polarization involves the bi-electronic operator of the Hamiltonian and requires
using either unrestricted mean-field formalisms or preferably multi-determinant
descriptions. This section briefly recalls the physics of this phenomenon and
illustrates its signatures.
In this chapter the molecular architectures, quantities and functions relative to
the free radicals will not receive ‘ nor “ upper symbols, those relative to ferromagnetic systems will be affected by the symbol ‘, and those concerning the
antiferromagnetic systems will be marked by “.
14.2 Spin Delocalization in Conjugated Free Radical
Hydrocarbons
14.2.1 Recall of Elementary Features
Hereafter, as we try to reach analytic conclusions, the π electrons of a conjugated
hydrocarbon are described by the Hückel Hamiltonian. The on-site energies are
assumed to be equal and define the zero of energy:
H =
X
ðp;qÞbonded
t pq ða
þ
p a q þ a
þ
q a p Þ
ð 14:1Þ
where the hopping integrals t pq on the bonds p–q are negative but may be of
different amplitudes, depending on the bond strength. The description makes also
use of a Hubbard Hamiltonian
H
0
¼
X
ðp;qÞbonded
t pq ða
þ
p a q þ a
þ
q a p Þ þ
X
p
U p n p" n p#
ð14:2Þ
where the second term accounts for the repulsion of two electrons occupying the
same site p. This simplified representation of the bi-electronic part of the
Hamiltonian keeps its leading qualitative effects.
In the strongly-correlated limit, when the electron delocalization (i.e. the t pq
terms) becomes smaller than the electron repulsion U, an appropriate description of
the lowest states is provided by the neutral VB determinants only, i.e. those in
which each carbon p bears one unpaired electron in its π atomic orbital (AO,
hereafter labelled χ p ). The π electron systems behaves as a pure spin system,
obeying a Heisenberg Hamiltonian [22, 23]. The inter-atomic delocalization, i.e. the
interaction between the neutral VB distributions and the ionic ones, results in an
antiferromagnetic spin coupling on each bond. One of the below-discussed rules,
known as the Ovchinnikov’s rule [21], has been derived from this magnetic
approach. Numerous works [24] have shown the relevance of magnetic descriptions
14 Magnetic Properties of Conjugated Hydrocarbons …
363
using either unrestricted mean-field formalisms or preferably multi-determinant
descriptions. This section briefly recalls the physics of this phenomenon and
illustrates its signatures.
In this chapter the molecular architectures, quantities and functions relative to
the free radicals will not receive ‘ nor “ upper symbols, those relative to ferromagnetic systems will be affected by the symbol ‘, and those concerning the
antiferromagnetic systems will be marked by “.
14.2 Spin Delocalization in Conjugated Free Radical
Hydrocarbons
14.2.1 Recall of Elementary Features
Hereafter, as we try to reach analytic conclusions, the π electrons of a conjugated
hydrocarbon are described by the Hückel Hamiltonian. The on-site energies are
assumed to be equal and define the zero of energy:
H =
X
ðp;qÞbonded
t pq ða
þ
p a q þ a
þ
q a p Þ
ð 14:1Þ
where the hopping integrals t pq on the bonds p–q are negative but may be of
different amplitudes, depending on the bond strength. The description makes also
use of a Hubbard Hamiltonian
H
0
¼
X
ðp;qÞbonded
t pq ða
þ
p a q þ a
þ
q a p Þ þ
X
p
U p n p" n p#
ð14:2Þ
where the second term accounts for the repulsion of two electrons occupying the
same site p. This simplified representation of the bi-electronic part of the
Hamiltonian keeps its leading qualitative effects.
In the strongly-correlated limit, when the electron delocalization (i.e. the t pq
terms) becomes smaller than the electron repulsion U, an appropriate description of
the lowest states is provided by the neutral VB determinants only, i.e. those in
which each carbon p bears one unpaired electron in its π atomic orbital (AO,
hereafter labelled χ p ). The π electron systems behaves as a pure spin system,
obeying a Heisenberg Hamiltonian [22, 23]. The inter-atomic delocalization, i.e. the
interaction between the neutral VB distributions and the ionic ones, results in an
antiferromagnetic spin coupling on each bond. One of the below-discussed rules,
known as the Ovchinnikov’s rule [21], has been derived from this magnetic
approach. Numerous works [24] have shown the relevance of magnetic descriptions
14 Magnetic Properties of Conjugated Hydrocarbons …
363