We examine in this chapter how the motion of the reactants as they approach each other governs chemical reactivity. This allows us to use a two-body point of view where the internal structure of the colliding species is not explicitly recognized. All that we can do therefore is lead the reactants up to a reaction. But we will not be able to describe the chemical rearrangement itself nor to address such questions as that of energy disposal in the products. Chapter 5 takes up these themes. On the other hand, without the approach of the reactants there cannot be a bimolecular reaction. The tools already at our disposal are sufficient to discuss this approach motion. As expected, the striking distance that we have called the impact parameter will be a key player. We do all of this in Section 3.2. What we will obtain is information about the dependence of the reaction cross-section on the collision energy.

In chemical kinetics one characterizes the role of energy in chemical reactivity by the temperature dependence of the reaction rate constant. In Section 3.1 we review the input from chemical kinetics – the Arrhenius representation of the rate constant – then go from the rate constant to the reaction cross-section. Next we go in the opposite direction, from the microscopic reaction cross-section to the macroscopic rate constant. What we obtain thereby is the Tolman interpretation of the activation energy as the (mean) excess energy of those collisions that lead to reaction.

This chapter explores how the directed nature of chemical bonding affects molecular collisions and chemical reactions. Whereas the concept of size or cross-section leads to numbers (scalar quantities), the concept of chemical shape leads to vectors (numbers tied to directions). Consequently, this topic is more mathematically challenging, but its study yields important insights into the nature of chemical transformations. As expected, just as the size of a molecule depends on the probe selected to measure this quantity, the chemical shape of a molecule also depends sensitively on the probe chosen for the measurement. It is important to stress that the physical shape of a molecule is most commonly determined from molecular spectroscopy; it is usually expressed in terms of bond angles and bond lengths. The chemical shape of a molecule refers to the apparent size and shape of a molecule as experienced by another atom or molecule that collides with it.

The steric factor and early history of stereodynamics

The notion of spatial requirements for a chemical reaction dates to the introduction of a steric factor p in simple versions of the collision theory of reactions, Section 3.2.5. This was forced upon the theory by the smaller than expected, or even much smaller, overall magnitude of the reaction cross-section. Unfortunately, the early theory gave no clue for determining the value of p, which has largely been treated as an adjustable parameter to make simple collision theory agree with observation.

“chem·i·stry (kem′ i strë), n., pl.-tries. The science that deals with or investigates the composition, properties, and transformations of substances and various elementary forms of matter.” The dictionary definition emphasizes chemical transformation as a central theme of chemistry.

By the end of the nineteenth century, the young science of physical chemistry had characterized the dependence of the rate of the chemical transformation on the concentrations of the reactants. This provided the concept of a chemical reaction rate constant k and by 1889 Arrhenius showed that the temperature dependence of the rate constant often took on the simple form k = A exp(−Ea∕RT), where A is referred to as the pre-exponential factor and Ea as the activation energy. Arrhenius introduced the interpretation of Ea as the energetic barrier to the chemical rearrangement. Only later did we understand that reactions also have steric requirements and that the Arrhenius A factor is the carrier of this information.

It was next realized that the net transformation often proceeds by a series of elementary steps. A key progress was the identification of the reaction mechanism, which is a collection of elementary processes (also called elementary steps or elementary reactions) that leads to the observed stoichiometry and explains how the overall reaction proceeds. A mechanism is a proposal from which you can work out a rate law that agrees with how the observed rate of the reaction depends on the concentrations.

Molecular reaction dynamics unfolds the history of change on the molecular level. It asks what happens on the atomic length and time scales as the chemical change occurs. This book is an introduction to the field.

Molecular reaction dynamics has become an integral part of modern chemistry and is set to become a cornerstone for much of the natural sciences. This is because we need a common meeting ground extending from nanoscale solid state devices through material and interface chemistry and energy sciences to astrochemistry, drug design, and protein mechanics. For some time now the quantitative understanding on the molecular level has provided this common ground. At first, the scaffolding was the concept of the molecular structure. Once we understood the spatial organization we felt that we had an entry to real understanding. The required input was provided by the different experimental methods for structure determination and, from the theory side, by quantum chemistry and by equilibrium statistical mechanics. But now we want more: not just the static structure, we also ask how this structure can evolve in time and what we can do to control this evolution. We want to write the history of the change or, better yet, to be a conductor and orchestrate the motion. This is what this book is about.

In going from statics to dynamics we need new experimental tools and also theoretical machinery that allows for the dependence on time.

In this chapter we follow the chemical change as it unfolds in time. We have three primary motivations. The first is that the time-dependent view matches the way most of us think about a reaction. We have an image of atoms moving, changing partners, etc., rather like a movie on a molecular scale. Time-resolved experiments provide insights that are intuitively appealing. On a more technical level, working in the time domain allows us not only to access the transition state region but also to probe the system as it exits from the transition state. We can see how things evolve rather than just the integrated effect of the dynamics as revealed by a post-collision analysis. Thirdly, time-resolved experiments reveal what happens at short times and this kind of information is otherwise hard to come by experimentally.

In earlier chapters we discussed experiments at a well-defined energy; we know where we start before the event and we can probe what happens well after it. But we have to infer what happens in the middle. In this chapter we discuss time-resolved experiments, experiments that are able to probe what the nuclei are doing throughout. This then paves the way for studies in the condensed phase where time-resolved experiments are a main tool.

To implement our program we have first to address two key issues, one of principle and one of practice. The uncertainty principle inherently imposes a loss of energy resolution when the time resolution becomes better.

We address the question that every chemist asks: given thermal reactants, how do you compute the rate of crossing the barrier toward the products? We seek to cast the answer using traditional tools and, specifically, the structure of the system at the barrier. Using just one, physically realistic, approximation, transition state theory enables us to do that. The theory identifies a bottleneck for the reaction and computes the rate of passage through it.

The success of transition state theory inspires us to do more. We shall, but we require additional assumptions to be made at each point where we seek a generalization. The most pressing reasons for doing this are that there may be more than one barrier separating the reactants and products and that there can be multiple reaction paths. The case of the O2 + C2H5 reaction, shown in Figure 5.6, represents the norm rather than the exception. Transition state theory allows us to compute the rate of barrier crossing, but to get to the products we may need to cross several barriers and/or take different paths. It is for this reason that quantum chemists have grown proficient in computing the structures at each barrier (and each hollow). But we still need to know how to compound the effects of multiple bottlenecks to obtain the overall reaction rate – and this task calls for either dynamical computations as in Chapter 5 or for an additional assumption as introduced in Section 6.2.

The collision of particles without internal structure is the simplest model for interacting molecules. For the bulk, the model can account for the deviations from ideal gas behavior all the way to the formation of clusters. For systems in thermal disequilibrium the model describes the relaxation back to equilibrium. What is missing from the model is chemistry, that is the internal atomic configuration of the molecules. We will not forget this key point but we need to develop a language for thinking about how reactions take place. In order to undergo a reaction, the two reactants need to get close to one another and it is this approach motion, unhindered by any environment, that is discussed in this chapter. The angular distribution of the particles as they exit from the collision serves as a probe for the forces that acted between them when they were close by. Knowledge of these interactions is also needed for the prediction of the properties of liquids and solids and for understanding the conformation of large molecules.

In this chapter we use a two-body, A + B, point of view. But we have to leave the familiar vibration and rotation of a bound diatomic AB and go to the unbound or continuum motion. It is the vibrational displacement that is unbound. The rotation of the bound diatomic remains a rotation of the A-B axis but we will have to recognize that during the collision the A-B center-to-center distance varies over a wide range.

Heterogeneous catalysis is responsible for a significant fraction of the output of the chemical industry. The kinetics of surface chemical reactions have therefore been studied extensively. One must not however think that surface processes are synonymous with catalysis. The study of lubrication, known under the modern name of tribology, is very active. For example, what determines the rate of information storage and retrieval is the speed with which the reading head can move over the hard disk at a very low elevation. Other familiar examples of surface processes, such as corrosion, come to mind. Microelectronics and nanostructures on surfaces are also benefiting from and contributing to progress in surface science. An important development is that imaging techniques initially introduced for the probing of surface structure on the atomic scale are revealing details about reaction dynamics. Scanning tunneling microscopy (STM) has been particularly useful.

Our intent in this chapter, as in the rest of this volume, is to examine the molecular-level description. We shall thus make no attempt to review the extensive literature on the macrolevel description but proceed immediately to the microlevel, considering first elastic and energy transfer collisions and then “pre-reactive” and reactive collisions. Many of the experimental and theoretical techniques are closely related to those used to study collisions in the gas phase, but both the experiments and the collision dynamics per se are more complicated because the surface is never really passive and may even be one of the reactants, as in the etching of silicon for microelectronics.

So far, the energy necessary for a change to take place had to be brought in by the reactants. In this chapter the required energy is provided by light. Often, light allows us to promote the system to an electronically excited state where the potential and hence the dynamics are different from those of the ground electronic state. But light-induced chemistry or “photochemistry” is not only a new way of driving chemical reactions. Photochemistry offers a degree of control – a selectivity – that is unique in its potential and variety. Our ability to tailor light toward specific applications is part of the secret. This control ranges from a simple requirement such as wavelength or polarization selection that allows us to access a particular excited state to more complex tasks such as manipulations that produce short light pulses that can freeze the motion and to pulse shaping in frequency and time for the purpose of guiding nuclear motions. Another important aspect is the ingenuity of scientists in matching up light and matter. In particular we have learned to take advantage of the changing character of the intramolecular dynamics of molecules as the energy increases. At low levels of excitation polyatomic molecules offer us a great individuality through their spectroscopic fingerprints, whereas at higher levels of excitation they have a quasi-continuum of vibrational states and can be made to take up lots of energy like a good heat sink. This saga is nowhere near to a closed chapter.

The presence of a solvent interacting with a system throughout its evolution from reactants to products brings about qualitative changes from the corresponding gas-phase reaction. There are changes in both the reaction rate and the dynamics. The energetic effects due to the solvent reflect the electronic reorganization that takes place as the system transverses the reaction path.The SN2 ion–molecule reaction, shown in Figure 11.1 for the generic X- + CH3X exchange reaction, provides an example in which the charge delocalization in the transition-state region causes qualitative changes in the energy profile along the reaction coordinate when the reaction is in the presence of a polar solvent. As becomes clear in this chapter, even the reaction coordinate itself is not exactly the same in solution as it is in the gas phase, because the solvent adaptation to the changing system must also be considered.

A solute molecule at room temperature undergoes of the order of 1013 collisions per second with solvent molecules. The solvent can therefore hinder the large-amplitude motions that often accompany chemical transformations (e.g., as in a twist isomerization, Figure 11.14). The cage effect, where the solvent hinders the separation of the products, Figure 1.8, or the approach of the reactants, was one of the first examples of the role of the solvent. The cage effect remains a major difference between gas-phase and solution dynamics. There can also be dynamical effects.

In this chapter we recognize that reactants (and hence products) have internal structure. We need to describe the potential energy that gives rise to the forces that act during the collision and to determine the dynamics. With this background we examine what features of the potential energy play a special role in the dynamics and how the internal states of the reactants participate. The rate of chemical reactions for polyatomic reactants is discussed in Sections 6.1 and 6.2. The derivation therein makes essential use of two features of the potential that we examine in this chapter. These are the potential energy barrier that separates reactants and products and the not unusual possibility that there are two (or more) barriers, between which there must be a hollow. To fully appreciate the approximations that are made in deriving expressions for the reaction rate that are based only on structural information, it is necessary to examine the dynamics, as we do in this chapter. We begin by reminding you about the Born–Oppenheimer separation of electronic and nuclear motions that underlies our work in this chapter, and the possibility of its failure.

The Born–Oppenheimer separation: a caveat

This chapter is based on generalizing the concept of an interatomic potential to the polyatomic case. Specifically, we are interested in many-atom systems that have enough energy to undergo a chemical change. It is therefore important to point out that the concept of a unique potential function governing the motion of the atoms is an approximation.