H. The Hückel Method
The Hückel method is a simple semi-empirical method for determining approximate
LCAO molecular orbitals to represent delocalized bonding in planar molecules. It
treats only pi electrons and assumes that the framework of sigma bonds has been
treated separately. As an example we consider the allyl radical, CH 2 CH−CH 2 ·.If
the plane of the molecule is thexyplane, each carbon atom has an unhybridized 2pz
orbital that is not involved in the sigma bonds, which are made from the 2sp^2 hybrids
in thexyplane with the appropriate rotation of the coordinate system at each atom
to provide maximum overlap. We construct linear combinations from the three 2pz
orbitals, as in Eq. (21.6-2).
φic 1 (i)ψ 1 +c( 2 i)ψ 2 +c( 3 i)ψ 3 (i1, 2, 3) (H-1)
whereiis an index specifying which LCAOMO is meant, and whereΨ 1 is the 2pz
orbital on carbon number 1, and so on. From three independent atomic orbitals, three
LCAOMOs can be made.
We assume that there is an effective one-electron Hamiltonian operator,̂Heff,in
which all attractions and repulsions are expressed in an approximate way such that
each electron moves independently of the other electrons. We apply the variational
method, seeking the lowest value of the variational orbital energy,
Wi
∫
φ∗iĤeffφid^3 r
∫
φ∗iφid^3 r
(H-2)
When the linear combination of Eq. (H-1) is substituted into this expression we have
Wi
∑^3
a 1
∑^3
b 1
ca(i)c(bi)Hab
∑^3
a 1
∑^3
b 1
c(ai)c(bi)Sab
N
D
(H-3)
where we abbreviate the numerator of this expression byNand the denominator byD.
This expression contains two types of integrals:
Sab
∫
ψ∗aψbd^3 r (H-4)
Hab
∫
ψ∗âHeffψbd^3 r (H-5)
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