Applications and Extensions of Statistical Theories

The RRKM rate constant as given by equation (6.73) in the previous chapter is expressed as a ratio of the sum of states in the transition state and the density of states in the reactant molecule. An accurate calculation of this rate constant requires that all vibrational anharmonicity and vibrational/rotational coupling be included in calculating the sum and density.

Potential Energy Surfaces

Properties of potential energy surfaces are integral to understanding the dynamics of unimolecular reactions. As discussed in chapter 2, the concept of a potential energy surface arises from the Born-Oppenheimer approximation, which separates electronic motion from vibrational/rotational motion. Potential energy surfaces are calculated by solving Eq. (2.3) in chapter 2 at fixed values for the nuclear coordinates R. Solving this equation gives electronic energies Eie(R) at the configuration R for the different electronic states of the molecule. Combining Eie(R) with the nuclear repulsive potential energy VNN(R) gives the potential energy surface Vi(R) for electronic state i (Hirst, 1985). Each state is identified by its spin angular momentum and orbital symmetry. Since the electronic density between nuclei is different for each electronic state, each state has its own equilibrium geometry, sets of vibrational frequencies, and bond dissociation energies. To illustrate this effect, vibrational frequencies for the ground singlet state (S0) and first excited singlet state (S1) of H2CO are compared in table 3.1. For a diatomic molecule, potential energy surfaces only depend on the internuclear separation, so that a potential energy curve results instead of a surface. Possible potential energy curves for a diatomic molecule are depicted in figure 3.1. Of particular interest in this figure are the different equilibrium bond lengths and dissociation energies for the different electronic states. The lowest potential curve is referred to as the ground electronic state potential. The primary focus of this chapter is the ground electronic state potential energy surface. In the last section potential energy surfaces are considered for excited electronic states. A unimolecular reactant molecule consisting of N atoms has a multidimensional potential energy surface which depends on 3N-6 independent coordinates. For the smallest nondiatomic reactant, a triatomic molecule, the potential energy surface is four-dimensional (three independent coordinates plus the energy). Since it is difficult, if not impossible, to visualize surfaces with more than three dimensions, methods are used to reduce the dimensionality of the problem in portraying surfaces. In a graphical representation of a surface the potential energy is depicted as a function of two coordinates with constraints placed on the remaining 3N-8 coordinates.

The Dissociation of Small and Large Clusters

Clusters are aggregates of loosely bonded molecules, in which each of the units retains the structure that it has as a free molecule. Because of the weak interactions among the molecules, clusters are stable only in cold environments such as are found in molecular beams. The weak intermolecular bonds provide an interesting testing ground for theories of intramolecular vibrational energy redistribution (IVR) and thus for theories of unimolecular dissociation. In addition, clusters constitute the bridge between the gas and liquid phases. Such phenomena as solvation, heat capacity, and phase transitions, which are ill defined for small clusters, become progressively more precise as the cluster size increases. Typical binding energies for neutral clusters are below 1000 cm-1. Ionic clusters, because of their ion-induced dipole forces, tend to be more strongly bonded with binding energies in excess of 5000 cm-1. Not infrequently, a neutral van der Waals dimer such as Ar2 with its binding energy of about 100 cm-1 (Tang and Toennies, 1986) changes its character upon ionization. The equilibrium bond distance is reduced from about 4 Å to 2.43 Å (Huber and Herzberg, 1979; Ma et al., 1993) and the binding energy increases to 10,000 cm-1 (Norwood et al., 1989; Furuya and Kimura, 1992). Clearly, the Ar2+ ion no longer meets our definition of a dimer. Rather, the neutral dimer is converted into a stable ion with a bond order of 1/2. A molecule that is frequently referred to as a cluster is C60. However, it is held together neither by weak bonds, nor is it composed of a collection of monomers. It is thus better classified as a large covalently bonded molecule. Table 10.1 summarizes some binding energies for various classes of dimers. When clusters comprise several loosely bound molecules, the atoms within each molecule are held together by strong bonds while the molecules themselves are attracted to neighboring molecules by weak bonds. This discrepancy in forces translates into disparities in the respective vibrational frequencies.

Product Energy Distributions

The measurement of product translational and rotational energies, and in some cases vibrational energy, is often more readily accomplished than the measurement of the dissociation rate. As a result there exists a considerable body of experimental information about product energy distributions (FED) for many classes of reactions. The only simple model for treating these FED is the statistical one; however, there is a considerable diversity in its application. In the dissociation of large molecules at moderate to large excess energies, the translational, rotational, and vibrational energy distributions can be treated as continuous functions. On the other hand, in the dissociation of triatomic molecules, it is often possible to measure the quantized rotational energy distribution for specific vibrational energy levels of the diatomic product. Just as in the determination of the dissociation rates, product energy partitioning is highly sensitive to the potential energy surface. If there is no reverse activation barrier, the product energies are often distributed statistically. That is, the distributions depend only upon the product phase space and are independent of the detailed shape of the potential energy surface. On the other hand, for reactions with a “tight” transition state located at the top of a reverse activation barrier, statistical redistribution of the product energies is often not possible. After passing through the transition-state region, the products move down the repulsive wall and rapidly dissociate with little chance to exchange and equilibrate the available energy. Often, such products are ejected with considerable translational energy. This happens in large as well as small molecules or ions. The resulting product energy partitioning is then highly nonstatistical, even though the dissociation rate is perfectly predicted by RRKM theory. That is, the dissociation rate and product energy partitioning are separate and uncoupled events. The rate is governed early in the reaction history by the structure of the transition state, while product energy partitioning is determined late in the reaction and is governed by the shape of the potential energy surface at large internuclear distances. The most effective model for treating product energy distributions (PEDs) of reactions with no reverse activation barriers is the statistical theory.

Theory of Unimolecular Decomposition— The Statistical Approach

The partition function and the sum or density of states are functions which are to statistical mechanics what the wave function is to quantum mechanics. Once they are known, all of the thermodynamic quantities of interest can be calculated. It is instructive to compare these two functions because they are closely related. Both provide a measure of the number of states in a system. The partition function is a quantity that is appropriate for thermal systems at a given temperature (canonical ensemble), whereas the sum and density of states are equivalent functions for systems at constant energy (microcanonical ensemble). In order to lay the groundwork for an understanding of these two functions as well as a number of other topics in the theory of unimolecular reactions, it is essential to review some basic ideas from classical and quantum statistical mechanics. As discussed in chapter 2, the classical Hamiltonian, H(p,q), is the total energy of the system expressed in terms of the momenta (p) and positions (q) of the atoms in the system.

Experimental Methods in Unimolecular Dissociation Studies

The experimental aspects of kinetic studies with state-selected reactants have played a critical role in the advancement of our understanding of unimolecular processes.

Vibrational/Rotational Energy Levels

The first step in a unimolecular reaction is the excitation of the reactant molecule’s energy levels. Thus, a complete description of the unimolecular reaction requires an understanding of such levels. In this chapter molecular vibrational/rotational levels are considered. The chapter begins with a discussion of the Born-Oppenheimer principle (Eyring, Walter, and Kimball, 1944), which separates electronic motion from vibrational/ rotational motion. This is followed by a discussion of classical molecular Hamiltonians, Hamilton’s equations of motion, and coordinate systems. Hamiltonians for vibrational, rotational, and vibrational/rotational motion are then discussed. The chapter ends with analyses of energy levels for vibrational/rotational motion. The Born-Oppenheimer principle assumes separation of nuclear and electronic motions in a molecule. The justification in this approximation is that motion of the light electrons is much faster than that of the heavier nuclei, so that electronic and nuclear motions are separable.

Dynamical Approaches to Unimolecular Rates

In the previous chapters theories were discussed for calculating the unimolecular rate constant as a function of energy and angular momentum. The assumption inherent in these theories is that a microcanonical ensemble is maintained during the unimolecular reaction and that every state in the energy interval E → E + dE has an equal probability of decomposing. Such theories are viewed as statistical since the unimolecular rate constant is found from a statistical counting of states in the microcanonical ensemble. A dynamical description of unimolecular decomposition is concerned with properties of individual states of the energized molecule. Of interest are the decomposition probabilities for the states as well as the rate of transitions between the states. Dynamical theories of unimolecular decomposition deal with the properties of vibrational/rotational energy levels, state preparation and intramolecular vibrational energy redistribution (IVR). Thus, the presentation in this chapter draws extensively on the previous chapters 2 and 4. Unimolecular decomposition dynamics can be treated using quantum and classical mechanics, and both perspectives are considered here. The role of nonadiabatic electronic transitions in unimolecular dynamics is also discussed. A molecule which can dissociate does not, strictly speaking, have a discrete energy spectrum. The relative motion of the product fragments is unbounded and, in this sense the motion of the unimolecular system is infinite, and hence the energy spectrum is continuous. However, it may happen that the dissociation probability of the molecule is sufficiently small that one can introduce the concept of quasi-stationary states. Such states are commonly referred to as resonances since the energy of the unimolecular fragments in the continuum is in resonance with (i.e., matches) the energy of a vibrational/rotational level of the unimolecular reactant. For unimolecular reactions there are two types of resonance states. The simplest type, a shape resonance, occurs when a molecule is temporarily trapped by a fairly high and wide potential energy barrier. The second type of resonance, called a Feshbach or compound-state resonance, occurs when energy is initially distributed between vibrational/rotational degrees of freedom of the molecule which are not strongly coupled to the fragment relative motion, so that there is a time lag for unimolecular dissociation.

State Preparation and Intramolecular Vibrational Energy Redistribution

The first step in a unimolecular reaction involves energizing the reactant molecule above its decomposition threshold. An accurate description of the ensuing unimolecular reaction requires an understanding of the state prepared by this energization process. In the first part of this chapter experimental procedures for energizing a reactant molecule are reviewed. This is followed by a description of the vibrational/rotational states prepared for both small and large molecules. For many experimental situations a superposition state is prepared, so that intramolecular vibrational energy redistribution (IVR) may occur (Parmenter, 1982). IVR is first discussed quantum mechanically from both time-dependent and time-independent perspectives. The chapter ends with a discussion of classical trajectory studies of IVR. A number of different experimental methods have been used to energize a unimolecular reactant. Energization can take place by transfer of energy in a bimolecular collision, as in . . . C2H6 + Ar → C2H6* + Ar . . . . . . (4.1) . . . Another method which involves molecular collisions is chemical activation. Here the excited unimolecular reactant is prepared by the potential energy released in a reactive collision such as . . . F + C2H4 → C2H4F* . . . . . . (4.2) . . . The excited C2H4F molecule can redissociate to the reactants F + C2H4 or form the new products H + C2H3F. Vibrationally excited molecules can also be prepared by absorption of electromagnetic radiation. A widely used method involves initial electronic excitation by absorption of one photon of visible or ultraviolet radiation. After this excitation, many molecules undergo rapid radiationless transitions (i.e., intersystem crossing or internal conversion) to the ground electronic state, which converts the energy of the absorbed photon into vibrational energy. Such an energization scheme is depicted in figure 4.1 for formaldehyde, where the complete excitation/decomposition mechanism is . . . H2CO(S0) + hν → H2CO(S1) → H2CO*(S0) → H2 + CO . . . . . . (4.3) . . . Here, S0 and S1 represent the ground and first excited singlet states.

Introduction

The field of unimolecular reactions has witnessed impressive advances in both experimental and theoretical techniques during the past 20 years. These developments have resulted in experimental measurements that finally permit critical tests of the major assumptions made more than 60 years ago when Rice and Ramsperger (1927, 1928) and Kassel (1928) first proposed their statistical RRK theory of unimolecular decay. At the heart of these advances is our ability to prepare molecules in narrow ranges of internal energy, even in single quantum states, at energies below and above the dissociation limit. This has led to detailed spectroscopic studies of intramolecular vibrational energy redistribution (IVR), a process that is intimately related to the assumption of random energy flow in the statistical theory of unimolecular decay. This book is devoted exclusively to the study of state- or energy-selected systems. However, in order to place these studies in the context of the much larger field of unimolecular reactions in general, we provide a brief background of the field up to about 1970. The experimental studies of unimolecular reactions developed in three stages. The early studies involved strictly thermal systems in which molecules were energized by heating the sample either in a bulb (Chambers and Kistiakowsky, 1934; Schlag and Rabinovitch, 1960; Flowers and Frey, 1962; Schneider and Rabinovitch, 1962), or by more sophisticated methods such as shock tubes which were applied to unimolecular reactions by Tsang (1965, 1972, 1978, 1981) and others (Astholz et al., 1979; Brouwer et al.,1983). The drawback of these studies is that molecules were prepared in a very broad (albeit well characterized) distribution of internal energy states. A major advance was the use of chemical activation in the early 1960s in which a species such as CH2 reacted with a molecule, thereby forming an energized species which could either isomerize or be stabilized by collisions (Rabinovitch and Flowers, 1964; Rabinovitch and Setser, 1964; Kirk et al., 1968; Hassler and Setser, 1966; Simons and Taylor, 1969). This approach permitted the reacting species to be prepared in a narrow range of internal energies.