They can be more easily obtained with the following rules, as alternative to
construct the complete representation matrices:
1. Any vector which is unchanged by the symmetry operations contributes +1.
2. Any vector which changes into its opposite by the symmetry operations contributes −1.
3. Any vector which changes in another one under the symmetry operations
contributes 0.
By these rules, it can be constructed the character of the total (reducible) representation. The character is used to decompose the reducible representation into
irreducible ones by the following theorem.
a i ¼
1
h
X
R
gv i ðRÞv T ðRÞ
where a i is the number of irreducible representations of type i contained in the total
one t, v i and v t are the characters under the different symmetry operations R added
each other for all the operations, h is the number of the group operations, g the
number of operations having the same character (class).
For the water molecule, having C 2v symmetry group
a A 1 ¼
1
4
gv A i ðEÞv T ðEÞ þ gv A i C 2
ð Þv T C 2
ð Þþgv A i r t
ð Þv T r t
ð Þþgv A i r
0
t
À Á v T r
0
t
À Á
Â
Ã
¼
1
4
½1  1  9 þ 1  1  ðÀ1Þ þ 1  1  1 þ 1  1  3Š ¼ 3
a A 2 ¼
1
4
½1  1  9 þ 1  1  ðÀ1Þ þ 1  ðÀ1Þ Â 1 þ 1  ðÀ1Þ Â 3Š ¼ 1
a B 1 ¼
1
4
½1  1  9 þ 1  ðÀ1Þ Â ðÀ1Þ þ 1  1  1 þ 1  ðÀ1Þ Â 3Š ¼ 2
a B 2 ¼
1
4
½1  1  9 þ 1  ðÀ1Þ Â ðÀ1Þ þ 1  ðÀ1Þ Â 1 þ 1  1  3Š ¼ 3
The total representation consists of the sum of 3A 1 1A 2 2B 1 3B 2 irreducible
representations, for a total number of nine representations, the same number as the
initial coordinates in the molecule.
1.6.3 Relation Between Molecular Wave Functions
and Irreducible Representations
Let us still consider that the energy of a molecule is unchanged by carrying out a
symmetry operation, as well as the Hamiltonian too. Thus, a symmetry operator
R commutes with the Hamiltonian operator.
20
1 The Electronic Structure Determination