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By George Davidson (auth.)

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Very few molecules exist that belong to the Cn point groups. Two other groups exist that possess only one symmetry element besides the identity. These have, respectively, one symmetry plane (point group C5 ) or a centre of symmetry (point group C;). Both of these groups have just two symmetry operations, since a and i, unlike Cn, generate only one operation each. 4, respectively. e. ,. Br: a molecule belonging to the point group C; 24 Group theory for chemists If we return to the Cn point groups, we can investigate the effect of adding symmetry elements to them.

Only a few of these have fundamental significance, however. These are the irreducible representations. We will define these in detail in the next chapter; all that we need to say for the moment is that these are (with very few exceptions) of order 1, 2 or 3 (that is, the transformation matrices are 1 x 1, 2 x 2 or 3 x 3) and that any representation that is not irreducible can be built up from irreducible representations of the relevant point groups. Such representations are reducible, and in the later chapters we will examine methods for converting reducible representations into their irreducible components.

If they are absent, look for (a) a horizontal plane (Cnh), (b) n vertical planes (Cnv), (c) an Szn axis coincident with Cn (Szn) or (d) no symmetry planes or other axes (Cn)· Groups ond their basic properties 31 4 If a Cn axis and n C2 perpendicular axes are present, check on the presence of (a) a horizontal plane (Dnh), (b) n vertical planes and no horizontal plane (Dnd), or (c) no symmetry planes or other axes (Dn). Following this standard procedure, we can assign the following molecules to their respective point groups: 1 Non-linear; does not belong to a highly symmetric group.

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