Point group antisymmetry

All the above examples applied to point groups. Antisymmetry and color symmetry, of course, may be introduced in space-group symmetries as well as examples illustrate in Figures 8-32, 8-37, and 9-46 (in the discussion of space groups). If we look only at the close-up of the tower in Figure 4-14b, it also has tranlational antisymmetry, specifically anti-glide-reflection symmetry together with similarity symmetry (these symmetries will be discussed in Chapter 8). 

A nondegenerate irrep that is symmetric with respect to the principl axis is denoted A, while B indicates antisymmetry with respect to this axis. In point groups with a horizontal plane of reflection, primes and " respectively indicate symmetry and antisymmetry with respect to the plane, while g and u indicate symmetry and antisymmetry with respect to inversion. For doubly degenerate irreps a subscript m indicates which spherical harmonics VJ, m form basis functions for that irrep. Numerical subscripts are used on nondegenerate irreps to distinguish them where necessary the numbers indicate the first of the vertical planes or perpendicular twofold axes (in the order specified in the character table) with respect to which the irrep is antisymmetric. 

For all point, axial rotation, and full rotation group symmetries, this observation holds if the orbitals are equivalent, certain space-spin symmetry combinations will vanish due to antisymmetry if the orbitals are not equivalent, all space-spin symmetry combinations consistent with the content of the direct product analysis are possible. In either case, one must proceed through the construction of determinental wavefunctions as outlined above. 

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