Interpolation surfaces, reaction paths

The kineties of eleetron-transfer reactions, which is also affected by the electrode potential and the metal-water interface, is more difficult and complex to treat than the thermodynamic aspects. While the theoretical development for electron transfer kinetics began decades ago, a practical implementation for surface reactions is still unavailable. Popular transition state-searching techniques such as the NEB method are not designed to search for minimum-energy reaction paths subject to a constant potential. Approximations that allow affordable quantum chemistry calculations to get around this limitation have been proposed, ranging from the electron affinity/ionization potential matching method to heuristic arguments based on interpolations. 

This fitting of a more or less global surface is not only unnecessary for applications such as VTST but very time consuming. Direct use of the ab initio data by interpolation methods promises to facilitate the application of ab initio calculations to reaction path dynamics [12, 44, 56, 57,... 

Schatz and Elgersma [161] have published an analytic surface for the reaction of Eq. (5.4). This surface has a relatively simple reaction path, somewhat like that of Figs. 4 and 5. To test the interpolation procedure thoroughly at reasonable cost, we pretend that this analytic surface supplies us with the ab initio data, and we require that our converged interpolated surface will give us the same classical trajectory result for the probability of reaction as the analytic surface [204]. 

Since there are very few energy surfaces which have been completely characterized by an exhaustive search for minima and TSs, comparisons of different optimization methods on the same surface are scarce. The interpolation technique.s, especially those which try to map the complete reaction path, tend to have somewhat higher computational requirements than the local methods. Consequently, interpolation methods have primarily been used in connection with force field and semiempirical electronic structure methods, while local methods have been associated mainly with ab initio-type calculations. 

The reaction coordinate is one specific path along the complete potential energy surface associated with the nuclear positions. It is possible to do a series of calculations representing a grid of points on the potential energy surface. The saddle point can then be found by inspection or more accurately by using mathematical techniques to interpolate between the grid points. 

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