Degenerate normal vibration

Y. Ideguchi, and K. Fukushima Molecular vibrations of high pol5nners. IV. A general method of treating degenerate normal vibrations of helical polymers and infrared active vibrations of isotactic polypropylene. J. Chem. Phys. 38, 2709-2720 (1963). 

The effect just discussed does not influence the expression for the rotational energy in the case of linear molecules. For here the nuclei in the degenerate normal vibrations, as we saw in the example illustrated by fig. 40, move in planes at right angles to the internuclear line so that these planes are parallel... 

Besides the molecules just discussed, ozone, O3, should be mentioned in this connection. It is of special chemical interest since it usually has been supposed to have an equilateral triangular form. If this is so, the molecule should have a doubly degenerate and a non-degenerate normal vibration, of which the former should be active in absorption or emission and inactive in the Raman effect, while for the latter the converse would be true. The experimental investigations, however, appear to show that there are three active vibrations in absorption. This can only be accounted for by the assumption that the nuclear equilibrium configuration is triangular but not equilateral. Further work will be required to clear up this question entirely. 

Coriolis coupling constant for the t-th degenerate normal vibration... 

The schemes of the vibrational modes can be derived from those of linear complexes. The main difference is that one of the n.v. originating in the Etu deformation of Ds,/ becomes a torsional mode during which the central atom vibrates perpendicularly to the y-axis and to the normal ring-metalring deformation. Since the doubly degenerate normal vibrations of D5J split up in the other three groups, seven n.v. will be infrared-active in C2 , while all nine vibrations are infrared-active for C, and Cj, and in the Raman spectra. 

In Table B. 1 in Appendix B are given formulae, analogous to those derived for the C2 point group, for determining the number of normal vibrations belonging to the various symmetry species in all non-degenerate point groups. 

Just as group theory enables one to find symmetry-adapted orbitals, which simplify the solution of the MO secular equation, group theory enables one to find symmetry-adapted displacement coordinates, which simplify the solution of the vibrational secular equation. We first show that the matrices describing the transformation properties of any set of degenerate normal coordinates form an irreducible representation of the molecular point group. The proof is based on the potential-energy expression for vibration, (6.23) and (6.33) ... 

Benzene thus has 10 nondegenerate and 20 doubly degenerate normal modes. The convention8 is to number the vibrational frequencies according to the order their symmetry species are listed in the character table modes of the same symmetry species are numbered in order of decreasing frequency. Thus the a2u frequency is called p4, the frequency of the two degenerate elg modes is called vn, the lower of the two e2u frequencies is called p20. 

In C70, because of its lower DSh symmetry, there are five kinds of non-equivalent atomic sites and eight kinds of non-equivalent bonds. This means that the number of normal vibrations increases for C70 in comparison to C60. Although there are now 204 vibrational degrees of freedom for the 70-atom molecule, the symmetry of C70 gives rise to a number of degenerate modes so that only 122 modes are expected. Of these 31 are infrared-active and 53 are Raman-active. 

A molecule composed of A atoms has in general 3N degrees of freedom, which include three each for translational and rotational motions, and (3N — 6) for the normal vibrations. During a normal vibration, all atoms execute simple harmonic motion at a characteristic frequency about their equilibrium positions. For a linear molecule, there are only two rotational degrees of freedom, and hence (3N — 5) vibrations. Note that normal vibrations that have the same symmetry and frequency constitute the equivalent components of a degenerate normal mode hence the number of normal modes is always equal to or less than the number of normal vibrations. In the following discussion, we shall demonstrate how to determine the symmetries and activities of the normal modes of a molecule, using NH3 as an example. 

In other words, among the six normal vibrations of NH3, two have Ai symmetry, two others form a degenerate E set, and the remaining two form another E set. 

For a doubly degenerate normal mode, both components must be used together as the basis of a two-dimensional irreducible representation. For example, the operations C2 and ctv on the two normal vibrations that constitute the i>6 mode lead to the character (sum of the diagonal elements of the corresponding 2x2 matrix) of -2 and 0, respectively, as illustrated below. Working through the remaining symmetry operations, the symmetry species of can be identified as Eu. 

The last two columns of the character table provide information about IR and Raman activities of normal vibrations. One column lists the symmetry species of translational motions along the x, y and z axes (Tx, Ty and Tz) and rotational motions around the x, y and z axes (Rx, Ry and Rz). The last column lists the symmetry species of the six components of polarizability. As will be discussed in Section 1.14, the vibration is IR-active if it belongs to a symmetry species that contains any T components and is Raman-active if it belongs to a symmetry species that contains any a. components. Pairs of these components are listed in parentheses when they belong to degenerate species. 

Figure 1-45 Normal vibrations of CH3X-type molecule (frequencies are given for X = Cl) v symmetric stretching vj, degenerate stretching 8S, symmetric bending 8j, degenerate bending pr rocking.

In general, nine (3x5-6) normal vibrations are expected for XY4-type molecules. As seen in Tables 4-1 and 4-2, this rule holds if we consider that vibrations belonging to E and F species are doubly and triply degenerate, respectively. 

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