Chemistry Reference
In-Depth Information
elements of the matrices for the full basis set must sum to the same value as those of the set
of irreducible representations from which it is composed. The value of
n
i
in Equation (4.25)
is set to the number of times we have to use irreducible representation
i
in forming the
reducible representation
.
In Chapter 5 we will exploit this relationship between the reducible and irreducible
representations further and find a general formula for obtaining the
n
i
values that control
the composition of
for any basis and any point group. This chapter finishes with a few
more examples of reducible and irreducible representations.
4.9
Classes of Operations
In the previous sections we have seen how the matrix representation allows us to follow
the transformation of a set of basis vectors under the operations of a point group. We have
also linked the characters that were introduced at the start of this chapter to the diagonal
elements of the matrix for a given operation. A particular basis vector can have characters
ranging from
1, corresponding to it being reversed by the operation or left alone,
and any value in between. Fractional values correspond to the transformed basis function
retaining part of their original form when we think about building the new vector from a
linear combination of the initial basis set.
The character tables listed in Appendix 12 give standard sets of these characters for the
irreducible representations of each point group. We saw in Chapter 3 that the top row of
the character table gives a list of the unique operations in the point group. In many cases
the operation symbol is preceded by a number which gives the number of equivalent oper-
ations of that type. These equivalent sets of operations are referred to as classes of opera-
tions, and now we can see how the same character arises for any operation within a class.
−
1to
+
4.9.1 [Ni(CN)
4
]
2
−
,
D
4h
Table 4.8 shows the standard character table for the
D
4h
point group. The top row of the
character table is the list of the operations valid in this point group. This list always begins
with the identity operator, and in groups like
D
4h
, which contains a rotational subgroup, the
rotational operations are given next. The principal axis is the
C
4
axis and the corresponding
Tab l e 4 . 8
The D
4h
character table.
D
4h
E
2
C
4
C
2
2
C
2
2
C
2
i
2
S
4
σ
h
2
σ
ν
2
σ
d
A
1g
1
1
1
1
1
1
1
1
1
1
x
2
+
y
2
,
z
2
A
2g
1
1
1
−
1
−
1
1
1
1
−
1
−
1
R
z
B
1g
1
−
1
1
1
−
1
1
−
1
1
1
−
1
x
2
−
y
2
B
2g
1
−
1
1
−
1
1
1
−
1
1
−
1
1
xy
E
g
2
0
−
2
0
0
2
0
−
2
0
0
(
R
x
,
R
y
)
(
xz
,
yz
)
A
1u
1
1
1
1
1
−
1
−
1
−
1
−
1
−
1
A
2u
1
1
1
−
1
−
1
−
1
−
1
−
1
1
1
z
B
1u
1
−
1
1
1
−
1
−
1
1
−
1
−
1
1
B
2u
1
−
1
1
−
1
1
−
1
1
−
1
1
−
1
E
u
2
0
−
2
0
0
−
2
0
2
0
0
(
x
,
y
)
