Page 45 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

30 C. CHAVY ET AL.

This equation is similar to the eq.(15) obtained in the atomic case. Thus one can
switch at will between the atomic and the molecular cases : if we give to the param-
eters the values determined for the atom as described in the
above section 1.4 (this implies ), then f is proportional to the RHF orbital of
the atom A with the quantum numbers l and m and the energy e ; if alternatively we
give to the parameters the values obtained for a molecular system as just explained,
then f is proportional to the l, m partial wave of the orbital of the molecular system
with the energy e.

We now use the Valley theorem : the atomic function f depends weakly of the
e parameter in a large region near the nucleus. It can be seen by inspection of
the eqs.(15),(16),(17) and (23) that the critical parameter in the molecular case is
an effective energy where are the
differences between the values of at the origin in the molecular
case and in the atomic case. Therefore, if is not too different from the atomic
orbital energy,then the two f functions obtained with the atomic and molecular values
of the parameters arc extremely close to be proportional one of the other in a finite
region near the nucleus. Stated differently : in a finite region near the nucleus of an
atom A, the partial waves of the optimum orbitals centered on A are proportional
to the corresponding RHF orbitals of the atom A , unless the atomic and molecular
parameters are very different from each other (i.e. unless the difference is much larger
than the variations mentionned in the section 1.5).

3. Asymptotic conditions
The Valley theorem leads to simple conditions for the optimised orbitals near the
nuclei. However these conditions are not sufficient to characterize these orbitals :
one needs in addition to take the asymptotic form of the equations into account.
In the asymptotic region, an electron approximately experiences a potential,
where is the charge of the molecule-minus-one-electron ( in the case of a
neutral molecule) and r the distance between the electron and the center of the charge
repartition of the molecule-minus -one-electron. Thus the orbital describing the
state of that electron must be close to the asymptotic form of the irregular solution
of the Schrödinger equation for the hydrogen-like atom with atomic number

(see for instance the Eq.13.5.2 of Ref.9) where e is the orbital energy. Since e is differ-
ent in the molecule and in the separated atoms, this asymptotic behaviour cannot be
represented properly if the molecular orbital is approximated by a linear combination
of the RHF orbitals of the separated atoms.

4. The case of
The interest of in the present context is that it provides a good test for the present
orbital optimisation theory because one knows the exact solution.