Page 119 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

104 J. P. MALRIEU AND J. P. DAUDEY
The most famous case concerns the symmetry breaking in the Hartree-Fock approximation.
The phenomenon appeared on elementary problems, such as when the so-called
unrestricted Hartree-Fock algorithms were tried. The unrestricted Hartree-Fock formalism,
using different orbitals for a and electrons, was first proposed by G. Berthier [5] in 1954
(and immediately after by J.A. Pople [6] ) for problems where the number of α and
electrons were different. This formulation takes the freedom to deviate from the constraints
of being an eigenfunction.
For problems, where the ground state is a singlet state, the use of such a wave
function appeared to give significantly lower energies than the orthodox symmetry-adapted
solution in many problems, as illustrated below. Later on other types of symmetry breaking
have been discovered and Fukutome [7] has given a systematics of the various HF
instabilities in a fundamental paper.

2.1. ATOMIC PHYSICS
In the Be atom, the two valence electrons occupy a 2s, 2p valence shell, the 2s and 2p
Atomic Orbitals (AO) having an important "differential overlap" (ie a good coincidence of
their spatial extension). The contribution of the 2p AO to the angular correlation of the
valence electrons is especially large (the Moller Plesset expansion from
being poorly convergent) and the proper valence function should be written

while the RHF approximation is reduced to the component. One obtains a much
lower energy using an UHF function which looses both the space and symmetry
constrainsts. The single determinant

is lower in energy than the best RHF solution due to the inclusion of some angular
correlation through the component, despite the contamination by the triplet
configuration This example illustrates wonderfully the physically suggestive
potentiality of the symmetry-broken solution. Since it tells us that when the α electrons is
on the right side of the nucleus the electron prefers to move into an
hybrid, ie on the left side of the nucleus. This is the best translation of the angular
correlation, and it is clear that superimposing and the degenerate non orthogonal
solution

into

will restore the singlet character of the wave function by eliminating the triplet
contamination but still disobeying the space-symmetry constraint [8].