Resonant periodic orbits

The eigenfunctions associated with the resonances have been obtained via wavepacket propagation. They appear to be localized along the symmetric-stretch periodic orbit 0, with a number of nodes equal to n and even under the exchange of iodine nuclei. Due to the relative stability of the symmetric-stretch orbit, we have thus here a system where the hypothesis of the orbit 12 representing the RPO, that is, resonant periodic orbit, does not hold. 

Marston, C.C., and Wyatt, R.E. (19846). Semiclassical theory of resonances in 3D chemical reations I Resonant periodic orbits for F+H2, J. Chem. Phys. 81, 1819. 

Consider then an adiabatic well in the hyperspherical coordinate system. Classically, the motion of the periodic orbit at the well would be an oscillation from a point on the inner equipotential curve in the reactant channel to a point on the same equipotential curve in the product channel. This is qualitatively the motion of what are termed "resonant periodic orbits" (RPO s). For example the RPO s of the IHI system are given in Fig. 5. Thus, finding adiabatic wells in the radial coordinate system corresponds to finding RPO s and quantizing their action. Note that in Fig. 5 we have also plotted all the periodic orbit dividing surfaces (PODS) of the system, except for the symmetric stretch. By definition, a PODS is a periodic orbit that starts and ends on different equi-potentials. Thus the symmetric stretch PODS would be an adiabatic well for an adiabatic surface in reaction path coordinates. However, the PODS in the entrance and exit channels shown in Fig. 5 may be considered as adiabatic barrieres in either the radial or reaction path coordinate systems. Here, the barrier in radial coordinates, has quantally a tunneling path between the entrance and exit channels. 

We compare the results in Table IV with the approximate resonant periodic orbit (RPO) calculations of Poliak and Wyatt (147) and with accurate quantal calculations (61). In the RPO calculations, the bending degrees of freedom are Included using... 

For the quasiperlodlc orbit shown In Figure 2, three projections of the orbit In the cylindrical coordinate space (R,p,y) ure Illustrated In Figure 3. In parts A and B, the direction of motion of the orbit between t and t end (14 turning points) Is also shown. Some of the turning points are numbered. Symmetry of the orbit about the Y>0 (colllnear) plane Is evident In part B. Fart C of this figure again shows that the (p,R) motion Is similar to a "blurred" colllnear resonant periodic orbit. 

Figure 7. A resonant periodic orbit at E=0,4 eV (y 29 ). Part A shows the orbit within the V 0.4 eV FH2 potential surface. Part B shows the orbit in the Internal coordinate space (R,x,y). Once again, projections of the orbit onto the three coordinate planes are also shown.

In Fig. 6 we show for the hydrogen exchange reaction a comparison between the well energies and locations as computed from resonant periodic orbits (RPO s) and the exact quantal averaged adiabatic surfaces E-(p) and Uj (p). Note that there is quantitative agreement with the wells of U (p). 

If the mapping (11.6.2) is the Poincare map of an autonomous system of differential equations, then the invariant curve corresponds to a two-dimensional smooth invariant torus (see Fig. 11.6.3). It is stable if L < 0, or it is saddle with a three-dimensional unstable manifold and an (m -h 2)-dimensional stable manifold if L > 0. Recall from Sec. 3.4, that the motion on the torus is determined by the Poincare rotation number if the rotation number v is irrational, then trajectories on the torus are quasiperiodic with a frequency rate u] otherwise, if the rotation number is rational, then there are resonant periodic orbits on a torus. 

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