Example 13.9
The following set of characters is for a linear combination of irreducible rep-
resentations for a system having C3vsymmetry (say, ammonia):E 2 C 3 3 v
combo 7 1 1Use equation 13.6 to determine what linear combination ofA 1 ,A 2 , and E
symmetry species is being represented.Solution
We need the characters for A 1 ,A 2 , and Efrom the character table of the C3vpoint
group. For the number of times A 1 appears in the linear combination,aA 1 1
6
(^)
all classes
NA 1 linear combo
where the summation has three terms. Solving:
A 1 ,from
character table
aA 1
1
6
1 1 7 2 1 1 3 1 1
N linear combo C 3 v
term for EaA 1 1
6
(7 2 3)
1
6
(12) 2
so that the A 1 symmetry species appears twice in the linear combination.
Similarly, for A 2 and E:
aA 2 ^16 [(1 1 7) (2 1 1) (3 1 1)] ^16 (6) 1
aE^16 [(1 2 7) (2 1 1) (3 0 1)] ^16 (12) 2
Therefore, this combois a sum of two A 1 , one A 2 , and two Esymmetry
species. This is how the great orthogonality theorem is applied to reduce
character sets into their unique set of irreducible representations.The mathematical way of illustrating the above combination of symmetry
species is to use the sign instead of a sign (which is how linear combina-
tions are usually expressed). One can therefore write combofrom above as
combo 2 A 1 1 A 2 2 E
The great orthogonality theorem is useful because it allows us to break
down any reducible representation into its irreducible representations. Once
any scheme for determining a representation for a wavefunction is applied, the
GOT can be used to reduce the representation into its irreducible components.
For example, in the case of the Rh(3)spherical point group, the characters
are given as expressions in terms of the angle of the particular rotation.
The characters for any one representation are transferable to the symmetry
operations of any other point group, which ultimately represent a reducible
representation in the new point group. Using the GOT, one determines the
13.7 The Great Orthogonality Theorem 439{
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