Chemistry Reference
In-Depth Information
The two permutation operators for the
E
,
Z
-isomerization, (18) and (1
0
8
0
), mov-
ing the first and the second moiety, respectively, do not correspond to any symme-
try element in any point group of the BAEs. This is due to the fact that this process
converts one form of the molecule (
E
) into another form (
Z
) and thus is a (feasible)
chemical reaction interconverting stereoisomers (or automers) rather than a point
group symmetry operation. An advantage of the molecular symmetry group for-
malism is that it allows inclusion of such processes, which may be rapid under
certain experimental conditions.
The permutation-inversion operators (18)*, (1
0
8
0
)*, (11
0
88
0
)(99
0
)*, and (18
0
81
0
)
(99
0
)* correspond to the reflection operators
˃
d
with the plane
x
¼
y
,
˃
d
0
with the
y
, and the rotation-reflection operators
S
4
3
(
z
) and
S
4
1
symmetry plane
x
¼
(
z
),
respectively. They are unique for
the
D
2d
symmetric orthogonally twisted
conformation.
The permutation operators (11
0
88
0
)(99
0
) and (18
0
81
0
)(99
0
) have no corresponding
symmetry operator in any point group of any conformation. They perform a forward
and backward cyclic permutation of the positions 1, 1
0
, 8, and 8
0
without inversion,
and correspond to the
C
4
3
and
C
4
1
rotation operators in the isomorphous point group
D
4h
. They are generated as products of two permutation operators or two
permutation-inversion operators as may be seen from the multiplication table
(Table
5
). Therefore, they are members of the molecular symmetry group due to
the basic postulate of 'closure', i.e., the postulate of group theory that for any two
operators that are members of a group, their product must also be a member of this
group [
279
].
It may be interesting to note that the permutation operators (18) and (1
0
8
0
)
corresponding to
E
,
Z
-isomerizations are the product of 'normal' symmetry opera-
tors, e.g., (18)
(18
0
)(81
0
)(99
0
)*. Where (11
0
88
0
)(99
0
)* corre-
sponds to
S
4
3
a symmetry operator characteristic of the
D
2d
orthogonally twisted
conformation, and (18
0
)(81
0
)(99
0
)* corresponds to
i
in, e.g., the
C
2h
(
y
)
anti
-folded
conformation. Thus, the fact that both an
anti
-folded conformation and an orthog-
onally twisted conformation are accessible to the molecule leads to the prediction
that
E
,
Z
-isomerizations are feasible, based on the group theoretical postulate of
'closure.' However,
E
,
Z
-isomerization of an
anti
-folded BAE does not necessarily
require (and thus prove) the existence of a
D
2d
symmetric orthogonally twisted
conformation as transition state or intermediate. The molecular symmetry group
will include the operators (18)*, (1
0
8
0
)*, (11
0
88
0
)(99
0
)*, and (18
0
81
0
)(99
0
)*, but these
permutation-inversion operators may simply be the product of the
E
,
Z
-operators
and some other operator and may not correspond to any symmetry operator in any
accessible conformation [
246
]. (For an illustrative example see the dynamic ste-
reochemistry of tetrabenzo[7,7
0
]fulvalene (8) described in Sect.
5
of [
3
].)
Likewise, accessibility of a planar
D
2h
conformation for any BAE with twisted
(
D
2
),
anti
-folded (
C
2h
(
y
)), or
syn
-folded (
C
2v
(
x
)) global minimum conformation,
leads to the prediction of feasible enantiomerizations or conformational inversions,
based on the fact that
E
* will be an element of the molecular symmetry group. By
the same reasoning, group theory predicts feasible enantiomerizations or inversions
if conformations with any two of the three point groups
D
2
,
C
2h
(
y
), and
C
2v
(
x
) are
(11
0
88
0
)(99
0
)*
¼
