The CH-Stretching Coordinates of Ethylene

1) evaluation of the direct sum i.e. the distribution of the appropriate linear combinations among the irreducible representations, and 

The number of times a particular irrep appears in the reduced representation of the coordinates is given by its scalar product (Section 2.2.5) with the reducible representation. The formal procedure can be bypassed as follows Since r Sri) has 4 under E and (t zx) and 0 everywhere else, whereas T 2h has a total of 8 sym-ops, an irrep can appear no more than once in the direct sum. Moreover, the irreps that make it up are necessarily the four that are symmetrical to reflection in the molecular plane, i.e have 1 rather than —1 under (r xy). The direct sum is therefore [a 62 b u big]., as shown in Table 4.1. 

A representative coordinate, say Sri is chosen and operated upon in turn by the projection operator of each of the irreducible representations, Fj  

Reading Equation 4.12 from right to left Sri is operated upon by each of the sym-ops of T 2h and the result is multiplied by the character of that operation 

Unlike a cartesian coordinate, cannot be converted to itself with change of sign, so it cannot contribute —1 to the character. 

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Table 4.1. Reduction of the representation of the CH-stretching coordinates of ethylene to the irreps of D2/1...

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