Each matrix is composed of three one-dimensional matrices, which lie on the
diagonal. If we are concerned with a one-dimensional vector like {x, 0, 0}, only the
first row of the total set of the four matrices is necessary to represent the behavior of
vector (1, −1, 1, −1). The row is called irreducible representation and takes the
denomination of B 1 . The irreducible representation for {0, y, 0} is (1, −1, −1, 1),
labeled B 2 and that for {0, 0, z} is (1, 1, 1, 1) labeled A 1 .
If a different object (function or operation) is selected to be base of the C 2v
symmetry operations, e.g., the C 2 rotation, also the behavior of this object can be
associated to an irreducible representation, in this case (1, 1, −1, −1), labeled A 2 .
1.6.1 Character Table of groups
The so-called table of characters associated to the C 2v group is
E
C 2
xz
yz
A 1
+1
+1
+1
+1
z
A 2
+1
+1
−1
−1
R z
B 1
+1
−1
+1
−1
x, R y
B 2
+1
−1
−1
+1
y, R x
where the first column reports the name of the irreducible representation of the
corresponding row and the last column indicates the base objects undergoing the
symmetry transformations. As for the label of the irreducible representations:
A and B refer to representations which have one single function as base of the
symmetry operations, A in particular has +1 character, and B has −1 for the C n
operation; 1 and 2 refer to the symmetric and antisymmetric role of the the r xz
operation.
Let us consider the NH 3 molecule and its symmetry operations
σ′ υ
σ υ
σ″ υ
H
(1.0)
y
x
(– 1/2, √3/2) H
(– 1/2, – √3/2) H
Symmetry planes of NH 3
1.6 Irreducible Representations
17