Microscopic reaction dynamics

Though statistical models are important, they may not provide a complete picture of the microscopic reaction dynamics. There are several basic questions associated with the microscopic dynamics of gas-phase SN2 nucleophilic substitution that are important to the development of accurate theoretical models for bimolecular and unimolecular reactions.1 Collisional association of X" with RY to form the X-—RY... 

We will make a reasonable (albeit brief) attempt to review the nonlinear dynamics literature, with an emphasis on some aspects that are relevant to microscopic reaction dynamics. However, our main goals are (1) to convey to the interested nonspecialist some of the interesting ideas that have arisen... 

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

The original microscopic rate theory is the transition state theory (TST) [10-12]. This theory is based on two fundamental assumptions about the system dynamics. (1) There is a transition state dividing surface that separates the short-time intrastate dynamics from the long-time interstate dynamics. (2) Once the reactant gains sufficient energy in its reaction coordinate and crosses the transition state the system will lose energy and become deactivated product. That is, the reaction dynamics is activated crossing of the barrier, and every activated state will successfully react to fonn product. 

The moving invariant manifolds determine the reactivity or nonreactivity of an individual trajectory under the influence of a specific noise sequence. They thus provide the most detailed microscopic information on the reaction dynamics that one can possibly possess. In practice, though, one is more often interested in macroscopic quantities that are obtained by averaging over the noise. To illustrate that such quantities can easily be derived from the microscopic information encoded in the TS trajectory, we calculate the probability for a trajectory starting at a point (q, v) in the space-fixed phase space to end up on the product side of the... 

Reaction dynamics deals with the intra- and intermolecular motions that characterize the elementary act of a chemical reaction. It also deals with the quantum states of the reactants and product. Since the dynamic study is concerned with the microscopic level and dynamic behaviour of reacting molecules, therefore, the term molecular dynamics is employed. 

Reaction dynamics is the part of chemical kinetics which is concerned with the microscopic-molecular dynamic behavior of reacting systems. Molecular reaction dynamics is coming of age and much more refined state-to-state information is becoming available on the fundamental reactions. The contribution of molecular beam experiments and laser techniques to chemical dynamics has become very useful in the study of isolated molecules and their mutual interactions not only in gas surface systems, but also in solute-solution systems. 

Molecular dynamics simulations yield an essentially exact (within the confines of classical mechanics) method for observing the dynamics of atoms and molecules during complex chemical reactions. Because the assumption of equilibrium is not necessary, this technique can be used to study a wide range of dynamical events which are associated with surfaces. For example, the atomic motions which lead to the ejection of surface species during keV particle bombardment (sputtering) have been identified using molecular dynamics, and these results have been directly correlated with various experimental observations. Such simulations often provide the only direct link between macroscopic experimental observations and microscopic chemical dynamics. 

As described above, silicon crystals can be grown from a variety of gas sources. Because the rate of growth can be modulated using these techniques, dopants can be efficiently incorporated into a growing crystal. This results in control of the atomic structure of the crystal, and allows the production of samples which have specific electronic properties. The mechanisms by which gas-phase silicon species are incorporated into the crystal, however, are still unclear, and so molecular dynamics simulations have been used to help understand these microscopic reaction events. 

Further large-deviation dynamical relationships are the so-called flucmation theorems, which concern the probability than some observable such as the work performed on the system would take positive or negative values under the effect of the nonequilibrium fluctuations. Since the early work of the flucmation theorem in the context of thermostated systems [52-54], stochastic [55-59] as well as Hamiltonian [60] versions have been derived. A flucmation theorem has also been derived for nonequilibrium chemical reactions [62]. A closely related result is the nonequilibrium work theorem [61] which can also be derived from the microscopic Hamiltonian dynamics. 

The detailed microscopic description of a chemical reaction in terms of the motion of the individual atoms taking part in the event is known as the reaction dynamics. The study of reaction dynamics at surfaces is progressing rapidly these years, to a large extent because more and more results from detailed molecular beam scattering experiments are becoming available. 

The link between the microscopic description of the reaction dynamics and the macroscopic kinetics that can be measured in a catalytic reactor is a micro-kinetic model. Such a model will start from binding energies and reaction rate constants deduced from surface science experiments on well defined single crystal surfaces and relate this to the macroscopic kinetics of the reaction. 

Achieving control over microscopic dynamics of molecules with external fields has long been a major goal in chemical reaction dynamics. This goal stimulated the development of quantum control schemes, which have been applied with spectacular results to unimolecular reactions, such as photodissociation or isomerization reactions. Attaining control over bimolecular reactions in a gas has proven to be a much bigger challenge. 

Understanding chemical reactions has been a major preoccupation since the historical origins of chemistry. A main difficulty is to reconcile the macroscopic description in which reactions are rate processes ruling the time evolution of populations of chemical species with the microscopic Hamiltonian dynamics governing the motion of the translational, vibrational, and rotational degrees of freedom of the reacting molecules. 

The knowledge of the two-minima energy surface is sufficient theoretically to determine the microscopic and static rate of reaction of a charge transfer in relation to a geometric variation of the molecule. In practice, the experimental study of the charge-transfer reactions in solution leads to a macroscopic reaction rate that characterizes the dynamics of the intramolecular motion of the solute molecule within the environment of the solvent molecules. Stochastic chemical reaction models restricted to the one-dimensional case are commonly used to establish the dynamical description. Therefore, it is of importance to recall (1) the fundamental properties of the stochastic processes under the Markov assumption that found the analysis of the unimolecular reaction dynamics and the Langevin-Fokker-Planck method, (2) the conditions of validity of the well-known Kramers results and their extension to the non-Markovian effects, and (3) the situation of a reaction in the absence of a potential barrier. 

The mechanisms of reactions that occur in condensed phases involve the participation of solvent degrees of freedom. In some cases, such as in certain ion association reactions involving solvent-separated ion pairs, even the very existence of reactant or product states depends on the presence of the solvent. Traditionally the solvent is described in a continuum approximation by reaction-diffusion equations. Kapral s group is interested in microscopic theories that, by treating the solvent at a molecular level, allow one to investigate the origin and range of validity of conventional continuum theories and to understand in a detailed way how solvent motions influence reaction dynamics. 

The transformation from reactants to products can be described at either a phenomenological level, as in classical chemical kinetics, or at a detailed molecular level, as in molecular reaction dynamics.1 The former description is based on experimental observation and, combined with chemical intuition, rate laws are proposed to enable a calculation of the rate of the reaction. It does not provide direct insight into the process at a microscopic molecular level. The aim of molecular reaction dynamics is to provide such insight as well as to deduce rate laws and calculate rate constants from basic molecular properties and dynamics. Dynamics is in this context the description of atomic motion under the influence of a force or, equivalently, a potential. 

Before we go on and discuss these objectives in more detail, it might be appropriate to consider the relation between molecular reaction dynamics and the science of physical chemistry. Normally physical chemistry is divided into four major branches, as sketched in the figure below (each of these areas are based on fundamental axioms). At the macroscopic level, we have the old disciplines thermodynamics and kinetics . At the microscopic level we have quantum mechanics , and the connection between the two levels is provided by statistical mechanics . Molecular reaction dynamics encompasses (as sketched by the oval) the central branches of physical chemistry with the exception of thermodynamics. 

The rearrangement of nuclei in an elementary chemical reaction takes place over a distance of a few angstrom (1 angstrom = 10 10 m) and within a time of about 10-100 femtoseconds (1 femtosecond = 10 15 s a femtosecond is to a second what one second is to 32 million years ), equivalent to atomic speeds of the order of 1 km/s. The challenges in molecular reaction dynamics are (i) to understand and follow in real time the detailed atomic dynamics involved in the elementary processes, (ii) to use this knowledge in the control of these reactions at the microscopic level, e.g., by means of external laser fields, and (iii) to establish the relation between such microscopic processes and macroscopic quantities like the rate constants of the elementary processes. 

In this chapter we consider bimolecular reactions from both a microscopic and a macroscopic point of view and thereby derive a theoretical expression for the macroscopic phenomenological rate constant. That is, a relation between molecular reaction dynamics and chemical kinetics is established. 

The main objective of statistical mechanics is to calculate macroscopic (thermodynamic) properties from a knowledge of microscopic information like quantum mechanical energy levels. The purpose of the present appendix is merely to present a selection1 of the results that are most relevant in the context of reaction dynamics. 

This book focuses on the basic concepts in molecular reaction dynamics, which is the microscopic atomic-level description of chemical reactions, in contrast to the macroscopic phenomenological description known from chemical kinetics. It is a very extensive field and we have obviously not been able, or even tried, to make a comprehensive treatment of all contributions to this field. Instead, we limited ourselves to give a reasonable coherent and systematic presentation of what we find to be central and important theoretical concepts and developments, which should be useful for students at the graduate or senior undergraduate level and for researchers who want to enter the field. 

In Chapter 7 we turn to the other basic type of elementary reaction, i.e., uni-molecular reactions, and discuss detailed reaction dynamics as well as transition-state theory for unimolecular reactions. In this chapter we also touch upon the question of the atomic-level detection and control of molecular dynamics. In the final chapter dealing with gas-phase reactions, Chapter 8, we consider unimolecular as well as bimolecular reactions and summarize the insights obtained concerning the microscopic interpretation of the Arrhenius parameters, i.e., the pre-exponential factor and the activation energy of the Arrhenius equation. 

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