Mutual exclusion, rule

For local symmetries with a center of symmetry (see Section 7.2), an infrared active vibration (phonon) is Raman inactive, and vice versa. This rule is usually known as the mutual exclusion rule. 

Recall the mutual exclusion rule stated in Section 6.7. The rule follows from the fact that the integral over all space of an odd ( ) function is zero. The functions x, y, and z belong to u representations of molecules with a center of symmetry, since inversion converts each to its negative. Hence one of the functions vjb and ib must belong to a g representation and one to a u representation if the integrand of (9.189) is not to be odd. Thus only g<->u IR transitions are allowed in molecules with a center of symmetry. In contrast, the functions (9.196) are all even (g), so that for centrosymmetric molecules only g<->g and u u Raman transitions are allowed. This proves the mutual exclusion rule. 

The vibrations of acetylene provide an example of the so-called mutual exclusion rule. The rule states that, for a molecule with a centre of inversion, the fundamentals which are active in the Raman spectrum (g vibrations) are inactive in the infrared spectrum whereas those active in the infrared spectrum (u vibrations) are inactive in the Raman spectrum that is, the two spectra are mutually exclusive. However, there are some vibrations which are forbidden in both spectra, such as the au torsional vibration of ethylene shown in Figure 6.23 in the Dlh point group (Table A.32 in Appendix A) au is the species of neither a translation nor a component of the polarizability. 

As stated in Section 1.7, selection rules are markedly different between IR and Raman spectroscopies. Thus, some vibrations are only Raman-active while others are only IR-active. Typical examples are found in molecules having a center of symmetry for which the mutual exclusion rule holds. In general, a vibration is IR-active, Raman-active, or active in both however, totally symmetric vibrations are always Raman-active. 

Due to the mutual exclusion rule, g modes (except Ajg) are Raman active, while u modes (except Ai ) are infrared active. We thus expect the internal modes to give rise to three bands in the Raman spectrum and to three bands in the infrared. In order to determine the lattice modes, we have to consider the carbonate anions with 6 degrees of freedom each and two calcium cations with three degrees of freedom. One obtains 2x6-1-2x3 3=15 lattice vibrations and 3 acoustic modes. These can be classified... 

On the other hand, when the unit cell is centrosymmetric the mutual exclusion rule is valid, so that Raman active modes are IR-inactive and vice versa. In practice, for centrosymmetric cells containing N oxo-anions, every internal vibrational mode of the oxo-anion gives rise to N/2 IR active modes and N/2 Raman active modes. 

In centrosymmetric molecules, HRS gains intensity via Herzberg-Teller term (the first vibronic B-term), indicating that IR-active modes and silent modes are enhanced. In the case of non-centrosymmetric molecules, however, Franck-Condon mechanism (A-term) dominantly contributes to the enhancement. Moreover, the mutual exclusive rules between HRS and RS are broken, and hence, some of RS-active modes selectively appear in the spectra. In the case of plasmonic enhancement, the spectral appearance is more sensitive to molecular orientations at the metal surface because of the surface selection rules [25]. 

Due to its high symmetry, benzene obeys the mutual exclusion rule, so that its infrared active bands are inactive in the Raman spectrum and vice versa. Since the aromatic centers in lignin do not have this symmetry, many more bands are active in both infrared and Raman spectra. However, the pattern remains that the highly polar vibrations are expected to be strongest in the infrared spectrum, whereas the least polar and most polarizable vibrations are expected to be most intense in the Raman spectrum. 

This conclusion may be drawn directly from observation of the mutual exclusion rule, which holds for D4/J but not for Tj. 

The three normal modes of linear X3(D /,)- and YXY(D ft)-type molecules were shown in Fig. 1.11 Vj is Raman-active but not infrared-active, whereas V2 and V3 are infrared-active but not Raman-active (mutual exclusion rule). However, all three... 

The D2/ structure may be confirmed if the infrared and Raman mutual exclusion rule holds. The D2 and D2 structures can be distinguished by comparing the number of fundamentals with that predicted for each structure 8 for D2 and 5 for D2J in the infrared, and 12 for D2 and 9 for D2d in the Raman. 

The distinction between cyclic dimers (M-OH)2 and infinite chains (M-OH) is based in the mutual exclusion rule for centrosymmetric cyclic dimers, i.e. infrared and Raman frequencies should not coincide. In the case of carboxylic acids R-COOH, the C=0 stretching mode is particularly sensitive and the difference between infrared and Raman frequencies can be as much as 60-70 cm . ... 

The mutual exclusion rule states that for certain molecules, vibrations that are IR-active are not Raman-active, and vice versa. Molecules must have a certain symmetry element in order for the mutual exclusion rule to apply. Examine the character tables in Appendix 3 and determine what that symmetry element is. 

---