Chemistry Reference
In-Depth Information
of the character table data requires the infinite number of operations to be noted, and this
is done by adding lines of dots to the title line to indicate that a sequence of related oper-
ations is present. The notation
C
∞
is used for the rotation operations about the infinite
axis, where
indicates the angle of rotation for the operation. There are two operations
for each angle, since clockwise and anticlockwise rotations belong to the same class.
Linear molecules are rather straightforward to visualize and so we will not provide illus-
trative examples here. Diatomic molecules, such as O
2
,Cl
2
and N
2
, provide good examples
of molecules in the
D
∞
h
point group, alongside the linear molecules ethylene, C
2
H
2
and
CO
2
. Diatomics formed from two different elements, such as CO, NO and HF, belong to
the
C
∞
v
point group, along with more complex linear molecules such as hydrogen cyanide
(HCN).
3.9 The Cubic Groups:
T
d
and
O
h
The point groups covered so far, with the exception of the simple
C
1
,
C
s
and
C
i
cases,
have had a principal axis which is straightforward to define and no other axes of order
higher than 2. Those point groups describe well the wide range of molecules that can be
considered to be axial or planar. In this section we turn to cases that are often regarded as
higher symmetry objects in which there are multiple axes of order higher than 2. This set
of related groups is best thought of as the symmetry groups for simple solid shapes. We
will deal with two common cases of tetrahedral and octahedral molecules and show how
these relate to the symmetry of the cube. There are some molecules of very high symmetry,
such as buckminsterfullerene (C
60
) in which all 60 C atoms are symmetry equivalent, but
the consideration of such cases is beyond the scope of this introductory text.
The tetrahedron is an important shape in chemistry. Methane, the complex [Ni(CO)
4
]
and many other molecules in which a central atom has four equivalent bonds take on this
geometry. The symmetry of the tetrahedron is best discussed with reference to a solid
model, and a paper template is provided in Appendix 3 from which a tetrahedron and the
related cube can be constructed. This paper model also has some representative symmetry
elements drawn on it which are also illustrated in Figure 3.25. The highest order axes are
C
3
which join each corner of the tetrahedron to the centre of the opposite face. There are
four corners, and so there are four axes; each gives rise to two operations (
C
3
1
and
C
3
2
),
and so 8
C
3
appears in the title line of the character table (Figure 3.26a).
Figure 3.25b shows that there are also
C
2
axes which run through the centres of opposite
edges of the tetrahedron; for methane, these axes bisect two of the H C H angles. The
tetrahedron has six edges and so there are 3
C
2
operations. The lines of the
C
2
axes are
shared by
S
4
axes; these are best seen by looking down an axis direction, as shown in
Figure 3.25c, in which the mirror plane used in the improper rotation is also given. The
S
4
2
operations are identical to the
C
2
rotations we have already identified, and so only
S
4
1
and
S
4
3
are counted at the head of the character table; with three axes this shows 6
S
4
. Finally,
there are six mirror planes, each of which contains an edge of the tetrahedron and bisects
the opposite edge, as shown in Figure 3.25d. This means that each plane contains two of
the
C
3
and one of the
C
2
axes. Since the
C
3
is the highest order axis, we would expect it to
define vertical; however, there is now no clear choice between the
C
3
axes to be made, as
they are all equivalent. This influences the designation of the six equivalent mirror planes,
