D3h character table

Atomic basis functions on B are straightforward to classify. Evidently, an s type function on B will be totally symmetric — an a orbital. A quick inspection of the D3h character table will show that a p set on B, which transforms like the three Cartesian directions, spans the reducible representation a 2 e. Functions centred on the F atoms require more effort. Since the operations in the classes containing C3 and S3 move all three F atoms, their character is necessarily zero for any functions centred on the F atoms. Consider first a set of s functions on each F. These span a reducible representation with character... 

The question might arise whether it would not be more straightforward for the MnF3 molecule to distort into a structure of C3v symmetry The D3h character table (Table 6-11) helps us to understand the reason. The out-of-plane vibration—that would make the molecule pyramidal—is of a symmetry. However, there is no irreducible representation with a symmetry among the irreducible representations that the direct product of E with itself reduces to (see, above). Therefore, a C3v-type of distortion is not possible—at least not in the Jahn-Teller manner. 

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