Tetrahedral MX4, Molecules and Degenerate Irreducible Representations

It may be useful to consider a definite numerical example, a distorted PO4 tetrahedron with internal coordinates  

The referenee bond length in a phosphate tetrahedron can be taken as 1.534 A [9]. The qualitative impression from this tabulation is that one bond length has been stretched, and another one shortened, both by large amounts the three shorter bonds have been bent towards the stretched bond, thus diminishing the three bond angles involving this bond. The numerical analysis confirms this impression but makes it more precise. For the linear combinations corresponding to the symmetry coordinates of Table 2.4 we obtain  

Thus we may say that as far as bond angles are concerned the fragment shows approximate C3V symmetry . 

For any given MX, molecule we have 5 M-X distances, 10 X-X distances and 5 = 120 ways of permuting the atomic labels. Because of the absence of any three-dimensional figure with the same symmetry properties as the permutation group P5, most structure correlation studies of molecules are based on those subgroups that do have an isomorphous counterpart among the point groups in three-dimensional space, e.g. D3h and 4 [10]. 

Octahedral MXs molecules with M = Cu (//), Ag (II), low-spin Co (//) or high-spin Ct(II) have an electronically degenerate ground state and are thus expected to undergo Jahn-Tbller distortion along displacement coordinates of Eg symmetry. Appropriate coordinates may be defined as  

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