Symmetric properties vibrational wave function

Just as the normal coordinates of a symmetrical molecule show certain symmetry properties, so do the vibrational wave functions. For example, the wave function for the lowest vibrational state of a molecule is (see Sec. 3-3)... 

Equation 144 is just the equation for the electronic energy which was discussed in Chapter XI. The electronic wave functions F x, y, z, r) therefore have the symmetry properties of the various irreducible representations of the groups Dooa or Coop according as the nuclei are identical or different. The vibrational wave function R r) depends only on the distance between the two nuclei and therefore belongs to the totally symmetrical representation. The complete wave fimction will thus have the symmetry properties of the product F x, y, z, r)U(6, x)-In order to discuss the nature of the solutions of 14 6 it will be con-... 

As discussed in preceding sections, FI and have nuclear spin 5, which may have drastic consequences on the vibrational spectra of the corresponding trimeric species. In fact, the nuclear spin functions can only have A, (quartet state) and E (doublet) symmetries. Since the total wave function must be antisymmetric, Ai rovibronic states are therefore not allowed. Thus, for 7 = 0, only resonance states of A2 and E symmetries exist, with calculated states of Ai symmetry being purely mathematical states. Similarly, only -symmetric pseudobound states are allowed for 7 = 0. Indeed, even when vibronic coupling is taken into account, only A and E vibronic states have physical significance. Table XVII-XIX summarize the symmetry properties of the wave functions for H3 and its isotopomers. 

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