Chemistry Reference
In-Depth Information
[
Fe
(
CO
)
5
]
,D
3
h
The statement of the selection rules given above also applies to degenerate modes of vibra-
tion. For example in Section 5.7 we show that the C
O vibrational modes of the
D
3
h
complex [Fe(CO)
5
] have the irreducible representations:
2
A
1
+
E
+
A
2
(6.7)
Inspection of the character table in Appendix 12 shows that the simplified selection rule
gives the
A
2
mode as IR active for a transition dipole moment in the
Z
-direction and the
E
doubly degenerate modes as both active: one for a transition dipole moment along
X
and
the other along
Y
.Thetwo
A
1
modes would not give rise to absorption and so would not
be seen as bands in the IR spectrum.
To check that the
E
representation conforms to the earlier statement of the selection
rule we need to form the products for the integrands:
E
×
E
×
ψ
1
μ
x
ψ
0
,
ψ
1
μ
y
ψ
0
⇒
A
1
E
×
A
2
×
ψ
1
μ
z
ψ
0
⇒
A
1
(6.8)
We have grouped the
μ
y
cases together because they form a degenerate pair within
the
E
representation. The first direct product is set out in Table 6.2. Under the identity
operator we have generated the character 4; since no irreducible representation contains 4
under this column, this product must be reducible. In Table 6.3, the reduction formula is
applied in the normal way to obtain
μ
x
and
E
×
E
=
A
1
+
A
2
+
E
(6.9)
Tab l e 6 . 2
The direct product of E
with itself in point group D
3h
.
μ
x
,
μ
y
(
D
3h
)
E
2
C
3
3
C
2
σ
h
2
S
3
3
σ
v
E
2
−
1
0
2
−
1
0
E
×
E
=
4
1
0
4
1
0
Tab l e 6 . 3
Application of the reduction formula to the E
×
E
direct
product from D
3h
.
μ
x
,
μ
y
(
D
3h
)
E
2
C
3
3
C
2
σ
h
2
S
3
3
σ
v
h
=
12
(
E
×
E
)
4
1
0
4
1
0
C
h
−1
C
g
c
χ
i
(
C
)
χ
(
C
)
A
1
4
2
4
2
12
1
A
2
4
2
4
2
12
1
E
8
−
2
8
−
2
12
1
A
1
4
2
−
4
−
2
0
0
A
2
4
2
−
4
−
2
0
0
E
8
−
2
−
8
2
0
0
