Polyad phase sphere

Figure 9.13 Phase space trajectories and polyad phase spheres for polyad N = 3 of H2O. H2O is an example of a molecule with two identical coupled anharmonic bond stretch oscillators. The local mode (Iz, ip) phase space trajectories in part (a) contain the same information as the normal mode (lz,ip) trajectories in part (b). The corresponding polyad phase spheres in parts (c) and (d) are identical except for a rotation by tt/2 about the y axis (from Xiao and Kellman, 1989).

Figure 9.13 displays the phase space structure of local and Hnormal for the N = vs + va = vr + vl = 3 [I = (N + l)/2 = 2]f polyad of H2O. Just as HloCAL and H qRMAL provide identical quality representations of the observed spectrum, so too do Wlocal and Tinormal- The phase space structures displayed in parts (a), LOCAL, and (b), NORMAL, of Fig. 9.13 are equivalent. The appearance of qualitatively different structures in the LOCAL and NORMAL representations is largely due to the mapping of the information onto an Iz, ip (or Iz, tp) planar rectangle rather than a polyad phase sphere. As shown in parts (c) and (d) of Fig. 9.13, the structures from parts (a) and (b) differ only by a rotation of the phase sphere by 7t/2 about the y axis. 

The central region of the local mode representation of the phase space trajectories is called the resonance region. In this resonance region of phase space the trajectories ( 3 and 4) are not free to explore the full 0 < ip < n range and are threfore classified as normal mode trajectories. Points A and B are fixed points which he at the maximum and minimum E extremes, Ea(I) and Eb(I), of the resonance region for a particular value of I. Point A at Iz = 0 (vr = vi) and ip = 7t/2 (out-of-phase motion of the R and L oscillators) is stable and corresponds to a pure antisymmetric stretch. Point B at Iz = 0 (vr = vl) and ip = 0 and 7r (in-phase motion) is unstable (because it lies on the separatrix) and corresponds to a pure symmetric stretch. Quasiperiodic trajectories that circulate about a stable fixed point resemble the fixed point periodic trajectory. At E > Ea(I) no trajectories of any type can exist. At E < Er(I) the B-like trajectories vanish and are replaced by trajectories that circulate about the Ca, Cb fixed points and are therefore C-like. The Ca and Cb lines (Iz = / = 2, 0 < ip < 7r) are actually the north and south poles on the local mode polyad phase sphere (Fig. 9.13(c)). The stable fixed points he near the poles and trajectories la and 2a circulate about the fixed point near Ca and trajectories lb... 

The normal mode representation of the phase space trajectories contains the same information as the local mode representation. However, the resonance region on the normal mode phase space map contains the local mode trajectories (la, lb 2a, 2b) and the stable fixed points Ca and C t,. The trajectories contained within the resonance zone are not free to explore the entire 0 < tp < n range whereas the trajectories outside the resonance zone do explore the 0 < ip < n range and are therefore classified as normal mode trajectories. The fixed point B (Iz = I = +2) is unstable, because it lies on a separatrix, and is located at the north pole of the normal mode polyad phase sphere. The stable fixed point A (7Z = — I = —2) is located at the south pole. 

Figure 9.16 shows the evolution of the local mode polyad phase spheres for H2O as I increases from 1 (N = vs + va = 1) to 3 (N = vs + va = 5). As I increases the lowest energy levels sequentially pass through the unstable fixed point B, depart from the normal mode resonance region, and become local mode states. At I = 1 (part (a)) there are only 2 levels and both are on the normal mode side of the separatrix. At I = 3 (part (e)) there are 6 levels and the lowest 4 of these have departed the resonance region and are local mode states. 

Figure 9.15 Evolution of the polyad phase sphere from the local mode to the normal mode limit as the strength of the 1 1 coupling term (antithetical to the local mode limit) is increased from 0 (part a) to oo (part f). As the coupling term, <5 in Eq. (9.4.174) increases from 0, first in part (c) one trajectory (level 4, at highest E) falls through the unstable fixed point into the normal mode region (antisymmetric stretch) eventually, in part (e), the resonance zone fills the entire phase sphere finally, in part (f), the normal mode limit is reached (from Xiao and Kellman, 1989).

Figure 9.17 The probability densities, ip 2, in the top row are found to resemble the configuration space trajectories in the bottom row. The states are from the JV = 5 polyad of H2O and are numbered in increasing energy order from 1 to 6 in correspondence with the numbered trajectories on the polyad phase sphere (from Xiao and Kellman, 1989).

Phase space trajectories and polyad phase spheres for polyad N —... 

Evolution of the polyad phase sphere from the local mode to the... 

There are many different forms that the transformed Hamiltonian K may take (43). A particularly convenient form is one where the Hamiltonian is block-diagonal. We have found this form to be an efficient approach for describing the triatomic molecules C02, HCN, H20, D20, and S02. It has also proven effective for the tetraatomics HCCH, DCCD, H2CO and D2CO and well as for the pentaatomic CHD3 (43,45-47,49). This structure is also the basis of the construction of the polyad phase spheres of Kellman and co-workers (69-71). 

---