Page 289 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

272 M. TADJEDDINE AND J. P. FLAMENT
alternative. In the stationary perturbation theory, the CPHF is the most known.
The CPHF originates in a perturbation development of the spin–orbitals and of their
energies on the expansion of the electric field . The polarizability (or the second
hyperpolarizability) is derived from the second– (or fourth–) order perturbation en-
ergy. The CPHF is akin to the finite–field method on the point that they treat the
bielectronic interactions in presence of the electric field in a self coherent way (26).
On the other hand, it is basically different with respect to the use of variational prin-
ciple : while the finite–field method variationaly treats the total energy in presence
of the field, the CPHF, by using the perturbation development, allows variational
approaches to the calculation of polarizabilities.

3.2. CHOICE OF TRIAL FUNCTION FOR THE POLARIZATION ORBITALS
In all the variational methods, the choice of trial function is the basic problem. Here
we are concerned with the choice of the trial function for the polarization orbitals
in the calculation of polarizabilities or hyperpolarizabilities. Basis sets are usually
energy optimized but recently we can find in literature a growing interest in the
research of adequate polarization functions (27).
By returning to the genuine meaning of the word ”polarization”, we propose polar-
ization functions suited to the calculation of the electric property of interest : our
polarization functions belong to the so–called field–induced ones (FIP) (28).
The foundation of our approach is the analytic calculations of the perturbed wave-
functions for a hydrogenic atom in the presence of a constant and uniform electric
field. The resolution into parabolic coordinates is derived from the early quantum
calculation of the Stark effect (29). Let us recall that for an atom, in a given Stark
eigenstate, we have :

The calculated perturbed wavefunctions have been rewritten in terms of a combina-
tion of normalized Slater orbitals in real form. Ref. 6 gives a detailed illustration for
the level 1s.
At the beginning it is necessary to describe the unperturbed system very well, inde-
pendently of the polarization functions : Let us assume that the unperturbed system
is reasonably well described by using some finite set of basis functions . As
shown by Hirschfelder et al. (30) we only need the first–order perturbed function for
and the second–order one for
We propose to construct the polarization functions from these perturbed wave func-
tions. The genuine basis set has to be enriched by :

• the Slater orbitals (STO) which form in order to calculate

• the STO which form
Thus, by following the hydrogenic model, we know not only the kind of angular
symmetry but also the value n of the quantum number of the suitable polarization
functions. In the case of a true hydrogenic atom these STO appear in a given linear
combination. To limit the size of the basis set, one could use an unique polarization