Homonuclear molecules, permutational

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+... 

As was shown in the preceding discussion (see also Sections Vin and IX), the rovibronic wave functions for a homonuclear diatomic molecule under the permutation of identical nuclei are symmetric for even J rotational quantum numbers in and E electronic states antisymmeUic for odd J values in and E elecbonic states symmetric for odd J values in E and E electronic states and antisymmeteic for even J values in Ej and E+ electeonic states. Note that the vibrational ground state is symmetric under pemrutation of the two nuclei. The most restrictive result arises therefore when the nuclear spin quantum number of the individual nuclei is 0. In this case, the nuclear spin function is always symmetric with respect to interchange of the identical nuclei, and hence only totally symmeUic rovibronic states are allowed since the total wave function must be symmetric for bosonic systems. For example, the nucleus has zero nuclear spin, and hence the rotational levels with odd values of J do not exist for the ground electronic state f EJ") of Cr. 

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. 

According to the argument presented above, any molecule must be described by wavefunctions that are antisymmetric with respect to the exchange of any two identical particles. For a homonuclear diatomic molecule, for example, thepossibility of permutation of the two identical nuclei must be considered. Although both the translational and vibrational wavefunctions are symmetric under such a permutation, die parity of the rotational wavefunction depends on the value of 7, the rotational quantum number. It can be shown that the wave-function is symmetric if J is even and antisymmetric if J is odd The overall... 

Next, we address some simple cases, begining with homonuclear diatomic molecules in 1S electronic states. The rotational wave functions are in this case the well-known spherical harmonics for even J values, yr(R) is symmetric under permutation of the identical nuclei for odd J values, y,.(R) is antisymmetric under the same permutation. A similar statement applies for any D.yjh type 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 ... 

As was shown in the preceding discussion (see also Sections VIII and IX), the rovibronic wave functions for a homonuclear diatomic molecule under the permutation of identical nuclei are symmetric for even J rotational quantum numbers in and electronic states antisymmetric for odd J values in and electronic states symmetric for odd J values in S7 and electronic... 

The nuclear permutation operator for a homonuclear diatomic molecule... 

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