13.7 Great Orthogonality Theorem
13.39.Reduce the following reducible representations using
the great orthogonality theorem.
(a)In the C 2 point group:
EC 2
5 1(b)In the C3vpoint group:
E 2 C 2 3 v
6 0 0(c)In the D 4 point group:
E 2 C 4 C 2 2 C 2
2 C 2
6 2 2 2 4(d)In the Tdpoint group:
E 8 C 3 3 C 2 6 S 4 6 d
7 2 3 1 113.40.Determine the resulting representations for the fol-
lowing products of irreducible representations.
(a)In C3v, A 1 A 2
(b)In C6v, E 1 E 2
(c)In D3h, A
2 A 1 E
(d)In D6h, B2gB2u
(e)In D6h, B1gB1g
(f)In Td, ET 1
(g)In Td, T 2 T 2
(h)In Oh, EgT2g
13.8 Using Symmetry in Integrals
13.41.Assume that you are evaluating the integral of prod-
ucts of functions having symmetry labels in exercise 13.40,
parts a–h. Which integrals, if any, are exactly zero due to sym-
metry considerations?
13.42.Assume that x-polarized light can be assigned an E
symmetry species in a system that has C4vsymmetry. Can a
transition from a Estate to a B 2 state occur? Why or why not?
13.43.Show that s→stransitions are not allowed in the hy-
drogen atom. To do this, show that the integral a(s)*Oˆb(s)
dis exactly zero where Oˆis the operator representing the
light. Assume that light has the symmetry species Du(1)in this
completely spherical system.
13.44.The five dorbitals in transition metals can be shown
to have the following characters under tetrahedral (Td) sym-
metry:
E 8 C 3 3 C 2 6 S 4 6 d
5 1 1 1 1(The character of Ebeing 5 for the five dorbitals is not a co-
incidence!) In Tdsymmetry, what symmetry species do the d
orbitals have? What are the degeneracies of the symmetry
species?13.9 SALC-MO Theory
13.45.Construct the symmetry-adapted linear combination
molecular orbitals for hydrogen sulfide, H 2 S.
13.46.Referring to exercise 13.45: How would the symme-
try-adapted linear combinations for the molecular orbitals of
H 2 S differ if the core atomic orbitals of S were included?
13.47.In Example 13.13, several of the molecular orbitals for
H 2 O were found to be simply atomic orbitals. Justify this in
light of the idea that molecular orbitals are orbitals of the mol-
ecule as a whole, and not orbitals of the atoms.
13.48.Should the molecular orbitals for H 2 O found in
Example 13.13 be orthogonal? Will this always be the case?
13.49.How many SALCs can be constructed for CH 4 using all
valence and core atomic orbitals?13.10 & 13.11 VB Theory and Hybrid Orbitals
13.50.Construct a list comparing and contrasting VB theory
with MO theory.
13.51.Why might one not be surprised to find that the first
excited state of H 2 is represented by three very closely spaced
lines in the spectrum?
13.52.Suppose you use p 0 , p 1 , and p 1 along with sorbitals
to construct hybrid orbitals. Will they be the same hybrid or-
bitals defined by the px, py, and pzorbitals? Justify your answer.
13.53.Show that the individual sporbitals, as written in equa-
tion 13.16, are orthogonal.
13.54.Show that the individual sp^3 orbitals, as written in
equation 13.15, are orthogonal.
13.55.What is the rough hybridization of the carbon orbitals
in the methyl carbonium ion, CH 3 , which is almost perfectly
planar triangular in shape?
13.56.Determine the symmetry species of the D3hpoint
group for the sp^2 hybrid orbitals, assuming that the C 3 axis is
coincident with the z-axis and that one of the orbitals lies
along the positive x-axis. (See example 13.16.)
13.57.Determine the D3hsymmetry species of the sp^3 dhy-
brid orbitals, assuming that the C 3 axis is coincident with the
z-axis and that one of the orbitals lies along the positive x-axis.
(See Example 13.16.)
13.58.Determine the Ohsymmetry species of the sp^3 d^2 hy-
brid orbitals, assuming that the hybrid orbitals are all coinci-
dent with the Cartesian axes.
13.59.In propene (CH 3 –CHCH 2 ), the first carbon has sp^3
hybrid orbitals and the second carbon has sp^2 hybrid orbitals.
These orbitals interact to make a bond. Why are these hy-
brid orbitals not orthogonal?Exercises for Chapter 13 459