The nuclear permutation operator for a homonuclear diatomic molecule

The nuclear permutation operator for a homonuclear diatomic 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  

The upper sign choice (symmetric behaviour) corresponds to the interchange of identical nuclei with integral spins such particles are called bosons. The lower sign 

Let us suppose that nuclei 1 and 2 have coordinates (X, Y, Z ) and (X2, Y2, Z2) in the translating but not rotating coordinate system. Since the origin of the axis system is at the nuclear centre of mass of the homonuclear molecule, we have 

so far as the nuclei are concerned, l 2 produces the same effect as E and, since the position of the molecule-fixed axes depends on the location of the nuclei only, P 2 has the same effect on a function of the Euler angles as does E  

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