Chemistry Reference
In-Depth Information
4.2.2 The Molecular Symmetry Group of Homomerous BAEs
The molecular symmetry group
G
16
of homomerous BAEs consists of following
permutation-inversion operators:
2. (18)(1
0
8
0
)
3. (11
0
)(88
0
)(99
0
)
4. (18
0
)(81
0
)(99
0
)
1. E
6. (1
0
8
0
)
7. (11
0
88
0
)(99
0
)
8. (18
0
81
0
)(99
0
)
5. (18)
10. (18)(1
0
8
0
)*
11. (11
0
)(88
0
)(99
0
)*
12. (18
0
)(81
0
)(99
0
)*
9. E*
14. (1
0
8
0
)*
15. (11
0
88
0
)(99
0
)*
16. (18
0
81
0
)(99
0
)*
13. (18)*
A multiplication table and a character table [
246
] of the permutation-inversion
operators of the molecular symmetry group
G
16
are given in Tables
5
and
6
,
respectively. The molecular symmetry group
G
16
of BAEs is isomorphous to
D
4h
(cf. Sect.
4.1.3
and Fig.
13
). Each of the above permutation-inversion operators may
be mapped onto a symmetry operator of
D
4h
with identical character such that
corresponding multiplication tables and character tables result. (For a multiplica-
tion table and character table of
D
4h
see, e.g., [
279
]). Note that there is no
D
4h
symmetric conformation of a BAE. In particular, there is no
C
4
axis in any
conformation. It should be noted that the molecular symmetry group includes two
operators corresponding to the
E
,
Z
-isomerization: (18) and (1
0
8
0
). They correspond
to rotating the first or the second tricyclic moiety about C
9
C
9
0
, respectively. This
feasible operation is included in the molecular symmetry group
G
16
in a consistent
and natural way.
The permutation-inversion operators of the molecular symmetry group of BAEs
were derived from topological considerations and a criterion of feasibility defined
by the axiom that no bonds may be broken. Alternatively, the molecular symmetry
group may be derived by analyzing the effect of the point group symmetry
operators on the labeled atoms and the handedness of the molecular framework,
thus correlating symmetry operators with permutation-inversion operators. All
operators of the point group(s) of a molecule in its different conformations are
combined and operators for known feasible processes (e.g., internal rotations of a
methyl group, etc.) are added to give the molecular symmetry group [
246
,
247
,
253
]. In both cases a multiplication table should be generated in order to verify that
the proposed set of operators is a group [
279
].
ΒΌ
4.2.3 Permutation-Inversion Operators and Corresponding Symmetry
Operators of BAEs
For visualization, it may be helpful to define a molecule-fixed coordinate system
orienting the molecule in a consistent way in all conformations and to point out the
conventional symmetry elements corresponding to the permutation-inversion oper-
ators, where appropriate. It should be noted, that the definition of a molecule-fixed
coordinate system is not required within the formalism of the molecular symmetry
group. Some permutation-inversion operators have no corresponding point group
