Vibrational modes, totally symmetric

If we carry out the same analysis on a homonuclear diatomic, such as Hj, then additional operations are present because the molecule is now in Dooh and so is unaltered if the two atoms swap over. This introduces the inversion centre i at the middle of the bond and the improper rotation class 2So. However, the vibration remains totally symmetric, so that for H2 the stretching mode belongs to the irreducible representation S+. 

Each such nonual mode can be assigned a synuuetry in the point group of the molecule. The wavefrmctions for non-degenerate modes have the following simple synuuetry properties the wavefrmctions with an odd vibrational quantum number v. have the same synuuetry as their nonual mode 2the ones with an even v. are totally symmetric. The synuuetry of the total vibrational wavefrmction (Q) is tlien the direct product of the synuuetries of its constituent nonual coordinate frmctions (p, (2,). In particular, the lowest vibrational state. 

In a symmetric top molecule such as NH3, if the transition dipole lies along the molecule s symmetry axis, only k = 0 contributes. Such vibrations preserve the molecule s symmetry relative to this symmetry axis (e.g. the totally symmetric N-H stretching mode in NH3). The additional selection rule AK = 0... 

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. 

Fig. 105. TO-LO splitting of totally symmetrical mode A/ (NbO vibration of RbsNbsOFig). Reproduced from [441], A. I. Agulyansky, R. Cavagnat, M. Couzi, J. Ravez, Phys. Stat. Sol. (a) 138 (1993) 327, Copyright 1993, with permission ofWiley-VCH.

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. 

Vibrations of the symmetry class Ai are totally symmetrical, that means all symmetry elements are conserved during the vibrational motion of the atoms. Vibrations of type B are anti-symmetrical with respect to the principal axis. The species of symmetry E are symmetrical with respect to the two in-plane molecular C2 axes and, therefore, two-fold degenerate. In consequence, the free molecule should have 11 observable vibrations. From the character table of the point group 04a the activity of the vibrations is as follows modes of Ai, E2, and 3 symmetry are Raman active, modes of B2 and El are infrared active, and Bi modes are inactive in the free molecule therefore, the number of observable vibrations is reduced to 10. 

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... 

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