E ¼
1 0 0
0 1 0
0 0 1
2
4
3
5 C 3 ¼
À
1
2 À
ffiffi
3
p
2
0
ffiffi
3
p
2
À
1
2
0
0
0
1
2
6
6
6
6
3
7
7
7
7
For r v
1
0 0
0 À1 0
0
0 1
2
4
3
5
X
Y
Z
2
4
3
5 ¼
X
0
Y
0
Z
0
2
4
3
5
For r
0
v and r
00
v
r t
0
¼
À
1
2
ffiffi
3
p
2
0
ffiffi
3
p
2
1
2
0
0
0 1
2
6
4
3
7
5 r t
00
¼
À
1
2
À
ffiffi
3
p
2
0
À
ffiffi
3
p
2
1
2
0
0
0
1
2
6
4
3
7
5
The table of characters for the C 3v group is
In the first column is the representation label, where E is a bidimensional representation that means it has a double base function.
In the second, there are the trace of matrices, in the third and forth the base
functions of the single representation.
1.6.2 Factoring the Total Representations into Irreducible
Representation
Let us consider the H 2 O molecule, having C 2v symmetry, and attribute 3 Cartesian
coordinates to each atom. Looking at the symmetry representations, for the E
operation, it results
18
1 The Electronic Structure Determination