Normal mode limit

The fact that both the local- and the normal-mode limits are contained within the algebraic approach allows one to study in a straightforward way the transition from one to the other. It is convenient to use, for this study, the local basis [Eq. (4.17)] and diagonalize the Hamiltonian for two identical bonds... 

The transition from the local- to the normal-mode limit is described by the parameter Xx2/A. When this parameter is zero, the Hamiltonian (4.28) is in the local limit, when the parameter is large the spectrum approaches the normalmode limit. It is convenient to define the dimensionless locality parameter as... 

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.

Although not much used in actual calculations, we also quote the case of the strict normal-mode limit. In this case, H can be written in terms of Casimir invariants of Eq. (4.48)... 

Figure 4.7 Vibrational quantum numbers in the normal mode limit of a linear tri-atomic molecule.

This causes a splitting of vibrational bands with different lh (Figure 4.9). A similar situation occurs in the normal mode limit with lh replaced by l2. I splitting is an important (at the level of accuracy of 10 cm-1) feature of linear tri-atomic molecules. 

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 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. 

Transformation Between Local and Normal Mode Limits... 

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).

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