Chemistry Reference
In-Depth Information
Any collection of basis vectors that complies with the molecular symmetry can generate a
character representation of the group, but in most cases it will be a reducible one and so
can be simplified. In this section we will show that the simplification of a reducible repre-
sentation
can be made using the data for the set of irreducible representations available
in the standard character tables.
Section 4.11 used the matrix representation to deal with a set of three basis vectors
x
,
y
,
z
on the central atom of a square planar
D
4h
complex. It was shown that this basis can be
reduced to
E
u
+
A
2u
by inspection of the matrices for the operations in the
D
4h
group. The
characters for the reducible and irreducible representations are shown in Table 5.2.
Tab l e 5 . 2
The reducible representation for the x, y, z basis on a central atom
in a D
4h
complex, and its composite irreducible representations.
D
4h
E
2
C
4
C
2
2
C
2
2
C
2
i
2
S
4
σ
h
2
σ
2
σ
v
d
31
−
1
−
1
−
1
−
3
−
1111
x
,
y
,
z
A
2u
11 1
−
1
−
1
−
1
−
1
−
1
1
1
z
E
u
20
−
200
−
20200
x
,
y
×
×
×
1 does not affect any of the
diagonal elements, which are the characters of our irreducible representations. This means
that the irreducible characters in each class must add up to the character in the reducible
representation they were derived from.
By inspection of Table 5.2, it can be seen that this is indeed the case. For any class
of operations in the group, the characters of the irreducible representations sum to give
that of
The breakdown of the 3
3 matrices into 2
2 and 1
.
Writing characters from the irreducible representations in class
C
as
χ
E
u
(
C
) and
χ
A
2u
(
C
)
and that for the reducible representation as
χ
(
C
)wehave
χ
(
C
)
=
χ
E
u
(
C
)
+
χ
A
2u
(
C
)
(5.14)
This is a special case for the basis used. In general, any of the irreducible representations
available in
D
4h
could have been present, and the sum for
χ
(
C
)is
χ
(
C
)
=
n
j
χ
j
(
C
)
(5.15)
j
where
n
j
is the number of times that the
j
th irreducible representation occurs. These
n
j
values may be 0; from Table 5.2 in our
D
4h
example, we expect all values to be 0 except
those for the
E
u
and
A
2u
representations, which will each be 1. In this case, Equation (5.15)
would become Equation (5.14).
So far, this is just a recap of the discussion leading to Equation (4.25). Now, Equa-
tion (5.15) will be used along with Properties 4 and 5 to obtain an expression for the set of
n
j
for the general case. This makes it possible to write down the make-up of any reducible
representation in terms of the irreducible ones.
