds of femtoseconds; transition states are amongst the most transient of chemical species, and studying them experimentally presents a huge challenge.

To date, two approaches have been developed that allow transition states to be probed directly. One is the femtosecond laser pump-probe technique pioneered by Zewail,¹²  introduced briefly in Section 1.3.4, and discussed more fully in Chapter 11. Femtochemistry experiments of this type allow the reaction to be followed in real time, and earned Zewail the 1999 Nobel Prize in Chemistry. The second approach, developed by Neumark and coworkers at Berkeley,¹³  is now widely known as transition state spectroscopy. Neumark's technique, illustrated schematically in Figure 1.13(a), takes advantage of the fact that for certain reactions, while the transition state is, by definition, extremely unstable, adding an electron results in a stable negative ion with a nuclear geometry often quite similar to that of the transition state of interest. This has the consequence that a transition state may be prepared by forming the corresponding stable negative ion and then using a UV laser pulse to photodetach an electron to form the desired transition state species. The total energy imparted to the electron is fixed by the wavelength of the laser pulse, and is partitioned between the energy required to reach the final quantum state of the neutral transition state species and the kinetic energy of the ejected electron. The kinetic energy distribution of the ejected electrons therefore mirrors the energy level structure of the transition state, and may be analyzed to reveal its nuclear configuration, vibrational frequencies, and lifetime.

One of the most celebrated examples of transition state spectroscopy in action is summarized in a joint experimental and theoretical study carried out in 1993 on the ‘benchmark’ reaction F+H₂→HF+F.¹⁴  As shown in Figure 1.13(b), the electron energy spectrum following photodetachment from the ion showed strong evidence for a bent transition state rather than the colinear transition state that had previously been assumed, shedding new light on the results of many earlier studies. Further experiments which help to reveal the nature of the transition state region are discussed in particular in Chapters 6 and 7.

In the preceding sections we have set the scene for the remaining Tutorials on molecular reaction dynamics. We have introduced some of the ‘language’ of reaction dynamics that you will find in the following pages, and have highlighted a number of the key concepts and themes to be investigated. We hope that the material covered in the following Tutorials will be of use to you in your studies and future research, and that it will inspire some of you to embark on new projects and initiate novel fields of research in the future.

1. 1. Determine the collision frequency of an OH radical in 100 mTorr of Ar. Assume that the collision cross section is 50 Ų.

   2. Use Poisson statistics to estimate the fraction of OH radicals that have undergone zero, one, or two collisions at a time delay of 50 ns after their photolytic production (e.g., via the photodissociation of hydrogen peroxide).

2. Use the data in the table below to estimate

   1. the total energy available to the products,

   2. the maximum orbital angular momentum quantum number, _max, and

   3. the rotational energy of the KI product if , where is the rotational angular momentum quantum number for KI.

   Identify any assumptions made in obtaining the estimate in (b).

   	I2	KI
   D0/kJ mol−1	149	319
   ωe/cm−1	214.5	186.5
   Be/cm−1	0.037	0.061

   	I2	KI
   D0/kJ mol−1	149	319
   ωe/cm−1	214.5	186.5
   Be/cm−1	0.037	0.061

   View Large

   [The mean vibrational energy of the I₂ reactants may be calculated assuming with θ_v=hcω_e/k_B.]

3. Ec/kJ mol−1	9	15	30	50
   σr(Ec)/10−20m2	0.5	1.25	2.0	2.2

   Ec/kJ mol−1	9	15	30	50
   σr(Ec)/10−20m2	0.5	1.25	2.0	2.2

   View Large

   1. Show that the cross section data are consistent with the line-of-centres model

      
      and determine the threshold energy, E₀, and the limiting, high collision energy cross section, πd².

   2. Use this equation to obtain a line-of-centres expression for k(T). Comment on the result you obtain. You may use the following integral without proof

      

4. Explain how the constraints imposed by the conservation of angular momentum influence the disposal of rotational energy in the reaction

   
   This reaction has been studied under crossed molecular beam conditions, at a reactant relative velocity, v_rel=976 ms⁻¹; the rotational state distribution in the product, BaI, was found to peak at the value j′=420. Given the orbital angular momentum of the reactants in this reaction can be written ||=μv_relb, estimate the most probable impact parameter, b, and the reaction cross section. [Take the masses to be m_Ba=137.3 u, m_H=1.0 u, and m_I=126.9 u.]