High symmetry groups

Simplified flow chart for assignment of a symmetry group. Special high symmetry groups (Td, Ot, Ii,) are not included in this chart, and need to be identified separately it applies to other and generally lower symmetry molecules, which are more often met in reality. 

Coordination Number Six. This is perhaps the most common coordination number, and the six ligands almost invariably lie at the vertices of an octahedron or a distorted octahedron. The very high symmetry, group Oh, of the regular octahedron, has been discussed in detail on page 20. 

The correlation tables for high-symmetry groups do not include all of the subgroups. It may be necessary to use more than one correlation table in order to obtain the symmetry connections. For instance, use 0(, Tj Cjv to connect the O, point group with its 2 subgroup. 

Does the molecule belong to a high-symmetry group such as Oh or Td ... 

The III character table is given in Table A.46 in Appendix A. The very high symmetry of this point group results in symmetry species with degeneracies of up to five, as in and... 

At higher frequencies (above 200 cm ) the vibrational spectra for fullerenes and their cry.stalline solids are dominated by the intramolecular modes. Because of the high symmetry of the Cgo molecule (icosahedral point group Ih), there are only 46 distinct molecular mode frequencies corresponding to the 180 6 = 174 degrees of freedom for the isolated Cgo molecule, and of these only 4 are infrared-active (all with Ti symmetry) and 10 are Raman-active (2 with Ag symmetry and 8 with Hg symmetry). The remaining 32 eigcnfrequencies correspond to silent modes, i.e., they are not optically active in first order. 

In all cases we start from a simple structure which has high symmetry. Every arrow (= -) in the preceding examples marks a reduction of symmetry, i.e. a group-subgroup relation. Since these are well-defined mathematically, they are an ideal tool for revealing structural relationships in a systematic way. Changes that may be the reason for symmetry reductions include ... 

In 1937 Jahn and Teller applied group-theoretical methods to derive a remarkable theorem nonlinear molecules in orbitally degenerate states are intrinsically unstable with respect to distortions that lower the symmetry and remove the orbital degeneracy.37 Although Jahn-Teller theory can predict neither the degree of distortion nor the final symmetry, it is widely applied in transition-metal chemistry to rationalize observed distortions from an expected high-symmetry structure.38 In this section we briefly illustrate the application of Jahn-Teller theory and describe how a localized-bond viewpoint can provide a complementary alternative picture of transition-metal coordination geometries. 

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