Chemistry Reference
In-Depth Information
is equivalent to a 90
◦
rotation anticlockwise (
C
4
3
). However, as noted earlier, the
C
4
2
operation is listed separately, as the first
C
2
rotation. The corresponding matrix is
⎛
⎝
⎞
⎠
−
100
0
−
C
2
=
10
001
(4.27)
which means that the 180
◦
rotation reverses
x
and
y
but leaves
z
unchanged and so has a
total character of
1 for this basis.
As a further example, we can see from Table 4.8 that the two vertical reflection planes
in
D
4h
are in the same class. Figure 4.11 shows the two
−
σ
v
planes, which are labelled
σ
v
A
and
σ
v
B
; from the diagram, we can write down the operation matrices as
⎛
⎞
⎛
⎞
100
0
−
100
010
001
⎝
⎠
⎝
⎠
σ
v
A
σ
v
B
=
10
001
−
and
=
(4.28)
In this case, either
x
or
y
is reversed by the reflection and the other two vectors are left
unchanged, so the trace of the matrix is 1 in both cases. This allows the two mirror planes
to be placed in the same class and gives the 2
σ
v
heading in the point group table.
Z
(a)
Z
σ
v
A
z'
z
σ
v
A
y'
y
x
x'
Y
Y
X
X
(b)
Z
Z
σ
v
B
z'
z
σ
v
B
x'
x
y
y'
Y
Y
X
X
Figure 4.11
The D
4h
complex [Ni(CN)
4
]
2−
, showing a basis of x, y, z functions representative
of the p-orbitals on Ni
2+
. The effects on the basis of (a) the
σ
v
A
operation and (b) the
σ
v
B
operation are illustrated.
Problem 4.9:
Draw sketches similar to Figures 4.10 and 4.11 showing the symmetry
elements for each of the remaining
D
4h
operations listed as classes in Table 4.8. From
your sketches, write down a matrix representation for each operation and then from the
matrices:
1. Confirm that each of the 2
C
2
,2
C
2
,2
S
4
,2
σ
v
and 2
σ
d
classes contain two operations
which in each case have the same character.
2. Show that the traces of the set of matrices give the character set shown in Table 4.9.
