Physical Chemistry Third Edition

21.6 Delocalized Bonding 887

Energy localization should turn all of the sigma bonding orbitals into localized orbitals,

but this is not the case with the pi orbitals.

We assume that the localized sigma bonds and the delocalized pi bonds can be

treated separately. We place the benzene molecule in thexyplane as in the valence-

bond treatment. We use the carbon 1sorbitals as nonbonding (inner-shell) orbitals.

As our basis functions for the sigma bonds we use the three carbon 2sp^2 hybrids

in thexyplane for each carbon atom and a 1sorbital on each hydrogen atom. We

orient the coordinate system at each carbon atom so that two of the 2sp^2 hybrid orbital

regions point in the directions of the C–C bonds and one points in the direction of the

C–H bond. Each C–C sigma bonding orbital is made from two 2sp^2 hybrid orbitals on

adjacent carbons. Each C–H bonding orbital is made from a 2sp^2 carbon hybrid orbital

anda1sorbital on a hydrogen atom. The benzene molecule has 42 electrons. Twelve

electrons occupy the nonbonding carbon 1sorbitals, twelve occupy the localized C–C

bonding orbitals, and twelve occupy the localized C–H bonding orbitals. This leaves

six electrons to occupy the pi orbitals.

We take the six unhybridized 2pzatomic orbitals as the basis functions for the

delocalized pi orbitals. We denote the 2pzorbitals byψ 1 ,ψ 2 ,...,ψ 6 where the subscript

indicates which carbon atom is involved. We construct delocalized LCAOMOs:

φc 1 ψ 1 +c 2 ψ 2 +c 3 ψ 3 +c 4 ψ 4 +c 5 ψ 5 +c 6 ψ 6 

∑^6

j 1

cjψj (21.6-2)

Since there are six basis functions, there can be six independent LCAOMOs.

The Hückel Method

The Hückel method is an approximation scheme for finding the coefficients in the

delocalized LCAOMOs. It is called asemiempirical method, because various quantities

are not calculated explicitly, but are assigned values to agree with experiment. We

assume that the Hamiltonian is a sum of effective one-electron Hamiltonian operators,

Ĥeff, in which all attractions and repulsions are expressed in some approximate way,

so that the wave function will be a product of orbitals and each orbital can be treated

separately. We apply the variation method, minimizing the orbital energyWobtained

from the wave functionφin Eq. (21.6-2):

W

∫

φ∗Ĥeffφd^3 r

∫

φ∗φd^3 r

(21.6-3)

This orbital energy is expressed in terms of theccoefficients. To minimize the energy

with respect to all of theccoefficients the variation energyWis differentiated with

respect to each of the coefficients and each derivative is set equal to zero. Six simulta-

neous equations are obtained. There is a condition on the simultaneous equations that

must be satisfied for them to have a usable solution. This condition is expressed in an

equation called asecular equation, which can be solved for six values of the variational

energyW. Each of these values is substituted into the simultaneous equations, leading

to a set of coefficients for each of the six delocalized orbitals.

The simultaneous equations and the secular equation contain a number of integrals

involving the effective Hamiltonian and the basis functions. In a semiempirical method