Non-totally symmetric vibrations

2(b) Non-totally symmetric vibrations The general vibronic selection rule in Equation (7.126) shows that many vibronic transitions involving one quantum of a non-totally symmetric vibration are allowed. For example, consider a molecule such as 

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 Franck-Condon approximation (see Section 7.2.5.3) assumes that an electronic transition is very rapid compared with the motion of the nuclei. One important result is that the transition moment for a vibronic transition is given by 

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. 

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). 

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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. 

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 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. 

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. 

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... 

C-H stretching motion in CH4 do not induce a dipole moment, and are thus infrared inactive non-totally-symmetric vibrations can also be inactive if they induce no dipole moment. 

The Jahn-Teller effect has been the subject of intense study in solid-state physics and molecular sciences for almost 80 years (45,46). In their seminal paper, Jahn and Teller proved that all systems in orbitally degenerate states are unstable with respect to non-totally symmetric vibrations which cause a... 

Non-totally symmetrical vibrations can be shown to involve no change in origin of the normal coordinates, thus the value of the integral (7) is identically zero for Av odd and small for Av even (zero excluded). Taken together with the expectation for totally symmetrical vibrations this leads to two vibrational selection rules, namely (Herzberg and Teller, 1933),... 

In case (b) the intensity is expected to fall off rapidly along the series, being much the greatest for Av — 0. Accordingly the vibrations expected in an electronic band system are primarily the totally symmetric vibrations, especially those vibrations which tend to deform the final structure so as to resemble the initial structure while non-totally symmetric vibrations should be largely inactive except in sequences (Av = 0 transitions). It should be emphasized, however, that these expectations are... 

Fig. 11. (a) Electronic configuration in the XAU state of CSH2 (Ingold and King, 1953). (6) Non-totally symmetrical vibrations of D A acetylene and totally symmetrical vibrations of C2A acetylene. 

If the molecule is performing a non-totally symmetric vibration, the polarizability ellipsoid changes its shape from a sphere to an ellipsoid during the... 

The 5-term involves two electronic excited states (e and 5) and provides a mechanism for resonance-enhancement of non-totally symmetric vibrations. The 5-term, can be expressed as... 

When the molecule is raised to the electronic excited state, the normal mode Q preserves the same decomposition with respect to the nuclear displacements, except for Jahn-Teller effects, which we exclude in this work. In these conditions, still in the harmonic approximation, two parameters, the equilibrium point and the potential curvature, will change in the excited state (in the particular case of non-totally-symmetric vibrations, the equilibrium point does not change). In the excited state e, the nuclear potential becomes... 

For a transition which does not satisfy the symmetry restriction imposed by equation 1, the transition moment integral can be non-zero if a second-order mechanism is invoked which necessitates the excitation of a non-totally symmetric vibration in one or other of the electronic states. This has the effect of reducing the magnitude of the transition moment integral compared with that for a symmetry-allowed transition, and also leads to the absence of the absorption and emission spectral features corresponding to transitions between the vibrationless grcxind and excited electronic states. 

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