Phase space polyad

Figure Al.2.12. Energy level pattern of a polyad with resonant collective modes. The top and bottom energy levels conespond to overtone motion along the two modes shown in figure Al.2.11. which have a different frequency. The spacing between adjacent levels decreases until it reaches a minimum between the third and fourth levels from the top. This minimum is the hallmark of a separatrix [29, 45] in phase space.

The phase space of a coupled, two-identical-anharmonic oscillator system is four-dimensional. Conservation of energy and polyad number reduces the number of independent variables from four to two. At specified values of E and N = vr + vl = vs+ v0 (in classical mechanics, N need no longer be restricted to integer values nor E to eigenenergies), accessible phase space divides into several distinct regions of regular, qualitatively describable motions and (for more general dynamical systems) chaotic, indescribable motions. Systematic variation of E and N reveals bifurcations in the number of forms of these describable motions. Examination of the classical mechanical form of the polyad Heff often reveals the locations and causes of such bifurcations. 

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. 

Trajectories in action/angle polyad phase space convey all of the most important qualitative relationships between a quantum spectrum and classical intramolecular dynamics. However, coordinate space trajectories are both more easily visualized and more directly comparable to quantum probability densities, ip(Qi,Q2,... ( 3jv-6) 2 Xiao and Kellman (1989) describe how the action/angle phase space trajectories for each eigenstate may be converted into a coordinate space trajectory. The key to this is the exact relationship between Morse oscillator displacement coordinates, rit and the action, angle variables, Ii,4>i (Rankin and Miller, 1971). Figure 9.17 shows, for the 6 eigenstates in the I = 3 (N = vs + va = 5) polyad of H20, the correspondences between the phase space trajectories, the coordinate space trajectories, and the probability densities. The resemblance between the classical coordinate space trajectories and the quantum probability densities is striking ... 

The phase space structures for two identical coupled anharmonic oscillators are relatively simple because the trajectories lie on the surface of a 2-dimensional manifold in a 4-dimensional phase space. The phase space of two identical 2-dimensional isotropic benders is 8-dimensional, the qualitative forms of the classifying trajectories are far more complicated, and there is a much wider range of possibilities for qualitative changes in the intramolecular dynamics. The classical mechanical polyad 7feff conveys unique insights into the dynamics encoded in the spectrum as represented by the Heff fit model. 

Analysis of this 7feff using the techniques of nonlinear classical dynamics reveals the structure of phase space (mapped as a continuous function of the conserved quantities E, Ka, and Kb) and the qualitative nature of the classical trajectory that corresponds to every eigenstate in every polyad. This analysis reveals qualitative changes, or bifurcations, in the dynamics, the onset of classical chaos, and the fraction of phase space associated with each qualitatively distinct class of regular (quasiperiodic) and chaotic trajectories. 

The 8-dimensional phase space of two 2-dimensional oscillators is reduced by the existence of two conserved actions, Ka and Kb, and by the absence of the conjugate angles, classical mechanical polyad 7feff. The conserved actions appear parametrically in 7feff, thus the phase space accessible at specified values of Ka and Kb is four dimensional. Since energy is conserved, in addition to Ka and Kb, all trajectories lie on the surface of a 3-dimensional energy shell. 

Figure 9.18 Surfaces of section for the HCCH [Nj, l] = [4,0] and [8,0] polyads. The surface section for the [4,0] polyad shows that all of phase space is divided between cis-bend and trans-bend normal modes. The phase space structure for the [8, 0] polyad contains large scale chaos as well as at least two new qualitative behaviors (from Jacobson, et al., 1999).

The surface of section for the [8,0] (Ka = 2.5) polyad contains several structures that are not present for the [4,0] polyad. Most importantly, a significant fraction of the surface of section is filled with the apparently random dots symptomatic of chaos. The fraction of phase space organized by the periodic trans-and eis-bending vibrations is considerably reduced relative to that for the [4,0] polyad. New closed loops have appeared (centered at Jf, ss 1.7, ipb = 7r and Jb = 0, ipb 7r/2), and the qualitatively new trajectories represent the first appearance of localized motions in which the two ends of the molecule are... 

Figure 9.19 Overview of the phase space and configuration space dynamics associated with the HCCH [JV = 22, l = 0] polyad. The top four plots are surfaces of section for four energies within the polyad. Only simple structures are found near the bottom (local bender) and top (counter-rotators) of the polyad. Chaos dominates at energies near the middle of the polyad, but two classes of stable trajectories coexist with chaos. The bottom four plots show the coordinate space motions of the left- and right-hand H-atoms that correspond to the two different periodic orbits (from Jacobson and Field, 2000b).

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

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).

---