Physical Chemistry Third Edition

21.6 Delocalized Bonding 891

Exercise 21.12

a.Describe the pi LCAOMOs in the cyclobutadiene molecule, C 4 H 4 , assuming a square struc-

ture. Use 2sp^2 hybrid orbitals for the sigma bonds, although they do not quite fit. There

are two ways to make a single node in a LCAOMO (either horizontal or vertical). Give the

electron configuration of the ground-state of the molecule.

b.Describe the bonding using the valence-bond method, using the resonance structures:

c.Describe the bonding using the valence-bond method, assuming the single structure with

alternating single and double bonds. The correct structure more nearly resembles one of

these resonance structures with bonds of unequal length. The molecule is not aromatic.^6

PROBLEMS

Section 21.6: Delocalized Bonding

21.24The motion of the pi electrons around the benzene

molecule is sometimes represented as de Broglie waves

moving around a circular ring. Take the carbon–carbon

distance in the ring as 139 pm, and take the

lowest-energy electron state to have a de Broglie

wavelength equal to the circumference of the ring, the

next to have a de Broglie wavelength equal to half

of the circumference, and so on. Find the energy

and wavelength of the photons absorbed in the

longest-wavelength ultraviolet absorption and

compare with the experimental wavelength of

180 nm.

21.25Using the valence-bond method with resonance, describe

the bonding in CO^23 −.

21.26Describe the bonding of the CO^23 −ion using

LCAOMOs. Place the nuclei in thex−ycoordinate

plane and make delocalized orbitals with the

unhybridizedpzorbitals on all four atoms, trying to

guess where the nodes are in the lowest-energy

delocalized orbitals.

21.27Sketch the orbital regions that you would expect for the

pi electrons in the allyl radical, C 3 H 5.

21.28Sketch the orbital regions that you expect for the pi

electrons in cyclobutadiene, C 4 H 4.

21.29Using the valence-bond method, with resonance where

appropriate, describe the bonding in the molecules:

a.HNO 3

b. SO 2

21.30Using the valence-bond method, describe the bonding

and molecular shape of

a.1,3,5-hexatriene

b.NO− 2

c.CH 3 (methyl radical)

21.31Carry out the Hückel calculation for the cyclopropenyl

radical, C 3 H 3 , by hand. Notice the similarity with the

analysis of the allyl radical in Appendix H. Sketch the

orbital region of the lowest-energy LCAOMO. Note that

the next-lowest LCAOMOs are degenerate so that

different linear combinations can be chosen. Choose two

that are eigenfunctions of a vertical reflection operator

passing through one of the carbon atoms.

21.32Carry out the Hückel calculation for the cyclopropenyl

cation, C 3 H+ 3. How does it differ from the analysis of the

cyclopropenyl radical?

21.33Use the energy level expressions for benzene and for

1,3-butadiene to obtain two different values for the

parameterβin the Hückel theory, using the fact that the

strongest ultraviolet absorption in benzene is at a

wavelength 180 nm, while the strongest ultraviolet

absorption in 1,3-butadiene is at 217 nm. Compare these

values with an accepted value ofβ− 2 .71 eV^7 and

explain why the values do not agree.

(^6) I. N. Levine,op. cit., p. 639 (note 2); T. H. Lowry and K. S. Richardson,Mechanism and Theory
in Organic Chemistry, 3rd ed., Harper and Row, New York, 1987, pp. 43–44.
(^7) D. J. Royer,Bonding Theory, McGraw-Hill, New York, 1968, p. 162.