Symmetry of normal vibration

As stated in Secs. 1-7 and 1-8, it is possible, by using group theory, to classify the normal vibration into various symmetry species. Experimentally, measurements of the infrared dichroism and polarization properties of Raman lines of an orientated ciystal provide valuable information about the symmetry of normal vibrations (Sec. 1-23). Here we consider the polarization properties of Raman lines in liquids and solutions in which molecules or ions take completely random orientations. ... 

To predict which bands wiU be Raman-active depend on the symmetry of the molecule. The symmetry elements of a molecule (i.e. centre, axes or planes) are associated with the symmetry operations which define all vibrational motions. For molecules with symmetry elements in its structure, the pattern of their normal vibrational modes will also have certain symmetry. Symmetry of normal vibrational modes influences a Raman tensor and consequently Raman activity of such vibrations. If we know the point group of the molecule and the symmetry labels for... 

Number of normal vibrations of each symmetry species... 

Table 6.5 Number of normal vibrations of each symmetry species (Spec.) in the C2 point group...

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. 

In order to apply group-theoretical descriptions of symmetry, it is necessary to determine what restrictions the symmetry of an atom or molecule imposes on its physical properties. For example, how are the symmetries of normal modes of vibration of a molecule related to, and derivable from, the full molecular symmetry How are the shapes of electronic wave functions of atoms and molecules related to, and derivable from, the symmetry of the nuclear framework ... 

The determination of molecular orbitals in terms of symmetry-adapted linear combinations of atomic orbitals is analogous to the determination of normal vibrational modes by forming symmetry-adapted linear combinations of displacements. Both calculations are in reality the reduction of a representa-... 

The infrared active v (CH2), v (CH2), 8 (CH2), and yr (CH2) fundamentals can be readily assigned as a result of the extensive spectroscopic studies on hydrocarbons which have been undertaken [Sheppard and Simpson (795)]. In addition, because of the polarized radiation studies on single crystals of normal paraffins [Krimm (95)], it is possible to assign uniquely the components of the doublets found in the spectrum for these bands to symmetry species. Similarly, the Raman active va(CH. ), vs(CH2), (CHg), v+ (0), and v+ (n) fundamentals can be unambiguously assigned, the latter two on the basis of normal vibration calculations... 

Since two resonance lines at 39.0 and 47.7 ppm that correspond to those observed in the ttgg form and a resonance line at 49.0 ppm that corresponds to that in the tttt form are recognized in the gel spectrum, a coexistence of these two forms in the gel might be supposed. In an attempt to determine the possibility of the coexistence of the two forms in the gel, we measured the IR spectrum that is sensitive to the molecular conformation. The number of normal vibrational modes depends sensitively on the molecular conformation based on the selection rule of the symmetry species. Kobayashi et al. confirmed the vibrational modes assignable to the ttgg conformation in the IR spectrum for the gel from a sPP/carbon disulfide system [117]. However, since we used o-dichlorobenzene as solvent, we examined whether the gel structure depends on the solvent. 

The set of molecular data required for statistical thermodynamic calculations includes molecular mass, structural parameters for determination of a point group, a symmetry number a and calculation of a product of principal moments of inertia IA IB Ic, as well as 3n - 6 frequencies of normal vibrations for an n-atomic molecule. 

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. 

Figure 6.6. Normal vibrations of tropolone molecule relevant for the tunneling tautomerization. The symmetry of each vibration is given in parentheses. The equilibrium bond length corresponds to the tropolone crystal. (From Redington et al. [1988].)...

In the present paper we assume that the molecule has the icosahedral symmetry. If one wants to consider a distortion of C 0+ or Cb0. the energy levels and their eigenvectors obtained here can be used as a starting point for the description of the Jahn-Teller effect in these systems. Indeed, the electron-phonon (or vibronic) coupling occurs if [.Tei]2 contains Fvib [19]. (Here Fd is the symmetry of an electronic molecular term, while J b is the symmetry of a vibrational normal mode.) Calculations using the terms in scheme of Ref. [4] have been performed in Ref. [20]. 

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. 

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. 

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