Physical Chemistry Third Edition

888 21 The Electronic Structure of Polyatomic Molecules

Orbital energy

 22 

 12 

 2 

 1 

6



5

3

1

2

4

Figure 21.8 Orbital Energies of the Delocalized Pi Orbitals in Benzene by the Hückel

Method.

such as the Hückel method, one does not evaluate such integrals, but assigns values to

them at the end of the calculation to obtain agreement with experiment. The approx-

imation scheme used for the various integrals is described in Appendix H, and the

calculations are carried out for the allyl radical. The benzene calculation is similar, but

more tedious.

The Hückel methodwasdevelopedin
the 1930s by Erich Hückel,1896–1980,
a Germanchemist who was also the
co-inventor of the Debye–Hückel theory
of electrolyte solutions.
Figure 21.8 shows the orbital energies of the six LCAOMOs.^5 The pattern of the
energy levels has the same shape as the molecule. This correspondence occurs in the
Hückel solution for all single-ring aromatic molecules. The integralβturns out to be
negative, so that the lowest value ofWfor benzene is

W 1 α+ 2 β (21.6-4)

When each value ofWis substituted into the simultaneous equations a different set of

cjcoefficients is obtained for each value ofW, so that there are six different delocalized

space orbitals. When the value ofWin Eq. (21.6-4) is substituted into the simultaneous

equations, we find that the normalized lowest-energy LCAOMO is

φ 1 

√

1

6

[ψ 1 +ψ 2 +ψ 3 +ψ 4 +ψ 5 +ψ 6 ] (21.6-5)

This orbital has no nodes between the carbon atoms, and an electron moves around

the entire molecule if it occupies this orbital. Other LCAOMOs are obtained from the

other values ofW, which actually represent either relative minima or relative maxima

inW. Figure 21.9 shows a view of the orbital regions of the six LCAOMOs. Since each

of the 2pzbasis functions has a nodal plane in thexyplane, each LCAOMO has this

nodal plane. The sign shown in the figure applies to the region above the plane. The

broken lines represent vertical nodal planes for the other LCAOMOs. A nodal plane

can occur between two carbon atoms or it can pass through a carbon nucleus if the

coefficient for the 2pzorbital of that carbon atom vanishes.

Our results follow the general rule that a higher energy corresponds to a larger

number of nodes. In addition to the nodal plane in the plane of the molecule, there

are no nodal surfaces in the lowest-energy orbital, one nodal plane in each of the next

two orbitals (which are degenerate), two nodal planes in the next two orbitals (also

degenerate), and three nodal planes in the highest-energy orbital. Without doing any

calculations, we might have been able to guess the pattern of the energy levels and the

pattern of nodes in the delocalized orbitals from the possible nodal patterns.

(^5) I. N. Levine,op. cit., p. 634ff (note 2).