Field Potential and Basis Functions

Next in importance to the cubic term in the ligand field potential is a term having tetragonal (axial) symmetry. For example, the ligands may be located at x = a, y = a, z = b, with a b. Such an environment, which departs from cubic s3onmetry by a distortion along the z-axis, is described by a potential, Vt, of the form (40) 

The potential Vt is invariant under the operations of the group D4h whose characters are listed in Table 6. However, since d-electrons generare representations of even parity only, we may refer to the simpler group D4 whose characters are also contained in Table 6. Moreover, since an irreducible representation in O is generally a reducible representation in D4, it is possible, with the help of the character table andEqs. (42 and 46), to decompose the e- and t2-representations in O into irreducible representations in D4. The results are 

Definitions (based on the z-axis along the 4-fold symmetry axis) 

E identity operation Cj a rotation of 180° about the z-axis C a rotation of 90° about the z-axis Cjt a rotation of 180° about the x- or y-axis 

Cj a rotation of 180° about an axis inclined at 45° to the x- and y-axis ih reflection in the xy-plane i inversion in the origin (ohCj) 

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