Potential energy surfaces gradient extremals

Starting at a saddle point, a path of steepest descent can be defined on the potential energy surface by using the gradient function 8W/8Qj the path of steepest descent is uniquely determined by extremal values of the gradient unless a stationary point is reached (55). Besides the minima corresponding to the reactant and product asymptotes, a potential energy surface may exhibit some additional minima due to, e.g., van der Waals (59) complexes or intermediates (see later). In such cases, the reactant and product asymptote can be interconnected by several steepest descent paths and the construction... 

Figure 2 Gradient extremal paths on the MUller-Brown mixlel potential energy surface. Solid lines contours dotted lines selected steepest descent lines bold lines gradient extremals. Reproduced with permission from J. Q. Sun and K. Ruedenberg, J. Chem. Phys., 1993, 98, 9707. Copyright (1993) American Institute of Physics...

If this relation is substituted into equations (2) and (3), it can be seen that the steepest descent path through any point on a gradient extremal has zero curvature. For a quadratic expansion of the energy about an arbitrary point on the potential surface, a step of Ax can be taken toward the gradient extremal path. 

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