Transition-state theory and the potential of mean force

2 Transition-state theory and the potential of mean force 

The purpose of this section is to give a detailed discussion of the material in Section 10.1, as well as to elaborate on the results. Equation (10.13) is a convenient expression from a computational point of view, but the simplicity of the expression is at the cost of hiding the complexity of the terms involved, including physical insights concerning solvent effects. 

We will have a closer look at the coordinate transformations leading to Eq. (10.13) from Eq. (10.3). Furthermore, we will consider some alternative forms of Eq. (10.13). In Eq. (10.13), the potential of mean force is related to the average force on the reaction coordinate. An alternative definition of the potential of mean force, which we will consider, is related to the average force on the atoms of the activated complex exerted by the solvent molecules. 

The structure of the activated complex may be perturbed compared to the gas phase when it is placed in a solvent. We derive an exact expression for the static solvent effect on the rate constant and an exact expression for the relation between the 

We consider in the following a bimolecular reaction, and the starting point for the evaluation of the solvent effects on the reaction is the situation where the reactants A and B have been brought together by diffusion in the same solvent cage . We then consider the reaction 

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From the potential of mean force the rate constant can be calculated. We first assume that transition-state theory is valid, and approximate the potential near the minimum and near the maximum by parabolas. The rate of escape of a particle from the well over the barrier is then [19] ... 

Solvent effects enter through the potential of mean force and the activation energy they may cancel or nearly cancel in the expression for kj (cf. Northrup and Hynes ). The collision frequency per unit density of B, ab(8 b /Mab) 1 expression for k° for a bimolecular reaction, takes the place of the frequency Wq in (3.23) for an isomerization reaction. This analysis shows that the transition state expression for the rate coefficient appears in this theory as a singular contribution to the rate kernel for the hard-sphere model of the reaction. 

A model of immiscible Lennard-Jones atomic solvents has been used to study the adsorption of a diatomic solute [71]. Subsequently, studies of solute transfer have been performed for atoms interacting through Lennard-Jones potentials [69] and an ion crossing an interface between a polar and a nonpolar liquid [72]. In both cases the potential of mean force experienced by the solute was computed the results of the simulation were compared with the result from the transition state theory (TST) in the first case, and with the result from a diffusion equation in the second case. The latter comparison has led to the conclusion that the rate calculated from the molecular dynamics trajectories agreed with the rate calculated using the diffusion equation, provided the mean-force potential and the diffusion coefficient were obtained from the microscopic model. 

When the potential of mean force is known, we can compute the transition state theory rate of barrier crossing in solution. It is customary to define A W(q) as the difference in the potential of mean force between the gas phase and solution and to write symbolically... 

V.2.1 Centroid transition state theory. A third methodology, is to construct approximate theories for dynamical properties, which make use of only thermodynamic quantities. In analogy with classical TST, Gillan, Voth and coworkers have formulated and studied a quantum TST which is based on the centroid potential of mean force Wc (q) ... 

Chapters 9-11 deal with elementary reactions in condensed phases. Chapter 9 is on the energetics of solvation and, for bimolecular reactions, the important interplay between diffusion and chemical reaction. Chapter 10 is on the calculation of reaction rates according to transition-state theory, including static solvent effects that are taken into account via the so-called potential-of-mean force. Finally, in Chapter 11, we describe how dynamical effects of the solvent may influence the rate constant, starting with Kramers theory and continuing with the more recent Grote-Hynes theory for... 

Potential of mean-force (PMF) calculations have been frequently used in the study of solvation at ambient and supercritical conditions [168,226-232], especially when researchers are interested in the behavior of solutes at high dilution, conditions at which solute-solute interactions become rare events and, consequently, cannot be accounted for by conventional distribution function calculations. The resulting PMF are in fact free energy profiles which can be used to determine the corresponding association constants, as well as the (transition state theory) kinetic rate constants for the conversion governing different solute-pair configurations separated by energy barriers. 

The motion in the reaction coordinate Q is, like in gas-phase transition-state theory, described as a free translational motion in a very narrow range of the reaction coordinate at the transition state, that is, for Q = 0 hence the subscript trans on the Hamiltonian. The potential may be considered to be constant and with zero slope in the direction of the reaction coordinate (that is, zero force in that direction) at the transition state. The central assumption in the theory is now that the flow about the transition state is given solely by the free motion at the transition state with no recrossings. So when we associate a free translational motion with that coordinate, it does not mean that the interaction potential energy is independent of the reaction coordinate, but rather that it has been set to its value at the transition state, Q j = 0, because we only consider the motion at that point. The Hamiltonian HXlans accordingly only depends on Px, as for a free translational motion, so... 

This review shows how the photochemistry of ketones can be rationalized through a single model, the Tunnel Effect Theory (TET), which treats reactions of ketones as radiationless transitions from reactant to product potential energy curves (PEC). Two critical approximations are involved in the development of this theory (i) the representation of reactants and products as diatomic harmonic oscillators of appropriate reduced masses and force constants (ii) the definition of a unidimensional reaction coordinate (RC) as the sum of the reactant and product bond distensions to the transition state. Within these approximations, TET is used to calculate the reactivity parameters of the most important photoreactions of ketones, using only a partially adjustable parameter, whose physical meaning is well understood and which admits only predictable variations. 

In the second part we study the ion speciation in infinitely dilute NaCI aqueous solutions by determining the constant of as.sociation by constraint molecular dynamics via mean-force potential calculations. We determine the temperature and density dependence of the extent of the ion association. In addition we analyze the kinetics of the interconversion between two ion pair configurations, the contact ion pair and the solvent-shared ion pair, by determining the transition state theory (TST) kinetic rates. 

In the case of a single test particle B in a fluid of molecules M, the effective one-dimensional potential f (R) is — fcrln[R gBM(f )]. where 0bm( ) is th radial distribution function of the solvent molecules around the test particle. In this chapter it will be assumed that 0bm( )> equilibrium property, is a known quantity and the aim is to develop a theory of diffusion of B in which the only input is bm( )> particle masses, temp>erature, and solvent density Pm- The friction of the particles M and B will be taken to be frequency indep>endent, and this should restrict the model to the case where > Wm, although the results will be tested in Section III B for self-diffusion. Instead of using a temporal cutoff of the force correlation function as did Kirkwood, a spatial cutoff of the forces arising from pair interactions will be invoked at the transition state Rj of i (R). While this is a natural choice because the mean effective force is zero at Rj, it will preclude contributions from beyond the first solvation shell. For a stationary stochastic process Eq. (3.1) can then be... 

It should be noted that nuclei and electrons are treated equivalently in //, which is clearly inconsistent with the way that we tend to think about them. Our understanding of chemical processes is strongly rooted in the concept of a potential energy surface which determines the forces that act upon the nuclei. The potential energy surface governs all behaviour associated with nuclear motion, such as vibrational frequencies, mean and equilibrium intemuclear separations and preferences for specific conformations in molecules as complex as proteins and nucleic acids. In addition, the potential energy surface provides the transition state and activation energy concepts that are at the heart of the theory of chemical reactions. Electronic motion, however, is never discussed in these terms. All of the important and useful ideas discussed above derive from the Bom-Oppenheimer approximation, which is discussed in some detail in section B3.1. Within this model, the electronic states are solutions to the equation... 

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