Prof. Keli Han worked at Dalian Institute of Chemical Physics, Chinese Academy of Sciences (CAS). He received his Ph. D. degree in physical chemistry from Dalian Institute of Chemical Physics, Chinese Academy of Sciences (CAS) in 1990. His research group is mainly interested in molecular reaction dynamics: chemical reaction rate computations, hydrogen-bond dynamics in intermolecular electron transfer, molecular dynamics with attosecond resolution, nonadiabatic dynamics in chemical processes, enzyme dynamics. To date, he has published more than 330 scientific papers with a high citation over 4000. Due to his excellent works, he was invited and published review articles in Int. Rev. Phys. Chem., Phys. Chem. Chem. Phys., J. Comput. Chem., J. Photochem. Photobio. C: Photochem. Rev., Exp. Opin. Drug. Disc. He is recipient of the National Outstanding Youth Foundation. In 1999, he received a first class Natural Science Award from CAS and the Young Chemist Award from the Chinese Chemical Society (CCS). He is the chair of the Virtual Laboratory for Computational Chemistry (VLCC), Supercomputing Center, Computer Network Information Center, CAS. He chaired many international meetings and presented many invited talks. He is a member of the editorial board of the Journal of Physical Chemistry. He is also a member of the advisory editorial board of the Journal of Theoretical & Computational Chemistry, Chinese Journal of Chemical Physics, Acta Physico-Chimica Sinica, Progress in Natural Science, Chinese Science Bulletin, Journal of Atomic and Molecular Physics.

Prof. Tianshu Chu is a visiting scholar in Dalian Institute of Chemical Physics, Chinese Academy of Sciences (CAS) and a professor in Qingdao University, China. She received her Ph. D. degree in physical chemistry from Dalian Institute of Chemical Physics, Chinese Academy of Sciences (CAS) in 2006. Her research interests focused on (i) developing quantum methods and dynamical codes for dynamics investigation of nuclei and electron motions in chemical and physical processes and (ii) growth dynamics of nanoparticles, kinetic and mechanistic analyse of chemical reactions. Worked with Prof. Keli Han, she carried out many rate computations based on quantum dynamics, and published review articles in Int. Rev. Chem. Phys. and Phys. Chem. Chem. Phys., each with a rather high citation of 165 and 94.

Reaction Rate Constant Computations

Theories and Applications

By Keli Han, Tianshu Chu

The Royal Society of Chemistry

Copyright © 2014 The Royal Society of Chemistry
All rights reserved.
ISBN: 978-1-84973-650-3

Contents

Chapter 1 Elementary Reactions: Rate Constants and their Temperature-Dependence Ian W. M. Smith, 1,
Chapter 2 Rate Constant Calculation of Benzylperoxy Radical Isomerization S. Canneaux, C. Hammaecher, F. Louis and M. Ribaucour, 34,
Chapter 3 Rate Constants and the Kinetic Isotope Effects in Multi-Proton Transfer Reactions: A Case Study of CIONO2 + HCl [right arrow] HNO3 + Cl2 Reactions with Water Clusters with Canonical Variational Transition State Theory using a Direct Ab Initio Dynamics Approach Yongho Kim, 55,
Chapter 4 Statisticodynamical and Multiscale Modeling of Cluster Dissociation F. Calvo and P. Parneix, 77,
Chapter 5 A Mixed Quantum-Classical View to the Kinetics of Chemical Reactions Involving Multiple Electronic States Aurélien de la Lande, Bernard Lévy and Isabelle Demachy, 99,
Chapter 6 Adiabatic Treatment of Torsional Anharmonicity and Mode Coupling in Molecular Partition Functions and Statistical Rate Coefficients: Application to Hydrogen Peroxide Zeb C. Kramer and Rex T. Skodje, 133,
Chapter 7 Dynamics of Chemical Reaction around a Saddle Point: What Divides Reacting and Non-Reacting Trajectories? Shinnosuke Kawai and Tamiki Komatsuzaki, 154,
Chapter 8 Derivation of Rate Constants from Accurate Quantum Wave Packet Theory for Nonadiabatic and Adiabatic Chemical Reactions Tianshu Chu and Keli Han, 180,
Chapter 9 Understanding Reactivity with Reduced Potential Energy Landscapes: Recent Advances and New Directions Bryan R. Goldsmith, Anthony Fong and Baron Peters, 213,
Chapter 10 Quantum-Classical Liouville Dynamics of Condensed Phase Quantum Processes Gabriel Hanna and Raymond Kapral, 233,
Chapter 11 Free Energetics and Kinetics of Charge Transfer and Shift Reactions in Room-Temperature Ionic Liquids Youngseon Shim and Hyung J. Kim, 260,
Chapter 12 Semi-Classical Treatments of Electron Transfer Rate from Weak to Strong Electronic Coupling Regime Yi Zhao, 283,
Chapter 13 Modified Zusman Equation for Quantum Solvation Dynamics and Rate Processes Hou-Dao Zhang, Jian Xu, Rui-Xue Xu and YiJing Yan, 319,
Chapter 14 Time-Dependent Treatment of SVRT Model for Polyatom–Polyatom Reaction John Z. H. Zhang, 337,
Chapter 15 Role of Water in Radical Reactions: Molecular Simulation and Modelling Dorota Swiatla-Wojcik, 352,
Chapter 16 Molecular Modelling of Proton Transfer Kinetics in Biological Systems Patrick Bertrand, 379,
Chapter 17 Putting Together the Pieces: A Global Description of Valence and Long-Range Forces via Combined Hyperbolic Inverse Power Representation of the Potential Energy Surface A. J. C. Varandas, 408,
Chapter 18 Extension of Marcus Rate Theory to Electron Transfer Reactions with Large Solvation Changes Guillaume Jeanmairet, Daniel Borgis, Anne Boutin and Rodolphe Vuilleumier, 446,
Chapter 19 Theoretical Studies on Mechanism and Kinetics of Atmospheric Chemical Reactions L. Sandhiya and K. Senthilkumar, 462,
Chapter 20 Computation of Intrinsic RRKM and Non-RRKM Unimolecular Rate Constants Amit Kumar Paul, Sujitha Kolakkandy, Subha Pratihar and William L. Hase, 494,
Chapter 21 Molecular Dynamics Simulation of Kinetic Isotope Effects in Enzyme-Catalyzed Reactions Jiali Gao, 530,
Subject Index, 550,

CHAPTER 1

Elementary Reactions: Rate Constants and their Temperature-Dependence

IAN W. M. SMITH

1.1 Introduction

This chapter considers the kinetics of elementary reactions. Unlike complex reactions, elementary reactions cannot be subdivided into processes of lesser molecular complexity, whereas complex reactions proceed through a network of elementary reactions. Elementary reactions necessarily involve the participation of a small integral number of atoms and/or molecules, and one can further define them by saying that the chemical change involves molecular processes which mimic the chemical equation that is used to represent the reaction. Thus, the reaction

F + H2 [right arrow] HF + H (R1)

occurs in binary collisions — though not all binary collisions — between fluorine atoms and molecules of di-hydrogen.

Elementary chemical reactions can be classified as collisional or decay processes. The former, of which reaction (R1) is an example, are generally referred to as bimolecular; two species (e.g. F and H2) collide in each microscopic event that leads to reaction and the formation of products (e.g. HF and H). Decay processes are unimolecular: chemical change occurs in processes where single molecules either dissociate to two new chemical species or isomerise, that is, change to a different form with the same chemical formula.

However, it is necessary to insert a cautionary note in the description of unimolecular processes as elementary since this designation disguises the fact that, although the elementary processes in which chemical change occurs are indeed unimolecular, they involve molecules of reactants that contain high internal energy compared with the great majority. Consequently, collisions in which energy is transferred but no chemical change occurs also play a vital role in the kinetics of these reactions. Finally, I note that association reactions are the reverse of dissociation reactions. They involve two reactant species, frequently two free radicals, coming together to form a collision complex, which is subsequently stabilised against re-dissociation — usually by collision with a third species which removes energy from the energised complex. In the limit of the reactants being atoms, for example, pairs of oxygen atoms, the lifetime of the diatomic complex is very short and the stabilising collision is essentially simultaneous with the radical-radical collision. In this case, the reaction can be considered as termolecular.

Reactions occurring in solution are inevitably affected to a greater or lesser degree by the close proximity of solvent molecules to the reactants. Consequently, one can argue that, by definition, elementary reactions only take place in the gas phase. Certainly, such reactions are simpler to treat theoretically. In this chapter, I shall consider only gas-phase reactions. For such reactions, a continuing synergy between experiment and theory has brought forth a remarkable improvement in our understanding, especially of the factors that influence the magnitude of rate constants and their dependence on temperature, and also of the dynamics of such reactions, that is, what factors control the motions of the atoms as chemical bonds rearrange and reactants are converted to products.

The improvement in our knowledge and understanding of elementary chemical reactions has been stimulated by two principal drivers. The first is the developments in experimental techniques and the ability to apply them over an ever-widening range of temperatures, coupled to a massive increase in computing power which has, inter alia, allowed potential energy surfaces to be calculated accurately for elementary reactions of increasing complexity (see section 1.3). The second driver has been the wish to model complex chemical environments: (a) in planetary atmospheres, especially that of Earth; (b) at high temperatures in pyrolytic and combustion systems; and (c) in interstellar and circumstellar media. The computer models contain a large number of ordinary differential equations in each of which the change with time of a particular chemical species (X) is represented as the difference in the sum of the rates of the elementary reactions in which X is formed and the sum of the rates of the elementary reactions in which X is consumed, that is:

dX/dt = Σ rates of formation processes – Σ rates of removal processes (1.1)

The temperatures in the three types of system, (a), (b) and (c), vary widely, from up to several thousand K in (a) to as low as 10 K in (c), emphasising the need either for measurements over a correspondingly wide range of temperatures, or for theoretical methods capable either of calculating the rate constants over similar ranges of temperature or, at least, of extrapolating the values of the experimental rate constants that may have been determined over a limited range of temperatures — or even a single temperature — to the temperatures appropriate to the models of a particular environment.

1.2 A Little History

The systematic study of chemical kinetics, that is, of the rates of chemical reactions and their dependence on temperature, dates back to the middle of the 19th century. During the next 60 years, a number of expressions were proposed to express the temperature-dependence of the rate constant, k(T). These efforts have been reviewed by Laidler, who pointed out the difficulty of distinguishing between the various proposals of how k(T) varies with temperature when the available values of k(T) cover only a small temperature range. After the early years of the 20th century, attention focused on what is generally referred to as the Arrhenius equation:

k(T) = A exp(-Eact/RT) (1.2)

where A is best referred to as the pre-exponential factor and Eact is the activation energy, and a modified form of this equation in which additional temperature-dependence is allowed for in the pre-exponential term:

[MATHEMATICAL EXPRESSION OMITTED] (1.3)

These equations came to be favoured over other temperature-dependent expressions for the rate constant largely because, in Laidler’s words, the other expressions were ‘theoretically sterile’, whereas eqn (1.2) could be rationalised on the basis of the reactants requiring some minimum amount of energy to undergo reaction.

Although the title of eqn (1.2) honours Arrhenius, its origin was in the work of van’t Hoff, who generously acknowledged still earlier work by Pfaundler. Van’t Hoff appreciated that, at equilibrium, the rate of forward and reverse reactions become equal so that the ratio of the rate constants, kf(T) and kr(T), for these reactions is equal to the equilibrium constant; i.e.

kf(T)/kr(T) = Kc(T)[dagger] (1.4)

Combining this equation with that from chemical thermodynamics which bears van’t Hoff’s name:

[MATHEMATICAL EXPRESSION OMITTED] (1.5)

yields

[MATHEMATICAL EXPRESSION OMITTED] (1.6)

Van’t Hoff then argued that the temperature-dependence of the rate constants kf(T) and kr(T) is influenced by two different energies, Ef and Er, whose difference corresponds to ΔUc°, so that:

[MATHEMATICAL EXPRESSION OMITTED] (1.7a)

and

[MATHEMATICAL EXPRESSION OMITTED] (1.7b)

Van’t Hoff recognised that ΔUc° is generally not independent of temperature. Of course, if it is, integrating either eqn (1.7.a) or (1.7b) recovers the Arrhenius equation (eqn (1.2)). Arrhenius’s contribution to this debate was to note that the effect of temperature on chemical reaction rates was much too large to be the result of changes in the translational energies of the reactants and, in a postulate reminiscent of transition state theory (see below), he suggested that an equilibrium is established between reactant molecules and ‘active’ ones that could react without further input of energy. If this equilibrium mirrors that for chemical equilibrium, and hence obeys an equation like eqn (1.5), then one obtains eqn (1.2).

More generally, it can be seen that eqn (1.6) can be used to obtain an expression for a temperature-dependent activation energy (and imply a temperature-dependent pre-exponential factor), usually written as:

[MATHEMATICAL EXPRESSION OMITTED] (1.8)

The modified form of the Arrhenius equation given in eqn (1.3) was apparently first suggested by Kooij, a student of van’t Hoff s. If that equation is operated on according to eqn (1.8), one obtains the following expression for the activation energy:

[MATHEMATICAL EXPRESSION OMITTED] (1.9)

In tables of rate constants, compiled for the purposes of combustion, atmospheric and astrochemical modelling, the recommended rate constants are often expressed using the following form of the Kooij equation:

[MATHEMATICAL EXPRESSION OMITTED] (1.10)

where it is sensible to view α, β and γ simply as parameters that define the temperature-dependence of a particular rate constant. Moreover, it should be noted that, because of correlations between these parameters, they are only accurately determined when k(T) has been measured (or calculated) accurately over a wide range of temperature.

Finally in this section, I refer to the insight into the activation energy provided by Tolman. This depends on the notion that for collisions between reactants at a specific relative velocity, u, one can define a rate coefficient as the product of u and the reaction cross-section, σ(u), with the result that the thermal rate constant, k(T), can (in principle) be found by first multiplying uσ(u), by a normalised function, f(u; T), describing the distribution of relative velocities at temperature T, and then integrating the resulting expression over u:

[MATHEMATICAL EXPRESSION OMITTED] (1.11)

This equation can be rewritten in terms of relative translational energies, Etrans = 1/2μu where μ is the collisional reduced mass, yielding:

[MATHEMATICAL EXPRESSION OMITTED] (1.12)

where the lower limit of integration, E°trans, is the threshold energy.

Tolman’s contribution was to realise that, if eqn (1.12) is substituted into the right-hand side of eqn (1.8), one finds that:

Eact = – (1.13)

That is the activation energy is the difference between the average translational energy in the collisions that lead to reaction and the average translational energy in all collisions. Although this treatment neglects any dependence of the reactivity on the internal states of the reactants, it demonstrates the important result that rate constants can decrease with temperature if the average collisional energy in reactive collisions is less than that in all collisions. This provides a rationale for the observation of negative activation energies for some reactions.

1.3 Potential Energy Surfaces and Transition State Theory

Within the Born–Oppenheimer approximation, the results of molecular collisions depend on the motion of the nuclear particles (atoms) on the potential energy (hyper) surface (PES) that describes how the electronic energy of the system depends on the relative position of the nuclei. (Remember that the derivatives of the PES with respect to the nuclear co-ordinates describe the forces acting on the atoms at any position on the PES.) In addition, dynamic and kinematic factors will determine the result (or, quantum mechanically, the probability of a given result) for any particular collision.

The calculation of a PES from first principles quantum mechanics is a formidable task, not least because it requires many individual calculations for different geometries if the PES is to be fully mapped. Even the simplest reactive system of interest involves three atoms, which I shall refer to as A, B and C, and requires three spatial co-ordinates, say rAB, rBC, and rCA, to define its instantaneous geometry. The potential energy can then be written as V (rAB, rBC, rCA) and the difficulty or ‘expense’ of the quantum chemical calculation of V at any single geometry depends strongly on the number of electrons in the system. From this point of view, a system of three H atoms is the simplest involving neutral atoms. A surface representing V(rAB, rBC, rCA) for three collinear atoms can be represented by the kind of contour plot shown in Figure 1.1. Imposing linearity means that V depends on only two independent co-ordinates so that V can be represented by a surface. Lifting this restriction so that V depends on three independent spatial co-ordinates means that a hypersurface is required.

A full characterisation of V(rAB, rBC, rCA) may be required for full scattering calculations, for example, for quasi-classical trajectory calculations. What might be termed traditional scattering calculations proceed through three stages. First, the energies at points on the PES are calculated by full quantum chemical methods, at a level of theory that is appropriate, or which can be afforded. Then these energies are fitted to a function that describes how V depends on the position co-ordinates of the system, and finally one calculates, by classical or quantum methods, the dynamics of collisions on the fitted PES. Monte Carlo methods can be used to select the initial parameters (position and momentum co-ordinates) for each collision in order to yield statistically meaningful results in a reasonably small sample of trajectories. Such calculations yield quantities such as scattering angles, product state distributions, etc., that is, the quantities that are measured to characterise the reaction dynamics of the reaction under examination.

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