13.1.In your own words, explain why an object that has
more symmetry elements is said to have “higher symmetry”
than an object with fewer symmetry elements.
13.2 & 13.3 Symmetry Operations
and Point Groups
13.2.Identify the symmetry elements present in the follow-
ing objects. (a)A blank sheet of paper, no holes. (b)A blank
sheet of three-holed paper. (c)A baseball, including the stitch-
ing. (d)A round pencil, sharpened, with cylindrical eraser. (e)
The Eiffel Tower. (You may have to look up a picture of it if
you don’t remember its shape!) (f)Any book. (g)A human
body, approximately. (Don’t consider internal organs, only
outward appearance.) (h)A perfect starfish. (i)An unpainted
stop sign.
13.3.Identify the point groups of the objects in the previous
problem, where possible.
13.4.Show that S 1 and S 2 i.
13.5.Without referring to Appendix 3, determine whether
each of the following combinations of symmetry operations
constitutes a complete group. For those that do not, supply
the missing symmetry operation(s). (a)E, C 2 (b)E, h(c)C 4 ,
C 2 , C^34 (d)E, C 3.
13.6.Any axis of symmetry Cnthat rotates an object by
360°/nabout the z-axis can be generalized by the matrix
cos sin 0Cnsin cos (^0) (13.18)
0 0 1
(See Table 13.1.) Determine the matrix forms of the C 2 , C 3 , C 6 ,
and C 1 rotation operators.
13.7.Use equation 13.18 in the above exercise to deduce the
general form for (a)the Snoperation and (b)the ioperation.
13.8.What are the number of classes and the order of the
following point groups? (a)C2v(b)D2h(c)D6h(d)S 4 (e)Cs
(f)Td(g)O(h)Oh(i)Rh(3)
13.9. (a)Show that the C3vpoint group satisfies the closure
property of a mathematical group. (b)Show that the C3v
point group satisfies the associative law by evaluating v(EC 3 )
and (vE)C 3.
13.10. (a)In the Tdpoint group, an S 4 ^1 improper rotation is
equivalent to what other improper rotation? (b)In the D6h
point group, the symmetry operation labeled C 2 ^1 is equiva-
lent to what other symmetry operation?
13.11.Determine which single symmetry operation of the
following point groups is equivalent to the given combination
of multiple symmetry operations. (a)In C2v, C 2 v? (b)In
C2h, iC 2 ? (c)In D6h, C 6 h? (d)In D2d, C 2 C
2 ? (e)In
Oh, iS 4 ?
13.12.Group theory requires that symmetry operations sat-
isfy the associative law.
(a)Do they satisfy the commutative law? That is, does a differ-
ent order of the same symmetry operations always yield the
same answer? Provide a specific example to support your answer.
(b)In matrix algebra, matrix multiplication is not commuta-
tive. For any two given matrices Aand B, it is not a certainty
that ABBA. Does this support or refute your conclusion in
part a, and why?
13.13.Figure 13.27 shows the structure of the molecule por-
phine. Determine the symmetry elements present in the mol-
ecule, and its point group. Does the point group change if an
Fe^2 ion is substituted for the two hydrogens in the center of
the porphine ring?
13.14.In Example 13.3e, we are assuming that resonance
structures of NO 3 are “averaging” out the symmetry to an
overall D3hpoint group. If resonance weren’t assumed, what
point group would define the structure of NO 3 ?
13.15.Identify all the symmetry elements present in the tetra-
hedron, the cube, and the octahedron.
13.16.Point groups are called such because all of the sym-
metry elements in the group intersect in one point in space.
For point groups that have ias a symmetry operation, why
must ibe at that point?
13.4 Molecules and Symmetry
13.17.Determine the point groups of the following molecules.
(a)Hydrogen peroxide, H 2 O 2 (It is notplanar.) (b)Allene,
CH 2 CCH 2 (c)D-glycine (Its absolute stereochemistry can
be found in any good organic or biochemistry textbook.)
(d)L-glycine, and compare with part c above (e)cis-1,2-
dichloroethylene (f)trans-1,2-dichloroethylene (g)Toluene,
C 6 H 5 CH 3 (h)1,3-Cyclohexadiene.
13.18.Determine the point groups of the following mole-
cules. (a)Hydrogen selenide, H 2 Se. (b)Partially deuterated
hydrogen sulfide, or HDS (c)The chair conformer of cyclo-
hexane, C 6 H 12 (d)The boat conformer of cyclohexane, C 6 H 12
(e)Fe(CO) 5 , which has a trigonal bipyramidal structure
(f)CO 32 , which has three resonance structures that con-
tribute to its overall shape (g)The perfectly staggered con-
former of ethane (h)The perfectly eclipsed conformer of
ethane (h)1,4-Cyclohexadiene
13.19.Identify the point group of the wavefunctions of the
following molecules. (a)Deuterium oxide, D 2 O (where D ^2 H)
(b)Boron trichloride, BCl 3 (c)Methylene chloride, CH 2 Cl 2
Exercises for Chapter 13 457
EXERCISES FOR CHAPTER 13
N HH N
N
N
Figure 13.27 The structure of porphine. (See exercise 13.13.)