Modes degenerate

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. 

The mode coordinates, tiansforming, respectively, as the — 1 and 1 components of the E (doubly degenerate) modes, have the form ... 

Let us then consider the case where the degenerate mode is doubly excited. In this case, V2 = V2a + V2i = 2 and the coiTesponding vibrational energy level will be triply degenerate with the associated wave functions being given by... 

The vibrational energy levels associated with a single normal mode have a degeneracy of one. However, molecules with high symmetry may have several normal modes with the same frequency. For example, COi has two bending modes, with the motion of one perpendicular to the motion of the other. Such modes are often referred to as degenerate modes, but there is a subtle difference... 

The effective spectroscopic classical Hamiltonian for Fermi resonance between a nondegenerate mode vi and a doubly degenerate mode V2 takes the form [30, 31]... 

The attribution of lines 1 and 2 in n-type GaAs Si is further substantiated by 1) the value of the ratio of the integrated absorptions A (1-H)/Aj(2-H), which is approximately equal to three (a ratio of about two is expected since lines 1-H and 2-H are attributed to doubly and singly degenerate modes respectively), and 2) the r-factor of 1.398 for line 1, explained by the small contribution of the Si atoms to the wagging motion. 

In contrast with 1, for which only four threefold degenerate modes are active, 30 and 33 exhibit 86 and 45 IR active modes, respectively, 58 and 31 of which are twofold degenerate. The lower symmetry of 30 and 33 as compared to 1 also results in a higher number of nonequivalent carbon atoms producing more expected signals... 

If two or more of the ks are equal, then (6.36) does not necessarily hold for the Qk s corresponding to the degenerate modes. Thus let suppose... 

Since we have abandoned the individual quantum numbers of the degenerate modes Qx and Qy, we will replace the summation over the 3N — 6 normal modes in energy expressions by a summation over the distinct vibrational frequencies. For a linear triatomic molecule there are four normal modes, but only three distinct vibrational frequencies. In the harmonic-oscillator approximation, the energy contribution of the doubly degenerate modes is... 

For nondegenerate modes, we saw that OrQ = Qj for doubly degenerate modes, we saw that Or QJ + Qj<) = (Qf + ) and so on. Hence 0R has no effect on the exponential factor in the vibrational wave function. The symmetry behavior of vib thus depends on the Hermite-polynomial factor ... 

Quantum mechanically, there is also the possibility of nuclear vibrational angular momentum for these degenerate modes. We denote the normal coordinates for the doubly degenerate vibrations by Qx and Qy. From Fig. 6.2, we have... 

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