common value J, its energy is −N b J. Moreover this determinant interacts with N b
determinants obtained by a spin exchange on the various bonds, which is the
maximum number of interactions which one may find on a line of the Hamiltonian.
Therefore this determinant only belongs to the lowest eigenvector which necessarily
has an M s = |n* − n| value, and an S = |n* − n| + 1 spin multiplicity.
Of course the Heisenberg Hamiltonian is supposed to be valid for systems where
the electron-electron repulsion prevails on the electron delocalization. It may be
established as an effective Hamiltonian from the Hubbard Hamiltonian by a
second-order expansion, using the quasi degenerate perturbation theory, provided
that the ratio |2t/U| < 1, which results in a value of J = 2t
2 /U. This inequality is
hardly satisfied in conjugated hydrocarbons, which are typically considered as
strongly delocalized (almost metallic) or weakly correlated. The validity of magnetic treatments of half-filled bands actually extends beyond the perturbative limit.
Using a non-perturbative estimate of the magnetic coupling obtained from the exact
solution of the two-center problem, the Heisenberg Hamiltonian has proved to be
extremely efficient in the treatment of the ground state and lowest excited states of
conjugated hydrocarbons, especially when the Hamiltonian adds a scalar potential
to reproduce the effect of the sigma bonds and when the magnetic coupling,
extracted from accurate ab initio calculations on the ethylene molecule, is
geometry-dependent [35]. Nevertheless it is desirable to produce a demonstration
valid whatever the value of the |t/U| ratio, and especially when U tends to zero.
Starting from the Hubbard Hamiltonian Lieb has given a general proof of the
same theorem (called Lieb’s theorem [36] in Solid State Physics community) for
regular alternant lattices. Our approach is different, it rests on the identification of
n* − n linearly independent non-bonding MOs, defined on the atoms of major
color, from the Hückel (U = 0) limit. Their existence does not depend on the values
of the inter-site hopping integrals. Then the interaction between the electrons in
these non-bonding MOs is purely ferromagnetic. As the exchange integrals between
the orthogonalized SOMOs are necessarily positive, the ground state is of major
spin multiplicity, which demonstrates the Ovchinnikov’s statement. Notice that the
exchange integrals between the SOMOs are necessarily positive, even for disjoint
diradicals, where it falls to zero in the crude Hubbard approximation. Then in full
generality the spin multiplicity of the ground state is necessarily equal to the
number n* − n of SOMOs plus one.
To summarize this section we may say that
• we have demonstrated the Ovchinnikov’s rule regarding the preferred spin
multiplicity of alternant graphs starting from the strong delocalization limit,
rather than from the strong correlation limit and an Heisenberg Hamiltonian, as
originally done. Our demonstrations are valid even when the |U/t| ratio tends to
zero,
• the Hückel picture enables us to predict the spin densities from
back-of-an-envelope calculations, while the solution of the Heisenberg
Hamiltonian are not accessible in such an easy manner (not to speak of ab initio
computations!),
14 Magnetic Properties of Conjugated Hydrocarbons …
375
determinants obtained by a spin exchange on the various bonds, which is the
maximum number of interactions which one may find on a line of the Hamiltonian.
Therefore this determinant only belongs to the lowest eigenvector which necessarily
has an M s = |n* − n| value, and an S = |n* − n| + 1 spin multiplicity.
Of course the Heisenberg Hamiltonian is supposed to be valid for systems where
the electron-electron repulsion prevails on the electron delocalization. It may be
established as an effective Hamiltonian from the Hubbard Hamiltonian by a
second-order expansion, using the quasi degenerate perturbation theory, provided
that the ratio |2t/U| < 1, which results in a value of J = 2t
2 /U. This inequality is
hardly satisfied in conjugated hydrocarbons, which are typically considered as
strongly delocalized (almost metallic) or weakly correlated. The validity of magnetic treatments of half-filled bands actually extends beyond the perturbative limit.
Using a non-perturbative estimate of the magnetic coupling obtained from the exact
solution of the two-center problem, the Heisenberg Hamiltonian has proved to be
extremely efficient in the treatment of the ground state and lowest excited states of
conjugated hydrocarbons, especially when the Hamiltonian adds a scalar potential
to reproduce the effect of the sigma bonds and when the magnetic coupling,
extracted from accurate ab initio calculations on the ethylene molecule, is
geometry-dependent [35]. Nevertheless it is desirable to produce a demonstration
valid whatever the value of the |t/U| ratio, and especially when U tends to zero.
Starting from the Hubbard Hamiltonian Lieb has given a general proof of the
same theorem (called Lieb’s theorem [36] in Solid State Physics community) for
regular alternant lattices. Our approach is different, it rests on the identification of
n* − n linearly independent non-bonding MOs, defined on the atoms of major
color, from the Hückel (U = 0) limit. Their existence does not depend on the values
of the inter-site hopping integrals. Then the interaction between the electrons in
these non-bonding MOs is purely ferromagnetic. As the exchange integrals between
the orthogonalized SOMOs are necessarily positive, the ground state is of major
spin multiplicity, which demonstrates the Ovchinnikov’s statement. Notice that the
exchange integrals between the SOMOs are necessarily positive, even for disjoint
diradicals, where it falls to zero in the crude Hubbard approximation. Then in full
generality the spin multiplicity of the ground state is necessarily equal to the
number n* − n of SOMOs plus one.
To summarize this section we may say that
• we have demonstrated the Ovchinnikov’s rule regarding the preferred spin
multiplicity of alternant graphs starting from the strong delocalization limit,
rather than from the strong correlation limit and an Heisenberg Hamiltonian, as
originally done. Our demonstrations are valid even when the |U/t| ratio tends to
zero,
• the Hückel picture enables us to predict the spin densities from
back-of-an-envelope calculations, while the solution of the Heisenberg
Hamiltonian are not accessible in such an easy manner (not to speak of ab initio
computations!),
14 Magnetic Properties of Conjugated Hydrocarbons …
375