Totally symmetric vibrations

In Figure 7.42 it is seen that the progression is built not on the Og but on the 6g band. The reason for this will become clear when we have seen, in the following section, how non-totally symmetric vibrations may be active in an electronic band system. 

The answer, very often, is that they do not obtain any intensity. Many such vibronic transitions, involving non-totally symmetric vibrations but which are allowed by symmetry, can be devised in many electronic band systems but, in practice, few have sufficient intensity to be observed. For those that do have sufficient intensity the explanation first put forward as to how it is derived was due to Herzberg and Teller. 

The first term on the right-hand side is the same as in Equation (7.128). Herzberg and Teller suggested that the second term, in particular (dRg/dQj), may be non-zero for certain non-totally symmetric vibrations. As the intensity is proportional to Rgy this term is the source of intensity of such vibronic transitions. 

Examples of vibronic transitions involving non-totally symmetric vibrations are in the system of chlorobenzene, a C2 molecule. One 2 vibration V29, with a wavenumber of 615 cm in the X state and 523 cm in the A state, is active in 29q and 29j bands similar to the case shown in Figure 7.43. There are 10 2 vibrations in chlorobenzene but the others are much less strongly active. The reason is that (9J g/9029)eq is much greater than the corresponding terms for all the other 2 vibrations. 

The A B2 — system of chlorobenzene is electronically allowed, since B2 = which satisfies Equation (7.122). The Ojj band, and progressions in totally symmetric vibrations built on it, obtain their intensity in the usual way, through the first term on the right-hand side of Equation (7.131). 

The A A2 X Ai, n -n system of formaldehyde (see Section 7.3.1.2) is also electronically forbidden since A2 is not a symmetry species of a translation (see Table A.l 1 in Appendix A). The main non-totally symmetric vibration which is active is Vq, the hj out-of-plane bending vibration (see Worked example 4.1, page 90) in 4q and d transitions. 

All the forbidden electronic transitions of regular octahedral transition metal complexes, mentioned in Section 7.3.1.4, are induced by non-totally symmetric vibrations. 

Although we have considered cases where (9/ g/90,)gq in Equation (7.131) may be quite large for a non-totally symmetric vibration, a few cases are known where (9/ g/90,)gq is appreciable for totally symmetric vibrations. In such cases the second term on the right-hand side of Equation (7.131) provides an additional source of intensity forAj orX vibronic transitions when Vx is totally symmetric. 

Nevertheless, 1,4-difluorobenzene has a rich two-photon fluorescence excitation spectrum, shown in Figure 9.29. The position of the forbidden Og (labelled 0-0) band is shown. All the vibronic transitions observed in the band system are induced by non-totally symmetric vibrations, rather like the one-photon case of benzene discussed in Section 7.3.4.2(b). The two-photon transition moment may become non-zero when certain vibrations are excited. 

It is experimentally easy to generate Raman spectra using polarized light and to observe the partial depolarization of the spectra. Bands of totally symmetric vibrations are strongly polarized in Hquid or solution spectra. AH other bands in Hquid or solution are depolarized. Polarization effects are essential to elucidate stmctures, but are usuaHy ignored in most other appHcations. Details of the theory and experimental procedure can be found in the Hterature (15,16). 

The strongest mode observed near 800 cm 1 is polarized along c and is a totally symmetrical vibration mode (Ai) corresponding to the niobium-oxygen vibrations vs (NbO) of infinite chains (NbOF4 )n running along the c -axis. The mode observed at 615 cm 1 is polarized perpendicular to c and corresponds to the NbF vibrations of the octahedrons of the same chains. The mode at 626 cm 1 is attributed to NbF vibrations of isolated complex ions - NbF 2 . The lines at 377, 390 and 272 cm 1 correspond to deformation modes 8(FNbF) of the two polyhedrons. 

The room temperature Raman spectrum excited in pre-resonance conditions [351 indeed shows bands at 169 cm-1 and 306 cm, which are in agreement with the modes observed in the fluorescence spectrum and that have been assigned by ab initio calculations to totally symmetric vibrations jl3). 

Several b-polarized sharp bands are assigned as ground slate totally symmetric vibrations at 699, 738, 1056, 1369, 1460 and 1504 cnT1 built on the fluorescence origin (see Fig. 6-18). These modes are in excellenl agreement with those obtained from the single crystal Raman spectra thal we measured exciting at 1064 and 632.8 nm [35]. 

Tabic 6-5. Comparison of (he aK vibrational modes in the ground and excited states. The totally symmetric vibrations of the ground stale measured in tire Raman spectrum excited in pre-resonance conditions 3S] and in the fluorescence spectrum ]62 ate compared with the results of ab initio calculations [131- The corresponding vibrations in the excited stale arc measured in die absorption spectrum. 

For nondegenerate vibrations all symmetry operations change Qj into 1 times itself. Hence Q/ is unchanged by all symmetry operations. In other words, Q and consequently y(O) behave as totally symmetric functions (i.e. the function is independent of symmetry). However, the wavefunction of the first excited state 3(1) has the same symmetry as Qj. For example, the wavefunction of a totally symmetric vibration (e.g. Qi of C02) is itself a totally symmetric function. 

A famous, yet simple example Is CO. CO tends to adsorb In highly symmetric positions on low Index surfaces, so that the point groups are C. and C. . The totally symmetric vibrations then... 

The shapes of both /w and 7hv lines are assumed to be represented by simple Lorentzians. For a totally symmetric vibration with a low polarization ratio as in the present case, the vibrational and reorientational relaxation times Tv and can be determined from the half-widths of the isotropic and anisotropic spectra. Since the value of /hv is much smaller than that of /w for the 1050 cm" line, the contribution of /gv to the isotropic intensity can be neglected ... 

The simplest intermediate of the nitrogen cation type is the nitronium ion, the active species in most aromatic nitration reactions. There is both cryoscopic and spectroscopic (Raman and infrared) evidence for its existence.802 On the other hand, it has a structure with quaternary rather than electron deficient nitrogen, a structure compatible with the centrosymmetric geometry demanded by the spectra. The Raman line at 1400 cm.-1 has been assigned to the totally symmetric vibration of the linear triatomic molecule. 

The term II scattering (equation 7) from vibronic activity in allowed electronic transitions mainly results in fundamental transitions of non-totally symmetric vibrations. This term corresponds to the B and C terms of the Albrecht theory25. 

Then the moment induced by the electric vector of the incident light is parallel to that vector resulting in complete polarization of the scattered radiation. The A lg i>(CO) mode of the hexacarbonyls provides a pertinent example08. Suppose we have a set of coupled vibrators, equidistant from some origin. Then it must be possible to express the basis functions for the vibrations in terms of spherical harmonics, for the former are orthogonal and the latter comprise a complete set. The polarization of a totally symmetric vibration will be determined by its overlap with the spherically symmetrical term which may be taken as r2 = x2 + y1 + z2. Because of the orthogo-... 

An additional concern arises in regard to any differences which may exist between the classical theory and the quantum-mechanical approach in the calculation of the Franck-Condon factors for symmetrical exchange reactions. In fact, the difference is not very large. For a frequency of 400 cm for metal-ligand totally symmetric vibrational modes, one can expect... 

Non-totally symmetric vibrations lower the symmetry of a molecule and previously forbidden bands may become allowed. The Hamiltonians considered up to now were all given for a fixed nuclear equilibrium geometry. A Taylor series expansion in the normal coordinates Q around this nuclear equilibrium geometry... 

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