Saddle points coordinate selection

Using the same method as for the first excited electronic state, we select a level shift in the region Xi < p < X2. This procedure may indeed lead to a transition state but in this way we always increase the function along the lowest mode. However, if we wish to increase it along a higher mode this can only be accomplished in a somewhat unsatisfactory manner by coordinate scaling. Nevertheless, this method has been used by several authors with considerable success.14 The problem of several first-order saddle points does not arise in electronic structure calculations since there is only one first excited state.15... 

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... 

Fig. 1.4a-e. Assignment of nondegenerate (a-c) and degenerate (d, e) critical points of the three-dimensional PES E ( 1, 2)—the coordinates qi, 2 selected such as to correspond to the directions of the main curvature axes, a shows the point of a minimum with d Ejdq > 0 d Ej > 0 (the stability region) b denotes the saddle point with d Ejdql < 0 d EJdql > 0 (the transition state structure) c the point of a maximum with d Ejdq < 0 d Efdql < 0 d the degenerate point of a minimum with d Ejdq = 0 d Ejdq > 0 e the degenerate second-order saddle point (monkey saddle) with d Ejdql = d Ejdql = 0... 

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