Vibrational wave function degenerate

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 ... 

The direct product enables one to find the symmetry of a wave function when the symmetries of its factors are known. For example, consider In the harmonic-oscillator approximation, the vibrational wave function is the product of 3N—6 harmonic-oscillator functions, one for each normal mode. To find the symmetry of we first examine the symmetries of its factors. Let the distinct vibrational frequencies of the molecule be vx>v2,..., vk,...,vn, and let vk be <4-fold degenerate let the harmonic-oscillator... 

The ti eatment of the Jahn-Teller effect for more complicated cases is similar. The general conclusion is that the appearance of a linear term in the off-diagonal matrix elements H+- and H-+ leads always to an instability at the most symmetric configuration due to the fact that integrals of the type do not vanish there when the product < / > / has the same species as a nontotally symmetiic vibration (see Appendix E). If T is the species of the degenerate electronic wave functions, the species of will be that of T, ... 

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... 

For states with degenerate vibrations excited, the degenerate Qk s are transformed into linear combinations of one another, and the symmetry of the wave function is best determined using group theory see Chapter 9. 

Spherical harmonics, 29, 38 Spherical polar coordinates, 21 Spherical top definition of, 199 degeneracy for, 210 degenerate vibrational modes in, 276 energies of, 209-210 microwave spectrum of, 217,225 polarizability, 202 wave functions of, 209, 211... 

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