Physical Chemistry Third Edition

892 21 The Electronic Structure of Polyatomic Molecules

21.34Obtain access to any of the common computer programs
that will solve the Hückel molecular orbital problem for
various molecules. You will have to find out how the
necessary information is put into the computer.
a.Run the program for benzene and for 1,3,5-hexatriene.
Compare the results and explain the differences.
b.Run the program for cyclobutadiene and for
1,3-butadiene. Compare the results and explain the
differences.

21.35Obtain access to any of the common computer programs

that will carry out the Hückel method. You will have to

find out how the necessary information is put into the

computer. Run the program for cyclooctatetraene and for

1,3,5,7-octatetraene. Compare the results and explain the

differences.

21.7 The Free-Electron Molecular Orbital Method

In the free-electron molecular orbital (FEMO) method delocalized molecular orbitals

are represented by particle-in-a-box wave functions. As an example, we discuss the

pi electrons in 1,3-butadiene, assuming that the sigma-bond framework has been

separately treated. The experimental carbon–carbon bond lengths are 146 pm for the

center bond and 134 pm for two outer bonds.^8 The orbital regions must extend beyond

the nuclei at the ends of the molecule, so we assign the length of the box in which the pi

electrons move to be the sum of the three bond lengths plus one additional bond length

at each end. If the extra bond length is taken as the average of the two bond lengths, a

total box length of 694 pm (6.94 Å) is obtained.

The energy eigenfunctions and energy levels of a particle in a box are given by

Eqs. (15.3-10) and (15.3-11):

ψψn

√

2

a

sin

(nπx

a

)

, (21.7-1)

EEn

h^2 n^2

8 ma^2

(21.7-2)

whereais the length of the box andnis a quantum number (a positive integer). The

energy values are nondegenerate. Since we have four electrons, the Aufbau principle

gives the ground-state wave function (including only the pi electrons) without anti-

symmetrization as

Ψgsψ 1 (1)α(1)ψ 1 (2)β(2)ψ 2 (3)α(3)ψ 2 (4)β(4) (21.7-3)

The ground-state pi-electron energy is

Egs

h^2

8 ma^2

(1^2 + 12 + 22 + 22 )

10 h^2

8 ma^2

(21.7-4)

(^8) K. Kuchitsu, F. Tsutomu, and Y. Morino,J. Mol Struct., 37 , 2074 (1962).