treatment is only one half of the total one [10, 53, 54]. Nevertheless it is possible to
estimate the contributions of the various physical effects entering in the energy
difference between the triplet and the singlet states from the unrestricted
single-determinant approaches [55, 56].
Exploiting the mirror theorem and the Hubbard Hamiltonian it is possible to
reach some qualitative conclusions from the analytic treatment of the spin polarization effect, which apply to conjugated hydrocarbons. We do not give here the
explicit derivations. They may be summarized as follows:
• in radicals and in ferromagnetic diradicals the spin polarization increases the
positive spin densities on the π orbitals of major color sites,
• it introduces negative spin densities on the π orbitals of minor color sites,
• this effect is non-local, the spin polarization introduces spin densities in regions
of the molecule where they were zero at the topological description level,
• the spin polarization usually contributes to increase the singlet triplet energy
gap, whatever its sign,
• the spin polarization of the doubly occupied π and σ MO fixes the preferred spin
multiplicity of the “undecidable” diradicals through the parity of the number of
Carbon atoms separating the disjoint radicalar regions. The ground state is
singlet when this number is even, triplet when it is odd.
All these statements will be illustrated in the next section.
14.5 Comparison Between Topological Assessments
and Numerical DFT Calculations
In order to assess the validity of our topological predictions, concerning both
energy differences and spin distributions, we decided to perform DFT calculations
on the previously introduced series of hydrocarbons. To make closer the comparison with topological Hamiltonians we used ideal geometries with equal CC (1.40
Å) and CH (1.05 Å) bond lengths, planar geometries and 120° angles. The basis set
was of triple zeta plus polarization quality, and the exchange correlation potential
was the B3LYP one. Both restricted and unrestricted calculations will be reported.
We did not compare with Hartree-Fock calculations since they are known [57–60]
to give a spurious concentration of the unpaired electron on the external site. Full
Configuration Interaction of the π electrons in the π valence MOs are usually
necessary to obtain reliable spin densities. The resulting natural MO, obtained at a
high computational cost, are close to the Hückel MOs and to the DFT MOs. The
numerical calculations used the B3LYP exchange correlation potential in the
Gaussian package [61]. The decomposition of the energy difference between the
singlet and the triplet has been performed according to the recently proposed
method [55, 56].
14 Magnetic Properties of Conjugated Hydrocarbons …
385
estimate the contributions of the various physical effects entering in the energy
difference between the triplet and the singlet states from the unrestricted
single-determinant approaches [55, 56].
Exploiting the mirror theorem and the Hubbard Hamiltonian it is possible to
reach some qualitative conclusions from the analytic treatment of the spin polarization effect, which apply to conjugated hydrocarbons. We do not give here the
explicit derivations. They may be summarized as follows:
• in radicals and in ferromagnetic diradicals the spin polarization increases the
positive spin densities on the π orbitals of major color sites,
• it introduces negative spin densities on the π orbitals of minor color sites,
• this effect is non-local, the spin polarization introduces spin densities in regions
of the molecule where they were zero at the topological description level,
• the spin polarization usually contributes to increase the singlet triplet energy
gap, whatever its sign,
• the spin polarization of the doubly occupied π and σ MO fixes the preferred spin
multiplicity of the “undecidable” diradicals through the parity of the number of
Carbon atoms separating the disjoint radicalar regions. The ground state is
singlet when this number is even, triplet when it is odd.
All these statements will be illustrated in the next section.
14.5 Comparison Between Topological Assessments
and Numerical DFT Calculations
In order to assess the validity of our topological predictions, concerning both
energy differences and spin distributions, we decided to perform DFT calculations
on the previously introduced series of hydrocarbons. To make closer the comparison with topological Hamiltonians we used ideal geometries with equal CC (1.40
Å) and CH (1.05 Å) bond lengths, planar geometries and 120° angles. The basis set
was of triple zeta plus polarization quality, and the exchange correlation potential
was the B3LYP one. Both restricted and unrestricted calculations will be reported.
We did not compare with Hartree-Fock calculations since they are known [57–60]
to give a spurious concentration of the unpaired electron on the external site. Full
Configuration Interaction of the π electrons in the π valence MOs are usually
necessary to obtain reliable spin densities. The resulting natural MO, obtained at a
high computational cost, are close to the Hückel MOs and to the DFT MOs. The
numerical calculations used the B3LYP exchange correlation potential in the
Gaussian package [61]. The decomposition of the energy difference between the
singlet and the triplet has been performed according to the recently proposed
method [55, 56].
14 Magnetic Properties of Conjugated Hydrocarbons …
385