Page 42 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

THEORY OFORBITALOPTIMIZATION INSCFANDMCSFCALCULATIONS 27
particular norm chosen here. In fact, it can be checked that deviations smaller than
10% of the value of the last extremum, are obtained for r values up to a limit close
to the covalent radius of the atom in all three cases ( for the 2p(C) orbital,
for the 3p(Si) orbital and for the 3d(Sc) orbital).
We conclude from these numerical examples that it is possible to give a quantitative
and probably rather general expression of the Valley theorem (weak e dependence
of the orbital in a finite volume around the nucleus) : a variation of the energy of
ca. 0.2 H results in a variation of the function smaller than 10% of the last
extremum of until a distance of the nucleus equal to ca. 90% of the covalent
radius of the corresponding atom.

2. Molecular systems
We arrive now at the main purpose of the present work : to find a qualitative descrip-
tion of the optimum orbitals (obtained by SCF or MCSCF calculations) of molecular
systems.
To that end, we will start with the same equation as the one used above in the case
of polyelectronic atoms,viz. the eq.(10), and we will try to use the equivalent of the
compensation between the kinetic energy and the nuclear attraction (T and -Z/r)
found in the atomic case.

In fact, it turns out that the compensation between the kinetic energy and the nuclear
attraction does lead to a qualitative description of the optimum orbitals in molecular
systems, but only in the frame of the following restrictive conditions.

i) Global versus local description. In the case of molecular systems, the one electron
part of the electronic hamiltonian includes a sum over the electron-nuclear attraction
of all the nuclei:

Therefore it appears that the above mentionned compensation takes place separatly
in the vicinity of each atom. We can arrive to a description of the optimum orbitals ;
however this description is not global, but local in the sense that it concerns separatly
the regions around each atom. Thus, we will hereafter consider only the region of a
single atom, say A, and study the effect of the compensation between T and
ii) Natural versus non natural orbitals. The factor is always combined in the
eq. (10) with the factor according to

If appears in several terms corresponding to
different orbitals, and it is difficult to demonstrate directly that the compensation
occurs separatly for each orbital. Therefore, we will consider here only the cases