Page 124 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

QUANTUM CHEMISTRY IN FRONT OF SYMMETRY BREAKINGS 109

energy. This solution is physically based on a good compromise between the electronic
repulsion, which keeps the electrons apart, one per cell, the kinetic energy which is higher
than in the delocalized RHF solution but lower than in the atom-centered UHF solution.
The benefit of that optimal balance compensates a weak diminution of nuclear attraction.
This discovery, confirmed by GVB [25] and later by CI calculations [26], led McAdon and
Goddard to propose a rather revolutionary picture of the metal, the "intersticial picture"
[27]. Lepetit et al., [26] have shown that
- for a 2n-electron problems there are different UHF solutions which differ
essentially by the distribution of the spin, the localized UHF MOs of the different UHF
being almost identical (except for small tails) and defining an unvariant vectorial space,
- there are similar solutions, up to the ferromagnetic one with similar content
of the localized MOs,
- the antiferromagnetic solution is the lower in energy, but the hierarchy of the energies
obeys the logics of an Heisenberg Hamiltonian. This means that the delocalization between
the interstices is small enough to be treated as a perturbation, through effective spin
couplings. From that hierarchy of energies of the various UHF solutions one may estimate
the amplitude of the spin coupling; and solving the Heisenberg Hamiltonian for the cluster
one obtains an energy quite close to the best CI estimates [26].
This strategy has been successfully applied to infinite periodic 1-D chains of Li atoms [28],
through the first symmetry-broken application of the ab-initio UHF version of the Torino's
CRYSTAL package [29]. The results of this work and of further treatments of 2-D lattices
of Li and even Mg ( Lepetit and coworkers, to be published) all confirm the validity of the
intersticial picture. This is a case where the symmetry-broken HF solutions have led to a
completely new picture of the electronic assembly.
When the symmetry breaking of the wave function represents a biased procedure to
decrease the weights of high energy VB structures which were fixed to unrealistic values
by the symmetry and single determinant constraints, one may expect that the valence
CASSCF wave function will be symmetry-adapted, since this function optimizes the
coefficients of all VB forms (the valence CASSCF is variational determination of the best
valence space and of the best valence function, i.e. an optimal valence VB picture). In most
problems the symmetry breaking should disappear when going to the appropriate MC SCF
level. This is not always the case, as shown below.

2.5. SYMMETRY BREAKING IN CASE OF WEAK RESONANCE BETWEEN
POLARIZED FORMS

In systems such as where an electron (or a hole) hesitates or oscillates between
two equivalent positions on subsystems A or A', symmetry breakings may occur when the
effective transfer integral between the two sites is weak. This will be the case when A and
A' are far apart, when they are bridged by an "insulating" ligand, or when the two localized
MOs concerned by the electron transfer have a very weak spatial overlap.
Actually in such problems the symmetry-adapted solutions should be

The two solutions may be reached independently since they belong to different symmetries.
Then one may define localized MOs, on A and A' respectively :