21.9 Groups of Symmetry Operators 903
molecule will leave the dipole moment unchanged, since it can at most exchange nuclei
of the same type. Only if the dipole moment vector is contained in all of the symmetry
elements can there be a nonzero dipole moment. Any molecule with a dipole moment
must belong to one of the groupsC 1 ,Cs,Cn,orCnv, wherencan be any integer. A
molecule with a center of symmetry or a horizontal mirror plane cannot have a dipole
moment.
Another immediate conclusion involves optical activity. Any molecule that cannot
be superimposed on its mirror image possesses anenantiomorph, which is an isomer
that is like the mirror image of the first molecule. Two enantiomorphs rotate plane
polarized light in opposite directions and are said to beoptically active. Any molecule
that has no symmetry elements or has only proper rotation axes can be optically active.
The only groups that meet these criteria areC 1 ,Cn, andDn, wherencan be any
integer. Any molecule with an inversion center, a reflection plane, or anSnaxis cannot
be optically active.
There are numerous more advanced applications of group theory. It is possible
to determine without calculation whether an overlap integral will vanish, what the
degeneracy of an energy level is, whether a transition between certain electronic or
vibrational states can be accompanied by emission or absorption of radiation, and so
on. These applications use the representations of groups, which are briefly introduced
in Appendix I.PROBLEMS
Section 21.9: Groups of Symmetry Operators
21.46Show that the group of operations belonging to the water
molecule is abelian (all operations commute with each
other).
21.47a.Obtain the multiplication table for theC 3 vpoint group.
b.Show that it satisfies the conditions to be a group only
ifC^23 is included in the group.
c.Show that the group is not abelian.21.48What is the symmetry operator that is equivalent to the
productˆîσh? Is this the same as the productσˆhˆi?
21.49What is the symmetry operator that is equivalent to the
product̂C 2 ˆi? Is it the same as the productˆîC 2?
21.50Write the function that is equal tôC (^4) zψ 2 px.
21.51List all of the symmetry operators that belong to the
formaldehyde (CH 2 O) molecule in its equilibrium
conformation. Assign it to a point group.
21.52Write the function that is equal tôC (^2) zψ 2 px.
21.53List all of the symmetry operations that belong to the BH 3
molecule. Assign it to a point group, using Figure 21.12.
21.54Construct the multiplication table for theD∞hpoint
group. Omit theC∞andS∞operations.
21.55Construct the multiplication table for theC 4 vpoint
group.
21.56Assign the following molecules to point groups:
a.Dichlorodibromomethane
b.Toluene
c.Naphthalene
21.57Assign the following molecules to point groups:
a.1,4-Dichlorobenzene
b.1,2-Dichlorobenzene
c.Tetrachloroethene
21.58Using the decision tree of Figure 21.12, assign the
following molecules to point groups:
a.Ethane (staggered)
b.Ethane (eclipsed)
c.Cyclohexane (boat conformation)
21.59Assign a tennis ball or a baseball to a point group.
21.60The N 2 O molecule is linear. Assign it to a point group
a.If it has the structure NON,
b.If it has the structure NNO.