Finally, some point groups have certain interesting symmetry elements. Any
linear system possesses a Caxis along the system axis. Therefore, there are
Cvand Dhpoint groups that have this element. Some systems have a large
number of symmetry elements that include several highfold principal axes.
Such point groups are called cubic groupsand consist of tetrahedral, octahe-
dral, and similarly named groups. All five of the Platonic solids (tetrahedron,
cube, octahedron, dodecahedron, and icosahedron, as shown in Figure 13.13)
have cubic symmetry. Finally, there is the symmetry group that defines a
sphere, which contains E,arbitrary Cand Saxes, and a center of inversion.
This group is labeled Rh(3)and is called the three-dimensional full rotation
group. It has particular application for atomic wavefunctions, since individual
atoms can be treated as if they were perfect spheres (which take on nonspher-
ical symmetry only when some other influence is imposed: bonding to other
atoms or being surrounded by ions in a crystal, for example). For real objects,
only a finite number of point groups are possible.
13.4 Molecules and Symmetry
All molecules have a structure belonging to one of the recognized point groups.
Although we have used molecules as examples of systems that have symmetry
elements, we have only assumed this idea so far. Most molecules actually pos-
sess very few symmetry elements, and so can be spoken of as having “low”
symmetry. All of them have at least E, and so can be recognized as having an
overall symmetry defined at least by the point group C 1. Many molecules,
especially smaller ones, have more symmetry elements and so are said to
have “higher” symmetry. For example, the molecule CH 4 has the shape of a
tetrahedron, which has a very high cubic symmetry. Whatever the molecule, its
structure can be assigned to a point group on the basis of its symmetry
13.4 Molecules and Symmetry 427TetrahedronOctahedronIcosahedronCubeDodecahedronFigure 13.13 The five Platonic solids—tetrahedron, cube, octahedron, dodecahedron, and
icosahedron. Collectively, these five shapes represent the cubic point groups.