Symmetry of Normal Vibrations and Selection Rules

To determine the symmetry of a normal vibration, it is necessary to begin by considering the kinetic and potential energies of the system. These were discussed in Sec. 1.4  

In this respect, all the normal modes of vibration shown in this book are only approximate. 

Consider a case in which a molecule performs only one normal vibration, Qi. Then, T = Qi and V = XiQj. These energies must be invariant when a symmetry operation, R, changes Qi to RQi. Thus, we obtain 

the normal coordinate must change either into itself or into its negative. If Qj = RQj, the vibration is said to be symmetric. If Qi = RQi, it is said to be antisymmetric. 

The symmetry properties of the normal vibrations of the H2O molecule shown in Fig. 1.12 are classified as indicated in Table 1.3. Here, +1 and —1 denote symmetric and antisymmetric, respectively. In the Vj and V2 vibrations, all the symmetry properties are preserved during the vibration. Therefore they are symmetric vibrations and are called, in particular, totally symmetric vibrations. In the V3 vibration, however, symmetry elements such as C2 and a (xz) are lost. Thus, it is called a nonsymmetric vibration. If a molecule has a number of symmetry elements, the normal vibrations are classified according to the number and the kind of symmetry elements preserved during the vibration. 

The symmetry properties of the normal vibrations of the H3O molecule shown in Fig. 1-7 are classified as indicated in Table 1-1. Here, +1 and -1 denote symmetric and antisymmetric, respectively. In the v, and V2 vibrations. 

To determine the activity of the vibrations in the infrared and Raman spectra, the selection rule must be applied to each normal vibration. From a quantum mechanical point of view, a vibration is active in the infrared spectrum ij the dipole moment of the molecule is changed during the vibration, and is active in the Rarnan spectrum if the polarizability cf the molecule is changed during the vibration. As stated in Sec. I-l, the induced dipole moment P is rdated to the strength of the electric field E by the relation 

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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 considerations on the symmetries of the ground and excited states and the above conditions lead to the selection rule for infrared spectroscopy A fundamental vibration will be infrared active if the corresponding normal mode belongs to the same irreducible representation as one or more of the Cartesian coordinates. 

Another class of techniques monitors surface vibration frequencies. High-resolution electron energy loss spectroscopy (HREELS) measures the inelastic scattering of low energy ( 5eV) electrons from surfaces. It is sensitive to the vibrational excitation of adsorbed atoms and molecules as well as surface phonons. This is particularly useful for chemisorption systems, allowing the identification of surface species. Application of normal mode analysis and selection rules can determine the point symmetry of the adsorption sites./24/ Infrarred reflectance-adsorption spectroscopy (IRRAS) is also used to study surface systems, although it is not intrinsically surface sensitive. IRRAS is less sensitive than HREELS but has much higher resolution. 

An assignment is the correlation between a measured vibrational spectrum and the normal vibrations of a molecule. The shapes and numbers of normal vibrations can be derived theoretically. In general, complete assignment is not possible for large, un.symmetrical molecules. On the other hand, the vibrations of small, highly symmetrical molecules can often be assigned easily by using selection rules, e.g., by their absence in the IR and/or the Raman spectrum. In this way the symmetries of small species can be identified, as described in the previous chapter. 

An isolated n-atom molecule has 3n degrees of freedom and in—6 vibration degrees of freedom. The collective motions of atoms, moving with the same frequency and which in phase with all other atoms, give rise to normal modes of vibration. In principle, the determination of the form of normal modes for any molecule requires the solution of equation of motion appropriate to the n-symmetry. Methods of group theory are important in deriving the symmetry properties of the normal modes. With the aid of the character tables for point groups and the symmetry properties of the normal modes, the selection rules for Raman and IR activity can be derived. For a molecule with a center of symmetry, e.g. AXe, octahedral molecule, a non-Raman active mode is also IR active, whereas for the BX4 tetrahedral molecule, some modes are simultaneously IR and Raman active. 

In Sec. 1.5, the symmetry and the point group allocation of a given molecule were discussed. To understand the symmetry and selection rules of normal vibrations in polyatomic molecules, however, a knowledge of group theory is required. The minimum amount of group theory needed for this purpose is given here. 

The fundamental vibrational spectra of crystals are properly interpreted on the basis of normal vibrations of the crystal as a whole. They are usually described as one- or multi-phonon transitions. These are governed by the laws of conservation of energy and wavevector in the photon-phonon system, e.g., for one phonon transition this latter rule allows only transitions creating phonons with the wavevector q = 0. In addition to these fundamental rules, there are selection rules based on symmetry considerations. For example, in elemental crystals which contain only 2 atoms per unit cell (such as Ge) the one optical phonon transition is forbidden (see Zallen (1968)). 

The frequency of any vibration is not dependent on amplitude. The displacement functions for these normal vibrations are described exactly as a sine-wave function. There are 3N-6 normal vibrations in a molecule, and their respective vibrational frequencies are called the fundamental frequencies of the molecule. Symmetry of the molecule is the single factor most important in determining both frequency and ampUtude of a molecular vibration. The selection rules discussed later will place a heavy emphasis on symmetry. 

The model fundamental to all analyses of vibrational motion requires that the atoms in the system oscillate with small amplitude about some defined set of equilibrium positions. The Hamiltonian describing this motion is customarily taken to be quadratic in the atomic displacements, hence in principle a set of normal modes can be found in terms of these normal modes both the kinetic energy and the potential energy of the system are diagonal. The interaction of the system with electromagnetic radiation, i.e. excitation of specific normal modes of vibration, is then governed by selection rules which depend on features of the microscopic symmetry. It is well known that this model can be worked out in detail for small molecules and for crystalline solids. In some very favorable simple cases the effects of anharmonicity can be accounted for, provided they are not too large. 

The integral < vib vib) maY vanish because of symmetry considerations. For example, the C02 normal mode v3 in Fig. 6.2 has eigenvalue — 1 for the inversion operation. Hence (Section 6.4), the v3 factor in the vibrational wave function is an even or odd function of the normal coordinate Q3, depending on whether v3 is even or odd. For a change of 1,3,5,... in the vibrational quantum number v3, the functions p vib and p"ib have different parities and their product is an odd function, so that ( ibl vib) vanishes. Thus we have the selection rule Ac3 = 0,2,4,... for electronic transitions in... 

Step 4. For a vibrational mode to be infrared (IR) active, it must bring about a change in the molecule s dipole moment. Since the symmetry species of the dipole moment s components are the same as rx, ry, and 1, a normal mode having the same symmetry as Ix, Ey, or 1 will be infrared active. The argument employed here is very similar to that used in the derivation of the selection rules for electric dipole transitions (Section 7.1.3). So, of the six vibrations of NH3, all are infrared active, and they comprise four normal modes with distinct fundamental frequencies. 

Symmetry selection rules for Raman spectrum can be derived by using a procedure similar to that for the IR spectrum. One should note, however, that the symmetry property of symmetry species of six components of polarizability are readily found in character tables. In point group C3V, for example, normal vibrations of the NH3 molecule (2A1 and 2E) are Raman-active. More generally, the vibration is Raman-active if the component(s) of the polarizability belong(s) to the same symmetry species as that of the vibration. 

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