Page 127 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

112 J.P.MALRIEU AND J.P. DAUDEY

2.7. MC SCF SYMMETRY-BREAKING
The precedings sections concentrated on single determinant variational functions. One may
wonder whether going to multiconfigurational SCF functions will restore symmetry and
when.
We have attributed the origin of the HF symmetry breaking of homopolar simple and
multiple bonds or in symmetric homoatomic clusters to unrealistic constraints on the
coefficients of the different VB components. Going to a valence CASSCF (ie an optimal
valence CI function) should restore the symmetry.
This will not be the case in the weak resonance systems. Notice that in these problems there
are only two dominant VB configurations (for instance and that the
minimal valence CAS function

reduces to the symmetry-adapted single determinant:

And actually the two-determinant function will be symmetry-broken for a symmetric
configuration when the resonance energy is weaker than the polarization energy. This has
been observed first by Ellinger et al. [36] in a problem with three electrons in two
equivalent orbitals on distant oxygen atoms. The authors restored the symmetry by
considering a local "antibonding" lone pair (or a 3p type orbital) in the active space in order
to reintroduce into the CAS function the instantaneous repolarization of the oxygen orbitals,
which are more diffuse when they are occupied by two electrons than by only one. But this
is by no means a universal recipe. If the two oxygen atoms were separated by more bonds,
the resonance would diminish and the rest of the dynamical polarization (the polarization of
the bonds between the oxygen atoms) would be larger than the resonance, inducing a new
symmetry-breaking of the enlarged CASSCF function.
The problem has already received a dramatic illustration on the LiF molecule, where the
avoided crossing between the ionic and the neutral VB configurations takes place at such a
large interatomic distance that the transfer integral is very
small The orbitals of are completely different of those of LiF., in spatial
extension and distortion (p-d mixing) and these relaxation phenomena bring much more
energy than the interaction between the ionic and the neutral VB structures. So that
Bauschlicher et al. [37] have never succeeded in making their CASSCF functions
continuous around the avoided crossing (as they should be) despite the enlargement of the
active space to all valence electrons in up to 12 active orbitals. As explained in ref. [38] this
failure is due to the fact that the part of the dynamical polarization which remains out of the
CAS space is still larger than the electron transfer integral.
CASSCF calculations are not a universal solution to symmetry-breaking of the wave
functions, and for such weak resonance problems it is far more reliable to start from state
average solutions which treat on an equal footing the two configurations which interact
weakly.