Static multipole polarizabilities of S atoms or ions using the Frozen-Core approximation

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Static multipole polarizabilities of S atoms or ions using the Frozen-Core approximation

S.I. Easa, G.C. Shukla

To cite this version:

S.I. Easa, G.C. Shukla. Static multipole polarizabilities of S atoms or ions using the Frozen-Core approximation. Journal de Physique, 1979, 40 (2), pp.137-138. �10.1051/jphys:01979004002013700�.

�jpa-00208892�

137

Static multipole polarizabilities ^{of S atoms}

or ions using ^the Frozen-Core approximation

S. I. Easa and G. C. Shukla Department ^of Physics, College ^of Science,

Basrah University, Basrah, Iraq

(Reçu ^{le 8 mai} 1978, révisé les 22 août et 18 octobre 1978, accepté ^{le 26 octobre} 1978)

Résumé.

Les polarisabilités statiques multipolaires ^{d’atomes ou d’ions S sont} calculées suivant deux méthodes variationnelles équivalentes. Les résultats sont en accord avec un travail précédent ^utilisant l’approximation

d’Hartree-Fock découplée.

Abstract.

Static multipole polarizabilities ^{of S atoms or} ions have been calculated by ^two equivalent variational methods. Our results are in harmony ^{with an} earlier work which used the uncoupled Hartree-Fock approximation.

LE

JOURNAL DE PHYSIQUE ^TOME 40, ^FÉVRIER 1979,

Classification Physics ^Abstracts

31.10

Recently ^we reported [1] ^a variational treatment for static multipole polarizabilities ^{of atomic} systems ; ^this

treatment was the generalization of Rivail and Rinal- di’s method [2] ^{of a} one-parameter variational calcu- lation to an arbitrary ^number of variational parameters until convergent ^values for static multipole polariza- bilities, a2L(o), ^{were achieved.} Subsequently ^{it was} applied ^{to obtain} OE2L(o) ^{for the} hydrogen-atom ^and

helium sequences. It ^was noted that the values of

a2L(o) ^obtained from this method [1, 2] ^{were identical}

with those obtained from the hydrodynamic ^{model to}

quantum ^mechanics [3]. ^{We have} already reported

our calculation of a2L(o) ^for atomic Li [4] ^{and its}

sequences [5] ^{from the} hydrodynamic analogy ^to

quantum ^mechanics using ^the frozen-core approxi-

mation whereby the effective Hamiltonian of Li- sequences becomes identical with a hydrogen-like

atom with a different nuclear charge ^{and outermost}

orbital.

In the frozen-core approximation ^Be, Mg ^{and Ca}

atoms become identical with the He atom with a

different nuclear charge ^{and outermost} orbital. This enables us to work out OE2L(o) ^{for such} systems ^and their _sequences with our model for ^{He atom as} reported ^earlier [1]. Also, oC2L(o) ^{for K atom has been} performed ⁱⁿ the frozen-core approximation.

By using either of our methods [1, 3], OE2L(o) ^is given by ^the following expression :

where

and we have chosen 5 variational constants [1, 3].

Note that expression (1) ^is applicable ^{to the Ca}

atom, Mg- ^and Be-sequences ^with Y’o(R ) being ^the

radial part ^{of 4} S2, ^{3 S2 and 2 S2 Clementi wave}

function [6] respectively. ^{On the other hand, for the}

K atom, ^Li- and Na-sequences ’P o(R) is ^{taken to be}

the radial part ^{of the 4} SI, ^{2 S1 and 3 S1 Clementi}

wave function [6] respectively ^{and formula} (2) ^is applicable ^{to such systems. Moreover, the} equations governing ^the variational constants C are

with matrix elements

The present calculations for a2L(o) ^{of S atoms or ions}

are listed in table I. Langhoff ^{and Hurst} [7] replaced

the unperturbed Hamiltonian for S atoms or ions by

the Hartree-Fock Hamiltonian. Then the time depen-

dent variational perturbation ^scheme developed by Karplus ^{and Kolker} [8] ^{was utilized to work out the}

static multipole polarizability ^{of the atomic} system under consideration. They also considered contribu- tions of all closed sub-shells. On the other hand our

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01979004002013700

138

Table 1.

Static multipole polarizabilities of ^{S atoms or ions : a - b} implies ^{a x l Ob} (in ^units aÕ(2L+ 1), for ^L

1, 2, 3, ⁴ referring ^to dipole, quadrupole, octupole ^and hexadecapole respectively).

Hamiltonian, ^{in the frozen core} approximation, behaves like ^{the Li or} He atom. In the latter system

we again ^neglected interelectronic repulsion whereby

the system ^{behaves like the} hydrogen ^{atom. With}

such a simple ^{model but} using ^{Clementi wave func-} tions, ^the multipole polarizabilities ^{of the S atoms}

or ions with Z

1 to 19 were calculated. We compared

our results with those of Langhoff ^{and Hurst because}

our scheme is parallel ^{to theirs, if not} identical. This

encouraged ^{us to} compare ^our values of a2L(o) ^with

theirs. It is observed that the present estimates of

a2L(o) (see ^table I) ^{are in} harmony ^{with those of} Langhoff ^{and Hurst} [7] despite ^our simple ^{model and} perturbation ^{effect on a} single bound orbital. We

are extending ^{our scheme to} incorporate ^the coupled

Hartree-Fock approximation.

The authors are grateful ^{to Dr. R.} Al-Rashid for his encouragement throughout ^the present ^investi-

gation.

References

[1] SHUKLA, ^{G. C. and} EASA, ^S. I., Solid State Commun. 26 (1978)

873.

[2] RIVAIL, J. L. and RINALDI, D., ^{C. R.} Hebd. Séan. Acad. Sci.

Paris 283B (1976) ^111.

[3] SHUKLA, ^{G. C. and} BARBARO, M., Phys. ^{Rev. 15A} (1977) ^23.

[4] SHUKLA, G. C., SURI, S. K. and MISHO, ^R. H., Solid State Commun. 23 (1977) ^67.

[5] TONDON, ^V. K., SURI, ^{S. K. and} SHUKLA, ^G. C., Solid State Commun. 25 (1978) ^931.

[6] CLEMENTI, E., Nucl. Data Atom. Data 14 (1974) ^1177.

[7] LANGHOFF, P. W. and HURST, ^R. P., Phys. Rev. 139A (1965)

1415.

[8] KARPLUS, ^{M. and} KOLKER, ^H. J., ^{J. Chem.} Phys. ³⁹ (1963)

2997.