Page 213 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

196 J. M. ANDRÉ ET AL.
we obtain:

In all the other schemes (UHF, PHF, EHF), the dissociation limit is the correct one
corresponding to two neutral hydrogen atoms (2H-); each FSGO-hydrogen atom energy is
thus obtained by the simple variational procedure:

dissociation limit for two hydrogens:
As a final comment, it is interesting to note that this FSGO study of the hydrogen
molecule offers a new and simple illustration of the behavior of sophisticated Hartree-
Fock schemes like UHF, PHF and EHF. Furthermore, it provides a very efficient
numerical example of instabilities in the standard Hartree-Fock method. It is important to
see that the UHF, PHF and EHF schemes all correct the wrong RHF behavior and lead to
the correct dissociation limit. However, the UHF and PHF schemes only correct the wave
function for large enough interatomic distances and the effect of projection in the PHF
scheme even results in a spurious minimum. The EHF scheme is thus the only one which
shows a lowering of the energy with respect to RHF for all interatomic distances.

3. Subminimal basis set Hartree-Fock-type calculations of the hydrogen molecule in
an external electric field.

When applying an external electrical field to the FSGO model of the hydrogen molecule,
one expects that the floating gaussian will be moved in accordance with the polarity of
the field, i.e., displaced towards the positive pole. Thus, near equilibrium internuclear
distances, a minimum should be obtained close to the middle of the molecule. On the
other hand, continuing to move the floating gaussian towards the positive pole, a barrier
should appear close to the hydrogen atom the gaussian is floating towards. After having
passed that barrier, the “energy catastrophe” of the unbound perturbing potential should
produce an infinitely negatively stable position. This is the type of behaviour which is
listed in Table 3 and illustrated in Figure 3. For the equilibrium internuclear distance (R =
1.474 a.u.) and the optimal exponent we compute the energy as a
function of the orbital position for various strength of the external electrical field (F
and 0.5 a.u.). The energy formulae can be obtained from the
Hamiltonian for the hydrogen molecule in the electric field (z being the axis
of the molecule):