Essentials of Computational Chemistry

SYMMETRY AND GROUP THEORY 559
partner related by inversion through the origin, and the inversion operation itself is usually
denoted i. An example of a molecule containing an S2 axis is the chair conformation of
2,5-dioxo-1,4-tetrahydropyran (Figure B.1). Lastly, note that the presence of higher order
improper axes implies the simultaneous presence of one or more proper rotation axes. In
particular, improper axes Sn where n is odd imply a coincident Cn axis and n perpendicular
C2 axes, and improper axes Sn where n is even imply a coincident Cn/2 axis.
Pointofinversion. The action of a point of inversion is described above in the context of
improper rotation axes. Note that planes of symmetry and points of inversion are somewhat
redundant symmetry elements, since they are already implicit in improper rotation axes.
However, they are somewhat more intuitive as separate phenomena than are Sn axes, and
thus most texts treat them separately.
B.2 Molecular Point Groups and Irreducible Representations
An individual molecular structure may possess no symmetry elements at all, or a single
symmetry element, or some combination of multiple symmetry elements. It turns out that
there are a finite number of possible combinations, and each such combination defines what
is referred to as a point group. The names of the various molecular point groups together
with a flow chart indicating how to assign a molecule to a point group are provided in
Figure B.2. [In crystallography, solids can be characterized by space groups, which are
analogous to point groups but more numerous as additional symmetry elements relating
different molecules in the crystal must also be considered. No further discussion of space
groups is provided here.]
There is a special algebra associated with the different point groups, and the mathematical
field of group theory is devoted to this topic. Group theory is a fascinating topic, but only
its most basic aspects are addressed here. To begin, all point groups other than the nonsymmetric
C1 group are characterized by two or more so-called irreducible representations,
or irreps for short. Operationally, an irreducible representation defines how a signed or
phased fragment (e.g., an orbital) of the symmetric structure ‘transforms’ under the various
possible symmetry operations that compose the point group.
For example, the Cs point group contains two irreps, usually called a ′ and a ′′ .Fragments
belonging to the a ′ irrep are unchanged upon reflection through the symmetry plane of
the molecule. Irreps leaving fragments unchanged under all symmetry operations of the
point group are referred to as ‘totally symmetric’ irreps. The a ′′ irrep of the Cs point
group, on the other hand, reverses the phase of a fragment on reflection through the mirror
plane.
A much more detailed example is provided in Figure 6.7, where the C2v point group to
which the water molecule belongs is characterized by four irreps. The totally symmetric
a1 irrep includes orbitals unchanged by rotation about the C2 axis or reflection through
either of the two vertical mirror planes. The a2 irrep includes orbitals that are unchanged
by rotation about the C2 axis but that are inverted by reflection through either of the
two vertical mirror planes (Figure 6.7 does not list any such orbitals since none exist in