Triply degenerate vibration

Figure 5. Normal modes for vibration of tetrahedral [Cr04] (chromate). There are four distinct vibrational frequencies, including one doubly-degenerate vibration (E symmetry) and two triply-degenerate vibrations (F2 symmetry), for a total of nine vibrational modes. Arrows show the characteristic motions of each atom during vibration, and the length of each arrow is proportional to the magnitude of atomic motion. Only F2 modes involve motion of the central chromium atom, and as a result their vibrational frequencies are affected by Cr-isotope substitution. The normal modes shown here were calculated with an ab initio quantum mechanical model, using hybrid Hartree-Fock/Density Functional Theory (B3LYP) and the 6-31G(d) basis set—other ab initio and empirical force-field models give very similar results.

In Fig. 3 we show a cross section of the Jahn-Teller surface around the tetrahedral geometry. The simplest Jahn-Teller intersection is of T t2 type (i.e., a triply degenerate electronic state whose components are coupled via a triply-degenerate vibration). Optimization of a triply degenerate conical intersection is not routinely possible at present. The vibrational modes of Td Fe(C0)4 span Qa 2 1 2E T 4T2, so there are clearly several candidate vibrations. This T t type of... 

Some texts use the symbol T for the triply degenerate vibration. 

Thus, the first overtone of the doubly degenerate vibration is IR- and Raman-active. The characters of overtones for triply degenerate vibrations are given by... 

A similar treatment of the spherical rotor can be carried out with the result that the triply degenerate vibrational levels are split by the energies... 

The degeneracy of a vibration is related to its character of the identity element of its irreducible representation label. Doubly degenerate vibrations always have an irreducible representation label having Xe 2. Triply degenerate vibrations always have an irreducible representation label having Xe 3. There are no higher degeneracies for vibrations. 

If we check the character table in Appendix 3, we find second-order functions listed with Aj, E, and T2 irreducible representations. Therefore, all of the vibrations of CCI4 will be Raman-active, and the Raman spectrum will consist of four signals representing one singly degenerate, one doubly degenerate, and two triply degenerate vibrations. 

F triply degenerate vibrations, i.e., a three-dimensional representation... 

In some cases where there is high point group symmetry, vibrations normally infrared inactive often appear as a weak band and doubly and triply degenerate vibrations can split into two and three compo-... 

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