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This chapter discusses an approximate approach — transition-state theory — to the calculation of rate constants for bimolecular reactions. A reaction coordinate is identified from a normal-mode coordinate analysis of the activated complex, i.e. the supermolecule on the saddle-point of the potential energy surface. Motion along this coordinate is treated by classical mechanics and recrossings of the saddle point from the product to the reactant side are neglected, leading to the result of conventional transition-state theory expressed in terms of relevant partition functions. Various alternative derivations are presented. Corrections that incorporate quantum mechanical tunnelling along the reaction coordinate are described. Tunnelling through an Eckart barrier is discussed and the approximate Wigner tunnelling correction factor is derived in the limit of a small degree of tunnelling. It concludes with applications of transition-state theory to, e.g. the F + H₂ reaction, and comparisons with results based on quasi-classical mechanics as well as exact quantum mechanics.

Starting from the concepts of empirical law, model, and scientific theory, a H-bond theory is defined as the encoding of its empirical laws in terms of a more fundamental theory, F. Though both VB and acid-base theories fulfill these condition and give acceptable H-bond theories, a better approach consists in considering the H-bond as a stationary point of a bimolecular reaction pathway leading from D···H-A to D-H···A through the D···H···A transition state. The operator F is now the transition-state theory with its paraphernalia (Leffler-Hammond postulate, Marcus rate-equilibrium theory, linear free-energy relationships, and so on). In this chapter, the validity of this newly-proposed transition-state H-bond theory (TSHBT) is successfully verified on a sample system (the N-H···O/O-H···N competition in the tautomeric ketohydrazone-azoenol system forming intramolecular RAHB) by variable-temperature X-ray crystallography (VTXRC) and Marcus analysis on the DFT-emulated stationary points along the proton-transfer pathway.

After establishing his credentials as a scientist, Polanyi was transferred to the Kaiser Wilhelm Institute for Physical Chemistry and Electrochemistry and was allowed to focus on reaction rates and transition state theory; the group employed gas-flame experiments to measure reaction rates and calculate the activation energies in them. Polanyi's interest in economics was stimulated by inflation, unemployment and social upheavals in Germany, debates with his brother, Karl Polanyi, who advocated a form of Christian socialism, economic conditions in the Soviet Union, and the rise of nihilism. Polanyi's second son, John Charles Polanyi, was born on January 23, 1929. Developments in quantum theory and dipole-dipole interactions confirmed Polanyi's theory of adsorption potential.

This chapter discusses static solvent effects on the rate constant for chemical reactions in solution. It starts with a brief discussion of the thermodynamic formulation of transition-state theory. The static equilibrium structure of the solvent will modify the potential energy surface for the chemical reaction. This effect is analyzed within the framework of transition-state theory. The rate constant is expressed in terms of the potential of mean force at the activated complex. Various definitions of this potential and their relations to n-particle- and pair-distribution functions are considered. The potential of mean force may, for example, be defined such that the gradient of the potential gives the average force on an atom in the activated complex, Boltzmann averaged over all configurations of the solvent. It concludes with a discussion of a relation between the rate constants in the gas phase and in solution.

This chapter reviews the microscopic interpretation of the pre-exponential factor and the activation energy in rate constant expressions of the Arrhenius form. The pre-exponential factor of apparent unimolecular reactions is, roughly, expected to be of the order of a vibrational frequency, whereas the pre-exponential factor of bimolecular reactions, roughly, is related to the number of collisions per unit time and per unit volume. The activation energy of an elementary reaction can be interpreted as the average energy of the molecules that react minus the average energy of the reactants. Specializing to conventional transition-state theory, the activation energy is related to the classical barrier height of the potential energy surface plus the difference in zero-point energies and average internal energies between the activated complex and the reactants. When quantum tunnelling is included in transition-state theory, the activation energy is reduced, compared to the interpretation given in conventional transition-state theory.

The theoretical treatment of cluster kinetics borrows most of its concepts and techniques from studies of smaller and larger systems. Some of the methods used for such smaller and larger systems are more useful than others for application to cluster kinetics and dynamics, however. This chapter is a review of specific approaches that have found fruitful use in theoretical and computational studies of cluster dynamics to date. The review includes some discussion of methodology; it also discusses examples of what has been learned from the various approaches, and it compares theory to experiment. A special emphasis is on microsolvated reactions—that is, reactions where one or a few solvent molecules are clustered onto gas-phase reactants and hence typically onto the transition state as well. Both analytic theory and computer simulations are included, and we note that the latter play an especially important role in understanding cluster reactions. Simulations not only provide quantitative results, but they provide insight into the dominant causes of observed behavior, and they can provide likelihood estimates for assessing qualitatively distinct mechanisms that can be used to explain the same experimental data. Simulations can also lead to a greater understanding of dynamical processes occurring in clusters by calculating details which cannot be observed experimentally. One interesting challenge that reactions in van der Waals and hydrogenbonded clusters offer is the possibility of studying specifically how weak interactions or microsolvation bonds affect a chemical reaction or dissociation process. In that sense, theoretical studies of weakly bound clusters have proved to be useful in learning about the "crossover" in behavior from that of an isolated nonsolvated molecule in the gas phase to that for a molecule in a liquid or solid solvent. It is very common to begin reviews with a disclaimer as to completeness. Such a disclaimer is, we hope, not required for this chapter because it is not a comprehensive review but a limited-scope discussion of selected work that illustrates some issues that we perceive to be especially important. The chapter is divided into three parts. Section 1.2 discusses collisional and statistical theories for cluster reactions.

This chapter discusses an approximate approach—transition-state theory—to the calculation of rate constants for bimolecular reactions. A reaction coordinate is identified from a normal-mode coordinate analysis of the activated complex, that is, the supermolecule on the saddle-point of the potential energy surface. Motion along this coordinate is treated by classical mechanics and recrossings of the saddle point from the product to the reactant side are neglected, leading to the result of conventional transition-state theory expressed in terms of relevant partition functions. Various alternative derivations are presented. Corrections that incorporate quantum mechanical tunnelling along the reaction coordinate are described. Tunnelling through an Eckart barrier is discussed and the approximate Wigner tunnelling correction factor is derived in the limit of a small degree of tunnelling. It concludes with applications of transition-state theory to, for example, the F + H₂ reaction, and comparisons with results based on quasi-classical mechanics as well as exact quantum mechanics.

This chapter discusses static solvent effects on the rate constant for chemical reactions in solution. It starts with a brief discussion of the thermodynamic formulation of transition-state theory. The static equilibrium structure of the solvent will modify the potential energy surface for the chemical reaction. This effect is analyzed within the framework of transition-state theory. The rate constant is expressed in terms of the potential of mean force at the activated complex. Various definitions of this potential and their relations to n-particle- and pair-distribution functions are considered. The potential of mean force may, for example, be defined such that the gradient of the potential gives the average force on an atom in the activated complex, Boltzmann averaged over all configurations of the solvent. It concludes with a discussion of a relation between the rate constants in the gas phase and in solution.

This chapter reviews the microscopic interpretation of the pre-exponential factor and the activation energy in rate constant expressions of the Arrhenius form. The pre-exponential factor of apparent unimolecular reactions is, roughly, expected to be of the order of a vibrational frequency, whereas the pre-exponential factor of bimolecular reactions, roughly, is related to the number of collisions per unit time and per unit volume. The activation energy of an elementary reaction can be interpreted as the average energy of the molecules that react minus the average energy of the reactants. Specializing to conventional transition-state theory, the activation energy is related to the classical barrier height of the potential energy surface plus the difference in zero-point energies and average internal energies between the activated complex and the reactants. When quantum tunnelling is included in transition-state theory, the activation energy is reduced, compared to the interpretation given in conventional transition-state theory.

Understanding chemical reactions in condensed phases is essentially the understanding of solvent effects on chemical processes. Such effects appear in many ways. Some stem from equilibrium properties, for example, solvation energies and free energy surfaces. Others result from dynamical phenomena: solvent effect on diffusion of reactants toward each other, dynamical cage effects, solvent-induced energy accumulation and relaxation, and suppression of dynamical change in molecular configuration by solvent induced friction. In attempting to sort out these different effects it is useful to note that a chemical reaction proceeds by two principal dynamical processes that appear in three stages. In the first and last stages the reactants are brought together and products are separated from each other. In the middle stage the assembled chemical system undergoes the structural/chemical change. In a condensed phase the first and last stages involve diffusion, sometimes (e.g. when the species involved are charged) in a force field. The middle stage often involves the crossing of a potential barrier. When the barrier is high the latter process is rate-determining. In unimolecular reactions the species that undergoes the chemical change is already assembled and only the barrier crossing process is relevant. On the other hand, in bi-molecular reactions with low barrier (of order kBT or less), the rate may be dominated by the diffusion process that brings the reactants together. It is therefore meaningful to discuss these two ingredients of chemical rate processes separately. Most of the discussion in this chapter is based on a classical mechanics description of chemical reactions. Such classical pictures are relevant to many condensed phase reactions at and above room temperature and, as we shall see, can be generalized when needed to take into account the discrete nature of molecular states. In some situations quantum effects dominate and need to be treated explicitly. This is the case, for example, when tunneling is a rate determining process. Another important class is nonadiabatic reactions, where the rate determining process is hopping (curve crossing) between two electronic states. Such reactions are discussed in Chapter 16.

As a polymer of many amino acids, any given protein can, in principle, adopt a huge number of configurations. In reality, however, the biologically stable protein adopts a single configuration that is stable over time. Thermodynamically, this configuration must represent a Gibbs free energy minimum. This chapter therefore explores how the thermodynamics and kinetics of protein folding and unfolding can be investigated experimentally (using, for example, chaotropes, heating or ligand interactions), and how these measurements can be used to enrich our understanding of protein configurations and stability.

This chapter outlines common mechanisms that contribute to wear, which is broadly defined to be any form of surface damage caused by rubbing one surface against another. Such wear mechanisms include delamination wear, adhesive wear (where adhesion followed by plastic shearing plucks the ends off the softer asperities, typically described by Archard’s law), abrasive wear (where hard particles or asperities gouge a surface and displace material), and oxidative wear (where surfaces react with atmospheric oxygen prior to being worn). Sliding conditions often determine which wear mechanism dominates, with the main factors being temperature, sliding velocity, oxidation, plasticity, loading force, and mechanical stresses. How wear rates respond to changes to these factors can be diagramed on a wear map. The last part of the chapter discusses how transition state theory can describe nanoscale wear by atomic attrition, and how plasticity and fracture occur at the nanoscale.

The development of techniques to simulate infrequent events has been an area of rapid progress in recent years. In this chapter, we shall discuss some of the simulation techniques developed to study the dynamics of rare events. A basic summary of the statistical mechanics of barrier crossing is followed by a discussion of approaches based on the identification of reaction coordinates, and those which seek to avoid prior assumptions about the transition path. The demanding technique of transition path sampling is introduced and forward flux sampling and transition interface sampling are considered as rigorous but computationally efficient approaches.

As was discussed in Chapter 2, stable and accurate numerical integration of the MD equations of motion demands a small time step. In MD simulations of solids, the integration step is usually of the order of one femtosecond (10−15 s). For this reason, the time horizon ofMDsimulations of solids rarely exceeds one nanosecond (10−9 s). On the other hand, dislocation behaviors of interest typically occur on time scales of milliseconds (10−3 s) or longer. Such behaviors remain out of reach for direct MD simulations. Time-scale limits of a similar nature also exist in MC simulations. For instance, the magnitude of the atomic displacements in the Metropolis algorithm has to be sufficiently small to ensure a reasonable acceptance ratio, which results in a slow exploration of the configurational space. This disparity of time scales can be traced to certain topographical features of the potential-energy function of the many-body system, typically consisting of deep energy basins separated by high energy barriers. The system spends most of its time wandering around within the energy basins (metastable states) only rarely interrupted by transitions from one basin to another. Whereas the long-term evolution of a solid results from transitions between the metastable states, direct MDand MC simulations spend most of the time faithfully tracing the unimportant fluctuations within the energy basins. In this sense, most of the computing cycles are wasted, leading to very low simulation efficiency. Because the transition rates decrease exponentially with the increasing barrier heights and decreasing temperature, this problem of time-scale disparity can be severe.