Stretching vibrations local 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. 

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). 

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.

Vaida et al. calculated the atmospheric /-values (the integrated value of the cross section, photon flux and quantum yield) for the vibrational overtone initiated visible photolysis and UV photolysis. The /-values were obtained assuming the local mode calculated third and fourth OH-stretching overtone cross sections for the vibrational transitions and the experimental upper limit for the UV cross section across the 179-224 nm range, and a unitary quantum yield for both regions [22]. These /-values showed that at altitudes below 45 km, the OH-stretching overtone initiated photolysis was the predominant process, whereas for higher alti-... 

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.

Kieffer has estimated the heat capacity of a large number of minerals from readily available data [8], The model, which may be used for many kinds of materials, consists of three parts. There are three acoustic branches whose maximum cut-off frequencies are determined from speed of sound data or from elastic constants. The corresponding heat capacity contributions are calculated using a modified Debye model where dispersion is taken into account. High-frequency optic modes are determined from specific localized internal vibrations (Si-O, C-0 and O-H stretches in different groups of atoms) as observed by IR and Raman spectroscopy. The heat capacity contributions are here calculated using the Einstein model. The remaining modes are ascribed to an optic continuum, where the density of states is constant in an interval from vl to vp and where the frequency limits Vy and Vp are estimated from Raman and IR spectra. 

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