Page 43 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

28 C. CHAVY ET AL.
where i.e. we will consider only natural orbitals.
iii) Strongly versus weakly occupied orbitals. It is then seen on the expression 19 that
appears in the eq.(10) multiplied by when natural orbitals are
used. Thus, if is small then dominate
the remaining terms of the eq.(10) only in a very small volume around the nucleus
of A. In the remaining part of the volume occupied by the molecular system the
description of this orbital cannot be deduced from the Valley theorem. Therefore,
we will consider here only strongly occupied orbitals with
In fact, a simple description of the weakly occupied orbitals resulting from valence
MCSCF calculations has already been presented (12) .
iv) Canonical versus non canonical orbitals. Let us now consider the right hand side
of the eq.(10) which depends of the off diagonal Lagrange multipliers through terms
like Such terms may present very steep variations with so that the
Valley theorem may lead to no special conclusion. Therefore, we consider here only
the cases where one can have . A similar restriction has not been made
in the atomic case (section 1.4 above) because it turns out that is very
small in all useful cases.
v) Partial waves versus orbital . Finally it is worth noting already that the present
approach will tell us nothing concerning the orbitals themselves! It will tell us some-
thing only on each of the partial wave around A separatly : the relative weights of
the different partial waves in the total orbital do not result from the local compensa-
tion between T and . It appears rather as a global property of the molecular
system .

Let us note that the two conditions can be satisfied
only with canonical SCF orbitals. Thus, in fact, the present theory can be applied
only in such cases. However it has been demonstrated (12) that in most systems,
the strongly occupied MCSCF orbitals and the SCF orbitals are extremely close one
to the others. Therefore, in practice, the present theory also applies to the strongly
occupied MCSCF orbitals.

On the all, the limitations coming from the above hypotheses i-v are :
- one can find a description of the partial waves of the optimum orbitals near each
atom separatly , not of the orbitals themselves;
- these descriptions concern only the strongly occupied canonical orbitals, not any
type of orbitals.

We now return to the eq.(10). In the frame of the hypotheses i-v it writes :