A&A 433, 745-750 (2005)
DOI: 10.1051/0004-6361:20042301

Energy levels and transition probabilities for nitrogen-like Fe XX[*],[*]

V. Jonauskas[*],1 - P. Bogdanovich2 - F. P. Keenan1 - M. E. Foord3 - R. F. Heeter3 - S. J. Rose4 - G. J. Ferland5 - R. Kisielius[*],1 - P. A. M. van Hoof1 - P. H. Norrington6

1 - Department of Pure and Applied Physics, The Queen's University of Belfast, Belfast BT7 1NN, Northern Ireland, UK
2 - Vilnius University Research Institute of Theoretical Physics and Astronomy, 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 2 November 2004 / Accepted 3 December 2004

Energies of the 700 lowest levels in Fe XX have been obtained using the multiconfiguration Dirac-Fock method. Configuration interaction method on the basis set of transformed radial orbitals with variable parameters taking into account relativistic corrections in the Breit-Pauli approximation was used to crosscheck our presented results. 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 from each level are also provided. Results are compared with data compiled by NIST and with other theoretical work.

Key words: atomic data

1 Introduction

The new generation X-ray telescopes on board the space observatories Chandra and XMM-Newton provide high resolution spectra of numerous astrophysical sources that are rich in emission and absorption lines from various iron ions, including Fe XX (van der Heyden et al. 2003; Mewe et al. 2003,2001). For example, the Fe XX emission line at 12.831 $\AA$ between the 2p2(1D)3d1 2P3/2 level and the first excited level of the ground configuration is prominent in the X-ray spectrum of Capella obtained with LETG spectrometer on Chandra (Mewe et al. 2001). In addition, several forbidden M1-type transitions among the levels of the ground configuration 2s22p3 of Fe XX have been identified in solar spectra obtained by Skylab and SOHO/SUMER spectrographs (Kucera et al. 2000).

Nahar (2004) report the largest calculations of radiative rates for N-like iron to date in the framework of the Iron Project (Hummer et al. 1993), an international project initiated to fulfill a demand for accurate atomic data for the analysis of spectra obtained from satellite-born telescope missions. They employed the SUPERSTRUCTURE (Eissner et al. 1974) and the Breit-Pauli R-matrix (BPRM) code (Berrington et al. 1995), where only one-electron Darwin and mass-velocity as well as spin-orbit operators are included. A total of 1792 bound fine-structure levels were considered, with atomic data for E1-type (electric dipole) transitions obtained using the latter code, and rates for forbidden E2 (electric quadrupole), E3 (electric octupole) as well as M1 (magnetic dipole) transitions calculated with the former one. In addition, Butler & Zeippen (2001) used the BPRM code to generate collisional data among 86 levels of the n=2 and n=3 complexes in Fe XX.

Calculations by Froese Fischer & Tachiev (2004) using the multiconfiguration Hartree-Fock (MCHF) code for C III show good agreement with the results of Nahar (2002) obtained from the BPRM code for allowed transition probabilities. However transition probabilities are larger by a factor of 3-5 for intercombination lines. Hibbert (2003) also observe different results comparing their CIV3 (Hibbert 1975) and MCHF (Froese Fischer & Tachiev 2004) data with the BPRM calculations of Berrington (2001) for some transitions in Na III. The differences in derived results for the various methods in the above papers demonstrate the need to perform calculations for Fe XX using other codes than employed by Nahar (2004). Also, the list of terms for levels presented by Nahar (2004) in their large-scale calculations is insufficient to unambiguously identify levels.

Earlier, Bhatia & Mason (1980) used computer packages (SUPERSTRUCTURE) developed at University College London (Eissner et al. 1974) to obtain radiative rates for transitions within the levels of the 2s22p3 and 2s12p4 configurations. Subsequently (Mason & Bhatia 1983), they supplemented their data by including 2s22p23s1 and 2s22p23d1 configurations. Later, Merkelis et al. (1997,1999) employed the stationary second-order many-body perturbation theory approach to calculate electric dipole, electric quadrupole and magnetic dipole transition data for the ions of the N I isolectronic sequence. These data include ions with $10 \le Z \le 30$ but are limited to configurations 1s22s22p3, 1s22s2p4 and 1s22p5. Large-scale calculations of oscillator strengths for Fe XX were also performed under the Opacity Project (The Opacity Project Team 1995,1997), but the relativistic effects were not included in their data.

The present results for Fe XX continue our series of calculations, which aim to provide highly reliable energy levels and radiative rates for iron ions up to the n=5 complex (Jonauskas et al. 2004b,a). 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). Accurate atomic data for highly excited levels of Fe XX are needed to properly interpret the high resolution spectra arising from Chandra and XMM-Newton, which have particularly large effective areas in the 6 to 18 $\AA$ wavelength range, covering Fe XX lines arising from the $n \ge 3$ complex.

Here we report MCDF calculations of level energies, E1, E2 and M1-type radiative transition probabilities, line and oscillator strengths for 700 levels of Fe XX . Calculated results are compared with data compiled by NIST, as well as results obtained by other authors. The agreement between the length and velocity forms of electric transition operators is checked as an additional measure of accuracy. In addition, total transition probabilities are provided, required for calculating branching ratios and the radiative lifetimes of levels.

2 Method of calculation

We perform two sets of calculations. In the first one we use the multiconfiguration Dirac-Fock (MCDF) method employed in the GRASP code of Grant et al. (1980) and Parpia et al. (1996) ( The second one adopts configuration interaction (CI) method on the basis of transformed radial orbitals (TROs) with variable parameters including relativistic effects in the Breit-Pauli approximation (Bogdanovich & Karpuskiene 1999). The latter one refered here as CITRO was used for crosschecking our MCDF result.

2.1 MCDF approach

In the MCDF method, relativistic orbitals with the same j but differing m quantum numbers have the same radial form:

\begin{displaymath}\phi (r)=\frac{1}{r}\left(
P_{nlj}(r) & \c...
...r{l}j}(r) & \chi _{\bar{l}jm}(\hat{r})%
\end{displaymath} (1)

The intermediate coupling wavefunctions are expanded on the basis of configuration state functions (CSFs) obtained for the jj-coupling scheme:

\begin{displaymath}\Psi_{\gamma} (J)=\sum\limits_{\alpha}c_{\gamma}(\alpha J)~\Phi (\alpha J)
\end{displaymath} (2)

where CSFs are expressed as antisymmetrized products of two-component orbitals, referred to as subshells. 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 wavefunctions by solving MCDF equations.

Intermediate coupling wavefunctions are eigenfunctions of the Dirac-Coulomb-Breit Hamiltonian in the relativistic approximation and the Coulomb-Breit-Pauli Hamiltonian in the nonrelativistic approximation. The relativistic Hamiltonian reduces to a nonrelativistic one, leaving terms up to the square of the fine-structure constant in the expansion for matrix elements.

The Breit operator presented in the Coulomb gauge:

\begin{displaymath}h_{ij}^{\rm Breit}=-\frac{\alpha _{i}\cdot \alpha _{j}}{2r_{i...
_{i}\cdot r_{ij})(\alpha _{j}\cdot r_{ij})}{2r_{ij}^{3}}
\end{displaymath} (3)

or written in the form of the sum of magnetic and retardation interactions:

\begin{displaymath}h_{ij}^{\rm Breit}=-\frac{\alpha _{i}\cdot \alpha _{j}}{r_{ij...
...ha_{i}\cdot \nabla_{i})(\alpha _{j}\cdot \nabla_{j})r_{ij}}{2}
\end{displaymath} (4)

is obtained in the limit $\omega \rightarrow 0$ from the transverse operator:
$\displaystyle h_{ij}^{\rm trans}=\frac{1}{r_{ij}}\left[ 1-\alpha _{i}\cdot \alp...
...\alpha _{j}\cdot \nabla _{j})~\frac{\cos (\omega r_{ij})-1}{\omega ^{2}}\right]$     (5)

where $\omega$ is the energy of a single photon exchanged between a pair of electrons i and j. The frequency-dependent transverse Breit interaction operator is used for the calculation of Breit matrix elements in the relativistic approximation. 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 2p orbital of the 1s22s22p3, 1s22s12p4 and 1s22p5 configurations up to the 8k orbital, as well as two-electron excitations from orbitals with n=2 to all possible combinations of two electrons in the shells with 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, additional 33 configurations are included, namely: 2s23p3, 2p34d2, 2p34f2, 2s22p13p14p1, 2s22p13p15p1, 2s22p13p16p1, 2s22p13p14f1, 2s12p23p14p1, 2s12p23p14d1, 2s12p24p14d1, 2s12p25p14d1, 2s12p23d14d1, 2s12p23s14d1, 2s12p24s14d1, 2s12p25s14d1, 2s22p13p14d1, 2s12p23p14f1, 2s22p13p15s1, 2s12p34f1, 2p33p14p1, 2p33p14f1, 2s22p14f2, 2s12p24f2, 2s23d14f2, 2s24d14f2, 2s22p15f2, 2s12p25f2, 2s12p24d2, 2s13p24d2, 2s22p14d2, 2s23p14d2, 2s23d14d2, 2s23s14d2.

2.2 CITRO method

In the nonrelativistic, multiconfiguration Hartree-Fock method, CSFs are obtained in the LSJ-coupling scheme and form intermediate coupling wavefunctions:

\begin{displaymath}\Psi_{\gamma} (J)=\sum\limits_{\alpha {\rm LS}}c_{\gamma}(\alpha {\rm LSJ})~\Phi (\alpha {\rm LSJ}).
\end{displaymath} (6)

One-electron orbitals as basis for CSFs have the form:

\begin{displaymath}\phi (r)=\frac{1}{r}P_{nl}(r)Y_{lm_{l}}(\vartheta ,\phi )\chi _{m_{\rm s}}.
\end{displaymath} (7)

Transformed radial orbitals with a variable parameters (Bogdanovich & Karpuskiene 1999) are employed to mimic the correlation effects of CSFs not introduced in the expansion for intermediate wavefunctions:

 \begin{displaymath}P_{nl}^{T}(r)=N\left\{ f(r)P_{nl}(r)-\sum_{n^{\prime \prime
}<n}c_{n^{\prime \prime },n}P_{n^{\prime \prime }l}(r)\right\}.
\end{displaymath} (8)

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

\begin{displaymath}f(r)=r^{k}\exp (-Ar^{m})
\end{displaymath} (9)

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 (8). Applications of CITRO to various atoms and ions (Bogdanovich & Karpuskiene 1999; Karpuskiene & Bogdanovich 2003; Bogdanovich et al. 2003b,a) demonstrate that such radial orbitals enables one to include correlation corrections in the CI method quite efficiently.

In the conventional Breit-Pauli (BP) approximation, the Hamiltonian includes mass-correction, one- and two-body Darwin, spin-spin contact, and orbit-orbit terms as well as spin-orbit, spin-other-orbit and spin-spin corrections (Karazija 1996). The former group of operators shifts energies of terms and the latter ones are responsible for the fine-structure splitting. Spin-other-orbit, orbit-orbit, spin-spin, spin-spin contact and two-body Darwin operators are derived from the Breit operator by expanding its matrix elements obtained with two-component relativistic orbitals in orders of the fine-structure constant. Orbit-orbit interaction, due to its complexity (Gaigalas 1999; Badnell 1997; Eissner et al. 1974), leads to a large consumption of computational time, and a small contribution to energies of levels is often omitted in calculations (Froese Fischer & Tachiev 2004). Our CITRO calculations include spin-orbit, spin-other-orbit and spin-spin corrections as well as orbit-orbit corrections within a shell of equivalent electrons. Orbit-orbit interactions between shells are usually smaller than within shells.

In CITRO calculations, we use Hartree-Fock radial orbitals for electrons with $n \le 5$ whose states are presented here. States with $6 \le n \le 10$ and $l \le 7$ employ TROs. Therefore, the number of radial orbitals used in calculations with CITRO totals 52. The method presented by Bogdanovich & Momkauskaite (2004); Bogdanovich et al. (2002) was adopted to reduce large number of admixed configurations leaving only configurations with significant influence on the energy of adjusted configurations. Number of CSFs with odd parity decreases from 249 252 to 132 746 and CSFs with even parity - from 243 104 to 124 217. Methods used for energy matrix diagonalization are provided by Bogdanovich et al. (2002).

3 Results and discussion

We present calculations for the 700 lowest energy levels of Fe XX, and radiative transition characteristics among these. Transition probabilities, oscillator and line strengths for electric dipole, electric quadrupole and magnetic dipole transitions are obtained in the fully relativistic MCDF approach. All 698 levels arising from the 1s22s22p3, 1s22s12p4, 1s22p5, 1s22s22p2nl, 1s22s12p3n'l, and 1s22p43l' (n=3,4,5, n'=3,4, l=0,..,n-1, l'=0,1,2) configurations are taken into account. Binding energies of the two lowest levels from the 1s22s12p35s1 configuration are lower than our chosen cut-off value, which corresponds to the highest level of the 1s22s22p25g1 configuration. Therefore those levels are also included here.

The energies of the above configurations calculated with the fully relativistic GRASP code are listed in Table 1. Indices for the levels in the first column of Table 1 are used in all tables except Table 2, where results obtained with the original CITRO code of Bogdanovich & Karpuskiene (1999) are presented. Energy levels are given in cm-1 relative to the ground state 1s22s22p3 4S3/2, along with the leading percentage compositions (where these exceed 10%) for intermediate wavefunctions. The LSJ-coupling CSFs with largest weights for the intermediate wavefunctions are provided in the second and third columns of the table. Intermediate coupling is strong for some excited levels, so the level assignments for some terms are ambiguous.

In Table 3 our energy levels obtained with the GRASP and CITRO codes are compared with values calculated by Mason & Bhatia (1983) with SUPERSTRUCTURE and Nahar (2004) BPRM results, as well as data compiled by NIST (National Institute for Standards and Technology: whose data are commonly used as reference set for atomic results. The energy levels are compared with respect to the ground level energies of the corresponding data sets. The Nahar (2004) values in the Table 3 are obtained with the BPRM code because it is used for E1-type transitions in their calculations. Their calculations with SUPERSTRUCTURE are in better agreement with NIST data than the BPRM results.

Fairly good agreement with NIST energy levels is obtained using the CITRO code. The highest deviation from the NIST energies does not exceed 0.4% and the average deviation is 0.2%. In addition, the ground state energy of -219 142 254 cm-1 is close to the NIST recommended value of $-219~167~600 \pm 112~300$ cm-1. There is also very good agreement between Mason & Bhatia (1983) and NIST data sets, showing an average deviation of only 0.4%. The highest deviations in the former calculations are for the excited levels of the ground configuration, but even then the discrepancy does not exceed 1.6%. Their scaling parameters for Thomas-Fermi potential were $\lambda_{\rm s}=1.255$, $\lambda_{\rm p}=1.150$ and $\lambda_{\rm d}=1.100$ for all cases. Mason & Bhatia (1983) includes only the 2s22p3, 2s12p4, 2s22p23s1 and 2s22p23d1 configurations for their results, while Nahar (2004) also use the SUPERSTRUCTURE for forbidden transitions and include 9 configurations. However, their results for energy levels are worse than the data of Mason & Bhatia (1983), while the latter calculations omitted much of the correlations. Nahar (2004) does not present values of the scaling parameters employed in their calculations.

The presented BPRM results of Nahar (2004) show an average discrepancy of 0.5% with NIST values. These authors obtain better agreement with NIST than data provided by Butler & Zeippen (2001) which are not presented in the Table 3. A maximum disagreement of 3.1% is obtained for the second excited level 1s22s22p3 2D5/2. One of the reasons for the discrepancy is that BPRM omits all two-electron corrections originating from the Breit-Pauli operator. In Table 4 we estimate the magnitudes of some corrections missed in their calculations using the Breit-Pauli code (without TROs). The largest discrepancies of energy computed with spin-orbit, spin-other-orbit, spin-spin and orbit-orbit (within a shell) interactions with those that include only spin-orbit corrections is obtained for the same level 1s22s22p3 2D5/2. Spin-other-orbit and orbit-orbit interactions have the largest influence to the shifts of energies compared with spin-spin corrections. Added spin-other-orbit and orbit-orbit corrections shift the level down by a similar amount relative to the ground level. The total shift caused by these corrections leads to 3438 cm-1. The influence of spin-spin interaction on the shift of the levels is smaller, and does not exceed 500 cm-1 for 1s22s12p4 4P3/2.

MCDF results show an average difference of 0.5% from NIST data for the energy levels displayed in Table 3. The largest deviation is for levels of the ground configuration, but it does not exceed 2.2% and is less than 1% for other energies. The total number of CSFs included in the CI basis is 10050, while the CITRO employs 256963 CSFs. After CI functions are supplemented by the above mentioned 33 configurations for the MCDF calculations, the average discrepancy changes from 0.6% to 0.5%. The discrepancy for the first excited level is reduced from 2.6% to 2.2%, and for the second excited level from 2.1% to 1.9%. It indicates that a larger set of CI wavefunctions would be required for our MCDF calculations to achieve higher accuracy.

To ensure the consistency of the spectroscopic dataset for levels presented in Table 1, new LSJ-coupling spectroscopic notations are proposed in Table 5 for the levels with similar contributions to intermediate wavefunctions. We use the same technique as presented in our earlier paper for Fe XIX (Jonauskas et al. 2004b).

Energy levels and intermediate coupling wavefunctions calculated with the configuration interaction method have been employed to derive matrix elements of transition operators, which subsequently are adopted for the calculation of transition probabilities, line and oscillator strengths. Our calculated wavelengths and line strengths using the two methods mentioned above, as well as values obtained by Nahar (2004) are listed in Table 6 along with data provided by NIST. Nahar (2004) use the BPRM code for E1-type transitions and SUPERSTRUCTURE for forbidden transitions, which correspond to those in the table within the n=2 complex. Only the ab initio calculations of Nahar (2004) are presented here, while their transition probabilities and oscillator strengths are corrected by the available transition energies from NIST. It is more expedient to compare calculated line strengths, as these do not explicitly depend on the transition energy and so do not contain errors arising from this quantity.

The CITRO results presented in Table 6 agree well with the wavelengths compiled by NIST, with differences of less than 1%. The largest discrepancies of up to 3.6% are obtained by the SUPERSTRUCTURE calculations of Nahar (2004) that correspond to forbidden transitions. MCDF wavelengths differ from NIST values by 2.4% and 1.9%, respectively, for transitions from the first and second excited levels to the ground state. Shorter wavelengths agree to better than 1% for our and the Nahar (2004) datasets. The agreement of the length and velocity forms (Babushkin and Coulomb gauges in the relativistic approach) is better for levels involving excited states, while weak transitions show the largest discrepancies.

Large discrepancies for line strengths are observed for transitions which include level 2s22p2(1D)3d1 2D5/2 (index 70). Due to strong mixing of the CSFs, the label of the level is ambiguous as the largest weight of the 2s22p2(1D)3d1 2D5/2 configuration state function amounts to less than 50% in both our calculations. NIST data report a weight of 54%, which is similar to our obtained values. On the other hand, agreement for weak transitions is never good due to mixing effects. It is interesting that all three calculations that incorporate different methods show similar discrepancies with the NIST line strengths. Similar discrepancies are also observed in all three calculations for transitions to the ground and second excited levels. Length and velocity forms of transitions from the level agree to better than 4% for both our results, indicating that major correlation effects are included in the intermediate coupling wavefunctions.

Line strengths obtained by Nahar (2004) using the BPRM code show the largest discrepancies with NIST data. In many cases it happens for intercombination spin-changing E1-type transitions. A similar effect was observed by Froese Fischer & Tachiev (2004) in Na III. However, some dipole allowed E1-type transitions of Nahar (2004) also differ by more than a factor of 4 from NIST values. As noted by Hibbert (2003), the large discrepancies can be understood by the fact that the BPRM code uses term-coupling coefficients to introduce relativistic effects, which lead to restrictions on the LSJ mixing coefficients. On the other hand, conventional atomic structure codes deal with the diagonalization of the full Hamiltonian matrix.

Wavelengths, transition probabilities, line and oscillator strengths obtained with the GRASP code for electric dipole, quadrupole and magnetic dipole transitions are presented in Tables 8, 9 and 10. Ratios between velocity and length forms for electric transitions are also provided. The total number of dipole allowed and intercombination E1-type transitions is 71 398, but only E1-type transitions with $f \ge 10^{-3}$ are included in Table 8. Tables 9 and 10 contain data for forbidden E2-type and M1-type transitions with $f \ge 10^{-8}$, yielding a total of 167 480 radiative rates.

The influence of forbidden transitions on the lifetimes of levels is prominent for excited levels of the ground configuration and highly excited 2s12p3(2D) 3d1 2G9/2 level (index 151). It can be seen from Table 7 that level 151 decays primarily through E2-type transitions, which reduces the lifetime of the level by more than a factor of 2. The E2-type transitions make a contribution of more than 10% to the decay of 2p4 (1D) 3d1 2G9/2 (261), 2s22p2 (3P) 3p1 4D7/2 (31), 2s12p3 (3D) 3p1 4F9/2 (107), 2p4 (1D) 3d1 2G7/2 (260), 2s22p2 (3P) 3p1 2S1/2 (25), 2s12p3 (3P) 3d1 4F9/2 (174), 2p4 (3P) 3d1 4F9/2 (245) and 2p4 (3P) 3d1 4D7/2 (241) levels. Most of these levels have large total quantum numbers, limiting the decay routes for strong dipole allowed transitions. Magnetic dipole transitions are responsible for finite lifetimes of excited levels of the ground configuration.

Finally, a comparison between the length and velocity forms of the electric dipole transitions shows an agreement of better than 10% for 790 transitions with $f \ge 0.1$, and an average deviation of only 6%. Two forms differ by up to 60% for some of the strong transitions, but their contributions to the lifetimes of the corresponding levels is negligible. For many E2 transitions, the two forms agree to better than 5%.

4 Conclusions

Multiconfiguration Dirac-Fock energy levels, as well as electric dipole, electric quadrupole and magnetic dipole transition probabilities, line and oscillator strengths have been computed for nitrogen-like Fe XX. The 700 lowest energy levels are considered. Calculated values have been compared with the data compiled by NIST and other theoretical results. Breit-Pauli energy levels and electric dipole transition characteristics on the basis set of transformed radial orbitals with variable parameters were used to crosscheck our MCDF result.

Leading percentage compositions for intermediate wavefunctions are presented in the basis of LSJ-coupling configuration state functions. Spectroscopic notations of levels identified by the largest weights of CSFs are checked for their completeness. Of the 700 levels, 203 have weights of LSJ-coupling CSFs of less than 50% due to strong mixing.

The 5 major radiative probabilities from each level and the total values obtained in the MCDF approximation have been provided, taking into account forbidden transitions. The largest contributions of forbidden M1-type transitions have been obtained for the lifetimes of fine-structure levels of the ground configuration. The electric quadrupole transitions are mainly noticeable for transitions from levels with large total quantum numbers. Their contributions to the lifetimes of levels exceed 10% for 9 highly excited levels. On the other hand, the 2s12p3(2D)3d1 2G9/2 level decay mainly trough E2-type transition. The influence of M2 and E3-type transitions that are not presented here is negligible.

Good agreement between our set of energy levels and radiative transition characteristics for Fe XX and the available NIST data, as well as our use of a large basis of configuration state functions, allows to conclude that the achieved accuracy of our calculations is higher than those available to date. We hope that our data will be useful in astrophysical and other applications.

F.P.K. and S.J.R. 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. We thank STScI for support through HST-AR-09923.01A.



5 Online Material

Table 1: MCDF calculated energy levels relative to the ground energy of Fe XX with spectroscopic identifications. The leading percentage compositions of levels which contributions exceed 10% are presented in the last column.

Table 2: CITRO calculated energy levels relative to the ground energy of Fe XX with spectroscopic identifications. The leading percentage compositions of levels which contributions exceed 10% are presented in the last column.

Table 3: Comparison of calculated energies for Fe XX levels with data compiled by NIST ( $E^{{\rm NIST}}$). $E^{{\rm SS}}$ - energies calculated by Mason & Bhatia (1983) with SUPERSTRUCTURE, $E^{{\rm BPRM}}$ - values obtained by Nahar (2004) with Breit-Pauli R-matrix code, $E^{{\rm GRASP}}$ and $E^{{\rm CITRO}}$ - our values. Indexes of levels in the first column and CSFs in the second column are taken from Table 1. Energies are in cm-1.
Index CSF $E^{{\rm NIST}}$ $E^{{\rm SS}}$ $E^{{\rm BPRM}}$ $E^{{\rm CITRO}}$ $E^{{\rm GRASP}}$
2 2s2 2p3 2D1.5 138620 140598 140903 138856 141715
3 2s2 2p3 2D2.5 176130 178989 181615 175952 179537
4 2s2 2p3 2P0.5 260270 257573 264577 260471 263209
5 2s2 2p3 2P1.5 323340 319877 328554 322368 325962
6 2s1 2p4 4P2.5 752730 747101 757407 750386 753649
7 2s1 2p4 4P1.5 820820 812698 824127 817703 821409
8 2s1 2p4 4P0.5 842740 834443 846184 839423 843536
9 2s1 2p4 2D1.5 1042570 1044277 1050625 1039244 1050924
10 2s1 2p4 2D2.5 1058360 1061216 1068073 1055467 1066222
11 2s1 2p4 2S0.5 1195260 1195284 1205245 1192125 1205489
12 2s1 2p4 2P1.5 1242430 1251404 1253310 1239208 1255768
13 2s1 2p4 2P0.5 1340040 1345312 1351305 1336259 1352396
14 2p5 2P1.5 1954520   1966690 1948441 1971784
15 2p5 2P0.5 2062200   2076076 2055642 2079179
16 2s2 2p2 (3P) 3s1 4P0.5 7194000 7193289 7162927 7182452 7155228
17 2s2 2p2 (3P) 3s1 4P1.5 7255000 7256327 7229209 7247014 7221726
18 2s2 2p2 (3P) 3s1 2P0.5 7287000 7286360 7259935 7277194 7252311
19 2s2 2p2 (3P) 3s1 4P2.5 7299000 7299603 7274815 7288611 7264310
20 2s2 2p2 (3P) 3s1 2P1.5 7331000 7330953 7306639 7320048 7296115
23 2s2 2p2 (1D) 3s1 2D2.5 7430000 7430336 7411460 7422476 7400563
24 2s2 2p2 (1D) 3s1 2D1.5 7440000 7439775 7421293 7432236 7410643
33 2s2 2p2 (1S) 3s1 2S0.5 7554000 7551182 7545340 7555745 7530985
39 2s2 2p2 (3P) 3d1 4F1.5 7672000 7650500 7646562 7663219 7638491
46 2s2 2p2 (3P) 3d1 4F3.5 7740000 7714100 7722324 7736076 7712951
47 2s2 2p2 (3P) 3d1 4D0.5 7752000 7721000 7727811 7743115 7720580
56 2s2 2p2 (3P) 3d1 4P2.5 7802000 7790100 7796375 7806999 7785892
58 2s2 2p2 (3P) 3d1 4P1.5 7802000 7803000 7808139 7818763 7798298
61 2s2 2p2 (3P) 3d1 2F3.5 7820000 7818200 7830306 7839918 7819869
62 2s2 2p2 (3P) 3d1 2D1.5 7859000 7850700 7853131 7864123 7844676
63 2s2 2p2 (3P) 3d1 2D2.5 7843000 7854500 7854097 7865251 7846514
69 2s2 2p2 (1D) 3d1 2D1.5 7919000 7909800 7925646 7933796 7915116
70 2s2 2p2 (1D) 3d1 2D2.5 7913000 7918200 7929859 7937908 7919464
71 2s2 2p2 (1D) 3d1 2P0.5 7964000 7931300 7947242 7953974 7936656
72 2s2 2p2 (3P) 3d1 2F3.5 7935000 7944100 7950051 7958832 7941761
79 2s2 2p2 (1D) 3d1 2S0.5 7995000 7960100 7979856 7986462 7969218
80 2s2 2p2 (3P) 3d1 2D2.5 7983000 7969000 7980602 7988008 7971406
82 2s2 2p2 (1D) 3d1 2P1.5 7967000 7966700 7984157 7989830 7974014
85 2s2 2p2 (1S) 3d1 2D2.5 8047000 8069800 8065978 8072444 8051967
86 2s2 2p2 (1S) 3d1 2D1.5 8061000 8080900 8075722 8083172 8063693

Table 4: Estimation of contribution spin-orbit (s-o), spin-other-orbit (s-o-o), spin-spin (s-s) and orbit-orbit (o-o) within shell interactions to energies for Fe XX levels within n=2 complex. $E_1=E^{{\rm s-o}}$, $E_2=E^{{\rm s-o}}+E^{{\rm s-o-o}}$, $E_3=E^{{\rm s-o}}+E^{{\rm s-o-o}}+E^{{\rm s-s}}$ and $E_4=E^{{\rm s-o}}+E^{{\rm s-o-o}}+E^{{\rm s-s}}+E^{{\rm o-o}}$.
Index E1 E2 E3 E4
2 141857 143122 143444 141910
3 182708 181082 181201 179270
4 264047 264411 264530 263975
5 327447 326489 326519 325670
6 755194 754560 754597 754662
7 821273 821367 821849 821936
8 843083 844242 843812 843945
9 1050595 1050928 1051144 1049976
10 1068168 1067125 1067206 1065988
11 1204074 1204221 1204372 1204963
12 1253448 1252420 1252540 1252569
13 1350217 1349293 1349347 1349601
14 1970331 1970400 1970517 1971268
15 2078969 2077652 2077773 2078527

Table 5: Suggested change of spectroscopic identifications of levels to ensure the completeness of spectroscopic dataset. The indexes 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. Indexes of levels in the first two columns are taken from Table 1.
Index Index Changed CSF
26 22 2s2 2p2 (3P) 3p1 4P3/2
42 48 2s2 2p2 (3P) 3d1 4D5/2
72 61 2s2 2p2 (1D) 3d1 2F7/2
80 63 2s2 2p2 (1D) 3d1 2F5/2
153 163 2s1 2p3 (1P) 3p1 2P1/2
180 192 2s1 2p3 (3S) 3p1 2P3/2
200 209 2s1 2p3 (3S) 3d1 2D5/2
211 218 2p4 (3P) 3s1 2P3/2
225 228 2p4 (3P) 3p1 2P1/2
250 254 2p4 (3P) 3d1 2F5/2
252 242 2p4 (3P) 3d1 2P1/2
256 243 2p4 (3P) 3d1 2D3/2
277 281 2s2 2p2 (3P) 4p1 4P3/2
293 309 2s2 2p2 (3P) 4f1 2D5/2
296 297 2s2 2p2 (3P) 4d1 2P3/2
300 286 2s2 2p2 (3P) 4d1 4P5/2
315 295 2s2 2p2 (3P) 4f1 4G7/2
317 333 2s2 2p2 (3P) 4f1 4D5/2
400 405 2s1 2p3 (3D) 4p1 2P3/2
445 479 2s1 2p3 (3D) 4f1 4F5/2
462 480 2s1 2p3 (3D) 4f1 4P3/2
488 487 2s1 2p3 (3P) 4d1 4P3/2
505 522 2s1 2p3 (3P) 4f1 4G7/2
506 536 2s2 2p2 (3P) 5p1 4P3/2
525 512 2s1 2p3 (3P) 4f1 4F5/2
545 571 2s2 2p2 (3P) 5f1 2G7/2
546 572 2s2 2p2 (3P) 5g1 2F7/2
547 574 2s2 2p2 (3P) 5g1 2H9/2
557 530 2s2 2p2 (3P) 5d1 2P3/2
558 541 2s2 2p2 (3P) 5p1 2S1/2
578 628 2s2 2p2 (3P) 5g1 2G7/2
579 618 2s2 2p2 (3P) 5f1 2D5/2
584 560 2s2 2p2 (3P) 5d1 4D7/2
588 532 2s2 2p2 (3P) 5d1 4P5/2
601 576 2s1 2p3 (3S) 4f1 2F5/2
604 625 2s2 2p2 (3P) 5g1 4G9/2
609 573 2s2 2p2 (3P) 5f1 2G9/2
624 605 2s2 2p2 (3P) 5g1 4G7/2
626 638 2s2 2p2 (3P) 5g1 4F5/2
631 629 2s2 2p2 (3P) 5g1 2F5/2
640 634 2s1 2p3 (1P) 4p1 2P1/2

Table 6: Comparison of calculated Fe XX wavelengths $\lambda $ and line strengths S with values presented by NIST. BPRM - data from Nahar (2004), GRASP and CITRO - our values. Indexes of levels in the first two columns are taken from Table 1.

Table 7: The five major spontaneous radiative transition probabilities Ar and total transition probabilities $\sum A^r$ for each level. Arrow marks the final level to which radiative transition happens from the level. The sum of all radiative probabilities from the corresponding level is given in the last column.

Copyright ESO 2005