Selection Rules for Infrared and Raman Spectra

According to quantum mechanics, the selection rule for the infrared spectrum is determined by the integral  

Here p is the dipole moment in the electronic ground state, fp is the vibrational eigenfunction given by Eq. 2.7, and u and v are the vibrational quantum numbers before and after the transition, respectively. Th activity of the normal vibration whose normal coordinate is Q, is being determined. By resolving 

If one of these integrals is not zero, the normal vibration associated with is infrared active. If all the integrals are zero, the vibration is infrared inactive. 

If one of these integrals is not zero, the normal vibration associated with is infrared- 

If one of these integrals is not zero, the normal vibration associated with is Raman-active. If all the integrals are zero, the vibration is Raman-inactive. As shown below, it is possible to determine whether the integrals of Eqs. 1.102 and 1.104 are zero or nonzero from a consideration of symmetry  

To determine whether the vibration is active in the IR and Raman spectra, the selection rules must be applied to each normal vibration. Since the origins of IR and Raman spectra are markedly different (Section 1.4), their selection rules are also distinctively different. According to quantum mechanics (18,19) a vibration is IR-active if the dipole moment is changed during the vibration and is Raman-active if the polarizability is changed during the vibration. 

The IR activity of small molecules can be determined by inspection of the mode of a normal vibration (normal mode). Obviously, the vibration of a homopolar diatomic molecule is not IR-active, whereas that of a heteropolar diatomic molecule is IR-active. As shown in Fig. 1-13, the dipole moment of the H2O molecule is changed during each normal vibration. Thus, all these vibrations are IR-active. From inspection of Fig. 1-11, one can readily see that V2 and v3 of the CO2 molecule are IR-active, whereas v is not IR-active. 

To discuss Raman activity, let us consider the nature of the polarizability (a) introduced in Section 1.4. When a molecule is placed in an electric field (laser beam), it suffers distortion since the positively charged nuclei are attracted toward the negative pole, and electrons toward the positive pole (Fig. 1-14). This charge separation produces an induced dipole moment (P) given by 

In actual molecules, such a simple relationship does not hold since both P and E are vectors consisting of three components in the x, y and z directions. Thus, Eq. (1-45) must be written as 

The first matrix on the right-hand side is called the polarizability tensor. In normal Raman scattering, this tensor is symmetric axy = ayz, a.xz = azx and V = v-zy- According to quantum mechanics, the vibration is Raman-active if one of these components of the polarizability tensor is changed during the vibration. 

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Both infrared and Raman spectra are concerned with measuring molecular vibration and rotational energy changes. However, the selection rules for Raman spectroscopy are very different from those of infrared - a change of polarisability... 

Donovan, B., Angress, F. Lattice vibrations. London Chapman and Hall 1971. Turrell, G. lniia.red and Raman spectra of crystals. New York Academic Press 1972. Fateley, W. G., Dollish, F. R., McDevitt, N. T., Bentley, F. F. Infrared and raman selection rules for molecular and lattice vibrations The correlation method. New York J. Wiley 1972. 

Derivation of selection rules for a particular molecule illustrates the complementary nature of infrared and Raman spectra and the application of group theory to the determination of molecular structure. 

Most of the sulfur rings have either no or only small dipole moments owing to the low or lacking polarity of the sulfur-sulfur bonds. Therefore, infrared spectra of these species are of low intensity as a result of the selection rule for infrared absorption. In contrast, the Raman scattering intensity of S-S bonds is very strong and Raman spectra are therefore the best technique to study sulfur melts and samples prepared from the melt like r-sulfur and /i-sulfur. In Fig. 1 a schematic comparison is made to demonstrate the differences in the Raman spectra of the homocycles with between 6 and 12 atoms. 

Some peaks in the INS spectrum are not seen in the infrared or Raman spectra for example the factor group splitting of all the modes in the INS is readily apparent because of the absence of selection rules. In the infrared and Raman spectra, some of the factor group components are either forbidden or have zero intensity. 

Fig. 10.2 shows the infrared, Raman and INS spectra of polyethylene in the 0-1600 cm" region. The infrared and Raman spectra show no coincidences since the crystal is centrosymmetric. The INS spectrum is very different the absence of selection rules means that the features present in both optical spectra are apparent, but the major difference is that the regions between the infirared and Raman bands are filled-in in the INS spectrum. This is because INS spectroscopy gives information at all values of k, not just those at zero as for the optical spectroscopies. 

There have been a number of reports of analogous surface selection rules for Raman spectra [27-29]. However, for SERS, the situation is complicated by the essential roughness of the metal surface and the mixture of enhancement mechanisms, in addition to the facts that the Raman effect depends upon the molecular polarizability tensor and the excitation frequencies are typically high enough to reduce the metal conductivity to levels where finite parallel, as well as perpendicular, electric vectors are established at the surface. An excellent recent review of this subject has been written by Creighton [30]. Suffice it to say here that surface selection rules evidently do exist for Raman spectroscopy but they are more complicated than the rule for infrared and EELS. 

As noted before, polyatomic molecules have 3 AT-6 or, if linear, 3JV—5 normal vibrations. For any given molecule, however, only vibrations that are permitted by the selection rule for that molecule appear in the infrared and Raman spectra. Since the selection rule is determined by the symmetry of the molecule, this must first be studied. 

If one of the Y atoms of a planar XY3 molecule is replaced by a Z atom, the symmetry is lowered to C2U. If two of the Y atoms are replaced by two different atoms, W and Z, the symmetry is lowered 10 C. As a result, the selection rules are changed, as already shown in Table 1-12. In both cases, all six vibrations become active in infrared and Raman spectra. Table II-4c lists the vibrational frequencies of planar ZXYj and ZXYW molecules. Although not listed in this table, the infrared spectra of binary mixed halides of boron" and aluminum " have been measured. The frequencies listed for ihe formate and acetate ions were obtained in aqueous solution. These frequencies are important when we discuss the vibrational spectra of metal salts of these anions (Sec. HI-7). 

Halford, R. S. Motions of molecules in condensed systems I. Selection rules, relative intensities, and orientation effect for Raman and infrared spectra. J. chem. Phys. 14, 8—15 (1946). 

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