Page 125 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

110 J. P. MALRIEU AND J. P. DAUDEY

and the amplitude of the transfer integral is given by

This quantity may be very weak. Now starting from the trial function

one may reach under certain circumstances an HF solution of spin-restricted or spin-
unrestricted character (this is not the main point) localizing the hole on site A or on site B

Now the core functions are adapted to the static electric field of the broken symmetry
situation, they are adapted for instance to the situation while the symmetry-adapted
solution optimized the cores in an field. We qualitatively understand the
physics of the problem :
The symmetry-breaking of the HF function occurs when the resonance between the two
localized VB form and is weaker than the electronic relaxation which one
obtains by optimizing the core function in a strong static field instead of keeping it in a
weak symmetrical field. If one considers for instance binding MOs between A and A' they
do not feel any field in the SA case and a strong one in the SB solution. The orbitals around
+
A concentrate, those around A' become more diffuse than the compromise orbitals of
and these optimisations lower the energy of the form. As a
counterpart, the energy of the which describes the situations in the
field of the core polarized in the situation grows up and the interaction between the
two resonant VB forms is now completely underestimated. This phenomenon was first
noticed for the ionization of the core levels of homonuclear diatoms ; the
ionization potentials from symmetry-adapted HF calculations give correct estimates of the
small energy splitting between the and ionization potentials but overestimate both of
them by several eV while the symmetry-broken solutions gave the correct mean energy and
miss the energy splitting. The mechanism was first elucidated by Snyder et al. [30], by
Denis et al., [31] in a more general and more correct analysis and later on by Cederbaum et
al. (32).
The HF symmetry-breaking also occurs in the valence shell for systems when the
overlap between the two electron donating groups is too low. One may quote for instance
recent works devoted to molecular architectures designed for Molecular Electronics. An
analytic treatment of the symmetry breaking of the HF function for such problems may be
found in ref. [33] where it is shown that the symetry breaking is a bifurcation.
Another interesting problem concerns the case of a weak resonance between two ionic VB
structures as occurs in the singlet state of ethylene (and longer polyenes). This state
(the allowed state) is of ionic VB content