Chemistry Reference
In-Depth Information
Fig. 3.6
The tetrahedral and octahedral symmetry groups: (
a
) tetrahedron, inscribed in a cube by
connecting four alternating corners; (
b
) orientation of the cube based on fourfold axes along the
Cartesian directions;
C
x
2
lies along the acute bisector of the
x
and
y
directions; the dashed triangle
is one face of the inscribed tetrahedron; (
c
) construction drawing of the octahedron in the same
fourfold setting; the
C
2
axis shown is the bisector of the positive
x
and negative
y
direction; and
(
d
) same drawing of the octahedron with a trigonal coordinate orientation: the
z
direction is along
the
C
3
axis, and the
x
-direction is along the bisector of the positive
x
- and negative
y
-directions
order as
T
d
and also contains
T
as its rotational subgroup. Molecular examples of
this group are quite rare and will mostly be encountered as symmetry lowering of
cubic or icosahedral molecules (see later, Fig.
3.8
).
The Cube and Octahedron
The group
O
h
contains 48 elements and is the symmetry group of the octahedron
and the cube (see Fig.
3.6
(b)). The system of Cartesian axes itself has octahedral
symmetry, and, as such, this symmetry group is the natural representative of 3D
space. It is ubiquitous in ionic crystals, where it corresponds to coordination num-
bers 6 (as in rock salt, NaCl) or 8 (as in caesium chloride, CsCl). It is also the
dominant symmetry group of coordination compounds. The rotational subgroup is
the group
O
with 24 elements. The octahedron provides us with an insight into the
basic architecture of polyhedra. There are three structural elements: vertices, edges,
and faces. Through each of these runs a rotational symmetry axis: a
C
4
axis through
the vertices, a
C
2
axis through the edge, and a
C
3
axis through the face center. In
