Dividing surface optimal

Optimal planar dividing surface VTST is thus reduced to finding the maximum of the free energy w[f]. 

PGH theory has been extended. It can be used in conjunction with VTST and optimized planar dividing surfaces,in which case, the energy loss is to be computed along the coordinate perpendicular to the optimal planar dividing surface. In the same vein it has been generalized to include the case of space and time dependent friction. ... 

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 ... 

Optimal planar dividing surface VTST has been used to study the effects of exponential time dependent friction in Ref. 93. The major interesting result was the prediction of a memory suppression of the rate of reaction which occurs when the memory time and the inverse damping time (y) are of the same order. When... 

A study of the effects of space and time dependent friction was presented in Ref. 68. One finds a substantial reduction of the rate relative to the parabolic barrier estimate when the friction is stronger in the well than at the barrier. In all cases, the effects become smaller as the reduced barrier height becomes larger. Comparison with molecular dynamics simulations shows that the optimal planar dividing surface estimate for the rate is usually quite accurate. 

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. 

Instead, classical trajectory simulations are performed to determine the fraction of trajectories crossing the dividing surface that actually contribute to the formation of product molecules [5,6]. If the surface is the optimal one corresponding to a minimum value for the rate constant, all trajectories crossing the dividing surface from the reactant side to the product side will lead to the formation of products. If not, a certain fraction of the trajectories crossing the dividing surface will turn around, recross the surface, and therefore not make a contribution to the formation of products. 

If a trajectory makes it to the product side, as sketched in Fig. 5.1.2, then it is also propagated backwards in time from the initial position to check if it originated on the reactant side. If so, the trajectory is marked as successful. In all other cases, that is, the trajectories do not make it to the product side, or if so, do not originate on the reactant side, then they are registered as unsuccessful. The fraction k of the total number of trajectories that are marked successful is now used to rectify the rate constant for not being calculated with the optimal choice of the dividing surface, and the final result is reported as... 

The formalism summarized above is well suited for bimolecular reactions with tight transition states and simple barrier potentials. In such cases we have found that the variational transition state can be found by optimization of a one-parameter sequence of dividing surfaces orthogonal to the reaction path, where the reaction path is defined as the MEP. However, although dividing surfaces defined as hyperplanes perpendicular to the tangent to the MEP (as described in Section 5.2.2) are very serviceable, a number of improvements have been put forth. 

The use of curvilinear coordinates and optimization of the orientation of the dividing surface are important for quantitative calculations on simple barrier reactions, but even more flexibility in the dividing surfaces is required to obtain quantitative results for very loose variational transition states such as those for barrier-less association reactions or their reverse (dissociation reactions without an intrinsic barrier). 

In the context of association reactions, an algorithm in which the reaction coordinate definition is optimized along with the dividing surface along a one-parameter sequence of paths is called variable reaction coordinate (VRC) variational transition state theory... 

If the reaction path and dividing surface are optimized in the gas phase, but the rate constant is calculated with the equilibrium solvation Hamiltonian, the resulting rate constant is called separable equilibrium solvation (SES) [57]. However, if the reaction path and dividing surface are optimized with the equilibrium solvation potential, the result is labeled equilibrium solvation path (ESP) [57,78]. 

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