Chemistry Reference
In-Depth Information
These do not involve
b
3
or
b
4
because these basis vectors are not symmetry related to
b
1
.
Another two functions can be obtained by projection of, say,
b
3
,togive
1
√
2
φ
(
A
1
)
=
(
b
3
+
b
4
)
(6.53)
1
√
2
φ
(
B
2
)
=
(
b
3
−
b
4
)
To get the overall picture of the vibrations we need to link together the modes for the two
sets of H atoms. This has to be done so that only modes of like symmetry are combined.
This ensures that in, say, an
A
1
mode for the molecule as a whole the movement of each
subset of atoms must conform to the same irreducible representation.
However, the subsets may move in phase with one another or out of phase with one
another. So, adding superscripts to indicate the subsets, the two pairs of basis functions
give four possible results:
1
√
2
φ
1
(
A
1
)
=
φ
12
(
A
1
)
+
φ
34
(
A
1
)
=
(
b
1
+
b
2
+
b
3
+
b
4
)
1
√
2
φ
2
(
A
1
)
=
φ
−
φ
=
(
b
1
+
b
2
−
b
3
−
12
(
A
1
)
34
(
A
1
)
b
4
)
(6.54)
1
√
2
(
b
1
−
φ
1
(
B
1
)
=
φ
12
(
B
1
)
+
φ
34
(
B
1
)
=
b
2
+
b
3
−
b
4
)
1
√
2
φ
2
(
B
1
)
=
φ
12
(
B
1
)
−
φ
34
(
B
1
)
=
(
b
1
−
b
2
−
b
3
+
b
4
)
Note that from a basis of four bond vectors we have again arrived at four vibrational
modes. The addition of two normalized functions in Equation (6.54) gives new functions
that are no longer of unit magnitude. However, it is straightforward to use the procedure
of Section 6.5 to renormalize, i.e. scale by a factor which restores the magnitude to one.
The renormalized vibrations are given in Figure 6.22, along with illustrations of the
modes. In contrast to 1,4-difluorobenzene, there is no cancellation of opposing dipoles for
any of the modes; so, in line with the selection rules, all are IR active. The earlier analysis
using the right-hand column of the character table also indicated that all four will also be
Raman active.
Problem 6.15:
Show, using the projection operator method, that combinations of the
mode using different symmetries such as
12
(
A
1
)
34
(
B
2
)
φ
+
φ
(6.55)
are inconsistent with either irreducible representation.
Hint
: write out the function explicitly in terms of the
b
vectors and tabulate the
projection operator results for each irreducible representation.
6.7.1
Sets of Atoms Differing in Mass or Chemical Bond Strength
The sets of inequivalent C H bonds in the above example are of very similar chemical
character, and so it is a good approximation to renormalize the bond vibrations to give
