Chemistry Reference
In-Depth Information
Table 7 Permutation-inversion operators of the molecular symmetry group
G
16
of homomerous
BAEs and their corresponding rotation, reflection, inversion and rotation-reflection operators
Reflection, inversion or
rotation-reflection
Permutation
Rotation
Permutation-inversion
E
E
E*
˃
(yz)
(18)(1
0
8
0
)
(18)(1
0
8
0
)*
C
2
(
z
)
˃
(
xz
)
(11
0
)(88
0
)(99
0
)
(11
0
)(88
0
)(99
0
)*
C
2
(
y
)
˃
(
xy
)
(18
0
)(81
0
)(99
0
)
(18
0
)(81
0
)(99
0
)*
C
2
(x)
i
(18)
E
,
Z
-Isomerization
(18)*
˃
(
x
¼
y
)
(1
0
8
0
)
(1
0
8
0
)*
E
,
Z
-Isomerization
˃
(
x
¼
-
y
)
(11
0
88
0
)(99
0
)
(11
0
88
0
)(99
0
)*
S
4
3
(
z
)
—
(18
0
81
0
)(99
0
)
(18
0
81
0
)(99
0
)*
S
4
1
(
z
)
—
(2), the
anti
-folded
C
2h
symmetric conformation a of dixanthylene (4), and the
syn
-
folded
C
2v
symmetric conformation of dixanthylene (4). The
S
4
rotation-reflection
axis in t
⊥
-
D
2d
is the
z
-axis. The center of symmetry,
i
,inp-
D
2h
and a-
C
2h
is at the
origin, where the
y
-axis intersects
C
9
0
). The symmetry operators
are given in permutation-inversion notation, as well as using point group symmetry
operator symbols following the Sch
¨
nflies notation [
282
].
The rotations (
C
n
), reflections (
(
xz
) (and C
9
˃
¼
), inversion (
i
), and rotation-reflections (
S
a
nk
)
corresponding to the permutation-inversion operators of the molecular symmetry
group
G
16
of homomerous overcrowded bistricyclic aromatic enes are listed in
Table
7
.
Some comments may be warranted at this point. It may be surprising that the
inversion operator,
E*
, of the molecular symmetry group does not correspond to the
center of symmetry,
i
. In fact, the treatment of improper rotations, particularly
questions related to the effect of
i
on the molecule-fixed axis system, represents one
area of sustained disagreement in the literature [
247
]. Here, the definitions of
Longuet-Higgins [
246
] and Hougen [
247
] will be followed: “To yield a mathemat-
ically consistent treatment of the symmetry operations, [
˃
] these improper rota-
tions must be associated with the nuclear permutation determined as above
[by comparing the original framework to the framework after the improper rotation]
followed by (or, equivalently, preceded by) an inversion of the coordinates of all
particles in the origin of the laboratory-fixed axis system.” [
247
] For a detailed
mathematical treatment see the earlier work by Hougen [
283
,
284
]. Only the
second, implicit inversion of the axis system allows superposition of the two
frameworks which otherwise would have different handedness. Thus, the inversion
at the center of symmetry,
i
(which is also termed the molecule-fixed inversion
[
247
]), e.g., of a
C
2h
symmetric
anti
-folded conformation, is interchanging atoms
1 and 8
0
, 8 and 1
0
, and 9 and 9
0
, etc., and consequently corresponds to the
permutation-inversion operator (18
0
)(81
0
)(99
0
)* (cf. [
247
]). On the other hand,
E*
(the laboratory-fixed inversion [
247
]) does not permute any atoms and thus should
be identified with the
...
(
yz
) operator of
D
2h
in, e.g., the planar conformation, which
reflects every atom onto itself (cf. [
247
]).
˃
