Chemistry Reference
In-Depth Information
rotation-reflection axes, which are not present in
S
5
. On the other hand, the rota-
tional subgroup
I
is indeed isomorphic to the alternating group
A
5
. The icosahedral
rotations act transitively on the set of five cubes. The icosahedron also contains six-
fold rotation-reflection axes,
S
6
. These symmetry elements act transitively on the
set of the six pentagonal directions. C
60
(Buckminsterfullerene) has perfect
I
h
sym-
metry and corresponds to a truncated icosahedron. Bonds that are adjacent to two
hexagons have pronounced double-bonding character. Metal fragments can coordi-
nate [
6
] to these bonds, as shown in Fig.
3.8
(b). A hexa-adduct is formed with near
T
h
symmetry.
Cylindrical Symmetries
Cylinders, Prisms, Antiprisms
The full symmetry group of an ideal cylinder is denoted by
D
h
.The
D
stands for
dihedral
6
and
h
for
horizontal
. Like the spherical group, the cylindrical symmetry
group is a continuous symmetry group, which has an infinite number of elements. It
contains any rotation or rotation-reflection about the
z
-axis, any
C
2
axis in the equa-
tor and any vertical symmetry plane,
∞
ˆ
σ
v
. In addition there are four singleton classes,
viz. the identity, the
C
2
rotation, the unique horizontal symmetry plane, and spatial
inversion. Cylindrical symmetry is met only in linear molecules such as homonu-
clear diatomics. Obviously, in nonlinear molecules the rotational symmetry of the
cylinder is replaced by a finite cyclic symmetry. Two shapes are realizations of max-
imal finite subgroups of the cylinder: prisms and antiprisms with respective symme-
tries
D
nh
and
D
nd
. In both structures the principal axis is a
C
n
axis, perpendicular
to which there are
n
twofold axes. The horizontal symmetry plane is conserved
only in prisms, and not in antiprisms. In Fig.
3.9
(a), the staggered configuration in
ferrocene exemplifies a pentagonal antiprism, while the eclipsed configuration in
ruthenocene is a pentagonal prism. The presence of inversion symmetry depends
on the parity of
n
. It is present only in
D
2
nh
prisms and
D
(
2
n
+
1
)d
antiprisms. The
cylindrical symmetries reach their lower limit when the principal rotation axis is
twofold. In the
D
2
h
case, the equatorial directions are no longer equivalent, and we
have a rectangular parallelepiped that is an
orthorhombic
structure with three dif-
ferent and mutually perpendicular directions. By contrast, in the twofold antiprism,
with symmetry
D
2
d
, we have a scalenohedron that has two directions perpendicular
to the
z
-axis, which are equivalent. In fact, the highest symmetry element in this
case is an
S
4
rotation-reflection axis, which is responsible for the equivalence of the
6
Dihedral means literally “having two planes.” The dihedral angle is an important molecular de-
scriptor. The dihedral angle of the central B
−
C bond in an A
−
B
−
C
−
D chain is the angle between
the ABC and BCD faces. In the present context, the term dihedral originates from crystallography,
such as when two plane faces meet in an apex of a crystal.
