Page 49 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

34 C. CHAVY ET AL.

distances in the case of the optimised orbital just comes from the fall off of gaussian
functions at large distances.
4.3. CONSTRUCTION OF THE ORBITAL OF

In the two preceding sections (4.1 and 4.2) we have presented numerical test of the
following description (resulting from the analysis of the sections 2 and 3) of the
optimised orbital of :
- near of a nucleus, the s wave (with origin on that nucleus ) of the optimised orbital
of is proportional to the s regular solution of the radial equation (eq.(2)) with
Z=l and a shifted energy given by (e=orbital energy, R=inter-
nuclear distance);

- outside the molecule, the s wave ( with origin at the midpoint of the two nuclei ) of
the optimised orbital of is proportional to the s irregular solution of the radial
equation with Z=2 and the actual energy of the orbital.

We examine now a numerical test of the reciprocal of this description : if a function
satisfies this description, then it is the optimised orbital of . If both the description
and the reciprocal are true we can conclude that the description is complete.

To that end we first introduce the following notations :
- the ’ internal zone ’ corresponding to the nucleus A is defined by the condition
(in there are two ’internal zones’, but , due to the symmetry, only one
of them will be considered here) ;
- the ’external zone’ is the region outside the molecule ;

is the orbital to be determined in the form of an expansion in a gaussian basis :

where is a gaussian function and the numerical coefficients to be determined;

is the s partial wave of with origin on the nucleus A;
is the σ partial wave of with origin at the middle of the bond;

and are the regular and irregular solutions of the two radial equations
corresponding to the internal and external zones (assuming and e to be known).
Then and are obtained by mean of the partial wave expansion of the gaussian
functions :