Page 39 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

24 C. CHAVY ET AL.

- finally we solve the eq.(15) with various values of e but always with the same
functions

The factor ensures that the solution of the eq.(15) is independent of a
multiplicative factor (if f is a solution, then is also a solution for any number )
and that f is proportional to when It turns out that no useful comparison
with the molecular case can be made in the absence of this factor.
The eq.(15) can be solved by mean of a power expansion of f, and of in the
same way as the eq.(5) :

Substituting the eq.(16) in the eq.(15) gives a recursion relation which allows to
determine the Owing to the factor it is possible here to choose as done
in the eq.(7), so that one gets :

etc ...

The main aspect of the eq.(17) is that the orbital energy e occurs only in the coeffi-
cients with Therefore we obtain here the same results as the one obtained
in the case of hydrogen-like atoms (§1.1 and §1.2) :

- the energy dependence of the RHF orbitals of polyelectronic atoms decrease faster
than in the region close to the nucleus (Valley theorem);
- and the corollary that these orbitals depend very weakly of the orbital energy in a
finite volume around the nucleus. The range of that volume, which depends of the
magnitude of and , will be now determined numerically.
1.5 NUMERICAL ILLUSTRATIONS

We present here numerical results illustrating that the solutions of the radial equa-
tions (eq.(5) for the hydrogen-like case and eq.(14) for polyelectronic atoms) are
’weakly’ dependent of e in a finite volume.

In the case of polyelectronic atoms we have calculated the and parameters
as described in the preceding section (see above, the §1.4) i.e. using the normalised
orbitals resulting from a RHF calculation of the atom in a gaussian basis (11).

The radial equations was then solved using the Runge-Kutta method (7).
We present in Fig. (1-6) the function defined in the eq.(2) (or defined in
the eq.(13)), in the case of the orbital 1s of Hydrogen (Fig.l), 2s and 2p of Carbon