Ellinger Y., Defranceschi M. (eds.) Strategies and applications in quantum chemistry (Kluwer, 200

To summarize, if the low–lying states connected to the ground state by allowed dipole transition are not valence states but present a predominant Rydberg character, we have to introduce a lot of states; if not, the value of dynamic polarizability near the first resonance is poor.

To circumvent this difficulty, we have developed a procedure which allows us to reach an extrapolate value of a from a finite number N of low–lying true spectral states

without the ”polynomial” contribution. The value of the exponent p is determined by a least– square fit and then the extrapolated polarizability is obtained by a linear regression. In the case of the dynamic polarizability, this extrapolation is done separately for the cases and

Figure (1) gives an illustration of this extrapolation procedure for the calculation of the static parallel polarizability in CO. In this case the extrapolated value

was obtained with the following equation

It is important to underline two points :

• The extrapolation procedure rests upon the hypothesis of exact or very accurate

and molecular basis sets with the extrapolation procedure.

• This extrapolation has been obtained with a finite number N (usually less than 10) of spectral states lying under the first ionization potential; thus, the continuum is not taken into account explicitly in our calculations. It has been simulated through the function and the extrapolation procedure as we are going to show it.

2.4. CONTINUUM CONTRIBUTION

Hydrogen atom, in its ground state, can be treated in an entirely analytic approach. The calculation of the second–order perturbed energy gives the well known values :

for the static polarizability of the ground state. Since we have used exact analytic wavefunctions which are the eigenstates of the electronic Hamiltonian, the continuum

268 M. TADJEDDINE AND J.P. FLAMENT

contribution has been taken into account, i.e.

On the other hand, the static polarizability can be calculated by a sum–over–states

polarizability, Rérat et al. (13) have carried out the calculation of the polarizability for the ground state of the hydrogen atom. This computation has been made with

Thakkar) corresponding respectively to 97.8% and 81.4% of the exact value. This result illustrates the fact that a large part of the continuum contribution is simulated

through the use of the dipolar factor. Moreover the convergence of the series is faster as we can see on table 1.

At last, the extrapolation procedure employed in that calculation gives the final value to be 4.503, i.e. 0.07% above the exact static value of

•the efficiency of the extrapolation procedure to obtain accurate values.

2.5.VIBRONIC CORRECTIONS

The theoretical method, as developed before, concerns a molecule whose nuclei are fixed in a given geometry and whose wavefunctions are the eigenfunctions of the electronic Hamiltonian. Actually, the molecular structure is vibrating and rotating and the electric field is acting on the vibration itself. Thus, in a companion work, we have evaluated the vibronic corrections (5) in order to correct and to compare our results with experimental values.

In the particular case of diatomic molecules, the molecular geometry can be described by the reduced coordinate

where R is the internuclear distance; Re, its equilibrium value in the electronic ground state. Energy and each component of the polarizability may be written as a power series in the reduced coordinate £ around their equilibrium values :

Moreover, the values obtained for the dynamic polarizability by varying the wavelength are in good agreement with experiment (1). Table 4 resumes the results obtained for the static polarizability of CO :

1. On the first line, we have reported our results (1) obtained with the spectrocopic states, the dipolar factor and the extrapolation procedure. In order to compare them with the experimental results (last line) we have corrected them by taking into account the vibronic coupling –temperature and electric field dependence– as developed before (second line). The parallel component,

is now in excellent agreement with experiment.

2. The two following lines present the results obtained later by Rérat et al. (17) : the method consists in adding one more term in the expression of given by Eq.14. He keeps the dipolar factor; from the summation on the spectroscopic states he retains only the first one of the symmetry of interest, thus there is no extrapolation procedure; on the other hand, he adds the Slater determinants

which contribute to the perturbation of the ground state by the operators

and he takes into account the non–orthogonality of the zeroth and first–order perturbed wavefunctions. Their results show an improvement for both a components, in particular for anisotropy.

3.These results are compared with those obtained by Oddershede and Svendsen

(18)using SOPPA or Sunil and Jordan (19) using MP4 or a coupled cluster approach, but without vibronic correction.

3.Determination of the polarization functions

In order to overcome the optimization process of the (hyper) polarizabilities calculations, we have been led to deeply study the perturbational and variational methods and in particular the variation–perturbation treatment introduced by Hylleras (20) since 1930. We will not develop here the theoretical framework of the recent study of N. El Bakali Kassimi (21). We propose criteria for generating adequate sets of polarization functions necessary to calculate (hyper) polarizabilities.

As our computations use the HONDO/8 program (22) which is based on the CPHF (Coupled Perturbed Hartree Fock) method (23) we begin by briefly recalling this method.

orbital which would be the relevant combination (a contracted STO). In fact, the hydrogenic model does not apply exactly to any polyelectronic atom, so we let the coefficients of the combination vary freely, as new variational parameters, in the CPHF equations.

Moreover according to such a model, the same exponent is used in each perturbed wavefunction, keeping the value of the non–perturbed wavefunction.

is taken in the literature once for all so as to describe the system for the best; let be this value. On the basis of the Hylleraas variation principle, we will determine the suitable value for the hydrogenic scale factor in the polarization functions derived

from and after optimization with respect to maximum polarizability.

are combinations of nd/ns, nf/np and ng/nd/ns respectively. Table 5 gives the orbitals with the pure spherical harmonics In programs using 6d, 10f or 15g cartesian functions, only the nd, nf or ng need be given since they include the corresponding ns for the nd shells; np for nf and nd/ns for ng.

3.3.APPLICATION TO He AND

Owing to their simplicity, the helium atom and the dihydrogen molecule have been the object of experiments (Ref. 31 for of He; Ref. 32 for of ) and calculations, some of them near the Hartree–Fock limit (Ref. 33 for He and Ref.

scribed through a Huzinaga’s CGTOs set (the 10 CGTOs one in Ref. 37). We use

the {2p, 3p} orbitals for and {3s/3d,4s/4d,5s/5d} orbitals for The

Table 6 clearly shows the effect of the polarization functions : the HF limits for energy, and are reached at the first, second and third levels of calculation. Moreover, we

have proved (6) that :

•the extension of the basis set does not produce further change, and

•if Table 6 shows the necessity of the d orbitals, any two of the three d shells give satisfactory values of

This last result which will be verified with the following applications is a consequence of our choice for the polarization functions. In effect, the STOs have nodeless radial

part and they all combine in phase in so that the resulting polarization function is also nodeless and can be approximately modeled by only one or two STOs with suitable exponents.

ization” what we are concerned with in this paper : the polarization produced by an electric field, and he calls ”valence polarization” : a kind of polarization du to quantum–mechanical valence forces. In order to correctly describe the chemical bond in it is necessary to include the ”valence polarization” function as soon as one calculates energy with the unperturbed function (i.e. the 2p orbital).

For this calculation we used the basis set l s , 2 s , 2 p of Fraga and Ransil (35) which gives near HF limit quality for energy The polarization functions were derived from the 1s orbital only, like in He calculations. Their expo- nent was optimized using the maximum probability criterion Table 7 presents the obtained results.

Now with the 2p valence polarization, it is possible to partly describe the polarizabil- ity since the first step of calculation with the unperturbed wavefunction, especially the parallel component which is generally easier to calculate in CPHF. The optimized

values of are excellent at the second step with