Chemistry Reference
In-Depth Information
then
Gz
. The corresponding
Gx
and
Gy
may then be found by cyclic permuta-
tion under the
C
xy
3
axis.
For the
H
irrep, one may make use of the
C
xyz
3
axis again. It resolves
H
Hζ
.We
can project the
Hζ
component out of this sum by using the
C
2
axis. Although
the
H
level subduces three totally symmetric irreps in
C
2
, there will be no
contamination with
Hθ
and
H
since these were already removed in the first
step by projecting out the trigonal
A
1
.
5.3 The total number of nuclear permutations and permutation-inversions for
CH
3
BF
2
is 24. This is the product of six permutations of the protons, two
permutations of the fluorine nuclei, and the binary group of the spatial inver-
sion. However, as the fluxionality of this molecule is limited to free rotations
of the methyl group, the operations should be limited to those permutations
or permutation-inversions that lead to structures that can be rotated back to
the original frame
or to a rotamer of this frame
. Only half of the operations
will comply with this requirement. As an example, the odd permutations of the
protons are not allowed since the resulting structure cannot be turned into the
original one by outer rotations or by rotations of the methyl group around the
C-B bond. The results are given in [
15
]. The corresponding symmetry group is
isomorphic with
D
3
h
.
5.4 Ferrocene is a molecule with two identical coaxial rotors. Its nuclear permu-
tation-inversion group consists of 100 elements. It has a halving rotational sub-
group of 50 proper permutations: for each of the cyclo-pentadienyl rings, there
are 5 cyclic permutation operations, yielding a total of 5
2
into
A
1
+
2
E
. This unique
A
1
component will be the sum
Hξ
+
Hη
+
25 operations, and
this number must be doubled to account for the permutation of the upper and
lower rings. In addition, there is a coset of improper permutation-inversions
containing the other 50 elements. This coset also contains two kinds of ele-
ments. In the table we summarize the structure of the group. The carbon atoms
are numbered 1
,...,
5 in the upper ring and 6
,...,
10 in the lower ring.
=
Nuclear permutation-inversion group for ferrocene (
u
and
l
refer to upper
and lower rotor)
R
#
C
5
×
C
5
(
12345
)
25
(ul)
(
16
)(
2
,
10
)(
39
)(
48
)(
57
)
25
σ
v
×
σ
v
(
25
)(
34
)(
7
,
10
)(
89
)
∗
5
ˆ
5
ˆ
25
(ul)
∗
(
16
)(
27
)(
38
)(
49
)(
5
,
10
)
∗
25
6.1 The
(t
1
u
)
2
configuration gives rise to 15 states. The direct product decomposes
as follows (see Appendix
D
):
T
1
u
×
T
1
u
=[
A
1
g
+
E
g
+
T
2
g
]+{
T
1
g
}
The symmetrized part will give rise to six singlet functions, while there are
nine triplet substates, forming a
3
T
1
g
multiplet. Since the 3-electron
Ψ
state is a
