A&A 455, 1157-1160 (2006)
DOI: 10.1051/0004-6361:20064822

Energy levels and transition probabilities for boron-like Fe XXII^,

V. Jonauskas^1,2 - P. Bogdanovich² - F. P. Keenan¹ - R. Kisielius^1,2 - M. E. Foord³ - R. F. Heeter³ - S. J. Rose⁴ - G. J. Ferland⁵ - P. H. Norrington⁶

1 - Department of Physics and Astronomy, The Queen's University of Belfast, Belfast BT7 1NN, Northern Ireland, UK
2 - Institute of Theoretical Physics and Astronomy of Vilnius University, A. Gostauto 12, 01108 Vilnius, Lithuania
3 - University of California, Lawrence Livermore National Laboratory, Livermore, CA 94551, USA
4 - Department of Physics, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, UK
5 - Department of Physics, University of Kentucky, Lexington, KY 40506, USA
6 - Department of Applied Mathematics and Theoretical Physics, The Queen's University of Belfast, Belfast BT7 1NN, Northern Ireland, UK

Received 6 January 2006 / Accepted 17 May 2006

Abstract
The Multiconfiguration Dirac-Fock method is used to calculate the energies of the 407 lowest levels in Fe XXII. These results are cross-checked using a suite of codes which employ the configuration interaction method on the basis set of transformed radial orbitals with variable parameters, and takes into account relativistic corrections in the Breit-Pauli approximation. Transition probabilities, oscillator and line strengths are presented for electric dipole (E1), electric quadrupole (E2) and magnetic dipole (M1) transitions among these levels. The total radiative transition probabilities, as well as the five largest values from each level be it of E1, M1, E2, M2, or E3 type, are also provided. Finally, the results are compared with data compiled by NIST.

Key words: atomic data

1 Introduction

Physical processes and conditions in astrophysical sources can be understood through the use of theoretical models to analyse high resolution spectral observations. In particular, X-ray spectra from iron L-shell ions are prominent candidates for astrophysics plasma diagnostics, as their emission lines are observed in the wavelength range from 6-18 Å, covered with large effective areas by X-ray telescopes on board the space observatories Chandra and XMM-Newton. Behar et al. (2001) identified Fe XXII lines in a spectrum of the Capella binary system obtained by the High Energy Transmission Grating Spectrometer on board Chandra, while Kinkhabwala et al. (2002) observed Fe XXII in the Seyfert 2 galaxy NGC 1068. To reliably analyse such spectra requires an accurate knowledge of wavelengths and radiative transition rates, obtained from calculations or experiment.

Numerous observations of boron-like iron have been previously reported in the X-ray spectra of the Sun (McKenzie et al. 1985; Phillips et al. 1996; Fawcett et al. 1987; Doschek et al. 1981,1973; Phillips et al. 1982). Wargelin et al. (1998) have presented an analysis of the density sensitivity of Fe XXII L-shell lines in plasma obtained in the Princeton Large Torus tokamak. They analysed the intensity ratios of 4d_5/2 - 2p_3/2 and 4d_3/2 - 2p_1/2 lines using the HULLAC atomic physics package (Bar-Shalom et al. 2001). Recently, Chen et al. (2004) used a radiative-collisional model to predict the density of a plasma produced using the Electron Beam Ion Trap at Lawrence Livermore National Laboratory. They showed that the line ratios of the 2p-3d transitions in boron-like iron are sensitive to the electron density in the range $n_{\rm e} = 10^{13}{-}10^{15}$ cm^-3. Earlier, electron-excitation rate calculations for 45 fine-structure levels, using the Breit-Pauli formulation of the R-matrix method, were performed by Zhang & Pradhan (1997). Badnell et al. (2001) used the AUTOSTRUCTURE code (Badnell 1997,1986) to generate radial orbitals and atomic structure for 204-level close-coupling calculations in Fe XXII. A large set of energy levels is presented in the CHIANTI database at http://wwwsolar.nrl.navy.mil/chianti.html.

The aim of the present paper is extend our series of calculations for iron ions (Jonauskas et al. 2004a,b,2005), by providing highly reliable energy levels and radiative transition rates for Fe XXII up to the n=5 complex. We note that in photoionized plasmas these high-lying levels will not in general be populated via electron impact (although collisional redistribution among the levels may play a role), but rather by a range of processes including recombination and charge transfer (Savin 2001). Transitions in Fe XXII from high-lying levels having 5d electrons were observed by Fawcett et al. (1987) in a solar flare spectrum. Wargelin et al. (1998) identified transitions from 2s²4d¹, 2s¹2p¹4p¹, 2s¹2p¹5p¹, 2s²5d¹, 2s¹2p¹5d¹ and 2s²6d¹ configurations which are populated in a radiative-collisional model by electron collisions from metastable state.

In the present paper calculations are performed using the multiconfiguration Dirac-Fock (MCDF) approach. Energies of the lowest 407 levels, plus E1, E2 and M1 type radiative transition probabilities, as well as line and oscillator strengths among the levels of Fe XXII, are provided. The calculated results are compared with data compiled by NIST, as well as results obtained from other codes. The agreement between the length and velocity forms of electric transition operators is checked for the strongest transitions as an additional measure of the accuracy for the data obtained. The total radiative probabilities and five strongest probabilities required for calculating branching ratios of levels are presented as well.

2 Method of calculation

Results were obtained using the multiconfiguration Dirac-Fock (MCDF) method employed in the GRASP code of Grant et al. (1980) and Parpia et al. (1996) (http://www.am.qub.ac.uk/DARC). To cross-check our MCDF data, we used a code developed by Bogdanovich & Karpuskiene (1999), which adopts the configuration interaction (CI) method on the basis of transformed radial orbitals (TROs) with variable parameters, including relativistic effects in the Breit-Pauli approximation ( CITRO).

2.1 MCDF approach

In the MCDF approach, intermediate coupling wavefunctions $\Psi_{\gamma} (J)$ are constructed by using an expansion of the form:

$$% \Psi_{\gamma} (J)=\sum\limits_{\alpha}c_{\gamma}(\alpha J)~\Phi (\alpha J),$$ (1)

where configuration state functions (CSFs) $\Phi (\alpha J)$ are expressed as antisymmetrized products of two-component orbitals

$$% \phi (r)=\frac{1}{r}\left( \begin{array}{rc} P_{nlj}(r) & ... ...bar{l}j}(r) & \chi _{\bar{l}jm}(\hat{r}) \end{array}\right).$$ (2)

Here P_nlj(r) and $Q_{n\bar{l}j}(r)$ are the large and small radial components of one-electron wavefunctions and  $\chi _{ljm}(\hat{r})$ are the two component Pauli spherical spinors. Two-component orbitals are obtained as the self-consistent solutions of the Dirac-Fock equations systematically increasing CI basis. Direct and indirect relativistic effects, when the contraction of inner orbitals leads to more effective screening of the nucleus for valence orbitals, are included in the relativistic wavefunctions by solving MCDF equations.

The configuration mixing coefficients $c_{\gamma}$ are obtained through diagonalization of the Dirac-Coulomb-Breit Hamiltonian:

$$% H^{{\rm DF}} = \sum\limits_{i}\ h_{i}^{\rm D} + \sum\limit... ...}\ h_{ij}^{{\rm e}} + \sum\limits_{i<j}\ h_{ij}^{{\rm trans}},$$ (3)

where $h_{i}^{{\rm D}}$ is one-electron Dirac Hamiltonian, $h_{ij}^{{\rm e}}$ is the instantaneous Coulomb repulsion. The effects of the transverse interaction  $h_{ij}^{{\rm trans}}$ that corresponds to the Breit interaction:

$$% h_{ij}^{{\rm Breit}}=-\frac{\alpha _{i}\cdot \alpha _{j}}{2... ...alpha _{i}\cdot r_{ij})(\alpha _{j}\cdot r_{ij})}{2r_{ij}^{3}}$$ (4)

in the low-frequency limit were evaluated in the first order of perturbation expansion. QED corrections, which include vacuum polarization and self-energy (known as the Lamb shift), are considered in the first order of perturbation theory.

One-electron excitations from the 2s and 2p orbitals of the 1s²2s²2p¹, 1s²2s¹2p² and 1s²2p³ configurations up to the 8k orbital, as well as two- and three-electron excitations from orbitals with n=2 to all possible combinations of two or three electrons in the shells up to n=3, were employed to generate one-electron wavefunctions as a basis set for CSFs in the MCDF method. Additionally, to extend the CI basis and obtain higher accuracy, an additional 14 configurations with electrons in n=4 are included, namely: 1s²2s¹4s², 1s²2s¹4p², 1s²2s¹4d², 1s²2s¹4f², 1s²2p¹4s², 1s²2p¹4p², 1s²2p¹4d², 1s²2p¹4f², 1s²2s¹3p¹4p¹, 1s²2s¹3p¹4f¹, 1s²3s¹3p¹4p¹, 1s²3s¹3p¹4f¹, 1s²2p¹3p¹4s¹, 1s²2p¹3p¹4d¹. The total number of CSFs in our MCDF calculations is 2253.

2.2 CITRO method

In the nonrelativistic, multiconfiguration Hartree-Fock approach, intermediate coupling wavefunctions  $\Psi_{\gamma} (J)$ being eigenfunctions of Coulomb-Breit-Pauli Hamiltonian are expanded in terms of CSFs $\Phi (\alpha LSJ)$ obtained in the LSJ-coupling scheme:

$$% \Psi_{\gamma} (J)=\sum\limits_{\alpha LS}c_{\gamma}(\alpha LSJ)~\Phi (\alpha LSJ)$$ (5)

CSFs are formed from a basis of one-electron spin orbitals:

$$% \phi (r)=\frac{1}{r}P_{nl}(r)Y_{lm_{l}}(\vartheta ,\phi )\chi _{m_{s}}.$$ (6)

The number of CSFs in the intermediate coupling wavefunction expansion (5) is limited. Hence transformed radial orbitals with variable parameters were proposed by Bogdanovich & Karpuskiene (1999) to mimic the correlation effects from omitted CSFs:

$$% P_{nl}^{{\rm T}}(r)=N\left\{ f(r)P_{nl}(r)-\sum_{n^{\prime ... ...<n}c_{n^{\prime \prime },n}P_{n^{\prime \prime }l}(r)\right\}.$$ (7)

Here N is a normalization factor, $c_{n^{\prime \prime },n}$ denotes the corresponding overlap integral, and f(r) is a transforming function:

$$% f(r)=r^{k}\exp (-Ar^{m})$$ (8)

with variable parameters k, m and A ($k \ge 0$, $k \ge l - l'$, m > 0, A > 0). The variation of all parameters ensure the largest corrections of correlation energies obtained in the second order of perturbation theory using admixed configurations with excited electrons. A Schmidt orthogonalization procedure is employed for TROs in (7).

The Coulomb-Breit-Pauli Hamiltonian is used for our configuration interaction calculations which lead to the expansion coefficients  $c_{\gamma}(\alpha LSJ)$ and the corresponding energies for levels. This Hamiltonian includes spin-orbit  $h_{i}^{{\rm so}}$, spin-other-orbit $h_{ij}^{{\rm soo}}$ and spin-spin $h_{ij}^{{\rm ss}}$ corrections as well as orbit-orbit corrections  $h_{ij}^{{\rm oo}}$ within a shell of equivalent electrons:

$$% H^{{\rm BP}} \sum\limits_{i}\ \left(-\frac{1}{2} \nabla^2_{i} - Z/r_{i}\right) + \sum\limits_{i<j}\ \frac{1}{r_{ij}} + \sum\limits_{i}\ h_{i}^{{\rm so}}$$ =

$$+ \sum\limits_{i<j}\ h_{ij}^{{\rm soo}} + \sum\limits_{i<j}\ h_{ij}^{{\rm ss}} + \sum\limits_{i<j}\ h_{ij}^{{\rm oo}},$$ (9)

where $H^{{\rm BP}}$ is expressed in atomic units. Orbit-orbit interactions between shells omitted in our CITRO calculations are usually smaller than within shells.

The frozen core approximation is used to obtain Hartree-Fock radial orbitals with $n \le 7$. To include correlation effects from higher CSFs, the obtained basis is supplemented by TROs with principal quantum numbers $8 \le n \le 10$ and orbital quantum numbers $l \le 7$. The large number of admixed configurations is reduced, leaving only configurations with significant influence on the energies of the adjusted configurations. For this, the method presented by Bogdanovich et al. (2002) and Bogdanovich & Momkauskaite (2004) is adopted. The total number of CSFs with odd parity is 2115, and the number with even parity is 1987 in our CITRO calculations.

3 Results and discussion

In Table 1 we list energies of the 407 lowest levels of Fe XXII obtained with the GRASP code. These levels correspond to 1s²2s²2p¹, 1s²2s¹2p², 1s²2p³, 1s²2s²3l, 1s²2s¹2p¹3l, 1s²2p²3l, 1s²2s²4l', 1s²2s¹2p¹4l', 1s²2p²4l', 1s²2s²5l'', 1s²2s¹2p¹5l'', 1s²2s²6l''' (l = 0,1,2, l' = 0,1,2,3, l'' = 0,1,2,3,4 and l''' = 0,1,2,3,4,5) configurations. Energy levels are given in cm^-1 relative to the ground state 1s²2s²2p^{1 2}P_0.5, along with the leading percentage compositions  $c_{\gamma}(\alpha J)^2$ (where these exceed 10%) from the expansion relation Eq. (1) for intermediate coupling wavefunctions. The expansion coefficients for the intermediate coupling wavefunctions with CSFs presented in the LSJ-coupling scheme (5) are obtained from the expansion terms of intermediate coupling wavefunctions with CSFs presented in jj-coupling scheme (1). Diagonalization of Dirac-Coulomb-Breit matrix gives the expansion coefficients for intermediate coupling wavefunctions with CSFs constructed from two-component orbitals. The indices for the levels provided in the first column of Table 1 are used in all subsequent tables.

Some excited levels due to strong intermediate coupling are assigned to the same CSFs. In Table 2 new LSJ-coupling spectroscopic notations are proposed for the levels which have the same CSFs with primary contributions to the intermediate coupling wavefunctions. For this, the technique presented in our earlier paper for Fe XIX (Jonauskas et al. 2004b) is used.

In Table 3 we compare our calculated energy levels with values compiled by NIST (National Institute for Standards and Technology: www.physics.nist.gov) whose data are commonly used as a reference set. Fairly good agreement is obtained for the MCDF calculations, with the many energy levels agreeing to better than within 1% with the NIST recommended values. However, some excited energy levels (indices 29, 55, 59, 69, 74, 174 and 195) differ by up to 2% from NIST data. As well, our absolute values of energies for two levels (55 and 174) show the largest deviations from the NIST recommended values, while another five levels (29, 59, 69, 74 and 195) have MCDF absolute energies lower than those from NIST. To cross-check our results for these levels, we present data from CITRO calculations. For both our calculations, the highly excited levels show the similar discrepancies from NIST values. On the other hand, the MCDF energies of levels 174 (1s²2s¹2p¹ (³P) 4d^{1 2}F_3.5) and 195 (1s²2s¹2p¹ (¹P) 4d^{1 2}F_3.5) show good agreement with NIST data after their values are interchanged. This suggests that there are some typographical errors in the NIST data for these levels. The calculated energies of level 29 (1s² 2s¹ 2p¹(³P)3p^{1 2}P_0.5) has similar discrepancies ($\sim$$1.5\%$) for both calculations presented in Table 3, but our MCDF result is closer to the NIST recommended value. As well, our CITRO and MCDF calculations show the same percentage composition for the intermediate coupling wavefunction of the level. The energy of the 1s² 2s¹ 2p¹ (³P) 3p^1 2P_0.5 level (29) is lower than that of 1s² 2s¹ 2p¹ (³P) 3p^1 2P_1.5 (28) in the NIST data, while the ordering of both levels in our calculations is opposite to the NIST order. It suggests that a bigger CI basis may be required for the MCDF calculations to obtain better agreement with the NIST values for the level. However, the energy for the level obtained by Zhang & Pradhan (1997) with SUPERSTRUCTURE (Eissner et al. 1974), and the value provided by Badnell et al. (2001) from their AUTOSTRUCTURE (Badnell 1997,1986) calculations show similar differences from NIST, and the same order as in our data.

For another five levels (55, 59, 69 and 74) with the largest discrepancies from NIST values, our MCDF and CITRO energies are in close agreement with the AUTOSTRUCTURE values of Badnell et al. (2001) and, in general, are slightly closer to the NIST data. Furthermore, as will be seen later, some transitions from those levels have largest discrepancies for line strengths compared with the NIST values. Additionally, our calculated lifetimes for the levels are presented in Table 3, to have an additional way to estimate the accuracy of our calculations. Differences between the calculated lifetimes for those levels is less than 10%, which confirm the accuracy of our results.

Energy levels and intermediate coupling wavefunctions calculated with the configuration interaction method have been employed to compute matrix elements of transition operators, which subsequently are adopted for the calculation of transition probabilities, line and oscillator strengths. The wavelengths and line strengths are presented in Table 4, where our transition data are compared with NIST results. All line strengths belong to E1 type transitions except one from the first excited level which corresponds to M1 type transition. E2 type transition from the level is much weaker (MCDF and NIST line strengths equal to 2.10 $\times$ 10^-3 and 2.15 $\times$ 10^-3 correspondingly). The length form is used for electric transitions, as they are less sensitive to the accuracy of wavefunction compared with results obtained in the velocity form of the transition operator. The wavelengths for many transitions agree to better than within 1% with the NIST data. The largest differences for the wavelengths are obtained for transitions between energy levels which showed largest discrepancies from NIST energies. The largest discrepancy for line strengths is observed for the weak transition (f = 3 $\times$ 10^-5) from the 2s¹2p^2 2P_0.5 level (8) to 2s²2p^1 2P_1.5 (2). The length and velocity forms differ by a factor of 3.5, while our calculated MCDF and CITRO line strengths coincide within 20% for the transition. Furthermore, the contribution of the transition to the lifetime of the level is very small. The line strengths for the transitions from 2s²4s^1 2S_0.5 (126) to the first and second levels of the ground configuration differ by 80% and 29%, respectively, from the NIST values, but are in the limits of uncertainties provided by NIST.

Additionally, large discrepancies are observed for some transitions from the above mentioned levels which showed the largest deviations for energy levels compared with the NIST values. The line strengths for transitions $55 \rightarrow 9$, $55 \rightarrow 10$, $59 \rightarrow 8$, $69 \rightarrow 8$, $74 \rightarrow 8$, $174 \rightarrow 7$ and $195 \rightarrow 7$ differ by 29% to up to a factor of 4. The contributions to the lifetimes of those dipole allowed transitions vary from 7% ( $74 \rightarrow 8$) to 84% ( $195 \rightarrow 7$) (Table 5). The large discrepancies observed for transitions from levels 174 and 195 can be explained by a typographical error in the NIST data. As in a case of energy levels, our line strengths and wavelengths for those transitions are in good agreement with the NIST line strengths, after the NIST values are interchanged. Finally, our MCDF results for all those transitions with large discrepancies with NIST line strengths are in close agreement with the CITRO ones. It is possible that the determination of all those energy levels is uncertain in the NIST database. Additionally, the fairly good agreement between length and velocity forms for all those transitions indicate that our transition characteristics are quite accurate and reliable.

Transition probabilities, line and oscillator strengths are calculated for radiative E1, E2, E3, M1 and M2 type transitions among the presented 407 energy levels of Fe XXII. We provide the transition wavelengths, probabilities, oscillator strengths and line strengths of E1, E2 and M1 type for Fe XXII in Tables 6-8. Only the strongest transitions are presented here for every type of transition: electric dipole transitions with $f \ge 0.001$, electric quadrupole with $f \ge 10^{-8}$ and magnetic dipole with $f \ge 10^{-8}$. The relation between length (Babushkin gauge in the relativistic approach) and velocity (Coulomb gauge) forms are provided for both electric transitions to have an additional indicator of accuracy of our MCDF results. The difference between both forms for many strong E1 type transitions ($f \ge 0.1$) does not exceed 20%. Only 4 transitions from highly excited levels have differences larger than 20% but less than 40%. On the other hand, the contribution of these transitions to the lifetimes of the levels is much smaller than 1%. Finally, the length and velocity forms of the strongest electric dipole transitions ($f \ge 0.1$) agree to better than within 10%, with an average deviation of only 1.8% for 457 transitions. For many E2 transitions, the two forms agree to better than within 10%.

In Table 5 we provide the five largest spontaneous radiative transition probabilities from each level, and the sum of all E1, E2, E3, M1 and M2 radiative transition probabilities from the corresponding level to all lower levels. The sums of radiative transition probabilities are important for branching ratios, while their inverse values are equal to the lifetimes of the levels. The contribution to the lifetimes of levels for many forbidden electric quadrupole and magnetic dipole transitions is negligible and does not exceed 1%. However, the highly excited 2s¹2p¹ (³P) 3p^1 4D_0.5 (33) and 2p² (¹D) 3d^1 2G_0.5 (114) levels decay primarily through E2 type transitions. For another two levels, i.e. 2p² (³P) 3d^1 4D_0.5 (97) and 2p² (³P) 3d^1 4F_0.5 (96), E2 type transitions contribute 48% and 46%, respectively. As well, the first excited level decays mainly due to magnetic dipole transition.

4 Conclusions

In this paper, energy levels, electric dipole, electric quadrupole and magnetic dipole radiative transition rates, oscillator and line strengths have been reported for the lowest 407 levels in Fe XXII in the MCDF approximation. The characteristics of the levels in the LSJ-coupling scheme have been provided and checked for their completeness. Calculated values have been compared with NIST recommended values. Breit-Pauli energy levels and electric dipole transition characteristics in the basis set of transformed radial orbitals with variable parameters were used to cross-check the accuracy of our results. Likely typographical errors for Fe XXII have been found in both the NIST energy levels and transition characteristics for the highly excited 2s¹2p¹ (³P) 4d^1 2F_3.5 and 2s¹2p¹ (¹P) 4d^1 2F_3.5 levels.

The forbidden transitions have been taken into account to obtain total radiative transition probabilities from levels. The total radiative probabilities allow estimates of the lifetimes of the presented states, and are also important for the calculation of branching ratios. It was found that the highly excited 2s¹2p¹ (³P) 3p^1 4D_0.5 and 2p² (¹D) 3d^1 2G_0.5 levels mainly decay due to forbidden E2 type transitions. Their contributions to the lifetimes of the 2p² (³P) 3d^1 4D_0.5 and 2p² (³P) 3d^1 4F_0.5 levels is 48% and 46%, respectively. The first excited level decays mainly through M1 type transition. For each level, the five strongest transition probabilities are presented.

To conclude, the comparison with the NIST recommended wavelengths and radiative transition data shows that all the presented theoretical results are reliable, and may be successfully used for the interpretation of astronomical and other spectral observations.

Acknowledgements

FPK and SJR are grateful to AWE Aldermaston for the award of William Penney Fellowships. This work was supported by PPARC and EPSRC, and also by NATO Collaborative Linkage Grant CLG.979443. We are also grateful to the Defence Science and Technology Laboratory (dstl) for support under the Joint Grants Scheme.

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Online Material

Table 1: MCDF results for energy levels of Fe XXII. The energies of levels are presented relative to the ground energy. The leading percentage compositions (>10%) of levels are provided in the last column.

Table 2: Suggested change of spectroscopic identifications of levels to ensure the completeness of spectroscopic dataset. The indices of levels for which spectroscopic identifications are changed are presented in the first column. The second column contains index of level with the same highest contribution of configuration state function (Table 1) as level from the first column before change. Indices of levels in the first two columns are taken from Table 1.

Table 3: Comparison of calculated energies  $E^{{\rm GRASP}}$ and  $E^{{\rm CITRO}}$ for Fe XXII levels with data compiled by NIST ( $E^{{\rm NIST}}$). Indices of levels in the first column and CSFs in the second column are taken from Table 1. Energies are in cm^-1. The lifetimes of levels are presented in the last two columns.

Table 4: Comparison of calculated wavelengths ( $\lambda ^{{\rm GRASP}}$, $\lambda ^{{\rm CITRO}}$) and line strengths ( $S^{{\rm GRASP}}$, $S^{{\rm CITRO}}$) for Fe XXII with values presented by NIST ( $\lambda ^{{\rm NIST}}$, $S^{{\rm NIST}}$). Indices of levels in the first two columns are taken from Table 1.

Table 5: The five major spontaneous radiative transition probabilities A^r from each level of Fe XXII and the sum of all radiative probabilities  $\sum A^{r}$ from the corresponding level. Electric dipole, quadrupole and octupole as well as magnetic dipole and quadrupole transitions are included. Arrow marks the final level to which radiative transition happens from the level.