Homonuclear molecules, permutational symmetry

Ptj, the symmetry operation involving interchange of identical particles (nuclei or electrons). All particles axe either Bosons or Fermions, and the total wavefunction must, respectively, be even or odd upon interchange of any pair of identical particles. The total wavefunction of a homonuclear molecule, exclusive of the nuclear spin part, is classified s or a according to whether it is even or odd with respect to nuclear exchange. Since electrons are Fermions, the total molecular wavefunction must be odd with respect to permutation of any two electrons. This requirement is satisfied by the determinantal form of the electronic wavefunction (see Section 3.2.4). 

For homonuclear molecules, the initial and final rotational states must belong to the same ortho/para nuclear permutation symmetry. For example, in the H2(X1E+) —> (X2E ) +e photoionization transition, the even No and N+... 

The final expression is the classical limit, valid above a certain critical temperature, which, however, in practical cases is low (i.e. 85 K for H2, 3 K for CO). For a homonuclear or a symmetric linear molecule, the factor a equals 2, while for a het-eronuclear molecule cr=l (Tab. 3.1). This symmetry factor stems from the indistinguishable permutations the molecule may undergo due to the rotation and actually also involves the nuclear partition function. The symmetry factor can be estimated directly from the symmetry of the molecule. 

A second useful symmetry operation exists for homonuclear diatomic molecules, namely the permutation of two identical nuclei, P 2. In the same way that E has two possible eigenfunctions 1 in equation (6.206), so there are two possible ways in which the molecular wave function can transform under P 2 ... 

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