Reactive fluxes

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... 

Den Otter, W.K., Briels, W.J. The reactive flux method applied to complex reactions using the unstable normal mode as a reaction coordinate. J. Chem. Phys. 106 (1997) 1-15. 

We counted the contribution of only those trajectories that have a positive momentum at the transition state. Trajectories with negative momentum at the transition state are moving from product to reactant. If any of those trajectories were deactivated as products, their contribution would need to be subtracted from the total. Why Because those trajectories are ones that originated from the product state, crossed the transition state twice, and were deactivated in the product state. In the TST approximation, only those trajectories that originate in the reactant well are deactivated as product and contribute to the reactive flux. We return to this point later in discussing dynamic corrections to TST. 

Using these distribution functions, we can write the reactive flux correlation function in the compact form... 

Figure 5 The transition state ensemble is the set of trajectories that are crossing the transition state from reactant to product at equilibrium (shown as black dots). There are four types of trajectories, shown top to bottom in the diagram. (1) Starting as reactant, the trajectory crosses and recrosses the transition state and is deactivated as reactant. It does not add to the reactive flux. (2) Starting as reactant, the trajectory is deactivated as product. It adds +1 to the reactive flux. (3) Starting as product, the trajectory crosses and recrosses the transition state and is deactivated as product. Such a trajectory must be subtracted from the ensemble, so it counts —1 to the reactive flux. (4) Starting as product, the trajectory is deactivated as reactant. It does not contribute to the reactive flux.

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

In most cases, the observables measured in the study of a chemical reaction are interpreted under the following (often valid) assumptions (1) each product channel observed corresponds to one path on the PES, (2) reactions follow the minimum energy path (MEP) to each product channel, and (3) the reactive flux passes over a single, well-defined transition state. In all of the reactions discussed in this chapter, at least one, and sometimes all of these assumptions, are invalid. 

One disadvantage of statistical approaches is that they rely on two of the assumptions stated in the introduction, namely, that reactions follow the minimum energy path to each product channel, and that the reactive flux passes through a transition state. Several examples in Section V violate one or both of these assumptions, and hence statistical methods generally cannot treat these instances of competing pathways [33]. 

The general criteria for an experimental investigation of the kinetics of reactions at liquid-liquid interfaces may be summarized as follows known interfacial area and well-defined interfacial contact are essential controlled, variable, and calculable mass transport rates are required to allow the transport and interfacial kinetic contributions to the overall rate to be quantified direct interfacial contact is preferred, since the use of a membrane to support the interface adds further resistances to the overall rate of the reaction [14,15] a renewable interface is useful, as the accumulation of products at the interface is possible. Finally, direct measurements of reactive fluxes at the interface of interest are desirable. 

Fig. 2. Schematic configuration space for the reaction AB + CD — A + BCD. R is the radial coordinate between the center-of-mass of the two diatoms, and r is the vibrational coordinate of the reactive AB diatom. I denotes the interaction region and II denotes the asymptotic region. The shaded regions are the absorption zones for the time-dependent wavefunction to avoid boundary reflections. The reactive flux is evaluated at the r = rB surface.

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. 

A number of reactions between metals (or metals and semi-metals) may be carried out by dissolving the elements in a suitable solvent , technically also termed a flux . The solvent may also act as a reactant and be involved in the formation of the compound (reactive solvent, reactive flux). Several fluxes, ranging from simple metallic elements up to complex substances have been used. 

Borides from metallic fluxes. The preparation of several binary borides of lanthanides, vanadium, tantalum, chromium has been performed by using aluminium as a flux. Aluminium has been used also as a reactive flux in the preparation of alumino-borides (of Mo, Fe) which are stable in concentrated HC1 solutions. 

Chandler D, Wolynes PG (1981) J Chem Phys 74 4078. Quantum Path Integral Monte Carlo can also provide a way to evaluhte the rate in non-adiabatic reactive flux correlation theory Wolynes PG (1987) J Chem Phys 87 6559... 

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. 

From a practical point of view, integrating trajectories for times which are of the order of eP is very expensive. When the reduced barrier height is sufficiently large, then solution of the Fokker-Planck equation also becomes numerically very difficult. It is for this reason, that the reactive flux method, described below has become an invaluable computational tool. 

The major advantage of the reactive flux method is that it enables one to initiate trajectories at the barrier top. instead of at reactants or products. Computer time is not wasted by waiting for the particle to escape from the well to the barrier. The method is based on the validity of Onsager s regression hypothesis/ which assures that fluctuations about the equilibrium state decay on the average with the same rate as macroscopic deviations from equilibrimn. It is sufficient to know the decay rate of equilibrimn correlation fimctions. There isn t any need to determine the decay rate of the macroscopic population as in the previous subsection. 

Taking the time deravitive of Eq.l2 with respect to t and setting t = 0 finds that the reactive flux obeys ... 

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