Exclusion rule

These limitations are largely eliminated in sophisticated defect calculations described in the following section. This approach can also include more sophisticated site exclusion rules, which allow defects to either cluster or keep apart from each other. Nevertheless, the formulas quoted are a very good starting point for an exploration of the role of defects in solids and do apply well when defect concentrations are small and at temperatures that are not too high. 

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. 

SOME IMPORTANT SPECIAL EFFECTS The Exclusion Rule... 

Another way of stating this result, the so-called exclusion rule, is as follows ... 

We have seen that not all molecules are like water in having all vibrational modes both (R and Raman active, (n fact, there is an extremely useful exclusion rule for molecules with a center of symmetry, i If a molecule has a center of symmetry. IR and Ranum active vibrational modes are mutually exclusive if a fibre ion is IR active, it cannot be Raman active, and vice versa.26... 

The domain of applicability of the QSAR should be explicitly defined. The QSAR should be associated with a description of the chemical classes for which it is applicable (inclusion rules) or inapplicable (exclusion rules). For QSARs, there should be an indication of the range of... 

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. 

As demonstrated by the spectra of 1,2-dichloroethane shown in Fig. 4.1-11C, two halogen atoms in 1,2-position also show in-phase as well as out-of-phase vibrations. The antiperiplanar conformation is subject to the exclusion rule op. Raman 750 cm Oa . IR 708 cm. In the case of the synclinal conformation, on the other hand, the in-phase vibration at 645 cm is stronger both in the IR and in the Raman spectrum than the out-of-phase vibration at 677 cm. Freely rotating 1,2-dihalogen compounds therefore show four different C-Cl stretching vibrations (Fig. 4.1-11C). 

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... 

Criteria and guidelines useful in network elucidation and supplementing the rules derived in this chapter include considerations of steric effects, molecularities of postulated reaction steps, and thermodynamic constraints as well as Tolman s 16- or 18-electron rule for reactions involving transition-metal complexes and the Woodward-Hoffmann exclusion rules based on the principle of conservation of molecular orbital symmetry. Auxiliary techniques that can be brought to bear include, among others, determinations of isomer distribution, isotope techniques, and spectrophotometry. 

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]. 

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