Polyad local mode

Dennison coupling produces a pattern in the spectrum that is very distinctly different from the pattern of a pure nonnal modes Hamiltonian , without coupling, such as (Al.2,7 ). Then, when we look at the classical Hamiltonian corresponding to the Darling-Deimison quantum fitting Hamiltonian, we will subject it to the mathematical tool of bifiircation analysis [M]- From this, we will infer a dramatic birth in bifiircations of new natural motions of the molecule, i.e. local modes. This will be directly coimected with the distinctive quantum spectral pattern of the polyads. Some aspects of the pattern can be accounted for by the classical bifiircation analysis while others give evidence of intrinsically non-classical effects in the quantum dynamics. 

Figure Al.2.10. Birth of local modes in a bifurcation. In (a), before the bifiircation there are stable anhamionic symmetric and antisymmetric stretch modes, as in figure Al.2.6. At a critical value of the energy and polyad number, one of the modes, in this example the symmetric stretch, becomes unstable and new stable local modes are bom in a bifurcation the system is shown shortly after the bifiircation in (b), where the new modes have moved away from the unstable syimnetric stretch. In (c), the new modes clearly have taken the character of the anliamionic local modes.

At this point it is useful to define creation (at) and annihilation (a) operators, which are analogous to angular momentum raising and lowering operators. These at, a operators profoundly simplify the algebra needed to set up the polyad Heff matrices, to apply some of the dynamics diagnostics discussed in Sections 9.1.4 and 9.1.7, and to transform between basis sets (e.g., between normal and local modes). They also provide a link between the quantum mechanical Heff model, which is expressed in terms of at, a operators and adjustable molecular constants (evaluated by least squares fits of spectra), and a reduced-dimension classical mechanical HeS model. 

The term (vr — vr)2 lifts the degeneracy of the members of the polyad, hence it provides the driving force toward the local mode limit. The overall zero-order energy spread of the TVth polyad is TV2. The zero-order states (vr,vl) and (vr + 1, vl — 1) are most nearly degenerate near the center of the polyad, where vr TV/2 and the adjacent level spacing is 2F. 

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).

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 9.16 Polyads N = 1 through N = 5 for H2O. As N increases, the new states that appear at the bottom of the polyad fall on the local mode side of the separatrix (from Xiao and Kellman, 1989).

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

Figure 13. Spectrum of a planar triatomic molecule in the local limit. N = 100, A = -3 cm , and A = 3 cm . (t)j, v, v,) is the degeneracy local mode notation (see the text). The degeneracy associated with the true local mode basis appears in brackets, p is the total number of stretching vibrational quanta in a given polyad.

Figure 34. Local-mode coupling (according to the three-dimensional algebraic model) in a bent triatomic molecule for the first two vibrational polyads.

The transformed (permuted) state is necessarily in the same polyad as the nontransformed one. This means that permutation operators act by changing the order of the basis states in a given polyad. This is very important because it allows one to obtain the transformed eigenvector explicitly (i.e., the linear combination of local modes) in terms of a simple linear (permutation) transformation. By applying the well-known formula of discrete group theory for the calculation of the character, we can schematically write... 

Figure 25 shows the scaled eigenvalues, E/(y/ /2), of the quantum equivalent of in order to illustrate four characteristic types of behavior. Panels (a) and (d) depict the simplest cases, for polyads in regions I and IV, in which the two modes are almost decoupled, due to a large local detuning, p, or small coupling, p. The two cases differ according to whether the point of the... 

We start by constructing local vibrational basis states for, say, CH stretching modes. We have the total vibrational (polyad) quantum... 

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