Free-energy surface saddle point

Describing the dissociative electron transfer step, RX e R + X involves determining the saddle point on the intersection of the two following free energy surfaces.22,31... 

Gao et al. located the saddle point of the two-dimensional free energy surface at Rc-n = 1.96 A and Rc-ci = 2.09 A, and used this point to define the transition state in solution. This way, their most striking finding was that on going from the gas phase... 

Strong interactions are observed between the reacting solute and the medium in charge transfer reactions in polar solvents in such a case, the solvent effects cannot be reduced to a simple modification of the adiabatic potential controlling the reactions, since the solvent nuclear motions may become decisive in the vicinity of the saddle point of the free energy surface (FES) controlling the reaction. Also, an explicit treatment of the medium coordinates may be required to evaluate the rate constant preexponential factor. 

Figure 19.19 Saddle point on free-energy surface.

In practical applications, this technique is applied by performing a meta-d3mamics reconstruction for each local minimum in the free energy surface. The walkers are forced to remain in their minimum by reflecting walls located just behind the saddle point region. In this manner the transition regions are explored in two independent metad3mamics, and the overlap is sufficient to make the solution to (36) reliable. 

We assume that the free-energy surface has a saddle point at p, a and that its parabolic expansion is... 

The first identity relates the saddle point frequencies to the parabolic barrier frequency. It results from the condition that the collective modes are normal modes of the free-energy surface such that - 0. The second identity means that the saddle point of the free-energy surface necessarily lies on the subspace q = q of the full coordinate space and the third identity relates the barrier height at the saddle point to the barrier height of the potential of mean force. 

This is offset somewhat by an increase in the surface area of the edge, u. As in traditional 2-D nucleation theories, the squared-in-radius term overweighs the linear-in-radius term and the F a, b a)) curve must have a maximum zX a = a. Hence, the free energy surface F(a, b) of the normal film must have a saddle point, a maximum versus a and a minimum versus b. The problem of finding the energy barrier against coalescence is then reduced to determining the saddle point of this surface ... 

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. 

One must recognize that TST is much simpler conceptually than VTST. Thus, there is one transition state in TST and that is located at the maximum energy on the MEP (the saddle point). In VTST the dividing surface is temperature dependent since the partition functions and consequently the free energy of activation are temperature dependent. 

Vq is the frequency of the small oscillation, and AG and AS are, respectively, the difference in Gibbs free energy and entropy of the adatom at the saddle point and the equilibrium adsorption site. Ed is the activation energy of surface diffusion, or the barrier height of the atomic jumps. 

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