Chemistry Reference
In-Depth Information
Fig. 3.7
(
a
) Icosahedron in a
D
2
h
setting, (
b
)
T
h
structure based on the icosahedral edges through
the Cartesian coordinate axes
a triangular face, the product of the three rotations equals the unit element. For the
triangle that is turned toward the viewer in Fig.
3.6
(c), this is
C
x
2
C
xyz
C
4
=
E
(3.32)
3
The symmetry elements are labeled by the indices
x
,
y
, and
z
, which refer to their
orientation in the Cartesian coordinate system, e.g.,
C
xy
3
indicates the
C
3
axis,
which is the diagonal of the positive Cartesian directions. This notation emphasizes
that symmetry elements are tied to the coordinate system and stay fixed in space.
The Icosahedron and Dodecahedron
Icosahedral symmetry is less obvious and thus more intriguing than cubic sym-
metry. Whilst tetrahedral and octahedral molecules were already known before the
turn of the nineteenth/twentieth century, the first structural study of an icosahedral
molecule was the closo-dodecaborane,
B
12
H
2
−
12
, in 1960. More examples would
follow: in 1984, dodecahedrane,
C
20
H
20
, and then the supermolecule Buckminster-
fullerene, C
60
, which has the shape of a truncated icosahedron [
2
-
4
]. Icosahedral
structures can be drawn inside a regular hexagon, as shown in Fig.
3.7
(a). The co-
ordinate axes in this figure are chosen in such a way that each of the Cartesian
directions coincides with a twofold axis. These axes together form a
D
2
h
subgroup.
This orientation thus corresponds to a
D
2
h
setting. Figure
3.7
(b) shows the orienta-
tion of the edges through the Cartesian directions. When going from one axis to a
