Vibration degeneracy

To evaluate Eq. 10.122, additional simplifying assumptions are necessary. Consider the vibrational degeneracy in the limit that v is very large. Applying Stirling s formula, we can rewrite Eq. 10.115 as... 

As a check, we calculate the total vibrational degeneracy from eq. (4) as 6, which is equal, as it should be, to 3 N — 6. The arithmetic involved in the reduction of the direct sum for the total motion of the atoms can be reduced by subtracting the representations for translational and rotational motion from T before reduction into a direct sum of IRs, but the method used above is to be preferred because it provides a useful arithmetical check on the accuracy of T and its reduction. 

More Complicated Molecules.—Calculations have been reported on a number of more complicated molecules, as indicated in Table 4. The work on BFa and SOa, and on NH3 and NF3, is of particular interest since these are the simplest symmetric top structures for which the calculation is practical, and for which there exist sufficient spectroscopic data to make it worthwhile for symmetric top molecules there are extra observable vibration-rotation interaction constants associated with the vibrational degeneracy that provide further information on the force field (see Table 3). Formaldehyde and ethylene, and their simple halogen derivatives, and also methane, are obvious candidates for further work. 

Finally the present treatment is restricted to cases where essential vibrational degeneracies between the level under consideration, Ev, and other levels are absent. 

In a series of Cr -alkylamine complexes with CrN skeletons, a correlation has been shown to exist between the radiationless transition rate and the number of active hydrogens attached to the donor atoms.The authors also report an expression that predicts the dependence of the non-radiative rates from the vibrational degeneracy and the displacement of the maximum frequency modes. 

Table 9.3 compares different ways of storing energy in one particular system (the HF molecule), but we see the same general trends that the correspondence principle predicts for all such systems. Vibrational, rotational, and translational energy levels become more closely spaced as the mass increases (remember that flg and (o both decrease with mass). The degeneracy increases at higher values of the quantum numbers /, n, tty, and n. For polyatomics, such as CO2, we have seen that the vibrational degeneracy also climbs with v. 

Fig. 6.1 S Time-dependent electronic populations of the C ( A2u) — B( E2g) — X CE g) manifold of Bz+ following a vertical transition to the C CA2u) state of the system. The full calculation, with all degeneracies retained (upper panel) is compared with an approximate one where the electronic and vibrational degeneracies are suppressed (lowerpanel)...

Unless there is an accidental near-vibrational degeneracy, the rotational spectrum of an asymmetric top in an excited vibrational state is similar to that obtained in the ground state, except that the spectrum is characterized by a slightly different set of rotation and distortion constants. Other nonrigid effects are often more important for asymmetric tops, such as internal rotation, and these are considered in Section VII. Similar statements apply to linear and symmetric-top molecules in excited nondegenerate vibrational states. For example, the rotational frequencies for symmetric tops in nondegenerate vibrational states are given by Eq. (54) with the rotation and distortion constants replaced by effective constants B ,... 

These results do not agree with experimental results. At room temperature, while the translational motion of diatomic molecules may be treated classically, the rotation and vibration have quantum attributes. In addition, quantum mechanically one should also consider the electronic degrees of freedom. However, typical electronic excitation energies are very large compared to k T (they are of the order of a few electronvolts, and 1 eV corresponds to 10 000 K). Such internal degrees of freedom are considered frozen, and an electronic cloud in a diatomic molecule is assumed to be in its ground state f with degeneracy g. The two nuclei A and... 

In general, at least three anchors are required as the basis for the loop, since the motion around a point requires two independent coordinates. However, symmetry sometimes requires a greater number of anchors. A well-known case is the Jahn-Teller degeneracy of perfect pentagons, heptagons, and so on, which will be covered in Section V. Another special case arises when the electronic wave function of one of the anchors is an out-of-phase combination of two spin-paired structures. One of the vibrational modes of the stable molecule in this anchor serves as the out-of-phase coordinate, and the loop is constructed of only two anchors (see Fig. 12). 

Figure 25 shows the results of the 2 distortion induced by a degenerate e 2 vibration that removes the Dsa degeneracy (compare Fig. 23). By symmetry, five...

Eqs. (D.5)-(D.7). However, when perturbations occur due to anharmonicity, the wave functions in Eqs. (D.11)-(D.13) will provide the conect zeroth-order ones. The quantum numbers and v h are therefore not physically significant, while V2 arid or V2 and I2 = m, are. It should also be pointed out that the degeneracy in the vibrational levels will be split due to anharmonicity [28]. 

In this paper, we review progress in the experimental detection and theoretical modeling of the normal modes of vibration of carbon nanotubes. Insofar as the theoretical calculations are concerned, a carbon nanotube is assumed to be an infinitely long cylinder with a mono-layer of hexagonally ordered carbon atoms in the tube wall. A carbon nanotube is, therefore, a one-dimensional system in which the cyclic boundary condition around the tube wall, as well as the periodic structure along the tube axis, determine the degeneracies and symmetry classes of the one-dimensional vibrational branches [1-3] and the electronic energy bands[4-12]. 

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