INSTITUTE OF PHYSICSPUBLISHING JOURNAL OFPHYSICSB: ATOMIC, MOLECULAR AND OPTICAL PHYSICS J. Phys. B: At. Mol. Opt. Phys. 36 (2003) 3457 3465 PII: S0953-4075(03)60551-1 The influence of core valence electron correlations on the convergence of energy levels and oscillator strengths of ions with an open 3d shell using Fe VIII as an example Jiaolong Zeng 1,2,Fengtao Jin 2,GangZhao 1 and Jianmin Yuan 2 1 National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing 100012, People s Republic of China 2 Department of Applied Physics, National University of Defense Technology, Changsha 410073, People s Republic of China Received 5 March 2003, in final form 7 July 2003 Published 5 August 2003 Onlineat stacks.iop.org/jphysb/36/3457 Abstract Accurate atomic data, such as fine structure energy levels and oscillator strengths of different ionization stages of iron ions, are important for astrophysical and laboratory plasmas. However, some important existing oscillator strengths for ions with an open 3d shell found in the literature might not be accurate enough for practical applications. As an example, the present paper checks the convergence behaviour of the energy levels and oscillator strengths of Fe VIII by systematically increasing the 3p n 3d n (n = 1, 2, 3 and 6) core valence electron correlations using the multiconfiguration Hartree Fock method. The results show that one should at least include up to 3p 3 3d 3 core valence electron correlations to obtain convergedresults. Large differencesare found between the present oscillator strengths and other theoretical results in the literature for some strong transitions. 1. Introduction Knowledge of the atomic data of iron in different ionization stages is crucial in modelling and understanding processes occurring in the Sun and in stellar coronae. For many years, spectral lines from iron ions have been observed in the Sun and other astrophysical objects [1 6]. There are a number of unidentified lines observed [7] and it is suspected that some of these lines could be Fe VIII lines. Accurate atomic data must be available for identification of various spectral lines and to infer the properties of solar and astrophysical plasmas. On the other hand, accurate radiative atomic data are also needed in opacity calculations. Winhart et al [8, 9] measured the spectrally resolved opacities and transmission of iron plasma in the energy range of 70 140 ev at a temperature of about 22 ev and a density of 0.01 g cm 3,respectively. Under these experimental conditions, Fe VIII is one of the most abundant ions. Springer et al 0953-4075/03/163457+09$30.00 2003 IOP Publishing Ltd Printed in the UK 3457
3458 JZenget al [10] carried out laboratory measurements of the opacity for stellar envelopes at a temperature of 20 ev and a density of 0.0001 g cm 3.Local thermodynamic equilibrium was achieved in their experiments. To simulate the experimental spectra, one needs a large amount of radiative atomic data. Recently, we [11 13] carried out detailed studies on the opacity of aluminium plasmas using accurate energy levels, oscillator strengths and photoionization cross sections. Afew physical effects, such as the effect of the autoionization widths of the K-shell excited states on the x-ray transmission, have been shown to be important for aluminium plasmas. There are a few calculations for the energy levels and oscillator strengths of Fe VIII, while most of them were carried out in the LScoupling scheme. Earlier, a few studies were carried out in the Coloumb Born [14, 15] and distorted wave [16] approximations. Saraph et al [17] and Lynas-Gray etal [18] carried out a comprehensive abinitio calculation to obtain the radiative data using close coupling approximations implemented by the R-matrix method. Their studies are part of the opacity project [19]. Tiwary et al [20 22] calculated the excitation energies and oscillator strengths for transitions of (1s 2 2s 2 2p 6 )3s 2 3p 6 3d 2 D 3s 2 3p 5 3d 22 P o, 2 D o and 2 F o using the CIV3 programme of Hibbert [23] or the multiconfiguration Hartree Fock (MCHF) method [24]. In their most accurate calculations, they included configuration interaction (CI) between the 3s 2 3p 6 3d, 3s 2 3p 6 4d, 3s 2 3p 4 3d 3,3s3p 6 3d 2,3s 2 3p 5 3d4f and 3s 2 3p 5 3d5f even configurations and the 3s 2 3p 5 3d 2,3s 2 3p 6 np(n = 4 7), 3s 2 3p 6 nf(n = 4 7), 3s 2 3p 5 3dns (n = 4 5), 3s 2 3p 5 3d4d, 3s 2 3p 3 3d 4, 3s3p 5 3d 3, 3s 2 3p 4 3d 2 4f, 3s 2 3p 4 3d 2 np(n = 4 7), 3s 2 3p 5 4fnp(n = 4 7) and 3s3p 6 3dnp (n = 4 7) odd configurations. Few studies have been carried out on intermediate coupling schemes. Fawcett [25] calculated the oscillator strengths and wavelengths for lines belonging to the transition arrays of 3s 2 3p 6 3d 3s 2 3p 5 3d 2,3s 2 3p 6 4p, 3s 2 3p 6 4f, 5f, 6f, 7f and 3s 2 3p 5 3d4s. CI was included between the 3s 2 3p 6 3d, 3s 2 3p 6 4d, 3s 2 3p 6 5d, 3s 2 3p 6 4s, 3s 2 3p 6 5s and 3s 2 3p 5 3d4p even configurations and the 3s 2 3p 5 3d 2,3s 2 3p 6 4p, 3s 2 3p 6 5p, 3s 2 3p 6 4f, 3s 2 3p 6 5f, 3s 2 3p 6 6f, 3s 2 3p 6 7f, 3s 2 3p 5 3d4s and 3s 2 3p 5 3d4d odd configurations. More recently, Bhatia and Eissner [26] reported the energy levels, oscillator strengths, radiative transition rates and collision strengths of Fe VIII for 73 fine structure energy levels obtained by including the configurations 3s 2 3p 6 3d, 3s 2 3p 5 3d 2, 3s 2 3p 6 4s, 3s 2 3p 6 4p and 3s 2 3p 5 3d4s using the superstructure programme [27]. From the work mentioned above, all past calculations, either in LS or in intermediate coupling, have not considered the complete 3p 2 3d 2 inter-shell core valence electron correlations. Due to the large overlap of the 3s, 3p and 3d orbitals of Fe VIII, 3p 2 3d 2, even 3p 3 3d 3 core valence electron correlations have large effects on the energy levels and oscillator strengths. Recently, we have tried to simulate the opacity of iron plasmas under the experimental conditions carried out by Winhart et al [8, 9]. The iron plasmas are in local thermodynamic equilibrium and at a temperature of about 22 ev and a density of 0.01 g cm 3. Fe VIII is one of the most abundant ions under these plasma conditions. The results show that calculated opacity tends to be larger than in experiment if adequate core valence electron correlations are not included in the expansion ofwavefunctions [28]. Therefore, it is necessary to clarify what kind of CI should be included to obtain converged results. The purpose of the present work is to check the effects of core valence electron correlations on the convergenceof energy levels and oscillator strengths of Fe VIII. The MCHF method [29] is used to obtain the radiative data with the relativistic effects implemented by using the Breit Pauli (BP) approximation. For simplicity, the present paper gives only the atomic data of the spectral lines belonging to the 3s 2 3p 6 3d 3s 2 3p 5 3d 2 transition array.
Convergence of energy levels and oscillator strengths 3459 2. Theoretical methods The CI wavefunction used in the non-relativistic MCHF approach takes the form (γ LS) = a i i (γ i LS) (1) i where the configuration state functions (CSFs) { i } are associated with a total spin and orbital angular momentum. The {γ i } denotes state labels such as orbital occupancy and angular momentum coupling schemes of the orbitals. γ denotes a further label for each level, typically the orbital occupancy and L, S of the dominant configuration (with the largest a i ), although in some cases the mixing between the { i } is so strong that such labelling is not particularly meaningful. The one-electron orbital from which the { i } are constructed takes the form u nlml m s = 1 r P nl(r)y ml (θ, φ)χ ms (σ ). (2) In the present work, we have obtained 18 radial orbitals: 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f, 5s, 5p, 5d, 5f, 6s, 6p, 6d and 6f. The variation principle is used to obtain the orbitals by minimizing the energyfunction of a given term or a weighted average of energy functions of one or more terms, where the weights can be user defined. 1s, 2s, 2p, 3s, 3p and 3d orbitals are chosen to be the Hartree Fock (HF) functions for the ground state 3s 2 3p 6 3d 2 DofFeVIII. The orbital functions of 4s, 4p, 4d and 4f are optimized on their respective terms of 3s 2 3p 6 nl (nl = 4s, 4p, 4d, 4f) configurations. The pseudo-orbitals 5s, 5p, 5d, 5f, 6s, 6p, 6d and 6f are obtained by optimizing, respectively, on the 3s 2 3p 5 ( 2 P o )3d 2 ( 3 F) 2 D o,3s 2 3p 5 ( 2 P o )3d 2 ( 3 P) 2 P o,3s 2 3p 5 ( 2 P o )3d 2 ( 3 F) 2 G o, 3s 2 3p 5 ( 2 P o )3d 2 ( 3 F) 2 F o, 3s 2 3p 5 ( 2 P o )3d 2 ( 1 D) 2 F o, 3s 2 3p 5 ( 2 P o )3d 2 ( 1 S) 2 P o, 3s 2 3p 5 ( 2 P o )3d 2 ( 1 G) 2 F o and 3s 2 3p 5 ( 2 P o )3d 2 ( 3 F) 2 F o terms. In the present work, we are mainly concerned with the transition array 3s 2 3p 6 3d 3s 2 3p 5 3d 2.Testcalculations show that inclusion of more orbitals with higher angular momenta, such as g or h orbitals, has negligible effects on the energies and oscillator strengths of this transition array, whereas it may have a definite contribution to higher levels. Once a set of radial orbitals has been obtained, the relativistic corrections can be taken into account within the BP approximation [29] to get the intermediate coupling wavefunction (γ J) = a i (LS) i (γ i LSJ). (3) LS i Thus the expansion is now the sum of the expansion over a set of terms. In constructing the CSFs, the core 1s 2 2s 2 2p 6 is frozen while single- and double-electron excitations are allowed among other included orbitals. The weighted oscillator strengths gf are calculated using the length and velocity formalisms: gf l = 2 E ( i 3 gf v = 2 ( i 3 E N 2 r p f ) (4) p=1 N 2 p f ), (5) p=1 where E = E f E i, E i and E f are, respectively, the energies of the initial and final states and g is the statistical weight of the lower state, i.e. g = 2J i +1forBPwavefunctions. Tachiev and Fischer [30] pointed out that, in the BP approximation, the length form is correct for O(α 2 )
3460 JZenget al (except for the omission of orbit orbit) while the velocity form requires a relativistic correction for the gradient operator [31]. Forsimple atoms or atomic ions, CI has been well understood and therefore the main task is to get accurate wavefunctions [32 35]. For complex ions with open d shells, the CI has not been understood so well, even for the simplest open 3d shell ion, Fe VIII. In the present paper, we study the convergence behaviour of the energy levels and oscillator strengths by systematically increasingthecore valenceelectroncorrelations. In thissystematicstudy, five cases are considered to show the effects of different core valence electron correlations on the calculated energy levels and oscillator strengths. In case A, a single configuration HF calculation is carried out, in which intrashell correlations have been included. In case B, CI between the 3s 2 3p 6 nl, 3s 2 3p 5 3dnl, 3s 2 3p 5 nln l and 3s3p 6 3dnl configurations are included for both even and odd parity levels. Here nl in 3s 2 3p 6 nl means 3d, 4s, 4d, 5s, 5d, 6s and 6d for even parity levels and 4p, 4f, 5p, 5f, 6p and 6f for odd parity levels. Other items such as 3s 2 3p 5 3dnl have similar meanings. In this case, it is equivalent to the inclusion of 3p 3d inter-shell electron correlations in the calculation. In case C, in addition to the CI included in case B, 3p 2 3d 2 electron correlations are also considered, that is, CI between 3s 2 3p 4 3d 2 nl and the configurations included in case B are taken into account. In case D, further inter-shell electron correlations, 3p 3 3d 3,areincluded. To show the convergence of the results, further 3p 4 3d 4,3p 5 3d 5 and 3p 6 3d 6 electron correlations are considered in case E. The orbital functions used in all five cases are the same throughout the paper; only the CI included in each case is different. 3. Results and discussion The results of the energy levels of Fe VIII are shown in table 1 for all five cases. The level designations shown in table 1 are selected automatically by the computer program. It names the first level according to the term linked with thelargest eigenvector component in the matrix. However, it should be noted that there is strong mixing for many levels. The experimental [36] and the most recent theoretical results obtainedbybhatia and Eissner [26] are also shown for comparison. The structure of the 3s 2 3p 5 3d 2 configuration is very complex and the energy levels of this configuration extend over a wide range from a little less than 400 000 to over 600 000 cm 1. In a single configuration calculation, there is good agreement between the experiment and the calculated energy levels whose energies are located in the intermediate region of the 3s 2 3p 5 3d 2 configuration. For example, the energies of the levels 3s 2 3p 5 3d 2 ( 1 G) 2 F7/2 o (no 20), 3s 2 3p 5 3d 2 ( 1 D) 2 F7/2 o (no 24) and 3s2 3p 5 3d 2 ( 1 D) 2 F5/2 o (no 28) are, respectively, 434 540, 449 870 and 459 950 cm 1 relative to the ground level 3s 2 3p 6 3d 2 D 3/2 (no 1), in excellent agreement with the corresponding experimental values of 434 555, 447 656 and 459 367 cm 1. For those lower levels, the calculated values are lower than the experimental results, while for higher levels, they are higher than the experimental results. For the results obtained in cases BandC,allthe calculated values are higher than the experimental results wherever available. Although there is generally better agreement between the experiment and the HF energy levels than between the experiment and those in cases B and C, one cannot say that the HF energy levels have been converged. Inclusion of 3p 3d and 3p 2 3d 2 core valence electron correlations cannot result in good energy levels, especially for the inclusion of 3p 2 3d 2 correlations in case C. However, good agreement can be obtained if 3p 3 3d 3 electron correlations are included (case D). The energy levels can be displaced downward more than 30 000 cm 1 by 3p 3 3d 3 core valence electron
Convergence of energy levels and oscillator strengths 3461 Table 1. Convergence behaviour of the energy levels (in cm 1 )for Fe VIII. The configuration with 1s 2 2s 2 2p 6 is truncated for simplicity. For comparison, the experimental [36] and the most recent theoretical results obtained by Bhatia and Eissner (BE) [26] are also shown. Key Configuration Term J A B C D E Expt. BE 1 3s 2 3p 6 3d 2 D 3/2 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2 5/2 1833 1 917 1 950 1 960 1 959 1 836 1 956 3 3s 2 3p 5 3d 2 ( 3 F) 4 D o 1/2 384 953 396 937 411 880 396 525 396 686 386 298 4 3/2 386 142 398 058 413 198 397 765 397 926 387 217 5 5/2 387 951 399 743 415 194 399 619 399 778 388 753 6 7/2 390 071 401 663 417 505 401 708 401 864 390 970 7 3s 2 3p 5 3d 2 ( 3 P) 4 P o 5/2 412 925 423 084 439 286 421 959 422 082 414 878 8 4 G o 11/2 407 890 418 091 438 405 422 009 422 138 409 019 9 9/2 411 034 421 222 441 557 425 136 425 261 410 651 10 3s 2 3p 5 3d 2 ( 3 P) 4 P o 3/2 412 925 426 539 442 733 425 393 425 515 416 686 11 1/2 418 495 428 649 444 779 427 432 427 553 418 960 12 3s 2 3p 5 3d 2 ( 3 F) 4 G o 7/2 413 609 423 787 444 097 427 655 427 776 412 676 13 5/2 415 614 425 784 446 047 429 590 429 708 413 636 14 4 F o 9/2 425 412 437 670 451 141 436 821 436 998 426 417 15 7/2 425 573 437 827 451 308 436 964 437 137 427 300 16 5/2 425 697 437 948 451 386 437 024 437 195 425 848 17 3/2 425 784 438 033 451 415 437 040 437 209 424 593 18 3s 2 3p 5 3d 2 ( 1 G) 2 F o 5/2 429 930 442 014 451 075 437 161 437 416 431 250 431 436 19 3s 2 3p 5 3d 2 ( 1 D) 2 D o 5/2 429 092 440 332 454 698 439 051 439 197 430 723 20 3s 2 3p 5 3d 2 ( 1 G) 2 F o 7/2 434 540 446 420 455 630 441 550 441 799 434 555 21 3s 2 3p 5 3d 2 ( 1 D) 2 D o 3/2 432 503 443 746 457 954 442 324 442 471 432 438 22 3s 2 3p 5 3d 2 ( 1 D) 2 P o 1/2 440 416 451 160 460 509 445 229 445 394 440 504 23 3/2 443 374 453 763 463 924 448 220 448 375 446 022 24 3s 2 3p 5 3d 2 ( 1 D) 2 F o 7/2 449 870 458 898 474 665 457 552 457 706 447 656 451 338 25 3s 2 3p 5 3d 2 ( 1 G) 2 H o 11/2 446 319 455 947 474 215 457 970 458 090 446 427 26 3s 2 3p 5 3d 2 ( 3 F) 2 G o 7/2 448 829 460 699 472 781 458 938 459 116 448 732 27 9/2 453 362 465 205 477 379 463 507 463 688 454 150 28 3s 2 3p 5 3d 2 ( 1 D) 2 F o 5/2 459 950 468 552 484 641 467 176 467 312 459 367 460 756 29 3s 2 3p 5 3d 2 ( 1 G) 2 H o 9/2 456 220 465 829 484 144 467 869 467 987 456 484 30 3s 2 3p 5 3d 2 ( 3 P) 4 D o 7/2 460 956 472 360 483 372 468 071 468 222 31 5/2 463 371 474 565 486 009 470 459 470 604 462 999 32 3/2 465 356 476 416 488 107 472 399 472 538 464 950 33 1/2 466 636 477 622 489 434 473 639 473 776 466 852 34 2 D o 3/2 482 917 494 428 500 999 486 410 486 582 483 374 35 5/2 486 204 497 746 504 164 489 622 489 797 487 278 36 2 S o 1/2 489 820 501 853 503 966 490 079 490 316 489 979 37 4 S o 3/2 489 820 501 853 507 753 492 992 493 144 489 874 38 3s 2 3p 5 3d 2 ( 1 G) 2 G o 9/2 492 355 504 787 511 509 498 344 498 531 492 856 39 7/2 494 017 506 417 513 156 499 946 500 133 493 626 40 3s 2 3p 5 3d 2 ( 1 S) 2 P o 3/2 519 080 527 574 542 856 518 338 518 338 508 518 516 021 41 1/2 528 931 535 904 552 774 530 166 530 147 520 822 528 523 42 3s 2 3p 5 3d 2 ( 3 F) 2 F o 5/2 562 012 563 039 573 995 548 671 548 359 535 909 560 158 43 7/2 567 259 567 925 579 161 553 730 553 420 541 755 565 012 44 3s 2 3p 5 3d 2 ( 3 P) 2 P o 1/2 614 628 622 137 630 978 606 590 606 127 591 964 613 650 45 3/2 617 385 625 069 633 728 609 702 609 264 595 152 616 596 46 3s 2 3p 5 3d 2 ( 3 F) 2 D o 5/2 628 631 637 055 649 201 613 394 612 328 596 463 625 940 47 3/2 628 933 637 354 649 430 613 643 612 577 597 065 626 155
3462 JZenget al correlations. For example, the relative energies of 3s 2 3p 5 3d 2 ( 3 F) 2 D o 5/2 (no 46) and 2 D o 3/2 (no 47) changed from 649 201 and 649 430 cm 1 in case C to 613 394 and 613 643 cm 1 in case D, lowering by 35 807 and 35 787 cm 1,respectively. Additionally inclusion of 3p n 3d n (n = 4, 5, 6) (case E) basically results in the same energy levels as in case D. We have also included more core valence CI, but the levels change little. In general, the BP approximation incorporates spin orbit coupling but omits some other relativistic terms. To estimate this effect, we have carried out calculations to take the contributions of other relativistic terms into account corresponding to case D. Take the transition of 3s 2 3p 6 3d 2 D 5/2 3s 2 3p 5 3d 22 D o 5/2 as an example. For the lower level, we include not only the CI among the 2 Dterm,butalsotheCI among other relativistic terms which have J = 5/2 levels,such as 2 F 5/2, 4 P 5/2, 4 D 5/2, 4 F 5/2, 4 G 5/2, 6 S 5/2, 6 P 5/2, 6 D 5/2, 6 F 5/2, 6 G 5/2 and 6 H 5/2.Similar considerations apply to the upper level. The final results demonstrate that this effect is not important for the transition array 3s 2 3p 6 3d 3s 2 3p 5 3d 2 of Fe VIII. Of course, one should in general take into account this kind of effect. These results show that converged results have been obtained when 3p 3 3d 3 electron correlations are included in the calculation. One can see that the theoretical level order is in accord with the experiment wherever the experimental values are available. However, as strong CI exists and the mixing coefficients of the CSFs are very sensitive to the CI, the order may not correspond exactly to the experimental one for the levels wherever the experimental values are unavailable. Unfortunately, more than half of the levels belonging to the 3s 2 3p 5 3d 2 configuration have not been experimentally measured. Most recently, Bhatia and Eissner [26] reported the energy levels using the CI scheme as well. For comparison, their results are also given in table 1. It can be easily seen that their calculated energy levels are somewhat similar to the HF results (case A). Compared with experimental levels, our converged results (cases D and E) are in better agreement with the experiment than those in case A and those obtained by Bhatia and Eissner. In case A, although there is a good agreement between the level nos 18, 20, 24 and 28, there are large deviations from the experiment for level nos 42 47. As will be demonstrated later from table 2, it is the transitions from the ground configuration to the level nos 42 47 which have large oscillator strengths. The weighted oscillator strengths forthe transitions belonging to the 3s 2 3p 6 3d 3s 2 3p 5 3d 2 transition array of Fe VIII are shown in table 2 for the five cases. Only transitions with gf values greater than 0.001 (calculated values in cases D or E) are listed. In case E, both the length and velocity forms of the oscillator strengths are given to show the quality of the calculation. The convergence behaviour can be easily seen from the results shown in cases A E with the successive addition of 3p n 3d n (n = 1, 2, 3, 6) core valence electron correlations. To have aclear understanding of the convergence behaviour, we discuss this problem according to whether the transition is strong or weak. First, let us pay attention to the strong transitions. For most of the strong transitions, the results obtained from cases A and B are very close. With more and more core valence electron correlations included, the oscillator strengths tend to decrease, in particular from case B to C. When the 3p 2 3d 2 electron correlations are taken into account, the oscillator strengths decrease dramatically. It shows that the 3p 2 3d 2 interaction is one of the most important correlations in a converged calculation of the oscillator strengths. With the addition of 3p 3 3d 3 correlations, the oscillator strengths further decrease a little. Further inclusion of the 3p n 3d n (n = 4, 5, 6) electron correlations results in little change in the excitation energies and oscillator strengths, as is shown in cases D and E. This shows that the oscillator strengths have been converged with the inclusion of the 3p 3 3d 3 correlation. In contrast, the oscillator strengths obtained by Bhatia and Eissner [26] and Fawcett [25] are somewhat similar to our results in case B, having large deviation from our converged results. For the
Convergence of energy levels and oscillator strengths 3463 Table 2. Convergence behaviour of the weighted oscillator strengths for Fe VIII. The transition column shows the lower and upper levels, where the numbers refer to the key listed in table 1. The excitation energies E (inryd), length and velocity forms of the oscillator strengths obtained in case E are given for convenience. The most recent theoretical oscillator strengths obtained by Bhatia and Eissner (BE) [26] and Fawcett [25] are also shown for comparison. A B C D E BE Fawcett Transition gf l gf l gf l gf l E gf l gf v gf gf 1 18 0.0427 0.0056 0.0295 0.0448 3.9861 0.0452 0.0428 1 21 0.0030 0.0042 0.0051 0.0033 4.0321 0.0033 0.0028 1 22 0.0042 0.0159 0.0163 0.0118 4.0588 0.0118 0.0102 0.011 0.013 1 23 0.0001 0.0031 0.0032 0.0023 4.0859 0.0023 0.0020 1 28 0.1205 0.1564 0.0772 0.1102 4.2585 0.1105 0.1020 0.134 0.109 1 34 0.0012 0.0039 0.0008 0.0017 4.4341 0.0018 0.0014 1 40 0.0001 0.0236 0.0021 0.0146 4.7235 0.0153 0.0142 0.031 0.023 1 41 0.0046 0.0273 0.0187 0.0596 4.8311 0.0565 0.0502 0.065 0.062 1 42 4.9089 3.7867 2.9707 2.8967 4.9971 2.8715 2.7823 5.147 4.119 1 44 2.6984 2.3621 2.1541 2.0460 5.5235 2.0431 2.1623 2.484 2.498 1 45 0.5433 0.4762 0.4336 0.4106 5.5521 0.4099 0.4368 0.499 0.473 1 46 0.0671 0.0680 0.5304 0.5135 5.5800 0.5078 0.5266 0.588 0.576 1 47 6.0357 6.1230 4.7750 4.6218 5.5823 4.5698 4.6812 6.027 6.035 2 18 0.0030 0.0039 0.0021 0.0031 3.9682 0.0032 0.0026 2 19 0.0039 0.0055 0.0069 0.0044 3.9844 0.0044 0.0041 2 20 0.0577 0.0911 0.0375 0.0654 4.0082 0.0661 0.0612 0.043 0.013 2 23 0.0066 0.0282 0.0287 0.0211 4.0681 0.0211 0.0198 0.017 0.022 2 24 0.1401 0.1620 0.0912 0.1156 4.1531 0.1152 0.1063 2 28 0.0086 0.0012 0.0055 0.0079 4.2406 0.0079 0.0072 0.020 0.026 2 40 0.0001 0.2087 0.0180 0.1322 4.7056 0.1385 0.1267 0.206 0.128 2 42 0.3495 0.2685 0.2107 0.2054 4.9792 0.2037 0.1924 0.263 0.199 2 43 7.0949 5.4368 4.2490 4.1674 5.0253 4.1323 4.0010 5.279 5.761 2 45 4.8754 4.2702 3.8926 3.6861 5.5342 3.6792 3.6912 4.495 4.557 2 46 9.3605 9.5030 7.4080 7.1707 5.5621 7.0900 7.1660 9.719 9.384 2 47 0.6687 0.6788 0.5292 0.5122 5.5644 0.5064 0.5179 0.681 0.630 strong transitions whose weighted oscillator strengths are greater than 1 (1 42, 1 44, 1 47, 2 43, 2 45, 2 46), the relative differences between our converged results and those obtained by Bhatia and Eissner [26] and Fawcett [25] range from over 20% to more than 80%. For the strongest transition, 3s 2 3p 6 3d 2 D 5/2 3s 2 3p 5 3d 2 ( 3 F) 2 D o 5/2 (2 46), the relative difference to the corresponding results of Bhatia and Eissner, and Fawcett are 37.1 and 32.4%, respectively. These strong transitions have large contributions to the absorption in the plasmas and therefore are important in opacity calculations. Opacities are crucial data for the radiative properties of astrophysical objects and inertial confinement fusion plasmas. The experimental progress [8 10] in the knowledge of iron opacity has explained different astrophysical situations. For example, iron opacity is important in solar physics. Though the iron abundance is very low compared with hydrogen, its opacity contributes largely to the radiative transfer in the interior of the Sun. The present study shows that it is necessary to make clear to what extent CI should be included in actual calculations to achieve accurate opacities. As has been shown in [8, 9], the theoretical opacity is larger than the experimental values, especially in the energy range of 3p 3d transitions. From the present result, one can conclude that, if adequate CI is included in the calculations, then the oscillator strengths will decrease dramatically and thetheoretical opacity should agree better with experiment.
3464 JZenget al Second, let us look at the relatively weak transitions. The oscillator strengths of some transitions are verysensitive to electron correlations. Different degrees of electron correlations can result in oscillator strengths differing by orders of magnitude. For example, the oscillator strength of 3s 2 3p 6 3d 2 D 5/2 3s 2 3p 5 3d 2 ( 1 S) 2 P3/2 o (2 40) is, respectively, 0.0001, 0.2087, 0.0180, 0.1322 and 0.1385 in cases A E. The HF result differs from the result with only 3p 3d correlations by more than four orders of magnitude and from the result with 3p 2 3d 2 correlations by more than three orders of magnitude. The final converged results in cases D and E are again larger than the HF value by more than three orders of magnitude. In this case, the convergence behaviour is much more complicated than those for strong transitions. One should be more cautious and include enough CI to achieve converged results. It is necessary that further studies be carried out to clarify the reason that the oscillator strength is sensitive to the CI. On the other hand, some transitions are not very sensitive to the different electron correlations and they may have some special applications, such as temperature and density diagnostics. The oscillator strength of 3s 2 3p 6 3d 2 D 5/2 3s 2 3p 5 3d 2 ( 3 F) 2 D o 3/2 (2 47) is 0.6687, 0.6788, 0.5292, 0.5122 and 0.5064 in cases A E. The biggest relative difference is only about 30%. Compared with other theoretical results, good agreement is found between our converged oscillator strengths and those obtained by Bhatia and Eissner [26] and Fawcett [25] for transitions such as 1 22, 1 28, 1 41, 2 40, 2 42, etc. The atomic data presented here are not only important in opacitycalculations, as mentioned above, but also helpful in line identification. A large number of unidentified lines exist in the solar or solar coronal spectra [6, 7]. Many of them may be from the iron ions with open 3d shells. To identify these lines, accurate atomic data, such as fine-structure levels and oscillator strengths, are required. From the present study, to obtain accurate basic atomic data, one should take elaborate CI into account. In this work, one can conclude that the relative difference between the theoretical and experimental energy levels wherever available is less than 3%. For the strong transitions, the oscillator strengths should be better than 15%, while for the weak transitions the uncertainties should, in general, be higher. In conclusion, a systematic study is carried out to investigate the effect of core valence electron correlations on the energy levels and oscillator strengths. The results show that up to 3p 3 3d 3 electron correlations should be included to obtain converged results. For the strong transitions, our calculated oscillator strengths are considerably less than other theoretical results in the literature. It suggests that some of the existing atomic data for the iron ions in the literature may not be accurate enough for practical applications. Independent experiments are urgently needed to accurately measure the oscillator strengths of iron ions, especially for the strong 3p 3d transitions, to clarify the differences between the theoretical results. Acknowledgments We are very grateful to the anonymous referees for their valuable comments and helpful suggestions. This work was supported by the National Science Fund for Distinguished Young Scholars under grant no 10025416, the National Natural Science Foundation of China under grant nos 10204024 and 19974075, the National High-Tech ICF Committee in China, and China Research Association of Atomic and Molecular Data. JZ acknowledges the support from the CAS K C Wong Post-Doctoral Research Award Fund. References [1] Widing K G and Sandlin G D 1968 Astrophys. J. 152 545 [2] Behring W E, Cohen L and Feldman U 1972 Astrophys. J. 175 493
Convergence of energy levels and oscillator strengths 3465 [3] Malinovsky M and Heroux L 1973 Astrophys. J. 181 1009 [4] Behring W E, Cohen L, Feldman U and Doschek G A 1976 Astrophys. J. 203 521 [5] Brosius J W, Davila J M and Thomas R J 1998 Astrophys. J. Suppl. 119 255 [6] Zhitnik I A, Kuzin S V, Oraevskii V N, Pertsov A A, Sobel man I I and Urnov A M 1998 Astron. Lett. 24 819 [7] Feldman U, Behring W E, Curd W, Schule U, Wilhelm K, Lemaire P and Moran T M 1997 Astrophys. J. Suppl. Ser. 113 195 [8] Winhart G, Eidmann K, Iglesias C A, Bar-Shalom A, Minguez E, Rickert A and Rose S J 1995 J. Quant. Spectrosc. Radiat. Transfer 54 437 [9] Winhart G, Eidmann K, Iglesias C A and Bar-Shalom A 1996 Phys. Rev. E 53 R1332 [10] Springer P T et al 1997 J. Quant. Spectrosc. Radiat. Transfer 58 927 [11] Zeng J, Jin F, Yuan J, Lu Q and Sun Y 2000 Phys. Rev. E 62 7251 [12] Zeng J, Yuan J and Lu Q 2001 Phys. Rev. E 64 066412 [13] Zeng J and Yuan J 2002 Phys. Rev. E 66 016401 [14] Czyak S J and Krueger T K 1966 Astrophys. J. 144 381 [15] Blaha M 1969 Astrophys. J. 157 473 [16] Pindzola M S, Griffin D C and Bottcher C 1989 Phys. Rev. A 39 2385 [17] Saraph H E, Storey P J and Taylor K T 1992 J. Phys. B: At. Mol. Opt. Phys. 25 4409 [18] Lynas-Gray A E, Seaton M J and Storey P J 1995 J. Phys. B: At. Mol. Opt. Phys. 28 2817 [19] Seaton M J 1987 J. Phys. B: At. Mol. Phys. 20 6363 [20] Tiwary S N 1982 Chem. Phys. Lett. 93 47 [21] Tiwary S N 1983 Astrophys. J. 269 803 [22] Tiwary S N, Kumar P and Roy R P 1996 Pramana 46 381 [23] Hibbert A 1975 Comput. Phys. Commun. 9 141 [24] Fischer C F 1991 Comput. Phys. Commun. 64 369 [25] Fawcett B C 1989 At. Data Nucl. Data Tables 43 71 [26] Bhatia A K and Eissner W 2000 At. Data Nucl. Data Tables 76 270 [27] Eissner W, Jones M and Nussbaumer H 1972 Comput. Phys. Commun. 8 270 [28] Zeng J, Jin F, Zhao G and Yuan J 2003 Chin. Phys. Lett. 20 862 [29] Fischer C F 1997 Computational Atomic Structure An MCHF Approach (Bristol: Institute of Physics Publishing) [30] Tachiev G and Fischer C F 1999 J. Phys. B: At. Mol. Opt. Phys. 32 5805 [31] Drake G W F 1972 Phys. Rev. A 5 1979 [32] Zeng J, Yuan J and Lu Q 2001 Phys. Rev. A 64 042704 [33] Zeng J, Yuan J and Lu Q 2001 J. Phys. B: At. Mol. Opt. Phys. 34 2823 [34] Zeng J and Yuan J 2002 J. Phys. B: At. Mol. Opt. Phys. 35 3041 [35] Zeng J and Yuan J 2002 Phys. Rev. A 66 022715 [36] Corliss C and Suger J 1985 J. Phys. Chem. Ref. Data 11 135