14.6 Conclusion
This work has shown how one may exploit the analytical properties of the Hückel
Hamiltonian to predict many properties of monoradicals, diradicals or polyradicals.
This topological model enables one to establish analytically the coefficients of the
singly occupied MOs on the different conjugated carbons of radicals. When the
molecule has an open-shell singlet ground state, it is possible to determine the
HOMO and LUMO energies from the coefficients of the SOMOs of the monoradicals obtained by subtracting from the graph successively the two atoms which
are expected to bear the unpaired electrons with major spin densities. Then introducing the bi-electronic repulsion through the Hubbard Hamiltonian, the
ground-state spin multiplicity is predicted by simply considering the topology. This
model affords a derivation of the Ovchinnikov’s rule in the weak correlation limit,
consistent with the physics of conjugated hydrocarbon (|t|/U is larger than ½). One
can note that this rule was originally derived from the Heisenberg Hamiltonian, i.e.
in the strong correlation limit.
Another advantage of this approach is that it provides a direct analytical estimate
of both singlet-triplet gaps of polyradicals and the coefficients of the SOMOs.
Estimates of both direct exchange integrals for ferromagnetic systems and kinetic
exchange contributions for antiferromagnetic systems are easily obtained. One may
note that at variance, the estimation of singlet-triplet gaps from the topological
Heisenberg Hamiltonian requires a matrix diagonalization. Moreover this analysis
gives a direct access to the spin-instability condition (broken-symmetry solution
lower in energy than the spin-restricted one). The reader may play with various
architectures, involving other branching of methylene groups on polyphenylene,
acenes or fused polycyclic aromatic hydrocarbons. Let us mention that these
analysis have been recently used to predict the conductance properties of polycyclic
hydrocarbons, as function of the sites of attachments to conducting sources [62].
Finally we would like to mention that the present topological analysis can be used
to rationalize the spin-symmetry breaking occurring in the series of polyacenes,
their diradical character, which is a matter of debate in the recent literature [63–67],
and the length dependence of the singlet-triplet energy gap.
The proofs of the here-formulated theorems and rules do not require equal
hopping integrals. As a consequence the conclusions may be relevant to systems
where the conjugated hydrocarbon is either weakly or strongly bonded to external
magnetic sites, such as open-shell metal ion complexes or organic radicals like
niytroxides [68] or nitrenes [69]. The topological conclusions remain valid, as far as
the on-site energies of the external magnetic sites are not too different from those of
the sp
2 carbons. Some of the conclusions are applicable to the dinuclear complexes
of Cu(II) or Ni(II) where the two magnetic ions are connected by long conjugated
hydrocarbons [70, 71]. Such architectures attract more and more attention in the
field of spintronic. Among the bridging ligands commonly used to connect transition metal ions, those which exhibit strong bond alternations (like polyenes or
chains of phenyls) are bad spin linkers. At variance, fused aromatic polybenzenic
392
J.-P. Malrieu et al.
This work has shown how one may exploit the analytical properties of the Hückel
Hamiltonian to predict many properties of monoradicals, diradicals or polyradicals.
This topological model enables one to establish analytically the coefficients of the
singly occupied MOs on the different conjugated carbons of radicals. When the
molecule has an open-shell singlet ground state, it is possible to determine the
HOMO and LUMO energies from the coefficients of the SOMOs of the monoradicals obtained by subtracting from the graph successively the two atoms which
are expected to bear the unpaired electrons with major spin densities. Then introducing the bi-electronic repulsion through the Hubbard Hamiltonian, the
ground-state spin multiplicity is predicted by simply considering the topology. This
model affords a derivation of the Ovchinnikov’s rule in the weak correlation limit,
consistent with the physics of conjugated hydrocarbon (|t|/U is larger than ½). One
can note that this rule was originally derived from the Heisenberg Hamiltonian, i.e.
in the strong correlation limit.
Another advantage of this approach is that it provides a direct analytical estimate
of both singlet-triplet gaps of polyradicals and the coefficients of the SOMOs.
Estimates of both direct exchange integrals for ferromagnetic systems and kinetic
exchange contributions for antiferromagnetic systems are easily obtained. One may
note that at variance, the estimation of singlet-triplet gaps from the topological
Heisenberg Hamiltonian requires a matrix diagonalization. Moreover this analysis
gives a direct access to the spin-instability condition (broken-symmetry solution
lower in energy than the spin-restricted one). The reader may play with various
architectures, involving other branching of methylene groups on polyphenylene,
acenes or fused polycyclic aromatic hydrocarbons. Let us mention that these
analysis have been recently used to predict the conductance properties of polycyclic
hydrocarbons, as function of the sites of attachments to conducting sources [62].
Finally we would like to mention that the present topological analysis can be used
to rationalize the spin-symmetry breaking occurring in the series of polyacenes,
their diradical character, which is a matter of debate in the recent literature [63–67],
and the length dependence of the singlet-triplet energy gap.
The proofs of the here-formulated theorems and rules do not require equal
hopping integrals. As a consequence the conclusions may be relevant to systems
where the conjugated hydrocarbon is either weakly or strongly bonded to external
magnetic sites, such as open-shell metal ion complexes or organic radicals like
niytroxides [68] or nitrenes [69]. The topological conclusions remain valid, as far as
the on-site energies of the external magnetic sites are not too different from those of
the sp
2 carbons. Some of the conclusions are applicable to the dinuclear complexes
of Cu(II) or Ni(II) where the two magnetic ions are connected by long conjugated
hydrocarbons [70, 71]. Such architectures attract more and more attention in the
field of spintronic. Among the bridging ligands commonly used to connect transition metal ions, those which exhibit strong bond alternations (like polyenes or
chains of phenyls) are bad spin linkers. At variance, fused aromatic polybenzenic
392
J.-P. Malrieu et al.