Chemistry Reference
In-Depth Information
The proof illustrates the connection between cosets and conjugacy classes. A spe-
cial example of this arises in the case of
normal
,or
invariant
, subgroups. A subgroup
H
is normal if its left and right cosets coincide, i.e., if
R
i
H
H R
i
. This implies
that all the elements of the group will map the subgroup onto itself or, for a normal
subgroup
H
,
=
∀
U
h
x
∈
:
U h
x
U
−
1
∈
G
&
H
∈
H
(3.31)
Normal subgroups are thus made up of entire conjugacy classes of
G
. As an exam-
ple, the group
C
3
is a normal subgroup of
C
3
v
since it contains the whole conjugacy
class of the trigonal elements. A molecular site that is stabilized by a normal sub-
group must be unique since it can be mapped only onto itself. Such is the case for
the nitrogen atom in ammonia. By contrast, the
C
s
subgroups stabilizing the hy-
drogen sites in ammonia are not normal since they are based on only one reflection
plane, while there are three symmetry planes in the corresponding conjugacy class.
Accordingly, the hydrogen sites are not unrelated.
3.7 Overview of the Point Groups
The highest point group symmetry is that of the spherical symmetry group. The
gradual descent in symmetry from the sphere provides a practical tool to determine
and classify molecular symmetry groups. Molecules with a symmetry that is closest
to the sphere are
isotropic
, in the sense that there is no unique direction to orient
them. The corresponding symmetry groups are the cubic and icosahedral groups.
Breaking spherical symmetry gives rise to symmetry groups based on the cylin-
der. Cylindrical symmetry splits 3D space into an axial 1D component and an equa-
torial 2D space, which remains isotropic. Molecules with cylindrical shapes have a
unique anisotropy axis, along which they may be oriented in space. Conventionally
this direction is denoted as the
z
-direction. Along this direction they have a principal
n
-fold rotation or rotation-reflection axis, which is responsible for the remaining in-
plane isotropy. Finally, further removal of all
C
n
or
S
n
with
n>
2 leads to molecules
that are completely anisotropic and have at most orthorhombic symmetry,
D
2
h
.In
this section we will provide a concise overview of the point groups, following a path
of descent in symmetry. We thereby refer to the list of point groups presented in the
character tables in Appendix
A
.
Spherical Symmetry and the Platonic Solids
Any plane through the center of a sphere is a reflection plane, and any axis through
the center is a rotation axis, as well as a rotation-reflection axis. In addition, the
sphere also is centrosymmetric, which means that the center is a point of inversion.
The resulting infinite-dimensional symmetry group of the sphere is usually denoted
