Issue 
A&A
Volume 623, March 2019



Article Number  A155  
Number of page(s)  12  
Section  Atomic, molecular, and nuclear data  
DOI  https://doi.org/10.1051/00046361/201834931  
Published online  26 March 2019 
Theoretical investigation of energy levels and transition data for S II, Cl III, Ar IV^{⋆}
^{1}
Institute of Theoretical Physics and Astronomy, Vilnius University, Saulėtekio av. 3, 10222 Vilnius, Lithuania
email: pavel.rynkun@tfai.vu.lt
^{2}
Group for Materials Science and Applied Mathematics, Malmö University, 20506 Malmö, Sweden
Received:
20
December
2018
Accepted:
28
January
2019
Aims. The aim of this work is to present accurate and extensive results of energy spectra and transition data for the S II, Cl III, and Ar IV ions. These data are useful for understanding and probing physical processes and conditions in various types of astrophysical plasmas.
Methods. The multiconfiguration Dirac–Hartree–Fock (MCDHF) and relativistic configuration interaction (RCI) methods, which are implemented in the generalpurpose relativistic atomic structure package GRASP2K, are used in the present work. In the RCI calculations the transversephoton (Breit) interaction, the vacuum polarization, and the selfenergy corrections are included.
Results. Energy spectra are presented comprising the 134, 87, and 103 lowest states in S II, Cl III, and Ar IV, respectively. Energy levels are in very good agreement with NIST database recommended values and associated with smaller uncertainties than energies from other theoretical computations. Electric dipole (E1), magnetic dipole (M1), and electric quadrupole (E2) transition data are computed between the above states together with the corresponding lifetimes. Based on internal validation, transition rates for the majority of the stronger transitions are estimated to have uncertainties of less than 3%.
Key words: atomic data
Tables 5–7 are only available at the CDS via anonymous ftp to cdsarc.ustrasbg.fr (130.79.128.5) or via http://cdsarc.ustrasbg.fr/vizbin/qcat?J/A+A/623/A155
© ESO 2019
1. Introduction
Atomic data are of importance in the understanding of the physical processes and conditions in various types of astrophysical plasmas, for example in the determination of chemical abundances of the elements and in the estimation of radiative transfer through stellar plasmas. The quality of atomic data have a large impact on the accuracy of chemical abundances (Juan de Dios & Rodríguez 2017; Kisielius et al. 2014). Juan de Dios & Rodríguez (2017) computed the ionic abundances of O II, O III, N II, Cl III, Ar III, Ar IV, Ne III, S II, and S III, and the total abundances of the corresponding elements. They observed that data for some of the studied ions need to be improved in order to derive more reliable values of chemical abundances in highdensity nebulae.
The studied elements (S II, Cl III, Ar IV) are in the astrophysically important group of lowionization ions which belongs to the phosphorous isoelectronic sequence. Fritzsche et al. (1999) used the multiconfiguration Dirac–Fock method to study forbidden transitions in the 3s^{2}3p^{3} configuration for phosphoruslike ions with low nuclear charge Z. Fischer et al. (2006) computed energy levels, lifetimes, and transition data for the sodiumlike to argonlike sequences using the multiconfiguration Hartree–Fock (MCHF) and multiconfiguration Dirac–Hartree–Fock (MCDHF) methods.
Tayal & Zatsarinny (2010) used the Bspline BreitPauli Rmatrix method to compute energy levels of the 3s^{2}3p^{3}, 3s3p^{4}, 3s^{2}3p^{2}3d, 3s^{2}3p^{2}4s, and 3s^{2}3p^{2}4p configurations and transitions in S II. Kisielius et al. (2014) presented the energy levels and transition data for S II using HF and quasirelativistic (QR) methods with transformed radial orbitals.
Sossah & Tayal (2012) used the MCHF and Bspline Breit–Pauli Rmatrix method to calculate transition probabilities and effective collision strengths for Cl III. Schectman et al. (2005) measured lifetimes and branching fractions with beamfoil techniques and derived oscillator strengths for transitions in Cl II and III.
Bredice et al. (1995) observed the spectra of Ar IV in the 280–5000 Å wavelength range and reanalyzed the 3s^{2}3p^{3}, 3s3p^{4}, 3s^{2}3p^{2}(3d + 4s) configurations. Djenize & Bukvić (2001) and Burger et al. (2012) studied transition probabilities in Ar III and Ar IV ions. Raineri et al. (2018) used a pulsed discharge light source to study the spectrum of Ar III and Ar IV in the 480–6218 Å region and predicted new levels using the Cowan code (Cowan 1981).
In this work energy spectrum calculations were performed for the 134 (72 even and 62 odd), 87 (52 even and 35 odd), and 103 (44 even and 59 odd) lowest states in the S II, Cl III, and Ar IV ions, respectively. Electric dipole, magnetic dipole, and electric quadrupole transition data were computed along with the corresponding lifetimes of these states. The calculations were done using the generalpurpose relativistic atomic structure package GRASP2K (Jönsson et al. 2013) with the modifications that are included in the newest version of the GRASP2018 package (Fischer et al. 2019).
2. Method
2.1. Computational procedure
The GRASP2K package used for the computations is based on the MCDHF and relativistic configuration interaction (RCI) methods. More details about these approaches can be found in Fischer et al. (2016) and Grant (2007).
In the MCDHF approximation atomic state functions (ASFs) are given as linear combinations of symmetry adapted configuration state functions (CSFs)
where J and M are the angular quantum numbers and P is parity. The CSFs Φ(γ_{i}PJM) are built from products of oneelectron Dirac orbitals. In the relativistic selfconsistent field procedure both the radial parts of the Dirac orbitals and the expansion coefficients were optimized to selfconsistency.
In RCI computations the wave function is expanded in CSFs and only the expansion coefficients are determined by diagonalizing the Hamiltonian matrix. The RCI method was used to include the transversephoton (Breit) interaction and quantum electrodynamic (QED) corrections: the vacuum polarization and the selfenergy.
In this work ASFs were obtained as expansions over jjcoupled CSFs. To transform these ASFs into an LSJcoupled CSF basis the method provided by Gaigalas et al. (2003, 2017) was used.
2.2. Computational scheme
For all three ions (S II, Cl III, Ar IV) similar computational schemes were used. As a starting point, MCDHF calculations were performed in the extended optimal level (EOL) scheme (Dyall et al. 1989) for the weighted average of the even and odd parity states. For the construction of the ASFs the multireferencesingledouble (MRSD) method (Fischer et al. 2016) was used. In this approach the CSF expansions were obtained by allowing SD substitutions from the configurations in the MR to active orbital sets. Only CSFs that have nonzero matrix elements with the CSFs belonging to the configurations in the MR were retained. No substitutions were allowed from the 1s, 2s, 2p shells, which define an inactive closed core. The MR and the active orbital sets for each of the ions are presented in Table 1. The MCDHF calculations were followed by RCI calculations, including the Breit interaction and leading QED effects. The RCI calculations were done separately for even and odd states.
Summary of active space construction.
At the last step, referred to as the RCI(CV) step in the tables, the MR was extended to include additional important configurations, and corevalence (CV) correlation effects were accounted for by allowing at most one substitution also from the 2p shell. The substitutions from the 2p shell increase the number of CSFs dramatically and for this reason these substitutions were restricted to a smaller orbital set.
The largescale calculations were performed with the MPI version of the GRASP code.
3. Results
The accuracy of the wave functions from the present calculation and some previous calculations was evaluated by comparing calculated energy levels with data from the NIST database (Kramida et al. 2018). In Table 2 a summary of this evaluation is presented: the number of computed energy levels (No. of levels in Ref.) and the average percentage difference between NIST and the different methods for the states covered by these methods (Av. difference).
Comparison of computed energy levels in the present work and other theoretical results with data from the NIST database for the S II, Cl III, and Ar IV ions.
The inclusion of the CV electron correlations and the extension of the MR set in the calculations improve the results. As is seen from Table 2 the averaged difference of the computed energy spectra (final RCI(CV) results) relative to the energies from the NIST database is 0.22%, 0.18%, and 0.21%, respectively, for the S II, Cl III, and Ar IV ions. Comparing the present results with results from other theoretical computations we obtain a better agreement with values given in the NIST database, except for the S II ion. The averaged uncertainties of energies presented in Tayal & Zatsarinny (2010) is only 0.06%, but they cover fewer energy levels. For the first time, levels of the 3s3p^{3}3d configuration are presented for the S II, Cl III, and Ar IV ions.
The mean contribution of the Breit and QED corrections to the final results is 0.05% for the studied ions. For the separate state the contribution of these effects can reach 0.1%.
The uncertainty of electric transition data was evaluated based on the quantity dT (Ekman et al. 2014), which is defined as
Here, A_{l} and A_{v} are transition rates in length and velocity forms. The mean dT for all presented E1 transitions is 12.00%, 5.95%, and 6.47%, respectively for the S II, Cl III, and Ar IV ions. The results for the different ions is discussed in more detail below.
3.1. S II
In Table A.1 energy spectra and wave function composition in LScoupling for 72 even states of the 3s3p^{4}, 3s^{2}3p^{2}3d, 3s^{2}3p^{2}4d, 3s^{2}3p^{2}4s, 3s^{2}3p^{2}5s configurations and for 62 odd states of the 3s^{2}3p^{3}, 3s^{2}3p^{2}4p, 3s3p^{3}3d, 3s^{2}3p^{2}4f, 3s^{2}3p^{2}5p configurations for S II are given. The states are given with unique labels (Gaigalas et al. 2017). The contribution was marked in bold for the states in which the labels were not assigned with largest contribution to the composition. In Table A.1 lifetimes in length and velocity gauges are also presented.
In Fig. 1 energy levels computed in this work and other theoretical calculations are compared with data from NIST (Kramida et al. 2018). From the figure we see that the relative uncertainties of energy levels obtained in this work in most cases are about 0.2%. Only for levels of the ground configuration the disagreements are larger, about 1.8%.
Fig. 1. Comparison of computed energy levels and other theoretical calculations with data from the NIST database for S II. The solid lines indicate 0.5% and the dashed lines 0.2% deviation from the NIST data. (Kisielius et al. 2014) a – HF data. (Kisielius et al. 2014) b – quasirelativistic data. 

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Transition data such as wavelengths; weighted oscillator strengths; transition rates of E1, M1, and E2 transitions; and the accuracy indicator dT are given in Table 5, and are available at the CDS. Generally, the uncertainty of transition data is small for the stronger transitions. To display this a scatterplot of dT versus the transition rate A for computed E1 transitions is given in Fig. 2. For most of the transitions, dT is well below 10%, and for the strongest ones dT is well below 3%. The weak transitions are either intercombination transitions, where in relativistic calculations the low rates result from strong cancellation of several large contributions to the transition moment (Ynnerman & Fischer 1995), or twoelectron onephoton (TEOP) transitions, where the rate is identically zero in the simplest approximation of the wave function and where the transition results from inclusion of correlation effects (Li et al. 2010). These types of transitions are still extremely challenging for theory and improved methodology is needed to further decrease the uncertainties.
Fig. 2. Scatterplot of dT vs. the transition rate A of E1 transitions for S II. The solid lines indicate the 4% and 10% deviations. 

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3.2. Cl III
In Table A.2 energy spectra, lifetimes, and wave function composition in LScoupling are presented for 52 even states of the 3s3p^{4}, 3s^{2}3p^{2}3d, 3s^{2}3p^{2}4s, 3s^{2}3p^{2}4d configurations and for 35 odd states of the 3s^{2}3p^{3}, 3s^{2}3p^{2}4p, 3s3p^{3}3d, 3p^{5} configurations in the Cl III ion. Energy levels are compared with results from NIST (Kramida et al. 2018). In the NIST database some levels of the 3s^{2}3p^{2}4p, 3s^{2}3p^{2}4d, and 3p^{5} configurations are flagged with question marks. Comparing these levels with present calculations and other computations there is a good agreement, except for states of the 3p^{5} configuration. Our energies for states of the 3p^{5} configuration are lower by 60 000 cm^{−1}. These levels in NIST are marked as levels that were determined by interpolation or extrapolation of known experimental values or by semiempirical calculations.
The uncertainty of the computed energy levels comparing with NIST data is less than 0.5%, and in most cases about 0.1%. Only for the first excited levels is the disagreement more than 1%. The averaged uncertainty of computed energy spectra comparing with NIST data is 0.18% (Table 2). The present energies are in better agreement with NIST than energies from previous theoretical computations. In addition, in this work more energy levels were studied, and for the first time levels of the 3s3p^{3}3d configuration are presented.
Transition data for E1, M1, and E2 transitions are given in Table 6, and are available at the CDS. In Fig. 3 the scatterplot of dT versus the transition rate A is displayed for all presented E1 transitions. The mean dT for the transitions is 5.95%. For most of the strongest transitions, dT is well below 2%. Table 3 gives the comparison of the theoretical and experimental results of wavelengths and oscillator strengths for the transitions in Cl III. From the table we see a very good agreement of wavelengths with the experimental values. Oscillator strengths are a little too large compared with experiment (Schectman et al. 2005).
Fig. 3. Scatterplot of dT vs. the transition rate A of E1 transitions for Cl III. The solid lines indicate the 2% and 10% deviations. 

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Comparison of wavelengths and oscillator strengths for the transitions in Cl III.
3.3. Ar IV
Table A.3 displays energy spectra, lifetimes, and wave function composition in LScoupling for 44 even states of the 3s3p^{4}, 3s^{2}3p^{2}3d, 3s^{2}3p^{2}4s configurations and for 59 odd states of the 3s^{2}3p^{3}, 3s^{2}3p^{2}4p, 3s3p^{3}3d, 3p^{5} configurations in Ar IV. The averaged uncertainty of energy levels obtained in this work compared with the NIST data is 0.21%. The largest disagreement (about 1%) is just for the first few excited levels.
Raineri et al. (2018) presented ten new energy levels of the 3s^{2}3p^{2}3d and 3s^{2}3p^{2}4p configurations for Ar IV. In Table 4 a comparison of these levels with this work and theoretical results by Fischer et al. (2006) is made. There is very good agreement between the new energy levels and the present calculations, except for the states for which the relative difference is about 14%. Such a large difference suggests that there is a misidentification and that further experimental analysis is needed.
Comparison of new levels by Raineri et al. (2018) with theoretical computations for Ar IV.
Transition data for E1, M1, and E2 transitions are given in Table 7, and are available at the CDS. In Fig. 4 a scatterplot of dT versus the transition rate A is displayed for all presented E1 transitions. The mean dT for the transitions is 6.47%. Again, for most of the strongest transitions, dT is well below 2%.
Fig. 4. Scatterplot of dT vs. the transition rate A of E1 transitions for Ar IV. The solid lines indicate the 2% and 10% deviations. 

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4. Conclusions
Energy spectra and transition data of E1, M1, and E2 transitions are presented for S II, Cl III, and Ar IV using MCDHF and RCI methods. The accuracy of the results is evaluated by comparing energy levels with data from NIST database and by the agreement of transition rates between length and velocity gauges. For the first time levels of the 3s3p^{33}d configuration are presented for the studied elements. The averaged uncertainty of computed energy levels compared with NIST data is 0.22%, 0.18%, and 0.21%, respectively for S II, Cl III, and Ar IV ions. The mean dT for all presented E1 transitions is 12.00%, 5.95%, and 6.47%, respectively, for the S II, Cl III, and Ar IV ions.
Acknowledgments
This research is funded by the European Social Fund under the No 09.3.3LMTK712 “Development of Competences of Scientists, other Researchers and Students through Practical Research Activities” measure.
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Appendix A: Wave function composition in LScoupling, energy levels, and lifetimes for the S II, Cl III, and Ar IV ions
Wave function composition (up to three LS components with a contribution >0.02 of the total wave function) in LScoupling and energy levels (in cm^{−1}) for S II.
Wave function composition (up to three LS components with a contribution >0.02 of the total wave function) in LScoupling and energy levels (in cm^{−1}) for Cl III.
Wave function composition (up to three LS components with a contribution >0.02 of the total wave function) in LScoupling and energy levels (in cm^{−1}) for Ar IV.
All Tables
Comparison of computed energy levels in the present work and other theoretical results with data from the NIST database for the S II, Cl III, and Ar IV ions.
Comparison of wavelengths and oscillator strengths for the transitions in Cl III.
Comparison of new levels by Raineri et al. (2018) with theoretical computations for Ar IV.
Wave function composition (up to three LS components with a contribution >0.02 of the total wave function) in LScoupling and energy levels (in cm^{−1}) for S II.
Wave function composition (up to three LS components with a contribution >0.02 of the total wave function) in LScoupling and energy levels (in cm^{−1}) for Cl III.
Wave function composition (up to three LS components with a contribution >0.02 of the total wave function) in LScoupling and energy levels (in cm^{−1}) for Ar IV.
All Figures
Fig. 1. Comparison of computed energy levels and other theoretical calculations with data from the NIST database for S II. The solid lines indicate 0.5% and the dashed lines 0.2% deviation from the NIST data. (Kisielius et al. 2014) a – HF data. (Kisielius et al. 2014) b – quasirelativistic data. 

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In the text 
Fig. 2. Scatterplot of dT vs. the transition rate A of E1 transitions for S II. The solid lines indicate the 4% and 10% deviations. 

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In the text 
Fig. 3. Scatterplot of dT vs. the transition rate A of E1 transitions for Cl III. The solid lines indicate the 2% and 10% deviations. 

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In the text 
Fig. 4. Scatterplot of dT vs. the transition rate A of E1 transitions for Ar IV. The solid lines indicate the 2% and 10% deviations. 

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In the text 
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