3.1 The Symmetry of Ammonia
23
Fig. 3.2 Applying ˆ
σ 1 ˆ
C 3 to the starting structure is equivalent to applying ˆ
σ 2
Table 3.1 Multiplication
table for the point group C 3v
C 3v
ˆ
E
ˆ
C 3
ˆ
C 2
3
ˆ
σ 1
ˆ
σ 2
ˆ
σ 3
ˆ
E
ˆ
E
ˆ
C 3
ˆ
C 2
3
ˆ
σ 1
ˆ
σ 2
ˆ
σ 3
ˆ
C 3
ˆ
C 3
ˆ
C 2
3
ˆ
E
ˆ
σ 3
ˆ
σ 1
ˆ
σ 2
ˆ
C 2
3
ˆ
C 2
3
ˆ
E
ˆ
C 3
ˆ
σ 2
ˆ
σ 3
ˆ
σ 1
ˆ
σ 1
ˆ
σ 1
ˆ
σ 2
ˆ
σ 3
ˆ
E
ˆ
C 3
ˆ
C 2
3
ˆ
σ 2
ˆ
σ 2
ˆ
σ 3
ˆ
σ 1
ˆ
C 2
3
ˆ
E
ˆ
C 3
ˆ
σ 3
ˆ
σ 3
ˆ
σ 1
ˆ
σ 2
ˆ
C 3
ˆ
C 2
3
ˆ
E
As has already been shown, these operations can also be performed directly in function space. Choosing the xy-plane 2p-orbitals on nitrogen, {p x ,p y }, as a suitable
basis set, we may represent all the symmetry elements by transformation matrices.
The resulting matrices are summarized in Table 3.2. Note that all six matrices are
different. The mapping between the symmetry elements and the matrices is therefore one-to-one, and the representation is said to be faithful.F o rt h e ˆ
C 3 axis, the
matrix corresponds to the one in Eq. (1.13), with rotation angle α = 2π/3, and for
the ˆ
C 2
3 axis, one has α = 4π/3, which is equivalent to the inverse angle α =−2π/3.
The ˆ
σ 1 element leaves p x unchanged and inverts p y . The other reflection planes are
similar to ˆ
σ 1 , which means that they can be obtained by a symmetry transformation
of this operator, using the results in Sect. 1.3; hence,
ˆ
σ 2 = ˆ
C 3 ˆ
σ 1 ˆ
C
−1
3
ˆ
σ 3 = ˆ
C 3 ˆ
σ 2 ˆ
C
−1
3
(3.3)
The set of the six matrices in Table 3.2 offers an alternative algebraic way of constructing the multiplication table by direct matrix multiplication. The product ˆ
C 3 ˆ
σ 1
is then replaced by the matrix multiplication D(C 3 ) × D(σ 1 ):
⎛
⎝
−
1
2
−
√
3
2
+
√
3
2
−
1
2
⎞
⎠ ×
10
0 −1
=
⎛
⎝
−
1
2
+
√
3
2
+
√
3
2
+
1
2
⎞
⎠
(3.4)
which yields the representation matrix for ˆ
σ 3 , in line with Eq. (3.1).