of conjugated hydrocarbons, despite the fact that the delocalization prevails over
bi-electronic repulsion. In the present work we shall enter the problem from the
opposite side, considering first the delocalization only and later the electronic
repulsion as a first-order perturbation.
Let us consider “alternant conjugated hydrocarbons” for which the molecular
graph, defined by the conjugated carbons, does not present odd-membered rings. As
a consequence the conjugate carbons can be separated in two classes to which one
may attribute “colors”, say red and blue. Each red atom is chemically linked to blue
atoms and vice versa. These graphs are also called “bipartite” by solid state
physicists. If one identifies the colors to spins, α or β, one may say that an alternant
hydrocarbon is a spin non-frustrated graph, in the sense that it accepts at least one
and at most two spin distributions for which each chemical bond presents a spin
alternation, and the determinants of lowest energy are the fully spin-alternant distributions (called Néel function in Solid State Physics). For sake of simplicity we
shall label p, q,… the atoms of a given color and p′, q′, … the atoms of the other
color.
A basic theorem has been established in the early days of Quantum Chemistry,
when it focused on the study of conjugated hydrocarbons. It is the so-called “mirror
theorem” [25], which says that for any alternant graph of 2n sites the 2n eigenvalues
of the Hückel Hamiltonian are in a mirror correspondence, namely to each bonding
MO φ k , of negative eigenvalue ε k one may associate an antibonding MO φ k* of
eigenvalue
e kà ¼ Àe k
ð14:3Þ
and the values of the coefficients in the corresponding eigenvectors on the atomic
orbitals χ p obeying
Hu k ¼ e k u k ; u k ¼
X
p
c kp v p
ð14:4Þ
Hu kà ¼e kà u kà ; u kà ¼
X
p
c kÃp v p
ð14:5Þ
are equal on the atoms p of a given color and opposite on the atoms q′ of the other
color
c kÃp ¼ c kp ; and c kÃq 0 ¼ À c kq
0 :
ð14:6Þ
The proof is straightforward and consists in projecting the eigenequations relative to u k and u
Ã
k on v p and on v q 0 . As an important point one should notice that
the theorem is satisfied whatever the values of the hopping integrals are, they do not
need to be equal. The key condition is the fact that one atom of a color only
interacts with atoms of the other color.
364
J.-P. Malrieu et al.
bi-electronic repulsion. In the present work we shall enter the problem from the
opposite side, considering first the delocalization only and later the electronic
repulsion as a first-order perturbation.
Let us consider “alternant conjugated hydrocarbons” for which the molecular
graph, defined by the conjugated carbons, does not present odd-membered rings. As
a consequence the conjugate carbons can be separated in two classes to which one
may attribute “colors”, say red and blue. Each red atom is chemically linked to blue
atoms and vice versa. These graphs are also called “bipartite” by solid state
physicists. If one identifies the colors to spins, α or β, one may say that an alternant
hydrocarbon is a spin non-frustrated graph, in the sense that it accepts at least one
and at most two spin distributions for which each chemical bond presents a spin
alternation, and the determinants of lowest energy are the fully spin-alternant distributions (called Néel function in Solid State Physics). For sake of simplicity we
shall label p, q,… the atoms of a given color and p′, q′, … the atoms of the other
color.
A basic theorem has been established in the early days of Quantum Chemistry,
when it focused on the study of conjugated hydrocarbons. It is the so-called “mirror
theorem” [25], which says that for any alternant graph of 2n sites the 2n eigenvalues
of the Hückel Hamiltonian are in a mirror correspondence, namely to each bonding
MO φ k , of negative eigenvalue ε k one may associate an antibonding MO φ k* of
eigenvalue
e kà ¼ Àe k
ð14:3Þ
and the values of the coefficients in the corresponding eigenvectors on the atomic
orbitals χ p obeying
Hu k ¼ e k u k ; u k ¼
X
p
c kp v p
ð14:4Þ
Hu kà ¼e kà u kà ; u kà ¼
X
p
c kÃp v p
ð14:5Þ
are equal on the atoms p of a given color and opposite on the atoms q′ of the other
color
c kÃp ¼ c kp ; and c kÃq 0 ¼ À c kq
0 :
ð14:6Þ
The proof is straightforward and consists in projecting the eigenequations relative to u k and u
Ã
k on v p and on v q 0 . As an important point one should notice that
the theorem is satisfied whatever the values of the hopping integrals are, they do not
need to be equal. The key condition is the fact that one atom of a color only
interacts with atoms of the other color.
364
J.-P. Malrieu et al.