Page 37 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

22 C. CHAVY ET AL.

is the extension providing a qualitative information (weak e dependence) valid inside
a finite volume . This last aspect (finite volume) is the one that allows the description
of the optimum orbitals in molecular systems.

The Cusp and the Valley theorems express the same aspect of the Schrödinger equa-
tion, eq.(l) : since has no pole for r=0, the pole of can be compensated
only by ; but a pole of with a residue equal to -Z implies the Cusp theorem
(at the origin) and the Valley theorem (inside a finite volume around the origin).

It should be noted that the weak energy dependence of the orbitals inside a finite
volume around the nucleus has already been noted and used in different contexts : the
numerical determination of atomic orbitals (4) as well as the scattering of electrons
by atoms (5).
1.3. ORBITAL OPTIMISATION IN POLYELECTRONIC SYSTEMS

The equation determining the optimum orbitals of polyelectronic systems in the case
of the SCF and MCSCF theories can be written in the form :

where
- are the creation operators corresponding to the orbitals and and j, l the
anihilation operators for the orbitals and
- h is the one electron part of the total Hamiltonian.
- is a local operator :

- the factors are the Lagrange multipliers that take care of the orthonormality
constraints.

1.4. POLYELECTRONIC ATOMS
We consider here only the SCF case where the off diagonal factors vanish.
In addition, we assume that the orbitals satisfy the usual symmetry constraint i.e.
that they are pure s, p, d ... functions (RHF approach). On the other hand, no spin
constraint is assumed. Then the eq.(10) is most conveniently written as :