The thermodynamic temperature

Equilibrium thermodynamics provides a very useful connection between mechanical and thermal properties of fluids and solids. The predicted relationships between different quantities measured under different thermodynamic conditions are a fundamental consequence of thermodynamics. It is natural to attempt to develop a similar thermodynamic treatment of non-equilibrium systems, at least for steady states. At present, there are a number of different treatments: the extended irreversible thermodynamics (Jou et al., 2001); the approach to microscopic relaxation processes (Öttinger, 2005); and the approach that we follow here. It is fair to say that, at present, there is no consensus on the correctness of any of these approaches, and indeed some debate about whether it is even possible to define the usual thermodynamic quantities for a nonequilibrium system. Clearly then, it is necessary to limit the types of nonequilibrium processes to which we apply thermodynamics. As an example of a system where a thermodynamic treatment may be successful, consider a steady-state Poiseuille flow system where we can define a local temperature and local shear rate at each point in the fluid. There will be gradients in both the shear rate and the temperature that determine the local streaming velocity profile and the conduction of heat to the boundary.

During the 1980s there have been many new developments regarding the nonequilibrium statistical mechanics of dense classical systems. These developments have had a major impact on the computer simulation methods used to model nonequilibrium fluids. Some of these new algorithms are discussed in the recent book by Allen and Tildesley (1987), Computer Simulation of Liquids. However, that book was never intended to provide a detailed statistical mechanical backdrop to the new computer algorithms. As the authors commented in their preface, their main purpose was to provide a working knowledge of computer simulation techniques. The present volume is, in part, an attempt to provide a pedagogical discussion of the statistical mechanical environment of these algorithms.

There is a symbiotic relationship between nonequilibrium statistical mechanics on the one hand and the theory and practice of computer simulation on the other. Sometimes, the initiative for progress has been with the pragmatic requirements of computer simulation and at other times, the initiative has been with the fundamental theory of nonequilibrium processes. Although progress has been rapid, the number of participants who have been involved in the exposition and development, rather than with application, has been relatively small.

The formal theory is often illustrated with examples involving shear flow in liquids.

Adiabatic linear response theory

In this chapter we will discuss how an external field Fe, perturbs an N-particle system. We assume that the field is sufficiently weak that only the linear response of the system need be considered. These considerations will lead us to equilibrium fluctuation expressions for mechanical transport coefficients such as electrical conductivity. These expressions are formally identical to the Green—Kubo formulae that were derived in the last chapter. The difference is that the Green—Kubo formulae pertain to thermal transport processes where boundary conditions perturb the system away from equilibrium — all Navier—Stokes processes fall into this category. Mechanical transport coefficients, on the other hand, refer to systems where mechanical fields which appear explicitly in the equations of motion for the system, drive the system away from equilibrium. As we will see, it is no coincidence that there is such a close similarity between the fluctuation expressions for thermal and mechanical transport coefficients. In fact one can often mathematically transform the nonequilibrium boundary conditions for a thermal transport process into a mechanical field. The two representations of the system are then said to be congruent.

A major difference between the derivations of the equilibrium fluctuation expressions for the two representations is that in the mechanical case one does not need to invoke Onsager's regression hypothesis.

Dissociative recombination is part of the broader field of electron–molecule scattering, which dates back to the famous Franck–Hertz experiment in 1914 (Franck & Hertz 1914). There are also similarities between dissociative recombination and photodissociation, and being a reactive process, recombination can also be considered as branch of chemical reaction dynamics. It is not possible to give a comprehensive presentation of all these topics, or this book would take an encyclopedic format. Instead we will focus on the topics that are most closely related to dissociative recombination.

A broad overview of the entire field of atomic collisions, including electron–ion recombination, is given in McDaniel (1989) and McDaniel, Mitchell, and Rudd (1993). The theory of electron–atom and electron–molecule collisions has been covered in Khare (2002). Since the recombination of atomic ions is not covered in the present book, the reader is referred to Dunn et al. (1984), Graham et al. (1992), McDaniel, Mitchell, and Rudd (1993), Hahn (1997), and Phaneuf et al. (1999) for reviews of this topic. Edited volumes by Christophorou (1984a, b), Märk and Dunn (1985), Ehrhardt and Morgan (1994), and Becker (1998) cover well the development in electron–molecule collisions during the 1980s and 1990s. Review articles covering dissociative recombination are given in Chapter 1.

This research monograph provides a single-volume description of the dissociative recombination of molecular ions with electrons. Since this is one of the most complex gas-phase processes, its study is a challenge to theorists and experimentalists alike. The theory, experiment, and applications of dissociative recombination are scattered in the scientific literature as original research articles, conference proceedings, and review articles. This book brings this information together in a single work for the first time.

The book is intended for researchers and Ph. D. students in the fields of atomic and molecular physics, chemical physics and physical chemistry, molecular astrophysics, atmospheric physics, and other areas of science where electrons and molecular ions are important.

This book was written during a period when each of us had several other commitments which slowed down the writing. One of us (AEO) was department chair at UC Davis essentially during the entire writing process, and ML chaired committees for the Swedish Space Board and the Swedish Research Council.

We are grateful for the hospitality of the Institute for Atomic and Molecular Physics (ITAMP) at the Harvard-Smithsonian Center for Astrophysics and Harvard University Physics Department (Kate Kirby, Hussein Sadeghpour), the Cluster Research Laboratory, Toyota Technological Institute, Tokyo (Tamotsu Kondow), and the University of Chicago (Takeshi Oka), all of which provided excellent working conditions for us when we needed to get away from our home institutions to focus on writing.

The applications of dissociative recombination have been described in the previous chapters in connection with discussions about different molecular ions. Thus, this chapter is short and serves to direct the reader to some key references.

Molecular astrophysics

The development of astrochemistry is well covered by the proceedings of the International Astronomical Union: Andrew (1980), Vardya and Tarafdar (1987), Singh (1992), Van Dishoeck (1997), Minh and Van Dishoeck (2000), and Lis, Blake, and Herbst (2006). Astrochemistry was also a topic at the Fifth International Chemical Congress of Pacific Basin Societies (PACIFICHEM) in December 2005 (Kaiser et al. 2006). A good introduction to the field of molecular astrophysics is the book of the same name edited by T. W. Hartquist in honor of the sixtieth birthday of Alex Dalgarno (Hartquist 1990). This was followed almost 10 years later by another book in honor of Alex Dalgarno (Hartquist & Williams 1998). During 2006, the Nobel Symposium No 133 was dedicated to cosmic chemistry and molecular astrophysics (URL: www.nobel133.physto.se), and PNAS published a special feature issue on interstellar chemistry (Klemperer 2006).

The role played by dissociative recombination in molecular astrophysics is particularly important to the chemistry of interstellar space. Eddington's 1926 Bakerian Lecture marks the starting point of the study of the interstellar medium (Eddington 1926), and the possibility of molecules in the interstellar medium was discussed even though at that time only atomic absorption lines had been observed.

This book is focused on a single molecular process, dissociative recombination, and it may at first seem surprising that this topic can fill a whole book. As we shall see, it is not surprising when the complexity and applicability of the process are taken into account, and when the formidable challenges the process has provided to both experimenters and theorists are considered. This book brings together all the information we have on dissociative recombination in a single source, something which so far has been missing from the scientific literature.

A free electron which has a positive kinetic energy recombines with a positive atomic or molecular ion if its energy can be removed, so that it can enter a bound state. In the absence of a third body that can absorb the excess energy, the energy can be carried away by a photon. This is the only option for an atomic ion, and the process is inefficient. A molecular ion can make use of its internal structure and transfer the electron to a bound state while breaking one or several chemical bonds. This is a very efficient process that has taken its name from the fact that the capture of the electron is stabilized by dissociation. It is a primary chemical process, but is rarely described in chemical textbooks. It is the most complex of gas-phase reactions leading to the production of neutral atoms and molecules.

Several experimental methods were mentioned in Chapter 1. In this chapter most of these methods will be described in more detail. They represent the experimental methods that have produced the vast majority of the dissociative recombination data.

Merged beams

Kinematics and resolution

It follows from Chapter 1 that the cross section for dissociative recombination is inversely proportional to the kinetic energy of the incident electron, and that resonances in the cross section may occur because of interferences between the indirect and the direct process. For these reasons, the ideal method for detailed studies of the cross section is the merging of beams of electrons and beams of molecular ions. The application of merged beams to the study of atomic and molecular collision processes has been comprehensively reviewed by Phaneuf et al. (1999).

A perfect experiment by means of merged beams would consist of a monoenergetic beam of ions interacting, at an angle of 0°, with a monoenergetic beam of electrons over a precisely known distance, and with no other atoms or molecules present. The beams overlap perfectly, and the velocity and currents of the beams are precisely known. The neutral reaction products are detected with 100% efficiency. This ideal situation can never be realized, of course. A beam of charged particles has an energy spread, it is difficult to keep the angle of intersection at exactly 0°, and no matter how good the vacuum, there will be residual gas molecules present in the interaction region.

Who, what, where, why, when and how – the elementary prescription for a news squib is also appropriate for a preface.

Who? The book is intended primarily as a text for advanced undergraduates and graduate students. It can also serve the needs of research workers in the wide area of nanochemistry, as molecular clusters and extended solid-state materials constitute the structural “bookends” of nanoparticles: species that are not large enough to be treated with solid-state concepts but too large to follow the simple rules of molecular clusters. Those interested in a wide-ranging introduction to models of electronic structure applicable to delocalized, three-dimensional systems will also find it useful.

What? This text circumscribes a non-traditional area of inorganic chemistry. The focus is on a class of compound that exhibits cluster bonding. Emphasis is on connections between the problems of small molecular clusters, where the vast majority of atoms are found at the surface, to large crystals, where most atoms are found in the bulk. A review of bonding in molecular compounds (Chapter 1) is followed by the fundamentals of cluster bonding in p-block clusters (Chapter 2) and transition-metal clusters (Chapter 3). After making connections with organometallic chemistry (Chapter 4), mixed p–d-block clusters are developed (Chapter 5). A bonding model for periodic extended structures (Chapter 6) is developed in the style of Chapter 1. Chapter 7 then illustrates some of the similarities and differences between the bonding of clusters and related solid-state structures.

To a large extent, we expect the cluster bonding principles established for main-group clusters in Chapter 2 to carry over to transition-metal clusters. However, the AO basis sets for building MOs differ for transition metals which means that the expression of the cluster bonding principles in geometric and electronic structure will also differ. That is, the observed cluster compositions and shapes differ and these differences can be associated with the participation of the metal d functions in cluster bonding. The d functions are the “wild cards” that make transition-metal chemistry so interestingly different from main-group chemistry. In writing Chapter 3 we have assumed that the reader has a basic understanding of the principles of cluster bonding as expressed by p-block clusters (Chapter 2). Emphasis here is placed on the varied expression of d-block metal character within a cluster context. A number of monographs on metal clusters are suggested at the end of this chapter as additional reading for those interested in pursuing a topic in more depth.

Three-connect clusters

Main-group clusters that exhibit three-connect shapes can often be described using localized two-center bonds. What is the situation for metal clusters?

Localized two-center bonds

Two-center–two-electron bonding and the eight-electron rule adequately rationalize three-connect clusters like P4; hence, we expect the 18-electron rule to suffice for three-connect transition-metal clusters like tetrahedral Ir4(CO)12 (Figure 3.1) with 12 terminal carbonyl ligands. Indeed it does.