Page 123 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

108 J. P. MALRIEU AND J. P. DAUDEY
distribution is random without any privilege neither for the spread of the three electrons in
three p AOs nor for the spin alignment, which would satisfy the atomic Hund's rules. The
UHF solution may be written :

where the MOs concentrate on different atoms and where the spin distribution is
antiferromagnetic (each a spin atom being surrounded by three spin atoms), and it
appears close to the equilibrium interatomic distance [21]. It is clear that it corrects the
spurious charge fluctuations on the atoms and satisfy their intrinsic preferences.
Symmetry breakings have been studied for systems with one electron per center such as the
systems of cyclic polyenes [22,23]. One finds here both charge-density-wave RHF
solutions , where the bond indexes are alternant (one strong bond (2i, 2i+i) between two
weak bonds (2i-i, 2i) and (2i+l, 2i+2) and spin-density-wave UHF solutions where the
electrons are spin-alternant (one electron on atom 2i surrounded by two electrons on
atoms The first one does not "dissociate" properly (when t/u tends to zero), since it
remains half neutral and half ionic but it reduces the weight of the most irrelevant VB
situations with respect to their importance in the symmetry-adapted solution. The charge-
density-wave solution tends to localize the electrons by pairs on the "strong bonds",
each one supporting a localized MO

with small delocalization tails. In such a function the probabilities to find one electron of
spin, one electron of spin, two electrons or zero electron on each atom remain equal,
as it occured in the symmetry-adapted HF function. But the probability to find two adjacent
positive or negative charges is now diminished (at least in the "strong bond") and the
avoidment of such high energy situation through the charge density wave RHF solution
lowers the energy (at least when PPP hamiltonian is prefered to the less realistic Hubbard
Hamiltonian which only counts the ionicity of each VB structure ). As a consequence of
that pairing of electrons in bonds, the probability to find two electrons of the same spin on
adjacent atoms is also diminished with respect to its probability of occurence in the
symmetry-adapted solution and this reduction is overestimated compared to the exact wave
function.
The UHF solution appears when the hopping integral t becomes small and leads to a spin
density wave. The localization of the MOs leads to a and b atom centered orbitals, localized
around odd and even labelled atoms respectively.

This solution can only be reached in linear or cyclic polyenes for rather unrealistic t/u ratios
(i.e. lengthened CC bonds) while it occurs in cyclic ideal clusters for realistic
interatomic distances in ab initio calculations [24]. But in that case another fascinating
symmetry breaking takes place, namely a bond-centered spin-density wave, as discovered
by Mc Adon and Goddard [24]. This UHF solution is much lower in energy, and it
consists in an antiferromagnetic distribution of the electrons, each electron occupying a MO
centered midway between adjacent atoms. In this solution the electrons have left the atoms
and each of them occupy its own cell, i.e. is delocalized into the largest intersticial zone.
This is made possible by the fact that a strong s-p hybridization does not require too much