13.20.Identify the point group of the wavefunctions of the
following molecules. (a)Hydrogen chloride, HCl (b)Sulfur
dioxide, SO 2 (c)Sulfur trioxide, SO 3
13.21. (a)What would be the formulas of possible molecules
that have carbon atoms in the positions of the vertices of the
five Platonic solids, in the cases where normal valence rules ap-
ply? (That is, C forms four and only four bonds. Any bonds not
made to other C atoms can be used by hydrogen atoms.)
(Note:Not all Platonic solids can be mimicked by carbon atoms
bonding together, because one requires five bonds at each
vertex.) (b)Verify that the two smallest molecules contain the
various symmetry elements of their respective cubic point
group. (Three of the possible hydrocarbons have actually been
synthesized. Can you find them in the chemical literature?)
13.22.Determine if the following species have permanent
dipole moments. (a)Dichloromethane, CH 2 Cl 2 (b)Chloro-
benzene, C 6 H 5 Cl (c)Ammonia, NH 3 (d)Carbon dioxide,
CO 2 (e)The carbonate ion, CO 32 (f)The phosphate ion,
PO 43 (g)Uranium hexafluoride, UF 6 (h)Bromine, Br 2
(i)Hydrogen deuteride, HD (D ^2 H)
13.23.Which of the following species will nothave permanent
dipole moments? (a)Hydrogen cyanide, HCN (b)Carbonyl
sulfide, OCS (c)Phosphorus pentachloride, PCl 5 (d)Tri-
methylamine, N(CH 3 ) 3 (e)Boron trifluoride, BF 3 (f)Diborane,
which has the following structure:
where the bridging hydrogens are perpendicular to the four
terminal hydrogens, which are all coplanar (g)Methane, CH 4
(h)Chloromethane, CH 3 Cl (i) Dichloromethane, CH 2 Cl 2
(j)Trichloromethane (or chloroform), CHCl 3 (k)Carbon tetra-
chloride, CCl 4 (l) 2,2-Dimethylpropane, CH 3 C(CH 3 ) 2 CH 3
(m)Cubane (see 13.21 above)
13.24. (a) Unlike methane, bromochlorofluoromethane
(CHBrClF) is chiral. Determine all symmetry elements that are
present in CHBrClF and identify its point group. (b)If the fluo-
rine in this molecule were substituted with a hydrogen, what
is the point group for the new molecule? Is it chiral?
13.5 Character Tables
13.25.Write out explicitly the 12 12 matrices that specify
the change in atomic coordinates of NH 3 upon operation of
the Esymmetry operation and any symmetry operation.
13.26.For NH 3 , write out explicitly the three 12 12 matri-
ces for all three planes of symmetry ,
, and . How are
they similar, and how are they different?
13.27.Show that the irreducible representations of the D 2
point group satisfy the closure requirement.
13.28.Show that the irreducible representations of the D2d
point group satisfy the closure requirement. You will have to use
the great orthogonality theorem to reduce one combination.
BHHHH HH
B13.29.Show that any two of the irreducible representations
of the following point groups are orthogonal to each other.
(a)C 2 (b)C2v(c)D2h(d)Oh(e)Td
13.30.Using the character tables in Appendix 3, can you de-
termine which symmetry element must be present in order to
have a two-dimensional irreducible representation (that is, one
that can be labeled using Einstead of Aor B)?
13.31.Why is it unnecessary to consider whether an irre-
ducible representation from C4his orthogonal to an irreducible
representation of D6h?
13.32.Explain why the characters for the proper and im-
proper rotations are mathematical expressions instead of num-
bers for Cvand Dh.
13.33.Use the expressions in Rh(3)to determine the charac-
ters of the forbitals in an octahedral (Oh) environment. For the
set of seven forbitals, the character of the identity symmetry
operation is 7.13.6 Wavefunctions and Symmetry
13.34. (a)What are the symmetry elements present in a plot
of the function F( ) sin? Assume that all symmetry ele-
ments intersect at the origin. (b)What are the symmetry ele-
ments present in a plot of the function F( ) cos , again as-
suming that all elements intersect at the origin?
13.35.What are the symmetry elements present in a plot of
the function F( ) sin , assuming that the point of inter-
section for all symmetry elements is on the x-axis at x/2?
(Another way of stating this is, What are the symmetry ele-
ments of the function about the point x/2?)
13.36.What point group(s) must the wavefunctions of all lin-
ear molecules belong?
13.37.The bonds in the ethylene molecule can be repre-
sented like this:where the two lobes are out of the plane made by the atoms.
(In ground-state ethylene, only the bonding orbital is filled.)
Determine the point group of ethylene and assign a symme-
try species to the pbonds.
13.38.Consider the bonds in ethylene shown in the previ-
ous exercise. In benzene (C 6 H 6 ), six porbitals from the six car-
bon atoms combine to make six molecular orbitals. In the
lowest-energy orbital, all of the phases assigned to the atomic
porbitals are the same on each side of the molecular plane; in
the highest-energy orbital, they alternate phases. Draw these
two orbitals and determine their symmetry species.CHHHHC++458 Exercises for Chapter 13