Local-mode limit

Figure 4.3 Correlation diagram between the local- and normal-mode limits as a function of the parameter E,. Note how the degeneracies typical of the local-mode limit are split and as % —> 1 become the almost harmonic spacings characteristic of the normalmode limit.

In diatomic molecules, T2 = 0, and thus the expectation value of C vanishes. This is the reason why this operator was not considered in Chapter 2. However, for linear triatomic molecules, t2 = / / 0, and the expectation value of C does not vanish. We note, however, that D J is a pseudoscalar operator. Since the Hamiltonian is a scalar, one must take either the absolute value of C [i.e., IC(0(4 2))I or its square IC(0(412))I2. We consider here its square, and add to either the local or the normal Hamiltonians (4.51) or (4.56) a term /412IC(0(412))I2. We thus consider, for the local-mode limit,... 

Figure 4.19 Normal-mode vibrational quantum numbers for a bent triatomic molecule. Contrast the results for water, which is (cf. Table 4.6) near the local-mode limit with that for S02, which is near the normal-mode limit.

A similar treatment can be done for bent molecules in the scheme of van Roosmalen. The lowest-order, local-mode limit is given by... 

Halonen, L., and Robiette, A. G. (1986), Rotational Energy Level Structure in the Local Mode Limit, 7. Chem. Phys. 84,6861. 

Michelot, F., Moret-Bailly, J., and De Martino, A. (1988), Dominant Rovibrational Interactions in the Local-Mode Limit, Chem. Phys. Lett. 148, 52. 

A model atom approximation is permitted if all of the stretching vibrations of the molecule are ascribed to the local-mode limit. In the normal-mode limit, using the effective Hamiltonian of the whole molecule is preferable, as was shown in the example of CH3CI and CH3F. 

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. 

The local mode trajectories on Fig. 9.13 come in pairs because there are two identical local oscillators. However, the near perfect superposition of the (la, 16) pair of trajectories with the (2a, 26) pair of trajectories illustrates a characteristic feature of local mode limiting behavior. This corresponds to the quantum mechanical signature of the local mode limit, the near perfect degeneracy of pairs of local mode eigenstates. 

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).

If the Hamiltonian now contains the Casimir operators of both G, and G[, which do not commute, then the labels of neither provide good quantum numbers. Of course, in general such a Hamiltonian has to be diagonalized numerically. In this way one can proceed to break the dynamical symmetries in a progressive fashion. In (61) all the quantum numbers of G, up to G remain good. If we add another subalgebra beside Gz only those quantum numbers provided by G, on will be conserved, etc. In applications, the different chains are found to correspond to different limiting cases such as the normal versus the local mode limits for coupled stretch vibrations (99). 

Fig.4 Correlation of the energy levels of two coupled oscillators between the normal mode and local mode limits...

Figure 2.1. Schematic stretching vibrational energy levels for the water molecule. The levels on the left side represent the normal mode Umit and are indicated with quantum numbers Vi, 1)3. The levels on the right side represent the local mode limit, and they are labeled with the quantum numbers m,n. The true energy levels are shown in the center of diagram. From reference 53. Reproduced by permission from Elsevier Science B.V.

Figure 2.1 shows that, for V = 2, the level of the combination band rii +H2 and the two overtone levels (02) and (20) are close but separated as within the normal mode concept, whereas in the local mode limit there are two degenerate levels (02) and (20). Between these two extremes, the real energy levels are shown in the center of the diagram. They may be obtained by introducing a nonzero anharmonicity constant Xm and a nonzero coupling constant k. The relationship of normal modes is illustrated with the detailed analysis of anharmonicity and coupling constants of the water molecule ... 

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