# Symmetry analysis of an impossible symmetry

I have a practical problem in chemistry. Consider a molecule MCp3 (Cp=C5H5) with the centroids of the Cp ring lying on an 60-60-60 triangle. The maximal symmetry you can get is C3h, because of the fivefold-axis Cp rings. (Orient one CH group into the centroid plane.) In praxis, though, the Cp rings rotate freely and the effective symmetry is D3h.

Now there are the usual tools of vibrational analysis, giving for C3h that the normal vibrations are 16A'+14A''+16E'+12E'' (modulo my usual typos :-). I want to know how the 16A'+14A'' split into A1'+A2'+A1''+A2'', the irreps of the fictive D3h symmetry. This is impossible by normal means - e.g. the first step of a standard tool is to apply a symmetry operation on the atom set and count the trace of the operation matrix. But the symmetry operations aren't...

I am quite sure that it is possible to work with half-integer characters and other abominations and in the end everything crosscancels and gives an answer that makes sense within the experimental data (which IS compatible with D3h).

So my question: can you do representation analysis for objects with pseudo symmetry? (In this special case, it might be even doable this way: Compute the result for 4- and 6-rings, where D3h is a valid symmetry, and do the mean. :-)

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You will get a better response if you use mathematician's terminology and notation; after all, this is a site for research mathematics. – Bruce Westbury Jan 11 '13 at 15:21
I don't see this as a research level question. It appears to be asking for someone to do a character calculation for you. – Bruce Westbury Jan 11 '13 at 15:23
(Mods, you can close the posting, I cracked the problem by unmathematical brute computing force. :-) For the record, the overall result is integer but as soon as you try to assign the found symmetries to specific atom movements, the half-integers rear their ugly head again.) – Hauke Reddmann May 10 '13 at 13:46