Page 209 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

192 J. M. ANDRÉ ET AL.

The “equilibrium” FSGO-hydrogen molecule is found for the parameters:
(could be compared to the experiment: 1.401 a.u.)
(center of the molecule)

One observes that the energy minimum at the calculated equilibrium internuclear distance
always corresponds to the symmetric solutions For the values given between
parentheses in the above table, the broken symmetry solution does not exist; the single
Gaussian remains centered at the middle of the H-H bond. However, for interatomic
distances greater than 5.6 a.u., the broken symmetry solutions (Gaussians centered near
give the absolute minimum while the symmetric solution has a higher
energy; this is a further example that, for approximate wave functions, the basic
symmetry properties do not follow automatically from the variation principle and
consequently do not have necessarily the full symmetry of the nuclear framework.
In the HF scheme, the first origin of the correlation between electrons of antiparallel spins
comes from the restriction that they are forced to occupy the same orbital (RHF scheme)
and thus some of the same location in space. A simple way of taking into account the
basic effects of the electronic correlation is to release the constraint of double occupation
and so use Different Orbitals for Different Spins
(DODS scheme which is the European way of calling UHF). In this methodology,
electrons with antiparallel spins are not found to doubly occupy the same orbital so that,
in principle, they are not forced to coexist in the same spatial region as is the case in usual
RHF wave functions.

A UHF wave function over different orbitals and is then:

The wave function obtained corresponds to the Unrestricted Hartree-Fock scheme and
becomes equivalent to the RHF case if the orbitals and are the same. In this UHF
form, the UHF wave function obeys the Pauli principle but is not an eigenfunction of the
total spin operator and is thus a mixture of different spin multiplicities. In the present
two-electron case, an alternative form of the wave function which has the same total
energy, which is a pure singlet state, but which is no longer antisymmetric as required by
the Pauli principle, is:

In both cases, the energy formula (E(UHF-FSGO)) is the same: