Page 118 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

Quantum Chemistry in Front of Symmetry-Breakings

J.P. MALRIEU and J.P. DAUDEY
Laboratoire de Physique Quantique, Université Paul Sabatier
118 route de Narbonne, 31062 Toulouse, France

1. Introduction
Symmetry breaking is a universal phenomenon, from cosmology to the microscopic world,
a perfectly familiar and daily experience which should not generate the reluctance that it
induces in some domains of Physics, and especially in Quantum Chemistry. In classical
physics, the symmetry breaking of an a-priori symmetrical problem is sometimes refered
to as the lack of symmetry of the initial conditions. But it may be a deeper phenomenon, the
symmetry-broken solutions being more stable than the symmetrical one.
Quantum chemistry experiences two types of symmetry breakings.
One is purely formal, it concerns the departure from symmetry of an approximate solution
of the Schrödinger equation for the electrons (ie within the Born-Oppenheimer
approximation). The most famous case is the symmetry-breaking of the solutions of the
Hartree-Fock equations[l-4]. The other symmetry-breaking concerns the appearance of
non symmetrical conformations of minimum potential energy. This phenomenon of
deviation of the molecular structure from symmetry is so familiar, confirmed by a huge
amount of physical evidences, of which chirality (i.e. the existence of optical isomers) was
the oldest one, that it is well accepted. However, there are many problems where the
Hartree-Fock symmetry breaking of the wave function for a symmetrical nuclear
conformation and the deformation of the nuclear skeleton are internally related, obeying the
same laws. And it is one purpose of the present review to stress on that internal link.

2. Symmetry breakings of the electronic wave function
The Schrödinger equation being linear, H commutes with the symmetry operations of space
and spin, and the wave function must be symmetry-adapted. This is the basic doxa which
we transmit to our students. If they are critical, they perhaps wonder why the atomic
orbital of the hydrogen atom is an eigenfunction, while symmetry-broken. Actually, we
usually do not take time to mention that for degenerate roots, it is the projector on the stable
subspace of these degenerate eigenvectors which commutes with the symmetry operators of
the problem. But the drama arises when the desired state is non degenerate and when an
approximate method delivers a symmetry-broken wave-function. The results is in general
considered negatively as spurious, contaminated and irrelevant, despite the fact that
meaningfull physics have been introduced in these solutions in a biased way, lowering the
energy with respect to the symmetry-adapted description obtained at the same level of
sophistication.
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Y. Ellinger and M. Defranceschi (eds.), Strategies and Applications in Quantum Chemistry, 103–118.
© 1996 Kluwer Academic Publishers. Printed in the Netherlands.