Chemistry Reference
In-Depth Information
vectors to visit any of the three positions in the original configuration of the molecule. This
means that the three basis vectors are indistinguishable; anything we can say about
b
1
will
also be true of
b
2
and
b
3
. Also, if we use one N H bond in a basis then we must include
all three; a basis of just
b
2
and
b
3
would involve transformations that would require us to
include
b
1
also.
We cannot tell the three N H bonds apart in the molecule, and so observables to do
with the individual bonds or H atoms, such as the proton NMR shifts, will be identical.
However, we show below that the vibrations due to N
H stretching motions need not
form three degenerate modes.
Problem 4.12:
Using the basis of N
H bonds defined in Figure 4.6, show that the
reducible representation for the three N
H bond vectors
b
1
,
b
2
and
b
3
is as shown in
Table 4.12.
Table 4.12
The reducible
representation for the basis
of N H bonds defined for
the C
3v
molecule ammonia
in Figure 4.6.
C
3v
E
2
C
3
3
σ
v
3
0
1
There is no need to use full matrix expressions for the operators in this case. Consider
an example operation from each class and arrive at the character using the rules:
1. if a basis vector is unchanged, add 1;
2. if a basis vector is swapped with another member of the basis, add 0.
In this case, no basis vectors which stay in position are ever reversed.
The reducible representation for the three N H bonds is shown in Table 4.12. The
irreducible representations from which this is composed must be from the standard set of
Table 4.11. To obtain the correct match we can use the property given by Equation (4.25):
that for each class of operations the appropriate standard irreducible characters sum to give
that of the reducible representation.
In relatively simple cases, such as the three N H bonds in ammonia, we can deduce
the values of
n
i
that govern how many of each irreducible representation are present by
inspection:
1. Under the identity operation
E
,
has a total character 3. There are several combinations
of the irreducible representations that could give this character; for example:
3
A
1
or 3
A
2
or
A
2
+
...
E
(4.32)
However, the sum must work with the same values of
n
i
for all classes of operation.
