Rate constants, reactive flux

In this chapter we consider a direct approach to the calculation of k(T) that bypasses the detailed state-to-state reaction cross-sections. The method is based on the calculation of the reactive flux across a dividing surface on the potential energy surface. The results are  

In the previous chapter, we have discussed the reaction dynamics of bimolecular collisions and its relation to the most detailed experimental quantities, the cross-sections obtained in molecular-beam experiments, as well as the relation to the well-known rate constants, measured in traditional bulk experiments. Indeed, in most chemical applications one needs only the rate constant—which represents a tremendous reduction in the detailed state-to-state information. 

An important recent theoretical development is the direct approaches for calculating rate constants. These approaches express the rate constant in terms of a so-called flux operator and bypass the necessity for calculating the complete state-to-state reaction probabilities or cross-sections prior to the evaluation of the rate constant [1-3]. This is the theme of this chapter. 

let us combine some key results from the previous chapters. For a reaction written in the form A + BC(n) — AB(m) + C, we have, according to Eq. (2.29), 

We insert Eq. (5.2) into Eq. (5.1), and note that the factors in front of the integral are related to the partition function associated with the relative translational motion of the reactants, see Eq. (A. 14)  

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The exact quantum expression for the activated rate constant was first derived by Yamamoto [6]. The resulting quantum reactive flux correlation fiinction expression is given by... 

Under the conditions (1.1) the rate constant is determined by the statistically averaged reactive flux from the initial to the final state. 

A number of papers are devoted to the effect of dissipation on tunneling.81"83,103,104 Wolynes81 was one of the first to consider this problem using the Feynman path integral approach to calculate the correlation function of the reactive flux involved in the expression for the rate constant,... 

The calculation of reaction rate constants with the transition path sampling methods does not require understanding of the reaction mechanism, for instance in the form of an appropriate reaction coordinate. If such information is available other methods such as the reactive flux formalism are likely to yield reaction rate constants at a lower computational cost than transition path sampling. 

It is perhaps wise to begin by questioning the conceptual simplicity of the uptake process as described by equation (35) and the assumptions given in Section 6.1.2. As discussed above, the Michaelis constant, Km, is determined by steady-state methods and represents a complex function of many rate constants [114,186,281]. For example, in the presence of a diffusion boundary layer, the apparent Michaelis-Menten constant will be too large, due to the depletion of metal near the reactive surface [9,282,283], In this case, a modified flux equation, taking into account a diffusion boundary layer and a first-order carrier-mediated uptake can be taken into account by the Best equation [9] (see Chapter 4 for a discussion of the limitations) or by other similar derivations [282] ... 

The Hamiltonian in Eq. (39) has bear used to calculate the adiahatic free energy as a function of the solvent coordinate using the umbrella sampling method, and reactive flux correlation function calculations have been used to determine the adiabatic rate constant. The results were qualitatively similar to the results based on the two-state model. 

Drozdov and Tucker have recently criticized the VTST method claiming that it does not bound the exact rate constant. Their argument was that the reactive flux method in the low barrier limit, is not identical to the lowest nonzero eigenvalue of the corresponding Fokker-Planck operator, hence an upper bound to the reactive flux is not an upper bound to the true rate. As aheady discussed above, when the barrier is low, the definition of the rate becomes problematic. All that can be said is that VTST bounds the reactive flux. Whenever the reactive flux method fails, VTST will not succeed either. 

The main advantage of the VTST method is that it can be applied also to realistic simulations of reactions in condensed phases.The optimal planar coordinate is determined by the matrix of the thermally averaged second derivatives of the potential at the barrier top. VTST has been applied to various models of the CP-i-CHsCl Sn2 exchange reaction in water, a system which was previously studied extensively by Wilson, Hynes and coworkers.Excellent agreement was found between the VTST predictions for the rate constant and the numerically exact results based on the reactive flux method. The VTST method also allows one to determine the dynamical source of the friction and its range, since it identifies a collective mode which has varying contributions from differ-... 

There is another more direct way of calculating the rate constant k(T), i.e., it is possible to bypass the calculation of the complete state-to-state reaction probabilities, S m(E) 2, or cross-sections prior to the evaluation of the rate constant. The formulation is based on the concept of reactive flux. We start with a version of this formulation based on classical dynamics and, in a subsequent section, we continue with the quantum mechanical version. It will become apparent in the next section that the classical version is valid not only in the gas phase, but in fact in any phase, that is, the foundation for condensed-phase applications will also be provided. 

In Chapter 5, the direct evaluation of k(T) via the reactive flux through a dividing surface on the potential energy surface was described. As a continuation of that approach, we consider in this chapter an—approximate—approach, the so-called transition-state theory (TST).1 We have already briefly touched upon this approximation, based on an evaluation of a stationary one-way flux, which implies that the rate constant can be obtained without any explicit consideration of the reaction dynamics. In this chapter, we elaborate on this important approach, in a form that takes some quantum effects into account. 

In Chapter 5, attention is directed toward the direct calculation of k(T), i.e., a method that bypasses the detailed state-to-state reaction cross-sections. In this approach the rate constant is calculated from the reactive flux of population across a dividing surface on the potential energy surface, an approach that also prepares for subsequent applications to condensed-phase reaction dynamics. In Chapter 6, we continue with the direct calculation of k(T) and the whole chapter is devoted to the approximate but very important approach of transition-state theory. The underlying assumptions of this theory imply that rate constants can be obtained from a stationary equilibrium flux without any explicit consideration of the reaction dynamics. 

Equations (359) and (361) indicate that the reason that the flux-flux autocorrelation formalism gives exact quantum reaction rate constants is simply that all the dynamical information from time zero to time inflnity has been included. Indeed, as shown by Eqs. (360) and (362), in the classical limit the flux-flux autocorrelation formalism requires us to follow all classical trajectories until t = +00 so as to rigorously tell which trajectory is reactive and which trajectory is nonreactive. Evidently, then, the flux-flux autocorrelation formalism is not a statistical reaction rate theory insofar as no approximation to the reaction dynamics is made. 

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