Relativistic MCDF oscillator strengths for the 4s2 1S0−4s4p 3P1, 1P1 transitions in zinc isoelectronic sequence

J. Quant. Spectrosc. Radiat. TransJer Vol. 42, No. 6, pp. 585-592, 1989 Printed in Great Britain. All rights reserved

0022-4073/89 $3.00 + 0.00 Copyright © 1989 Pergamon Press plc

RELATIVISTIC M C D F OSCILLATOR STRENGTHS F O R THE 4s 2 ISo-4S4p 3PI, 1PI TRANSITIONS IN ZINC I S O E L E C T R O N I C S E Q U E N C E J. MIGDALEK and M. STANEK Institute of Physics, Pedagogical University of Krakow, Podchorazych2, 30-084 Krakow, Poland (Received 9 December 1988; receivedfor publication 14 April 1989) Abstract--Relativistic multiconfiguration Dirac-Fock (MCDF) transition energies and oscillator strengths are computed for both the allowed 4s 2 ISo-4S4p ~Pl and forbidden 4s 2 tSo-4S4 p 3P l transitions in the ZnI through RbVIII spectra. In the computational approach, the intravalence correlation is represented through limited relativistic configuration mixing while the valence-core correlation is approximated by a core-polarization model. The results agree favourably with available experimental data, except for Se V and Br VI. Comparison is also made with other theoretical data and particularly with nonrelativistic, multiconfiguration Hartree-Fock results where both intravalence and core-valence correlation is treated through elaborate configuration mixing. The multiconfiguration Dirac-Fock calculations are performed by use of an "optimal level" scheme; for neutral zinc, an "average level" scheme is also employed, which allows a thorough comparison between these two schemes in oscillator-strength computations.

INTRODUCTION During

the last few years, relativistic oscillator strengths have been reported for the

ns 2 ISo-nsn p 3 p l , 1P 1 transitions in the first few members of the mercury ~ and cadmium 2 iso-

electronic sequences. The computations combined a limited multiconfiguration approach to represent at least the leading part of the intravalence electron correlation with a polarization model to account for valence-core correlation. Choice of the mercury and cadmium sequences was dictated by severe discrepancies observed between existing theoretical results and accurate experimental data. The proposed approach turned out to be very successful in eliminating or at least greatly reducing these discrepancies. For the zinc isoelectronic sequence, there already existed theoretical results, 3 obtained by treating both intravalence and core-valence electron correlation within the nonrelativistic M C D F scheme (some approximate relativistic corrections were also introduced in the final stage of calculations) in good agreement with experiments. Unfortunately, such an approach is very difficult to apply for heavy systems like mercury and cadmium for which relativistic effects have to be included accurately and simultaneously (cf. Migdalek and Baylis4) with electron correlation effects, an approach that would lead to very long basis sets because few relativistic configurations usually correspond to a single nonrelativistic counterpart and, in consequence, lead also to serious convergence problems. In the present study devoted to the zinc sequence, we wish to compare the approach used previously for the mercury and cadmium sequences with that of Froese-Fischer and Hansen, 3 who treated both intra- and valence-core correlation within a multiconfiguration scheme. Zinc and its isoelectronic ions are well suited for comparing different ways to treat electron-correlation effects in oscillator-strength calculations because they are much lighter than their homologous systems belonging to mercury or cadmium sequences and, therefore, interference caused by different treatments of relativistic effects is less important. Nevertheless, it should be stressed that the present calculations are, to our knowledge, the first to account simultaneously for relativistic effects and both intravalence and core-valence electron correlation in the zinc isoelectronic sequence. The second aim of this study is to compare two different methods of carrying out the relativistic M C D F calculations, i.e., the "optimal level" (OL) and the "average level" (AL) methods. The comparisons performed so far were almost exclusively restricted to energy-structure calculations, whereas we wish to give our attention to oscillator-strength computations. 585

586

J. MIGOALEK and M. STANEK CALCULATIONS

Calculations were performed with a modified Desclaux's M C D F code 5 in the "optimal level" (OL) scheme, i.e., convergence is achieved separately for the ground IS0 state and for each of the two excited states with J = 1 (corresponding to 3p~ and ip~), assuming complete relaxation of the core.

The ground IS0 state was represented as 14 IS0) = a114s~/2 J = 0 ) + aal4p~/: J = O) + a314p~/2 J = 0 ) ,

(1)

whereas the upper 3 p l , ip~ states are the mixtures of two pure jj configurations, viz. 14

,p,>d=[c,Jl4s,/:4p,/2J=1> +

14st/24p3/:J= 1).

(2)

Two types of calculations were performed. In the first, the valence-core electron correlations are entirely neglected. In the second type, this contribution is described as polarization of the core by valence electrons. The details of the core polarization approach were presented in our earlier papers ~'2'4 and we summarize them briefly here. According to this approach, the core-polarization potential (in atomic units) is

and describes interaction of the core through its static dipole polarizability ~ with the resultant of the electric fields Fi produced by each valence electron at ri. The expression for the electric field F~ = r;(r02 + r/2) -3/2 is chosen to ensure that the interaction remains finite at small r. The r0 cut-off parameter in F i is considered to be a measure of the core radius. The values of the static dipole polarizabilities calculated for core-like ions by Fragga et al 6 were used as core polarizabilities ~. The mean radii of the outermost spin-orbital of the unpolarized core-like ions computed in our earlier paper 7 were employed as values for the cut-off parameter r0. The ~ and r 0 parameters are given in Table 1. The one-electron terms of Eq. (3) are included in the one-electron Hamiltonian of each valence electron and the two-electron terms are added to the direct Coulomb repulsion between valence electrons. In calculations of oscillator strength, the "dipole length" form is used, with the dipole moment d = - r of each valence electron replaced by d + de, where d c = ~F is the additional dipole moment induced in the core by the valence electron. The computations presented here were performed assuming complete relaxation of the core and the largest contributions, which stem from overlap integrals between orbitals involved in Eqs. (1) and (2) were included in oscillator-strength calculations. Because our earlier computations were carried out in the "frozen core" approximation, we repeated, for comparison, our calculations for neutral zinc using the ls2... 3d I° core frozen in the ground ~S0 state. The very small differences that appeared will be discussed in the following section. Table 1. The values of core polarizabilities ct and cut-off radii r 0 (in atomic units) used in the core-polarization model. 5poctEum

Core

polarizability c~

Zn

I

2.2S~5

Cut-o~

radius rO

0,8656

Ga

II

1.2822

0.77S0

Ge

III

0.8058

0.7121

As

IU

Se U

0.5355

0.6580

0.~0~5

0.G125

BE UI

0.2655

0.57~

Kr UII

0.2024

0,5~10

Rb UIII

0.1350

0.$115

Relativistic MCDF oscillator strengths

587

Almost all computations performed with the M C D F method employ the "average level" (AL) or even the "extended average level" (EAL) scheme proposed by Grant et al, 8 instead of the "optimal level" (OL) scheme used by us. In these two schemes (AL or EAL), the spin-orbitals are chosen to minimize the average energies of the configurations involved, rather than the energy of an individual level (OL). As Grant et al 8 point out, the "average level" scheme is most attractive because of simplicity and computational ease. Based on our own experience, we add that it allows us to avoid most of the convergence problems which are so often encountered in "optimal level" calculations. In the present study, we wish to compare, with neutral zinc as an example, the OL and AL schemes of calculations assuming the same description of states [Eqs. (1) and (2)] in both schemes. In order to determine the Si/2,Pl/2 , 3/2 orbitals for the excited 4 3p~ and 4 Ip¿ states, two different expressions were used for the average energy. The first was suggested by Grant et al 8 and gives all configurations equal weights, viz. G = ~ CW2 Eav 1 av Ea., l (sl/2Pl/:) + IE _ , . , _ , = IE (Sl/2P3/2), av

i

(4)

whereas the second is a weighted average proposed by Desclaux 9 as follows: Ea~ = ~l E av(S|/2P¿/2) + ~E 2 ~v(sl/2P3/2),

(5)

where Eav is the average energy of a single jj configuration. The s~/2, P~/2, 3/2 orbitals for the ground 4 ~S0 state were determined through minimization of one of the following expressions for the average energy:

E aG v_-_~1E

2 I av 2 (s~/2)+~E ( P 2t / 2 ) + ~I l t T a v (~P3/2s,

(6)

ED _ 1 ~" jL-'av £ 2 "~ I av 2 61[Tav/" 2 av--~ ~,si/21-~t-$E (Pl/2)-l-g~-, ~,P3/21,

(7)

1

av

av

2

Ear = ~E (sl/2)

1 1

av

2

6

av

2

+i[vE (P L,2)q- ~E (P3/:)].

(8)

The expressions (6) and (7) correspond to Grant's and Desclaux's suggestions, respectively, whereas Eq. (8) yields results midway between these two and ascribes equal weights to the nonrelativistic configurations s 2 and p2, with the latter averaged over its relativistic counterparts. The relativistic Pl/2P3/2 configuration is excluded because it has no state with J = 0. The results of the specified computations described will now be discussed. D I S C U S S I O N OF RESULTS In Table 2, the 4 1S0-4 ip) and 4 150--4 3p! transition energies computed in the present study are compared with other theoretical and also with experimental data. The theoretical data used for comparison include the results of different approaches: nonrelativistic single-configuration Hartree-Fock 3 (HF) calculations, single-configuration Hartree-Fock calculations with perturbative relativistic corrections )° (HFR), nonrelativistic multiconfiguration Hartree-Fock 3 (MCHF) calculations, relativistic multiconfiguration Dirac-Fock calculations including only intravalence electron correlation H (MCDF), relativistic multiconfiguration Dirac-Fock calculations employing an "extended average level" schemer and including only an intravalence electron correlation ~: ( M C D F - E A L ) and a relativistic random-phase approximation ~3 (RRPA). As can be seen from Table 2, the 4 1S0-43P 1 forbidden transition energies computed here agree very well with the experimental data particularly when they are corrected for core polarization. This excellent agreement, stems at least partially, from the fact that inclusion of electron correlation improves both the 4 IS0 and 4 3p~ states similarly and that the remaining deviations from the experimental data are of the same size for both these states. This is not the case for the allowed 4 IS0-4 ~P~ transition, for which the correlation corrections improve the ionization energy of the ~P~ state much less than the energy of the ~S0 state. As a result, our present values corrected for both intravalence correlation and core polarization overestimate the experimental transition energies (except for neutral zinc). The uniform treatment of the intravalence and valence-core electron correlation through configuration mixing employed by Froese-Fischer and Hansen, 3 who also used much tin the EAL scheme,averagingof the energyexpressionis extendedto configurationstates with differenttotal angular momentum J and differentparity.

588

J. MIGDALEK and M. STANEK

Table 2. Comparison of calculated transition energies (in atomic units) with experimental and other theoretical data. For descriptions o f different theoretical approaches, see the text. TheorW Spectrum

3 HF

i0 HFP

3 ~ICHF

*

ll

12

rlCOF f

MCDF-E~L t 1

E

(4

13 RRPA

HEOFmOL f

Experiment

§

1 S 0

P

4

) ! 15

Zn

I

0.1759

0.2021

Ga

II

0.3011

0.3155

Be

III

0.4018

~s

!U

0.2122

0.2183

5.2130

0.3351

0.34S5

0.3~ii

0.3077

0.3221

0.4108

0.431~

0.~538

0.4438

0.4053

O.ilB5

O.~S2B

0.4529

0.5257

0.5531

0.5~00

0.5!0~

Se U

0.570~

0.5224

0.8157

O.TqB8

0.8327

0.8002

BE WI

0.882S

0.881

0.8855

0.7085

0.7522

0.7287

Kr U!I

0.74~5

0.770

0.7!80

0.7850

0.8352

0.8190

0.7784

0.9283

0.2138

5.8580

0.1234

0.1422

0.i151

15

15

15

15

iS 0.8888

O.SBS3 15

12 Rb U I i I

0.8835 1 E

(4

3 S

-

4

O

P

)

l 15

Zn

I

5.1175

8a

!I

0.2050

0.2120

0.2179

Se

Ill

0.2722

0.2252

0.2R48

ms

IU

0.2223

0.2514

0.3507

O. ~ m = =

0.4175

0.4152

0,470B

0,48!5

0.4248

0,535S

0.2!70

0.5471

0.5025

0.5112

0.5122

15

15

iS

15 Se U

0.3~53

IE 2r U!

0.~51

12

Kr

Ull

0.5288

12 Rb U!II

$ * §

Intrava!mncm Intravalence Intravale~ce

0.2240

correlation and valen=e ccrrela:i~n

onlw. - core correlation. and valence - core ccrrelatl~n

(~h~

la~=er

:n C ~ r e - p o i a r l z a = I D n

ap~rDxlma~icn)

Relativistic MCDF oscillator strengths

589

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Relativistic MCDF oscillator strengths

591

longer basis sets, yields more accurate energies for the allowed 4 I S 0 - 4 ~Pt transitions in ionized systems, whereas for neutral zinc, as well as for the forbidden 4 ~S0-4 3p~ transitions, our combined approach leads to the best agreement with experimental data. Differences between our MCDF-OL energies (without core-polarization correction) and the MCDF data of Anderson and Anderson t~ or the MCDF-EAL data of Biemont '2 result from the use of different basis sets, as well as from different energy expressions employed in the minimization process. Our present results show that relaxation of the core in the MCDF-OL computations has a rather small influence on the transition energies: the corresponding changes are 0.8 and 0.9% for the allowed 4 IS0-4 IP 1 and forbidden 4 ~S0-4 3p~ transition energies, respectively. The present MCDF-AL calculationst (with core polarization) for neutral zinc yield the following transition energies: 0.1440 and 0.2278 a.u. for 4 ~S0-4 3p~ and 4 ~S0-4 ~P~ transitions, respectively, with slightly worse agreement with experiment (0.148 l, 0.2130) than those of the MCDF-OL calculations (0.1489, 0.2193). The dependence of the AL transition energies on the form of the energy functional that is minimized in the computations is entirely negligible. Tables 3 and 4 show oscillator strengths computed in the present study for the allowed 4 I S 0 - 4 IP l and forbidden 4 ~S0-4 3PI transitions, respectively, compared with both experiments and the results of different ab initio theoretical approaches described above (HF, 3 HFR, ~° MCHF, 3 MCDF, ~ MCDF-EAL, ~2 RRPAI3), as well as with the following semiempirical results: HXR + LSQ (least-square-fit analysis of experimental energy levels plus HXR radial transition integrals) of Biemont ~2 and model potential calculations (MP) of Victor and Taylor/4 Our M C D F - O L values of the oscillator strengths, corrected for core polarization, compare favourably with the available experimental data, except for the Se V and Br VI spectra. In these last two cases, there is no other theoretical result that is in agreement with the experiment. This, as well as the good agreement between our result and the experimental oscillator strength for KrVII corrected for cascading effects, suggests that the lack of this correction in the experimental data for SeV and BrVI is responsible for disagreement between theory and experiment. There is also good agreement between our MCDF-OL (a) oscillator strengths which include only a limited amount of intravalence correlation and the corresponding MCHF results of Froese-Fischer and Hansen, 3 despite a considerable difference in the length of basis set used (differences in the accuracy of inclusion of relativistic effects are less important for such light systems as zinc and its isoelectronic ions). The only larger discrepancy occurs for neutral zinc, for which intravalence correlation seems to be more important than for ionized systems and this discrepancy may be due to the difference in the basis length. Similarly, our corepolarization-corrected MCDF-OL (b) oscillator strengths agree well with the corresponding MCHF results for which the core-valence correlation is included through proper configuration mixing. Similarly, for the transition energies, the slight discrepancies between the relativistic oscillator strengths of different authors can be attributed to differences in the basis length and in the forms of the energy functional minimized in the computations, as well as to the inclusion or neglect of core-valence correlation. It is interesting to note that the MCHF calculations of Froese-Fischer and Hansen 3 predict a much larger influence of the core-valence correlation on the 4 iS0-4 ~P~ oscillator strength in neutral zinc than our core-polarization model. The influence of core relaxation on the 4 ~S0-4 3p~, tpi oscillator strengths in neutral zinc studied in this paper is small and amounts to < 1.5% if theoretical transition energies are used and to 0.8% for calculations employing experimental transition energies. In comparison with the MCDF-OL approach, the MCDF-AL calculations overestimate oscillator strengths computed with theoretical transition energies and core-polarization correction for the allowed 4 ~S0-4 3P l transition in neutral zinc [1.67 vs 1.33 resulting from MCDF-OL (b) method] but underestimate the forbidden 4 ISo-3p 1 transition (8.9 x 10-5 vs 11.8 x 10-5). The form of the energy functional minimized in calculations has a negligible influence on the resulting M C D F - A L oscillator strengths. An interesting feature of MCDF-AL oscillator strengths is their enhanced sensitivity to the core-polarization effect. Inclusion of core polarization reduces the difference between MCDF-OL and MCDF-AL oscillator strengths computed with the same basis sets. tWe employ the forms of the energy functional given by Eqs. (4) and (6).

592

J. MIGDALEKand M. STANEK

Comparison of the M C D F - O L and M C D F - A L approaches seem to show that, at the same length of the basis sets, the M C D F - O L method yields more reliable transition energies and, especially, oscillator strengths. However, the great advantage of the M C D F - A L method is that it allows us to extend the basis set considerably, without running into problems with convergence and, as a result, to obtain a more adequate description of the correlation effect at low computational effort. This result may be very important for oscillator-strength calculations of transitions involving strongly perturbed energy levels. Further studies in this direction are planned.

Acknowledgement--The support of this study by the Central Programme for Fundamental Research CPBP 01.06 is gratefully acknowledged. REFERENCES I. 2. 3. 4. 5. 6. 7. 8. 9. 10. l l. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

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