Page 207 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

190 J. M. ANDRÉ ET AL.
With respect to correlation, the behaviour of the hydrogen molecule studied in a
subminimal FSGO basis set is still more striking than the one observed in a minimal basis
set. By symmetry arguments, the single FSGO which describes the electron pair of the
hydrogen molecule is centred at the middle of the H-H bond. As the internuclear distance
increases and ultimately when the molecule dissociates, such a description would lead to
a physical nonsense. Indeed, at the dissociation limit, this would correspond to two
protons and an isolated pair of electrons

Thus, we understand that, in the FSGO model, for some critical distance, the single
Gaussian will jump from its symmetric position at the middle of the H-H bond to a
dissymetric one represented below. Thus, the FSGO dissociation scheme corresponds to
one electron pair on one of the proton and no electron on the second proton

This behaviour is a nice example of the symmetry dilemma in the conventional Hartree-
Fock scheme and is intimately connected with the question of Hartree-Fock instability.
In this paper, we analyze the instabilities which appear in the Hartree-Fock method. Our
analysis is made in the framework of basis sets. In the hydrogen molecule, the single
floating Gaussian orbital (FSGO) desribing the electron pair has its optimal position in
the middle of the hydrogen molecule only for small internuclear distances. For large
enough distances its optimum position is close to one of the nuclei and a broken-
symmetry solution is thus preferred. Application of an external electric field along the
molecular axis induces some additional instabilities in the lowest-energy solution with
respect to the electric field value. Here, both types of instabilities are investigated
analytically as well as numerically.
This paper is, thus, a double tribute to Professor Berthier. On one side, G. Berthier has
provided excellent analysis of quantum mechanical instabilities [4], while additionally
being at the origin of the interest of the Namur group for studies of (hyper)polarizabilities
in organic molecules and chains.

2. Subminimal basis set Hartree-Fock-type calculations of the hydrogen molecule
In a FSGO basis set, the Gaussian orbital is defined by:

where both, the Gaussian exponent and the Gaussian center are optimized in a FSGO
calculation. Depending on the complexity of the calculation, we could have to compute
the following integrals:
one-electron integrals: