α+ 2 β
α
α− 2 β
E
1E
2E
3E
4E
.................................................................................................................................................................................................................................................................................
...
...
..
...
...
..
...
...
...
...
..
...
...
..
...
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The following important theorem for the eigenvectors of symmetric matrices follows
from Properties 1–3:
The neigenvectors of a real symmetric matrix of order nform (or
can be chosen to form) a system of northogonal unit (orthonormal)
vectors:
(19.23)
EXAMPLE 19.9Hückel theory of cyclobutadiene
In the molecular-orbital theory of π-electron systems, the states of the πelectrons are
described by a matrix eigenvalue equation
HC 1 = 1 EC
in which the matrix Hrepresents the ‘effective Hamiltonian’ for a πelectron in the
system, the eigenvalues Eof Hare the orbital energies of the πelectrons, and the
eigenvectors Crepresent the corresponding molecular orbitals (the components of C
are the coefficients in a ‘linear combination of atomic orbitals’ (LCAO) description
of a molecular orbital). In the Hückel theory of cyclobutadiene (C
4H
4), His the real
symmetric matrix
(the Hückel parameters αand βare real negative scalars with the dimensions of
energy) with characteristic equation
= 1 (α 1 − 1 E)
2(α 1 − 1 E 1 + 12 β)(α 1 − 1 E 1 − 12 β) 1 = 10
The eigenvalues (orbital energies) are therefore
E
11 = 1 α 1 + 12 β, E
21 = 1 E
31 = 1 α, E
41 = 1 α 1 − 12 β
and the eigenvalue spectrum is shown in Figure 19.1.
det(HI−=)
−
−
−
−
E
E
E
E
E
αβ β
βα β
βα β
ββα
0
0
0
0
H=
αβ β
βα β
βα β
ββα
0
0
0
0
xx
kl klkl
kl
T==
=
≠
δ
1
0
if
if
19.3 Properties of the eigenvectors 541
Figure 19.1