Variational dividing surfaces

The final system considered in Table 5 is the Br + HCl reaction for the extended LEPS surface of Douglas et dl. Although this system has its saddle point well into the product channel, it differs from other systems with this property in that it does not have the location of its variational dividing surface dominated by the bending... 

We will use the superscript a to denote surface quantities calculated on the preceding assumption that the bulk phases continue unchanged to an assumed mathematical dividing surface. For an arbitrary set of variations from equilibrium. 

The hypersurface fomied from variations in the system s coordinates and momenta at//(p, q) = /Tis the microcanonical system s phase space, which, for a Hamiltonian with 3n coordinates, has a dimension of 6n -1. The assumption that the system s states are populated statistically means that the population density over the whole surface of the phase space is unifomi. Thus, the ratio of molecules at the dividing surface to the total molecules [dA(qi, p )/A]... 

In both solvents, the variational transition state (associated with the free energy maximum) corresponds, within the numerical errors, to the dividing surface located at rc = 0. It has to be underlined that this fact is not a previous hypothesis (which would rather correspond to the Conventional Transition State Theory), but it arises, in this particular case, from the Umbrella Sampling calculations. However, there is no information about which is the location of the actual transition state structure in solution. Anyway, the definition of this saddle point has no relevance at all, because the Monte Carlo simulation provides directly the free energy barrier, the determination of the transition state structure requiring additional work and being unnecessary and unuseful. 

Different prescriptions to choice the set of dividing surfaces in order to apply the Variational Transition State Theory, should be analyzed and compared. 

Of course, one is not really interested in classical mechanical calculations. Thus in normal practice the partition functions used in TST, as discussed in Chapter 4, are evaluated using quantum partition functions for harmonic frequencies (extension to anharmonicity is straightforward). On the other hand rotations and translations are handled classically both in TST and in VTST, which is a standard approximation except at very low temperatures. Later, by introducing canonical partition functions one can direct the discussion towards canonical variational transition state theory (CVTST) where the statistical mechanics involves ensembles defined in terms of temperature and volume. There is also a form of variational transition state theory based on microcanonical ensembles referred to by the symbol p,. Discussion of VTST based on microcanonical ensembles pVTST is beyond the scope of the discussion here. It is only mentioned that in pVTST the dividing surface is... 

The overestimation of the TST rate constant leads to a variational principle for the optimization of the position of the dividing surface constituting the transition state. In general, one can write ... 

The optimal choice of the dividing surface S(pj,r) is, according to the Wigner theorem, the surface that gives the smallest rate constant k(T). In principle, it can be determined by a variational calculation of k(T) with respect to the surface such that 6k(T) = 0. 

The approach described above will, in general, not give the exact rate constant, since it is based on a quite arbitrary choice of the dividing surface we do not know if the choice is valid according to the Wigner theorem, namely that the rate constant is at a minimum with respect to variations in the choice of dividing surface. A variational determination of the rate constant with respect to the position of the dividing surface is usually not done directly. 

The central idea in the variational treatment of TST [15,38,39] is to consider all possible dividing surfaces that partition coordinate space into two separate regions, one associated with reactants and the other with products. One then considers the flux across each of these surfaces and chooses the surface with the minimum flux as the TS. The logic is as follows We are interested in the rate at which states cross the TS. If, during the course of a reaction, the trajectory of each initial reactant state only crosses the TS once, then the rate of reaction can... 

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