Advance Access publication 2018 June 11
Multiconfiguration Dirac–Hartree–Fock calculations of energy levels and
radiative rates of Fe
VII
Yang Li,
1Xiaokai Xu,
1Bowen Li,
1‹Per J¨onsson
2and Ximeng Chen
11School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China
2Group for Materials Science and Applied Mathematics, Malm¨o University, SE-205 06 Malm¨o, Sweden
Accepted 2018 June 8. Received 2018 June 8; in original form 2017 October 24
A B S T R A C T
Detailed calculations are performed for 134 fine-structure levels of the 3p63d2, 3p63d4s, 3p53d3 and 3p63d4p configurations in Fe VII using the multiconfiguration Dirac–Hartree– Fock (MCDHF) and relativistic configuration interaction (RCI) methods. Important electron correlation effects are systematically accounted for through active space (AS) expansions. Our results compare well with experimental measurements, emphasizing the importance of a careful treatment of electron correlation, and provide some missing data in the NIST atomic database. The data obtained are expected to be useful in astrophysical applications, particularly for the research of the solar coronal plasma.
Key words: Atomic data – Atomic processes.
1 I N T R O D U C T I O N
Accurate atomic data for Fe VII are important in astrophysical
plasma modeling (e.g. Aller, Bowen & Minkowski 1955; Aller et al.1973; Thackeray1977; Holczer, Behar & Kaspi2005). For example, The Solar Ultraviolet Measurements of Emitted Radiation (SUMER) spectrometer installed on the Solar and Heliospheric Ob-servatory (SOHO) recorded a large number of solar coronal spectra in the 500–1610 Å region. Feldman et al. (1997) and Ekberg & Feldman (2003) identified some lines of the 3d4s− 3d4p and 3d4p − 3d4d transitions in FeVII. Emission lines in Fe VIIformed at
tem-peratures around 300,000 K in the solar atmosphere in the range 150–300 Å, mainly from 3p63d2– 3p53d3transitions, have been
observed by the Hinode EUV Imaging Spectrometer (Culhane et al.
1994). Plasma diagnostics of an optically thin astrophysical plasma require accurate data including level energies, transition probabili-ties, and rate coefficients for various dynamic processes. However, in the NIST Atomic Spectra Database (ASD), there are many levels missing for the 3p53d3configuration.
The earliest theoretical calculations of five strong transitions from the 3s23p53d3 configuration have been presented by Fawcett &
Cowan (1973) using self-consistant field (SCF) Hartree X methods, which helped to analyze a laboratory vacuum spark spectrum. Sub-sequently, Ekberg (1981) determined many lines of the 3s23p63d4p
configurations. Faulkner et al. (1981) measured the 3d4p− 3d4d and 3d4d− 3d4f transitions in MnVI, FeVII, and CoVIIIusing
collision-based spectroscopy, and identified these lines with the aid of multi-configuration Hartree–Fock (MCHF) calculations and isoelectronic comparison. Because of its importance, Fe VII has been further
E-mail:libw@lzu.edu.cn
studied using a number of different theoretical methods (see latest paper by Tayal & Zatsarinny (2014), and references therein); e.g., by Zeng et al. (2005) (Z05) using the flexible atomic code (FAC), by Witthoeft & Badnell (2008) (WB08) using the AUTOSTRUCTURE code and by Tayal & Zatsarinny (2014) (T14) using the MCHF-BP program with non-orthogonal orbitals. Atomic data for Fe VIIhave
been evaluated by Del Zanna (2009) (D09) based on Hinode so-lar observations and calculations by Witthoeft & Badnell (2008). Moreover, the photorecombination rate coefficients of Fe VIIand
of Fe VIIIwere measured by Schmidt et al. (2008) by employing
the merged electron-ion beams method at the Heidelberg heavy-ion storage-ring TSR and analyzed with the aid of data from the AUTOSTRUCTURE code. Extreme-ultraviolet (EUV) spectra of Fe VI– FeXIVwere recorded in the 125–265 Å region at the Hei-delberg electron beam ion trap (EBIT) by Liang et al. (2009) and they performed collisional-radiative model calculations using FAC. Young & Landi (2009) analysed emission lines in FeVII, Fe VIII,
and FeIXobserved in 2007 by the Hinode EUV Imaging
Spectrom-eter. They compared Fe VIIlines with predictions and proposed that
four new line ratios can be used to diagnose plasma temperature and density. However, they also found that there were a number of disagreements between theory and observed line ratios insen-sitive to density and temperature, implying that new benchmark calculations are required. Using the NIST EBIT, Beiersdorfer et al. (2014) measured several lines of Fe between 200 and 205 Å. They identified some weaker lines of iron and also found that there are some clear disagreements between the CHIANTI database and the NIST database. They found large discrepancies between observed and theoretical line positions and intensities for strong EUV lines involving the 3s23p53d3configuration. They thought these
discrep-ancies were due to incorrect line identifications, which implies that more experimental data and theoretical calculations are still needed.
Table 1. Expansion schemes of computational models for the even and odd states. The model MR is the minimal basis set model while other models include the electron correlation contributions to a different extent. The configuration space was generated by single (S) and double (D) substitutions from the configurations in the MR to increasing active sets of orbitals.
States AS set Number of CSFs
J= 0 J= 1 J= 2 J= 3 J= 4 J= 5 J= 6 MR 2 2 5 2 2 – – Even 4SD 2970 7935 11270 12023 10910 – – states 5SD 14474 39789 57491 64024 60966 – – 6SD 34817 96470 139975 157596 151833 – – 7SD 63999 177978 258722 292739 283511 – – MR 8 22 29 27 19 11 5 Odd 4SD 11837 32924 47054 52212 49008 40114 28966 states 5SD 61095 171521 250061 286054 280129 242489 188088 6SD 149121 419634 614821 708576 700966 614690 484455 7SD 275915 777263 1141334 1319778 1311519 1156717 918067
Table 2. The size (M) of the 7SD expansion and the size of the zero-order space (m). The zero-order approximation was obtained that consisted of the CSFs of the 5SD expanstion that accounted for 99.9 percent of the normalized expansion. (See text).
States J M m 0 63999 520 Even 1 177978 2273 states 2 258722 5003 3 292739 3257 4 283511 2568 0 275915 5313 Odd 1 777263 19796 states 2 1141334 30149 3 1319778 31943 4 1311519 25571 5 1156717 17012 6 918067 8896
Recently, Ryabtsev (2017) observed a vacuum spark spectrum and as a result of the analysis of the FeVII3p63d2– (3p63d4p + 3p53d3)
transitions, 185 lines from the region 158–266 Å were identified, 55 lines being new. Furthermore, Ryabtsev claimed that the Fe VII
line identifications in the solar spectrum performed by Del Zanna (2009) are quite questionable. Therefore, accurate calculations of atomic data for Fe VIIare still of importance.
In this paper, we used the Multiconfiguration Dirac–Hartree– Fock (MCDHF) and relativistic configuration interaction (RCI) method implemented in GRASP2K (J¨onsson et al.2013) to cal-culate 134 fine-structure levels of the 3p63d2, 3p63d4s, 3p53d3and
3p63d4p configurations in Fe
VII. These calculations provide more
accurate atomic data for astrophysical applications, particularly for the research of the solar coronal plasma.
2 T H E O RY 2.1 MCDHF theory
The MCDHF method has been described in detail by Grant (2007). Here, we only briefly outline the theoretical methods used for the calculation of energy levels and transition probabilities.
In the MCDHF method, an atomic state wavefunction (ASF) is given as an expansion over configuration state functions (CSFs)
with the same parity P and angular momentum (J, M)
(γ P J M)= nC
j=1
cj(γjP J M), (1) where ncis the number of CSFs and cjare configuration mixing co-efficients. The CSFs are antisymmetrized products of a common set of orthonormal orbitals that are optimized based on the relativistic Dirac–Coulomb Hamiltonian.
According to the time-dependent perturbation theory, the sponta-neous emission transition rates of Einstein from an initial state γJ
to a final state γ J can be written
AγJ→γ J = 2π 2J+ 1 M M | Mγ J M,γJM |2. (2) In the expression above, Mγ J M,γJM is the transition matrix ele-ment
Mγ J M,γJM= (γ P J M) | T | (γPJM), (3)
where T is the transition tensor operator.
For electric multipole transitions, there are two forms of transition operator: Babushkin and Coulomb gauges, which correspond to the length form (Al) and the velocity form (Av) in non-relativistic limit, respectively (Grant2007). The agreement between two gauges can be expressed as
dT = |Al− Av|
max(Al,Av)
, (4)
is a parameter related to the estimated uncertainty of the radiative rates (Ekman, Godefroid & Hartman2014).
2.2 Computational details
As demonstrated by Zeng et al. (2005) and by Tayal & Zatsarinny (2014), both valence correlation among the outer 3d electrons and core–valence correlation involving the 3s23p5 or 3s23p6 are
ex-pected to be important because there are significant overlaps be-tween the 3s, 3p, and 3d orbitals. In the MCDHF method, electron correlation effects may be treated by building the CSF expansion space systematically using the active set approach (see Froese Fis-cher & J¨onsson1994).
The first step was to perform Dirac–Hartree–Fock (DHF) cal-culations using the extended optimal-level (EOL) scheme for all states associated with the 3s23p63d2 and 3s23p63d4s even
par-ity states and the 3s23p53d3 and 3s23p63d4p odd parity states.
Table 3. Level energies (in cm−1) for FeVIIfrom calculations with expansions obtained by SD substitutions to increasing active sets of orbitals. Level energies from the NIST database are shown for comparison.
Index Configuration Term NIST 4SD 5SD 6SD 7SD
1 3s23p63d2 3F 2 0.0 0.0 0.0 0.0 0.0 2 3s23p63d2 3F 3 1051.5 1027 1021 874 864 3 3s23p63d2 3F 4 2331.5 2285 2271 2340 2347 4 3s23p63d2 1D 2 17475.5 18273 17967 17771 17728 5 3s23p63d2 3P 0 20040.3 20194 20313 20328 20330 6 3s23p63d2 3P 1 20430.1 20572 20690 20631 20610 7 3s23p63d2 3P 2 21278.6 21454 21527 21525 21516 8 3s23p63d2 1G 4 28927.3 30349 29610 29297 29249 9 3s23p63d2 1S 0 67078.3 69351 68606 68154 68034 10 3s23p63d 4s 3D 1 344463.3 342752 343954 343790 343866 11 3s23p63d 4s 3D 2 345028.7 343331 344524 344155 344214 12 3s23p63d 4s 3D 3 346262.2 344530 345712 345559 345639 13 3s23p63d 4s 1D2 350332.6 349455 350441 349831 349869
Figure 1. Percentage differences of calculated level energies based on CSF expansions to active sets MR, 4SD, 5SD, 6SD, and 7SD relative to energies from the NIST database.
Figure 2. Percentage differences between Zeng et al. (2005) (Z05), Wit-thoeft & Badnell (2008) (WB08), Del Zanna (2009) (D09), Tayal & Zat-sarinny (2014) (T14), and present calculated-level energies relative to ener-gies from the NIST database.
The above configurations define the multireference (MR) for the even and odd parities, respectively. This calculation determined the inactive core 1s, 2s, and 2p orbitals for all subsequent calcu-lations. Then, sequentially, correlation expansions were obtained by allowing single (S) and double (D) excitations from the the MR configurations to an extended active set of orbitals, with maximum principal quantum numbers up to n = 7 and l = 3. These active sets are denoted by MR, 4SD, 5SD, 6SD, and 7SD, respectively.
Table1summarizes the size of CSFs as a function of the ex-pansion model. Note that the size of CSFs increases rapidly as the principal quantum number increases from 4 to 7. To handle such large expansions, parallel programs were employed with 20 processors. Moreover, for n= 6 and n = 7, an approximation com-putational scheme is employed in which the CSF is divided into zero- and a first-order spaces (Froese Fischer, Gaigalas & J¨onsson
2017) (γ1P J M),· · · , (γmP J M) zero-order space , (γm+1P J M),· · · , (γMP J M) first-order space , (5)
where M is the total number of CSFs. The zero-order space con-tains the most important CSFs. For n= 6 and n = 7, a zero-order approximation was obtained that consisted of the CSFs of the n= 5 SD expansion that accounted for 99.9 percent of the normalized expansion. All other terms of the n = 6 and n = 7 expansions were treated as first-order corrections and can be regarded as mi-nor corrections. The sizes of the zero-order spaces (m) are given in Table2.
Finally, the Breit interaction and the dominant quantum-electrodynamical (QED) contributions such as the vacuum polar-ization and self-energy corrections are also included in the RCI calculations as suggested by Grant (2007).
In order to match the computed states against the NIST database and other calculations, the representation of CSFs were trans-formed from jj – to LSJ – coupling (Gaigalas, Z ˘Zalandauskas & Rudzikas 2003; J¨onsson et al. 2013). The labels obtained with this approach are, however, not unique, an extra index was then added to obtain unique labels (J¨onsson et al.2016; Gaigalas et al.
2017).
Table 4. Level energies (in cm−1) relative to the 3s23p63d2(3
2F)˜3F2ground state in FeVII.
Index Configuration Term NIST Z05 WB08 D09 T14 S01 Present
1 3s23p63d2(3 2F) 3F2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2 3s23p63d2(3 2F) 3F3 1051.5 994 1284 1192 1129 1044 864 3 3s23p63d2(3 2F) 3F4 2331.5 2279 2853 2635 2581 2325 2347 4 3s23p63d2(1 2D) 1D2 17475.5 18057 18271 17436 17018 17054 17728 5 3s23p63d2(3 2P) 3P0 20040.3 21429 21311 19949 19841 19936 20330 6 3s23p63d2(3 2P) 3P1 20430.1 21768 21783 20389 20244 20324 20610 7 3s23p63d2(3 2P) 3P2 21278.6 22568 22474 21373 21212 21140 21516 8 3s23p63d2(1 21G) 1G4 28927.3 32099 32504 28933 28632 28140 29249 9 3s23p63d2(1 0S) 1S0 67078.3 68949 67905 67081 65814 66065 68034 10 3s23p63d˜2D 4s 3D 1 344463.3 363226 341228 344398 339960 343866 11 3s23p63d˜2D 4s 3D2 345028.7 363785 341876 344990 345364 344214 12 3s23p63d˜2D 4s 3D3 346262.2 364951 343258 346310 346735 345639 13 3s23p63d˜2D 4s 1D2 350332.6 369583 347955 350355 349880 349869
NIST: level energies from NIST database (Kramida et al.2018). Z05: from Zeng et al. (2005) using the FAC.
WB08: from Witthoeft & Badnell (2008) using the AUTOSTRUCTURE code. D09: compiled data by Del Zanna (2009).
T14: from Tayal & Zatsarinny (2014) by MCHF-BP method. S01: from Safronova et al. (2001) by RMBPT method. Full table is available on-line.
3 R E S U LT S 3.1 Energies
In Table3, we present the computed level positions for the 13 lowest levels in FeVIIbelonging to the 3p63d2and 3p63d4s configurations
as a function of the increasing active sets. Observed energies from the NIST ASD (ver. 5.5.6; Kramida et al.2018) are also are given for comparison. The excitation energies are also shown in Fig.1. The agreement between the computed excitation energies and the observed energies is generally very good. Except for the first excited state 3d2 3F
3, the relative differences between theory and
observa-tion is 22.69 percent, 4.91 percent, 2.81 percent, 2.02 percent, and 1.58 percent for calculations based on the MR and 4SD, 5SD, 6SD, and 7SD, respectively. It is obvious that convergence is pro-gressively refined except for the first excited states. From Fig.1,
we find that the level energies calculated between 6SD and 7SD are less then 0.3 percent (except for 3d2 3F
3) for the whole energy
region, which imply that a fair convergence is reached.
To further show the accuracy of the present calculations, we com-pare, in Fig.2and Table4(full table is available online), our theoret-ical results based on the largest orbital set n= 7 with other available calculated data from FAC produced by Zeng et al. (2005), from AU-TOSTRUCTURE produced by Witthoeft & Badnell (2008), from compiled data by Del Zanna (2009), from MCHF-BP calculations by Tayal & Zatsarinny (2014) and from relativistic many-body per-turbation theory (RMBPT) by Safronova et al. (2001). In Table4, we also provide data from NIST ASD. In their RMBPT calculations, Safronova et al. (2001) start from a 1s22s22p63s23p6Dirac–Fock
potential and the model space consists of nine states in 3d3d. The relative differences between present calculations and RMBPT val-ues of the 3p63d2is less then 4 per cent except for the first excited
states, this difference is mainly due to omitting the contribution from the 3s and 3p electron correlation effects in the latter. This issue is also demonstrated by another work by Ding et al. (2017) for Ca-like tungsten using MCDHF method. Tayal & Zatsarinny (2014) pointed out that the 3p43d3nl configurations were omitted
by Zeng et al. (2005) and by Witthoeft & Badnell (2008) result in less accurate energies for the 3p64l levels in their data. The
present calculations compare well with the compiled data by Del Zanna (2009) although we note that some level orders belonging to 3s23p53d3states are inconsistent between present calculations and
NIST data and D09.
3.2 The transition rates and lifetimes
In Table5, we list transition wavelengths (in Å), weighted oscilla-tor strengths (gf), and radiative rates (A in s−1) for dipole-allowed (E1) transitions. Again, we make comparison of our data with other calculations from Witthoeft & Badnell (2008), Del Zanna (2009), and Tayal & Zatsarinny (2014). Because of limited space, transi-tions are restricted to those lines that can be compared (full ta-ble is availata-ble online along with the uncertainty indicator dT). In general, the transition wavelengths, oscillator strengths. and radia-tive rates of Del Zanna (2009) show very good agreement with our results, although there are individual differences. For most of the strong transitions with A-value 108dT is below 10 percent,
while for the weaker transitions, the uncertainty dT is somewhat larger. It should be noted that wavelengths from MCHF calcula-tions by Tayal & Zatsarinny (2014) in general differ more with other calculations.
Forbidden transitions such as 3p63d2– 3p63d4s are of particular
importance because they can be used for plasma temperature di-agnostic (e.g. Del Zanna2009). These decays have been observed by Hinode EUV Imaging Spectrometer. In Table 6, we compare a few of E2 transitions of the type 3d2– 3d4s with the compiled
data by Del Zanna (2009), the AUTOSTRUCTURE calculations by Witthoeft & Badnell (2008), and from the NIST database which are based mainly on the work of Nussbaumer & Storey (1982) as well as E2 and M1 transitions within 3d2in Table7(full table for
forbidden E2, M1, and M2 transitions is available online). The SU-PERSTRUCTURE calculation by Biemont et al. (1992) are shown for comparison. The A-values agree well with each other.
The lifetimes of the excited states were calculated from dipole allowed E1 and forbidden E2 transition rates in both the length and velocity forms. Contributions from forbidden M1 and M2 transitions are also included. In Table8, we compare calculated lifetimes in length and velocity forms with lifetimes obtained with the MCHF
Table 5. Comparison of wavelengths, oscillator strengths, and transition rates for some E1 transitions of FeVII.
i− j Wavelength(Å) gf A (s−1)
Present NIST D09 T14 Present D09 WB08 Present D09 T14
6-133 163.86 165.764 165.75 155.36 1.53 1.49 1.5 1.265× 1011 1.2× 1011 1.43× 1011 6-117 187.93 189.450 188.18 182.82 1.05 1.35 1.2 3.955× 1010 5.1× 1010 5.10× 1010 1-35 235.23 235.223 233.04 235.20 0.40 0.45 0.51 1.244× 1010 1.9× 1010 2.03× 1010 1-130 163.79 165.489 165.57 160.75 3.22 3.44 3.5 2.658× 1011 2.8× 1011 2.30× 1011 4-56 233.69 234.757 234.71 234.52 0.31 0.29 0.30 1.256× 1010 1.2× 1010 1.17× 1010 4-134 160.94 163.183 163.17 158.06 2.45 2.84 3.0 2.101× 1011 2.4× 1011 2.44× 1011 7-133 164.10 165.997 166.02 155.59 2.35 2.30 2.4 1.939× 1011 1.9× 1011 2.09× 1011 1-28 238.88 238.76 231.18 2.2× 10−2 2.1× 10−2 8.7× 10−4 5.054× 108 4.8× 108 1.62× 109 1-124 175.25 177.171 177.06 170.90 3.50 3.19 3.8 1.521× 1011 1.4× 1011 8.08× 1010 4-36 245.43 245.152 244.51 245.76 0.24 0.37 0.39 5.332× 109 8.3× 109 6.94× 109 4-119 185.02 186.656 183.99 179.10 2.00 2.24 2.3 7.805× 1010 8.8× 1010 1.04× 1011 7-51 239.88 240.223 240.62 241.14 0.42 0.54 0.32 9.672× 109 1.2× 1010 7.53× 109 7-123 182.35 183.825 182.32 177.51 1.78 2.26 2.3 7.133× 1010 9.1× 1010 3.99× 1010 1-98 207.32 207.712 207.69 200.39 0.72 0.75 0.75 1.589× 1010 1.7× 1010 1.69× 1010 1-105 195.34 196.045 196.96 189.43 1.21 1.36 1.1 3.014× 1010 3.3× 1010 3.82× 1010 4-120 184.07 185.547 186.80 178.80 1.73 2.01 2.3 4.861× 1010 5.5× 1010 6.31× 1010 4-118 185.84 187.233 185.24 180.98 1.45 0.78 0.56 3.999× 1010 2.2× 1010 2.43× 1010 7-120 185.37 186.866 188.19 180.14 1.41 0.73 0.52 3.910× 1010 2.0× 1010 2.35× 1010 7-118 187.16 188.576 186.60 182.35 1.33 2.0 2.3 3.613× 1010 5.5× 1010 5.06× 1010 2-37 234.33 234.338 233.20 235.35 0.31 0.46 0.59 7.423× 109 1.1× 1010 1.65× 1010 2-129 164.21 165.919 166.00 161.20 5.25 5.85 6.0 2.598× 1011 2.8× 1011 2.81× 1011 2-47 231.63 232.257 232.00 233.93 0.13 0.23 0.11 2.261× 109 4.1× 109 3.96× 109 2-125 174.99 176.927 176.97 170.86 5.48 5.03 5.2 1.705× 1011 1.5× 1011 1.57× 1011 2-43 232.54 235.28 227.34 0.06 0.11 0.09 8.759× 108 1.5× 109 1.30× 109 2-96 208.99 211.931 209.52 202.28 0.77 0.73 0.74 1.304× 1010 1.2× 1010 1.11× 1010 2-107 194.62 196.423 196.16 188.88 2.11 1.24 2.1 4.136× 1010 2.4× 1010 4.90× 1010 3-42 233.41 233.308 233.53 235.75 0.64 0.90 0.80 1.121× 1010 1.6× 1010 1.61× 1010 3-128 164.71 166.365 166.38 161.55 7.68 8.55 8.8 2.697× 1011 2.9× 1011 2.91× 1011 8-53 242.46 243.378 243.37 243.60 1.71 1.7 1.71 2.770× 1010 2.8× 1010 2.63× 1010 8-132 166.25 164.34 160.49 8.45 9.85 10 2.912× 1011 3.5× 1011 3.04× 1011 3-19 254.98 254.26 252.68 4.2× 10−4 8.5× 10−4 2.9× 10−4 4.733× 106 9.3× 106 2.72× 106 3-48 231.49 231.728 231.97 233.63 0.46 0.48 0.46 6.352× 109 6.6× 109 4.94× 109 3-126 174.85 176.745 176.81 170.71 7.74 7.15 7.4 1.876× 1011 1.7× 1011 1.72× 1011 8-131 170.70 173.442 171.10 167.52 10.59 8.52 8.6 2.695× 1011 2.2× 1011 1.90× 1011 3-20 250.11 254.26 247.68 7.4× 10−5 1.7× 10−5 1.4× 10−5 7.680× 105 1.6× 105 8.42× 104 3-45 233.11 235.83 227.76 1.0× 10−1 9.2× 10−2 9.5× 10−2 1.135× 109 1.0× 109 1.08× 109 3-97 209.64 212.663 210.59 203.56 0.55 0.62 0.60 7.540× 109 8.5× 109 1.11× 1010 3-108 194.46 195.388 196.13 188.75 3.07 2.90 3.0 4.930× 1010 4.6× 1010 5.78× 1010 8-115 195.19 196.22 191.01 2.59 2.22 2.3 4.117× 1010 3.5× 1010 4.04× 1010
D09: compiled data by Del Zanna (2009).
T14: from Tayal & Zatsarinny (2014) by MCHF-BP method.
WB08: from Witthoeft & Badnell (2008) using the AUTOSTRUCTURE code.
Table 6. Comparison of wavelengths and transition rates for some E2 transitions of the type 3d2– 3d4s in FeVII.
Wavelength (Å) A (s−1)
index Lower-Upper Exp D09 Present D09 WB08 Present
3–12 3d2 3F4– 3d 4s˜3D3 290.756 290.97 291.30 1.6× 105 1.57× 105 1.61× 105 2–11 3d2 3F3- 3d 4s˜3D2 290.717 290.87 291.25 1.0× 105 9.97× 104 1.03× 105 1–10 3d2 3F2- 3d 4s˜3D1 290.307 290.36 290.81 1.4× 105 1.36× 105 1.41× 105 1–11 3d2 3F2- 3d 4s˜3D2 289.831 289.86 290.52 6.2× 104 5.95× 104 5.95× 104 2–12 3d2 3F3- 3d 4s˜3D3 289.678 289.76 290.04 4.6× 104 4.43× 104 4.56× 104 3–11 3d2 3F4- 3d 4s˜3D2 291.803 292.09 292.51 4.5× 104 4.31× 104 4.42× 104 2–10 3d2 3F3- 3d 4s˜3D1 291.196 291.37 291.54 7.1× 104 6.80× 104 7.00× 104
Exp and D09: measured wavelengths using the Hinode EUV Imaging Spectrometer and identifications by Del Zanna (2009). WB08: from Witthoeft & Badnell (2008) using the AUTOSTRUCTURE code.
method (Tayal & Zatsarinny2014) and with the RMBPT method (Safronova et al.2001) and with experimental lifetime by Tr¨abert et al. (2003) using the heavy-ion storage ring for the lowest 12 excited states belonging to 3p63d2and 3p63d4s configurations. The
recommended lifetime value of 3p63d2 1S
0 is 33.1 ms in NIST
database while the experiment value is 29.6± 1.8 ms (Tr¨abert et al.
2003) The length form of the lifetime is in very good agreement with the measured value.
Table 7. Comparison of transition rates for some M1 and E2 transitions within the 3d2configuration in FeVII.
M1 E2
index Lower-Upper B92 NIST Present B92 NIST Present
1 - 2 3F 2−3F3 3.26× 10−2 2.98× 10−2 1.651× 10−2 1.03× 10−9 2.803× 10−10 - 3 3F 2−3F4 1.64× 10−9 1.872× 10−9 - 4 3F 2−1D2 3.50× 10−1 3.6× 10−1 1.806× 10−1 2.96× 10−4 2.9× 10−4 2.744× 10−4 - 5 3F 2−3P0 9.84× 10−2 9.4× 10−2 9.219× 10−2 - 6 3F 2−3P1 1.59× 10−3 1.7× 10−3 6.615× 10−3 3.57× 10−2 3.4× 10−2 3.371× 10−2 - 7 3F 2−3P2 1.63× 10−2 1.6× 10−2 1.492× 10−1 3.27× 10−3 3.0× 10−3 2.715× 10−3 - 8 3F 2−1G4 6.10× 10−4 5.6× 10−4 3.870× 10−4 - 9 3F 2−1S0 1.71× 10−1 1.7× 10−1 9.239× 10−2 2 - 3 3F 3−3F4 4.65× 10−2 4.24× 10−2 6.572× 10−2 2.20× 10−9 3.464× 10−9 - 4 3F 3−1D2 5.58× 10−1 5.8× 10−1 3.565× 10−1 4.62× 10−4 4.3× 10−4 6.576× 10−4 - 6 3F 3−3P1 5.48× 10−2 5.3× 10−2 5.356× 10−2 - 7 3F 3−3P2 5.71× 10−2 5.9× 10−2 1.377× 10−1 1.93× 10−2 1.8× 10−2 1.853× 10−2 - 8 3F 3−1G4 3.04× 10−1 3.1× 10−1 4.098× 10−1 1.20× 10−5 1.484× 10−5 3 - 4 3F 4−1D2 1.61× 10−3 1.6× 10−3 2.028× 10−3 - 7 3F4−3P2 5.18× 10−2 5.0× 10−2 4.752× 10−2 - 8 3F4−1G4 4.38× 10−1 4.5× 10−1 6.056× 10−1 2.24× 10−4 2.2× 10−4 2.739× 10−4 4 - 5 1D2−3P0 1.98× 10−7 2.932× 10−7 - 6 1D2−3P1 4.21× 10−2 4.4× 10−2 5.340× 10−2 4.28× 10−7 4.315× 10−7 - 7 1D2−3P2 1.55× 10−1 1.6× 10−1 1.967× 10−1 1.29× 10−7 1.484× 10−7 - 8 1D 2−1G4 1.37× 10−3 1.2× 10−3 1.192× 10−3 - 9 1D 2−1S0 2.32× 101 2.2× 101 2.147× 101 5 - 6 3P 0−3P1 1.16× 10−3 1.06× 10−3 3.961× 10−4 - 7 3P 0−3P2 1.72× 10−8 1.027× 10−8 6 - 7 3P 1−3P2 8.58× 10−3 7.62× 10−3 8.991× 10−3 5.83× 10−9 5.725× 10−9 - 9 3P 1−1S0 6.86× 100 6.9× 100 6.062× 100 7 - 8 3P 2−1G4 1.98× 10−5 2.319× 10−5 - 9 3P 2−1S0 1.47× 100 1.4× 100 1.831× 100
B92: from Biemont et al. (1992) using the SUPERSTRUCTURE code.
NIST: from NIST database (Kramida et al.2018) which are based mainly on the work of Nussbaumer & Storey (1982).
Table 8. Comparison of lifetimes in s.
Configuration Term τl τv S01 T14 3s23p63d2 3F 3 6.056× 101 6.056× 101 3.42× 101 2.42× 101 3s23p63d2 3F 4 1.522× 101 1.522× 101 2.35× 101 1.68× 101 3s23p63d2 1D 2 1.852× 100 1.854× 100 2.51× 100 1.00× 100 3s23p63d2 3P 0 1.085× 101 1.531× 101 1.12× 101 1.15× 101 3s23p63d2 3P 1 6.772× 100 7.983× 100 8.76× 100 7.67× 100 3s23p63d2 3P 2 1.781× 100 1.843× 100 5.07× 100 3.08× 100 3s23p63d2 1G 4 9.830× 10−1 9.834× 10−1 2.84× 100 1.31× 100 3s23p63d2 1S0 3.395× 10−2 4.239× 10−2 4.34× 10−2 3.54× 10−2 3s23p63d 4s 3D1 3.597× 10−6 3.504× 10−6 3.64× 10−6 3s23p63d 4s 3D2 3.590× 10−6 3.481× 10−6 3.59× 10−6 3s23p63d 4s 3D3 3.589× 10−6 3.486× 10−6 3.62× 10−6 3s23p63d 4s 1D 2 3.008× 10−6 2.830× 10−6 2.94× 10−6
τland τv: present calculations in length and velocity form, respectively.
S01: from Safronova et al. (2001) by RMBPT method. T14: from Tayal & Zatsarinny (2014) by MCHF-BP method.
4 S U P P L E M E N TA RY M AT E R I A L
The 134 levels are indexed in Table4. The complete set of E1 tran-sition data (of which the Table5above is a small part) is given in Table5, which contains transition energies (E in cm−1), wave-lengths (λ in Å), transition rates (A in s−1), and weighted oscillator strengths (gf), along with the uncertainty indicator dT, while Table6
contains the equivalent material for E2, M1, and M2 transitions.
5 S U M M A RY A N D C O N C L U S I O N S
We have calculated energy levels, radiative rates, and oscillator strengths for 134 levels of the 3p63d2, 3p63d4s 3p53d3, and 3p63d4p
configurations in FeVIIby MCDHF and subsequent RCI methods.
Valance–valance, valance–core and core–core electron correlation effects are included by increasing the active orbital set layer by layer. To handle such large expansions, CSFs are divided into a
zero- and first-order space. The zero-order space contains CSFs that account for the major parts of the wave functions, while the first-order space contains CSFs that represent minor corrections. The first-order space can be treated as first-order corrections, i.e. allowing interaction between CSFs in the zero-order space and between CSFs in the zero- and first-order spaces and only diagonal interactions between CSFs in the first-order space. This restriction results in a huge reduction in the total number of matrix elements and hence a reduction in computational time. Systematic inclusion of electron correlation effects is very important to obtain accurate energy and transition data. Comparisons between new calculated data show very good agreement with the observed energies from the NIST ASD and compiled data of Del Zanna (2009), which implies the reliability of present correlation model. Further work on the other Ca-like ions of iron group elements is in progress. We hope these data will provide useful information for astrophysical applications.
AC K N OW L E D G E M E N T S
The LZU group acknowledges support from the National Natu-ral Science Foundation of China under Grant Nos. 11404152 and U1732269 and Fundamental Research Funds for the Central Uni-versities Grant No. lzujbky-2017-94.
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S U P P O RT I N G I N F O R M AT I O N
Supplementary data are available at 0:italic MNRAS/0:italic online.
Table4-online.dat Table5-online.dat Table6-online.dat
Table 4. Level energies (in cm−1) relative to the 3s23p63d2(3 2F)
3F 2
ground state in Fe VII.
Table 5. Oscillator strengths and transition rates for E1 transitions
of Fe VII.
Table 6. Oscillator strengths and transition rates for E2, M1 and M2
transitions of FeVII.
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