Chemistry Reference
In-Depth Information
Fig. 3.8
(
a
) Dodecahedron, the square is the top face of an inscribed cube; (
b
) supramolecu-
lar complex of C
60
Buckminsterfullerene with six
η
2
-platina-bis(triphenyl)phosphine adducts on
double bonds adjacent to two hexagons; the adducts adopt a
D
2
h
setting and reduce the icosahedral
symmetry of the buckyball to
T
h
neighboring axis, one always rotates the edges by 90
◦
. In fact, there are two ways to
realize this
D
2
h
setting either with the top edge in the
xz
plane, as in Fig.
3.7
(b), or
in the
yz
plane [
5
]. The reason is that the
O
h
symmetry of the Cartesian frame is not
a subgroup of
I
h
since
. Their highest common subgroup
is the 24-element group
T
h
, as indicated in Fig.
3.5
. The structure of the six edges in
Fig.
3.7
(b) does indeed have
T
h
symmetry. Note that
T
h
is a normal subgroup of
O
h
,
but not of
I
h
. The dual of the icosahedron is the dodecahedron, shown in Fig.
3.8
(a).
In a dodecahedron, one can select a set of eight vertices that form the corners of a
cube. The dashed square in the figure is the top face of such an inscribed cube. Eu-
clid already demonstrated that a cube could be drawn inside a dodecahedron in such
a way that its twelve edges each lie in one of the twelve faces of the dodecahedron.
In turn, in the center of each of the faces of this cube there lies a dodecahedral edge,
which breaks the fourfold symmetry axes of the cube. The stabilizer of the cube in-
side
I
h
is the subgroup
T
h
. The icosahedral group can be partitioned into five cosets
of
T
h
, which can be generated in a cyclic orbit by a pentagonal generator:
|
O
h
|
is not a divisor of
|
I
h
|
5
( C
5
)
n
T
h
I
h
=
(3.33)
n
=
1
Each of the cosets refers to an inscribed cube, exactly as the three
C
s
cosets in
C
3
v
ammonia refer to the three equivalent hydrogen sites. Hence, in a dodecahe-
dron there will be five inscribed cubes. Since the order of the
I
h
group is 120,
which equals 5
, it is tempting to think of this group as isomorphic to
S
5
, permuting
the five inscribed cubes. This, however, is not the case since
I
h
contains 10-fold
!
