Symmetry of the vibrations

The most common type of sequence, one with Av = 0 shown in Figure 7.18, is always allowed by symmetry because, whatever the symmetry of the vibration involved. 

Bordiga et al. [48,52] explained the experimental evidence reported in Fig. 6 in terms of symmetry once the [Ti(H20)204] or [Ti(NH3)204] complexes are formed, the T -like symmetry of Ti(IV) species is destroyed in such a way that the symmetry of the vibrational modes is no longer the same as that of the LMCT, and the Raman resonance is quenched. 

We now address the fact that the symmetry of the vibrational modes must be adapted to the nuclear spin multiplicity. Since 7Li is a fermion with nuclear spin... 

When deriving selection rules from character tables it is noted that vibrations are usually excited from the ground state which is totally symmetric. The excited state has the symmetry of the vibration being excited. Hence A vibration will be spectroscopically active if the vibration has the same symmetry species as the relevant operator. 

The requirements for Raman resonance that must be fulfilled are the following (120,121) (a) total symmetry of the vibrations with respect to the absorbing center, and (b) same molecular deformation induced by the electronic and vibrational excitations. Quantum chemical calculations (41) of the vibrational frequencies and the electronic structure of shell-3 cluster models allowed the assignment of the main vibrational features, as shown in Fig. 7. The 1125 cm-1 band is unequivocally assigned to the symmetric stretching of the Ti04 tetrahedron. 

This is, of course, the same result as obtained above by analyzing Che symmetries of the vibrational modes. 

Theoretically, 0=s p =s 3/4, depending on the nature and symmetry of the vibration. Nonsymmetric vibrations give depolarizations of 3/4. Symmetric vibrations give p ranging from 0 to 3/4. Accurate values of p are important for determining the assignment of a Raman line to a symmetric or an asymmetric vibration. 

Figure 3.12. Schematic drawings of Raman modes of vibration in rhombohedral carbonates. Notation and symmetry of the vibration and the range of Raman shift for all rhombohedral carbonates are included for each mode. Hexagonal unit cell axes are shown for orientation. (After White, 1974.)...

The electron-vibration coupling V has the same symmetry of the vibration. This is because the Hamiltonian is totally symmetric under transformations of the point group of the ensemble molecule plus substrate. In Appendix we give further details in particular Eq. (A4) shows that in order to preserve the invariance of the Hamiltonian under transformation of the nuclear coordinates, the electronic coordinates must transform in the same way [37]. Hence, if a symmetric mode is excited, the electron-vibration coupling will also be symmetric in the electronic-coordinate transformations. Thus only electronic states of the same symmetry will give non-zero matrix elements for a symmetric vibration. This kind of reasoning can be used over the different vibrations of the molecule. 

This molecule has a square-planar structure with >4h symmetry. With reference to the coordinate system displayed in Fig. 7.3.6, the symmetry of the vibrational modes may be derived in the following manner ... 

Finally, it is useful to mention here a systematic way to derive the symmetries of the vibrational modes for a linear molecule. Even though formally carbon suboxide has T>ooh symmetry, its vibrational modes may be derived by using the l)2u character table. The procedure is illustrated below for the carbon suboxide molecule ... 

Raman spectra of 2-deuteriothiophene, 2-deuterioselenophene, and 3-deuterioselenophene were useful for assigning the vibrations of type A because fourteen of the fifteen vibrations of the type are polarized and intense. A deuterium atom distorts the symmetry of the vibrations which are of type B in selenophene or thiophene. The assignment of the in-plane vibrations fits the isotopic product rules and the sum rules (see Table III). These data confirm the similarity of the vibrational spectra of thiophene and selenophene. 

The allowed changes in the rotational quantum number J are AJ = 1 for parallel (2 ) transitions and A7= 0, 1 for perpendicular (II ) transitions. Parallel transitions such as for acetylene thus have P i J= 1) and R(AJ = +1) branches with a characteristic minimum between them, as shown for diatomic molecules such as HCl in Fig. 37-3 and for the HCN mode in Fig. 2. However, perpendicular transitions such as Vs for acetylene and V2 for HCN (Fig. 2) have a strong central Q branch (AJ = 0) along with P and R branches. This characteristic PQR-Yersus-PR band shape is quite obvious in the spectrum and is a useful aid in assigning the symmetries of the vibrational levels involved in the infrared transitions of a hnear molecule. 

In Sec. 2.13 A it is demonstrated that the depolarization ratio may be used to determine the symmetry of the vibrations of molecules in the liquid state, see also Long (1977). 

Spectra of samples in the liquid state (Fig. 2.6-lB) are given by molecules which may have any orientation with respect to the beam of the spectrometer. Like in gases, flexible molecules in a liquid may assume any of the possible conformations. Some bands are broad, since they are the sum of spectra due to different complexes of interacting molecules. In the low frequency region spectra often show wings due to hindered translational and rotational motions of randomly oriented molecules in associates. These are analogous to the lattice vibrations in molecular crystals, which, however, give rise to sharp and well-defined bands. The depolarization ratio p of a Raman spectrum of molecules in the liquid state (Eqs. 2.4-11... 13) characterizes the symmetry of the vibrations, i.e., it allows to differ between totally symmetric and all other vibrations (see Sec. 2.7.3.4). 

We will, throughout this section, refer to symmetry force constants (K(F) where F is the symmetry of the vibrational mode) derived directly from observed frequencies. Since we will be dealing with all- C O molecules as a matter of course, these are simply found via a relationship analogous to equation (6) ... 

In the gas phase, the vibrational transitions couple with the rotational ones, giving rise to rotovihrational spectra. The different rotovibrational contours depend on the symmetry of the vibration in relation to the symmetry of the molecule, and on the resolution of the rotational components. In some cases, the energy of the pure vibrational transition corresponds to the minimum of the absorption band ... 

Whether 0 = x or y for the e,/3> state is determined by the symmetry of the vibrational co-ordinate Q. The Wigner-Eckart theorem allows the component-dependence of the vibronic-coupling matrix elements to be expressed, using the V coefficients of Griffith (47), by... 

The definition of the depolarisation ratio, pi, is illustrated in Fig. 1 for linearly-polarised incident radiation and a 90° scattering geometry. In the normal Raman effect it is well known that the measurement of p may identify the symmetry of the vibrational mode responsible for a given Raman band pi < 3/4 (pi = 0 in cubic or higher symmetries) for totally symmetric modes and Pi= 3/4 for non-totally symmetric modes. In the resonance Raman effect, the value of P, and its dependence on the exciting frequency, may be more informative. This is because the symmetries of... 

Classically, in a polyatomic molecule, a normal mode of vibration occurs when each nucleus undergoes simple harmonic motion with the same frequency as and in phase with the other nuclei. The possible normal modes of vibrational of a non-linear polyatomic molecule with A atoms are 3A— 6 and are categorized as arising from stretching, bending, torsional and non-bonded interactions. Whether a particular mode is IR active depends on whether the symmetry of the vibration corresponds to one of the components of the dipole moment. 

Useful information on the scattering process and on the symmetry of the vibration involved can be obtained by measuring the depolarization ratio, p, of a Raman band. 

The depolarization ratio depends on the symmetry of the vibrations re.sponsible for the scattering. For example, the Kand for carbon tetrachloride at 459 cm (Figure 18-2) arises from a totally. symmetric brcaih-ing vibration involving the simultaneous movement of the four Ictrahcdrally arranged chlorine atoms toward... 

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