The agreement is similar regarding the antiferromagnetic diradicals, reported in
Table 14.8, although we may only compare the topological model to the UDFT
calculations, which incorporate spin polarization effects. In these cases the
approximate spin-decontamination process of Yamaguchi was used [50, 51]. The
first striking result is the smallness of the K ab exchange integral, calculated at the
RODFT level, using the magnetic MOs of the triplet and localizing them. This
result is in agreement with the topological analysis. One may see a dramatic contrast between the values of the exchange integral in meta- versus para-dimethylene
acenes (series 1′a, 2′a, 3′a in Table 14.5, versus series 1″a, 2″a, 3″a in
Table 14.8). While the distances between the methylene groups are similar the
exchange integral almost vanish in the antiferromagnetic series. The order of
magnitude of the singlet to triplet energy gap is again correctly predicted by the
topological model. Of course this model predicts a zero value for the disjoint
diradicals 2″c and 3″c, but the DFT calculations confirm the extremely low value
of the corresponding excitation energies (0.078 and 0.018 eV respectively).
The quantitative impact of the spin polarization on the energy difference between
the singlet and the triplet states is significant but it is not directly accessible in DFT
calculations. One may of course compare the restricted DFT and unrestricted DFT
gaps, for instance in the dimethylene naphtalene 2′a which has a triplet ground
state. The RDFT values of the excitation energies are 1.0 eV while the UDFT
calculation (corrected for the spin contamination) gives 0.38 eV. This overestimation is essentially due to the closed-shell constraint imposed to the restricted
DFT calculation of the Singlet, which is not physical in these ferromagnetic systems, rather than to the spin polarization itself. The core spin-polarization free
description should be the two-electron in two-orbital CASSCF solution. This
description properly reduces the ionic component of the wave function which is
exceedingly large in the restricted Ms = 0 wave function.
It is interesting at this point to return to the cases which appeared as unpredictable from the spin-delocalization only, such as the tetra-methylene ethylene
(TME). The spin polarization contribution of the π doubly occupied MOs acts in
favor of the singlet, according to the preceding arguments, despite the fact that the
kinetic exchange was not decisive, due to the smallness of the hopping integral
between the localized singly occupied MOs. The spin-polarization contribution
strictly obeys the parity determinism expressed by the Ovchinnikov’s rule. This is
observed in compounds 2″c and 3″c (Table 14.8). It is worth noticing that the σ
bond spin polarization works in the same direction as that of the π system, the CC σ
bond connecting the two allyl groups in TME prefers a complete spin alternation in
the π system. Notice that the spin-polarization mechanism represents a step in the
MO picture toward the Heisenberg description. Both treatments introduce for
instance negative spin densities in the triplet state.
390
J.-P. Malrieu et al.
Table 14.8, although we may only compare the topological model to the UDFT
calculations, which incorporate spin polarization effects. In these cases the
approximate spin-decontamination process of Yamaguchi was used [50, 51]. The
first striking result is the smallness of the K ab exchange integral, calculated at the
RODFT level, using the magnetic MOs of the triplet and localizing them. This
result is in agreement with the topological analysis. One may see a dramatic contrast between the values of the exchange integral in meta- versus para-dimethylene
acenes (series 1′a, 2′a, 3′a in Table 14.5, versus series 1″a, 2″a, 3″a in
Table 14.8). While the distances between the methylene groups are similar the
exchange integral almost vanish in the antiferromagnetic series. The order of
magnitude of the singlet to triplet energy gap is again correctly predicted by the
topological model. Of course this model predicts a zero value for the disjoint
diradicals 2″c and 3″c, but the DFT calculations confirm the extremely low value
of the corresponding excitation energies (0.078 and 0.018 eV respectively).
The quantitative impact of the spin polarization on the energy difference between
the singlet and the triplet states is significant but it is not directly accessible in DFT
calculations. One may of course compare the restricted DFT and unrestricted DFT
gaps, for instance in the dimethylene naphtalene 2′a which has a triplet ground
state. The RDFT values of the excitation energies are 1.0 eV while the UDFT
calculation (corrected for the spin contamination) gives 0.38 eV. This overestimation is essentially due to the closed-shell constraint imposed to the restricted
DFT calculation of the Singlet, which is not physical in these ferromagnetic systems, rather than to the spin polarization itself. The core spin-polarization free
description should be the two-electron in two-orbital CASSCF solution. This
description properly reduces the ionic component of the wave function which is
exceedingly large in the restricted Ms = 0 wave function.
It is interesting at this point to return to the cases which appeared as unpredictable from the spin-delocalization only, such as the tetra-methylene ethylene
(TME). The spin polarization contribution of the π doubly occupied MOs acts in
favor of the singlet, according to the preceding arguments, despite the fact that the
kinetic exchange was not decisive, due to the smallness of the hopping integral
between the localized singly occupied MOs. The spin-polarization contribution
strictly obeys the parity determinism expressed by the Ovchinnikov’s rule. This is
observed in compounds 2″c and 3″c (Table 14.8). It is worth noticing that the σ
bond spin polarization works in the same direction as that of the π system, the CC σ
bond connecting the two allyl groups in TME prefers a complete spin alternation in
the π system. Notice that the spin-polarization mechanism represents a step in the
MO picture toward the Heisenberg description. Both treatments introduce for
instance negative spin densities in the triplet state.
390
J.-P. Malrieu et al.