Chemistry Reference
In-Depth Information
but is reversed by either
σ
v
. The new set of characters is given the label
A
2
, and this
completes the standard set of representations for the
C
2v
point group.
The use of a character by character multiplication of two representations to obtain the
characters for a new function is referred to as a
direct product
. In this example we have
found that the representation for the d
xy
function is the direct product of the representations
for p
x
and p
y
. This is another property of groups that we will make use of in future chapters;
in general, the direct product of any two representations in a group can be expressed as a
sum of representations from the group. In this case we have found that
σ
v
or
B
1
×
B
2
=
A
2
(4.1)
so the resulting summation requires only a single term,
A
2
. We will find in Chapter 6 that
direct products of this type are useful in calculating the properties of integrals required
by spectroscopic selection rules. In that chapter we will meet more complex examples of
direct products.
Problem 4.2:
The O(p
z
) orbital in the H
2
O example has a representation
A
1
. Show that
this implies that the O(d
xz
) and O(d
yz
) orbitals will have the same representation as the
O(p
x
) and O(p
y
) functions respectively. Check your solutions by drawing diagrams for
the result of each
C
2v
operation with the O(d
xz
) and O(d
yz
) orbitals.
Problem 4.3:
By considering the direct products show that:
1. The direct product of
A
1
with any representation in
C
2v
simply gives the representa-
tion back again.
2. The direct products
A
2
×
B
1
=
B
2
and
A
2
×
B
2
=
B
1
are correct in
C
2v
.
We will look more closely at the origin of the representation labels in Section 4.11, but
first we have to consider the types of character that can occur and their relationship to the
more complete matrix representations for point group operations.
4.3
Multiplication Tables for Character Representations
To be valid representations, the sets of numbers we have produced for the
A
1
,
A
2
,
B
1
and
B
2
representations must each act in the same way as the symmetry operations themselves. For
example, we found in Chapter 2 that
C
2
σ
v
=
σ
v
for the
C
2v
point group of H
2
O. In the
A
1
representation, both
C
2
and
σ
v
have character 1 and so the character product is 1, which
is also the character of
σ
v
. This must hold for the whole multiplication table between the
symmetry operators. For
A
1
, the table is quite trivial, since it contains only 1s. However,
the other representations must also give consistent multiplication tables.
The
C
2v
group multiplication table was constructed from the group operations in
Problem 2.1 and the result is reproduced in Table 4.4a. The same table using the
B
1
rep-
resentation characters is given in Table 4.4b. In this representation version, the operation
symbols for the row and column headings are replaced by the corresponding
B
1
characters.
The result of a product of operations then becomes a simple numerical multiplication of
