### Abstract

We consider a triatomic system with zero total angular momentum and demonstrate that, no matter how complicated the anharmonic part of the potential energy function, classical dynamics in the vicinity of a saddle point is constrained by symmetry properties. At short times and at not too high energies, recrossing dynamics is largely determined by elementary local structural parameters and thus can be described in configuration space only. Conditions for recrossing are given in the form of inequalities involving structural parameters only. Explicit expressions for recrossing times, valid for microcanonical ensembles, are shown to obey interesting regularities. In a forward reaction, when the transition state is nonlinear and tight enough, one-fourth of the trajectories are expected to recross the plane R = R * (where R * denotes the position of the saddle point) within a short time. Another fourth of them are expected to have previously recrossed at a short negative time, i.e., close to the saddle point. These trajectories do not contribute to the reaction rate. The reactive trajectories that obey the transition state model are to be found in the remaining half. However, no conclusion can be derived for them, except that if recrossings occur, then they must either take place in the distant future or already have taken place in the remote past, i.e., far away from the saddle point. Trajectories that all cross the plane R = R * at time t = 0, with the same positive translational momentum can be partitioned into two sets, distinguished by the parity of their initial conditions; both sets have the same average equation of motion up to and including terms cubic in time. Coordination is excellent in the vicinity of the saddle point but fades out at long (positive or negative) times, i.e., far away from the transition state.

Received 21 January 2014
Accepted 20 March 2014
Published online 04 April 2014

Acknowledgments:
I am very grateful to Professor Bernard Leyh and Professor David Wales for helpful comments on a first draft of the paper.

Article outline:

I. INTRODUCTION
II. GENERIC PROPERTIES OF THE ATOM-DIATOM INTERACTION
III. A SPECIFIC HAMILTONIAN
A. Reaction coordinate
B. Vibration of the diatomic
C. Angular degree of freedom
D. Three-body interaction term
IV. ENERGY EQUIPARTITION
V. AVERAGE RECROSSING TIMES
A. The cubic approximation
B. Forward reactions
1. The first set of initial conditions
2. The second set of initial conditions
3. The remaining two sets of initial conditions
4. Conclusions
5. Numerical example
C. Reverse reactions
VI. INDIVIDUAL RECROSSING TIMES AT LOW ENERGIES
A. Forward reactions for a soft TS
B. Forward reactions when the TS is stiff
C. Conclusions
VII. THE MICROCANONICAL ENSEMBLE
VIII. MOTION COORDINATION
IX. CONCLUDING REMARKS

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