Chemistry Reference
In-Depth Information
Using the result from Equation (6.50) in a comparison of the
b
2
coefficients in
Equation (6.49) now gives
1
2
√
2
+
q
√
2
1
2
0
=
i.e.
q
=−
(6.51)
which is the character required for the second function in the pair derived for the
E
irreducible representation.
The sum of the characters obtained for
1, which agrees with the
entry under the 2
C
3
class of the standard
C
3v
character table (Appendix 12). This, along
with the solution to Problem 6.14, confirms that the two vibrations shown in Figure 6.20b
and c as a pair are consistent with the
E
irreducible representation, at least for the 2
C
3
class.
φ
1
(
E
) and
φ
2
(
E
)is
−
Problem 6.14:
Under the identity class of
C
3
v
modes,
φ
2
(
E
) clearly give a
character of 2. To complete the check that this pair of modes follows the characters of
the
E
irreducible representation, use the above procedure for the 3
φ
1
(
E
) and
σ
v
class. Note: this is
most easily done by using the
σ
v
A
plane (containing H
1
in Figure 6.20) as the example
operation.
6.7 Linking Results for Symmetry-Inequivalent Sets of Atoms
The projection operator results provide SALCs for symmetry-related basis vectors.
Because it depends on the symmetry operations of the point group, the method does not
provide information on the relative motion of symmetry-
inequivalent
atoms. For example,
the basis of four C H bond vectors shown in Figure 6.21 could be used to investigate
the C H stretch modes of the
C
2v
molecule 1,2-difluorobenzene. The four basis vectors
easily split into two subsets (
b
1
with
b
2
and
b
3
with
b
4
) because none of the point-group
operations interchange vectors between these pairs (e.g.
b
1
and
b
3
cannot be swapped by
an operation). Projection of the
b
1
vector would give the two functions already seen with
the simple H
2
O example:
1
√
2
φ
(
A
1
)
=
(
b
1
+
b
2
)
(6.52)
1
√
2
φ
(
B
2
)
=
(
b
1
−
b
2
)
C
2
Z
F
b
1
F
H
1
b
3
b
2
Y
H
3
H
2
b
4
H
4
Figure 6.21
A basis of four C H bond vectors for the C H stretch modes of the C
2v
molecule 1,2-difluorobenzene.
