Page 51 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

36 C. CHAVY ET AL.
of the system, approximatly the region extending from the nucleus to the middle
of the bonds starting from that nucleus. In that region each partial wave of the
optimum orbital is proportional to the atomic orbital with the same value of the
quantum number l, unless the molecular potential differs too much from the atomic
potential, or unless the coupling term with other waves of the same orbital is too
strong (polarisation orbilals).
This description results from the fact that the optimum orbitals are essentially deter-
mined in the region surrounding each atom by the compensation between the kinetic
energy T of the electron and the Coulomb attraction of the electron by the nucleus
of that atom. This compensation implies that the orbital is very weakly dependent
of the environment of the atom in the molecular system so that it is essentially de-
termined by atomic conditions (Valley theorem).
A special aspect of this description appears if one starts the orbital optimisation
process with orbitals obtained by linear combinations of RHF orbitals of the isolated
atoms (LCAO approximation s.str.). Let and be the starting and final
orbitals of such a calculation. Then the difference between and in the
vicinity of each atom merely consists in a distortion of the atomic orbitals of each
atom. This distortion just compensates the contribution of the orbitals of the other
atoms to in order to restore the proportionality between the partial waves of
and the appropriate atomic orbital.

This description is completed by describing what happens outside the molecule : the
partial waves of the optimum orbital are there proportional to the irregular solution
of a radial equation involving the actual energy of the orbital .

We have checked, using as a test case, that the description of the optimum orbital
of the molecular system is then complete in the sense that it allows (assuming that
the orbital energy is known) to construct by a fit process an optimum orbital which
is very close to the one obtained by a diagonalisation process in a gaussian basis.

Clearly, several aspects of the orbital optimisation remain to be clarified. Firstly a
numerical test using a system more complex than should be made. What happens
to orbitals or strongly hybridized orbitals should be also examined. It would be also
interesting to explain how the optimisation - as described here - is related to an energy
lowering, as well as the practical use of the present description in actual calculations,
etc ... These different aspects will be examined in forthcoming publications.

References

1. W.Moffit, Proc. Roy. Soc London A 210,224,245 (1951).
R.Parr and J.Rychlewski, J.Chem.Phys 84, 1 (1986).

2. L.D.Landau and E. M. Lifshitz, Quantum Mechanics, Fcrgamon Press, Oxford,
1977.
3. I.N.Levine, Quantum Chemistry, Allyn and Bacon, Boston, (1093).