Transition rate for a multidimensional system

More insight into the nature of TST can be obtained from the generalization of the above treatment to a multidimensional system. Consider an (N + l)-dimensional system defined by the Hamiltonian 

Herex + denotes the set (xo,xi. xn ). It is assumed that the potential, C(x ), has a well whose minimum is at some point and which is surrounded by a domain of attraction, separated from the outside space by a potential barrier. 

As before, TST assumes that (1) thermal equilibrium exists within the reactant space, and (2) trajectories that cross the dividing surface from the reactant to the product space do not recross on the timescale of thennal relaxation in the product space. A straightforward generalization of the calculation of Section 14.3.2 then leads to 

This happens for the practical reason that the position of the saddle point on a multidimensional potential surface is not always known. 

While Eq. (14.21) seems much more complicated than its one-dimensional coun-teipart (14.12), a close scrutiny shows that they contain the same elements. The -function in (14.21) defines the dividing surface, the term V/ is the component of the momentum normal to this surface and the 0 function selects outwards going particles. If, for physical reasons, a particular direction, say xq, is identified as the reaction coordinate, then a standard choice for f in the vicinity of the saddle point x is) = vo — xso, where %so is the value ofthe reaction coordinate at that saddle point. This implies V/ = pQ, that is, the component of the 

This re-derivation of the one-dimensional TST result emphasizes the effective character of the potential used in one-dimensional treatments of barrier crossing problems. The one-dimensional model, Eq. (14.11), will yield the correct TST result provided that the potential V (x) is taken as the effective potential of the reaction coordinate, that is, the potential of mean force along this coordinate where all other degrees of freedom are in thermal equilibrium at any given position of this coordinate. It should be stressed, however, that this choice of the one-dimensional effective potential assumes that such a coordinate can be identified and that a point along this coordinate can be identified as the transition point that separates reactants from products. 

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