Molecular Symmetry and Group Theory |
The C3v character table is given opposite: |
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Note that the numbers in a character table are not necessarily restricted to 1 or -1. We shall consider the meaning of the character 2, which describes the effect of the identity operation for the irreducible representation labelled E, in Workshop 4. |
Suppose we have the following reducible representation (reducible representations are normally given the label Γ):
E | 2C3 | 3σv | |
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Γ | 4 | 1 | -2 |
This may be reduced to its constituent irreducible representations by applying the reduction formula:
n(A1) | — | is the number of times the irreducible representation A1 occurs in the reducible representation, |
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h | — | is the order of the group i.e. the total number of operations in the group, |
Σ | — | the sum is taken over all classes, |
χR | — | is the character of the reducible representation, |
χI | — | is the character of the irreducible representation, |
N | — | number of symmetry operations in the class. |
Fortunately this formula is much easier to use than to remember! Consider the reducible representation given above. For A1 we apply the following procedure:
E | 2C3 | 3σv | |
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Γ | 4 | 1 | -2 |
A1 | 1 | 1 | 1 |
1×4×1 | 1×1×2 | 1×(-2)×3 |
4 | 2 | -6 |
Full Example of A1:
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Repeat the procedure above for A2 and E, and thus determine the composition of Γ.
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