The behaviour of long-lived classical trajectories through saddle points and their contribution to the semi-classicalS-matrix

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Abstract
The properties of long-lived classical trajectories on two-dimensional repulsive surfaces with a saddle point have been investigated using two model potentials. In the first the motion in the region of the saddle point is separable in the two coordinates (one internal and one reaction coordinate), and an analytic form is given for the transition probability contributions from trajectories which exhibit many oscillations at the saddle point. The second potential includes a quartic coupling term between the two degrees of freedom, and this potential is shown to simulate the properties of a surface studied by Marcus and Stine. Their method of summing contributions of long-lived trajectories to the S-matrix is discussed.