Essentials of Computational Chemistry

H
1 C 3
C 2 C
H
H
p C
H
H
4.4 HÜCKEL THEORY 117
f3 =
f 2 =
f 1 =
a − √2b
a
a + √2b
Figure 4.2 Hückel MOs for the allyl system. One pC orbital per atom defines the basis set. Combinations
of these 3 AOs create the 3 MOs shown. The electron occupation illustrated corresponds to
the allyl cation. One additional electron in φ2 would correspond to the allyl radical, and a second
(spin-paired) electron in φ2 would correspond to the allyl anion
The basis set size of three implies that we will need to solve a 3 × 3 secular equation.
Hückel conventions (b)–(e) tell us the value of each element in the secular equation, so that
Eq. (4.21) is rendered as
α − E β 0
β α − E β
= 0 (4.24)
0 β α − E
The use of the Krönecker delta to define the overlap matrix ensures that E appears only
in the diagonal elements of the determinant. Since this is a 3 × 3 determinant, it may be
expanded using Cramer’s rule as
(α − E) 3 + (β 2 · 0) + (0 · β 2 ) − [0 · (α − E) · 0] − β 2 (α − E) − (α − E)β 2 = 0 (4.25)
which is a fairly simple cubic equation in E that has three solutions, namely
E = α + √ 2β, α, α − √ 2β (4.26)
Since α and β are negative by definition, the lowest energy solution is α + √ 2β. Tofind
the MO associated with this energy, we employ it in the set of linear Eqs. (4.20), together
E