Extended theoretical transition data in C I - IV
MMNRAS , 1–21 (2020) Preprint 25 January 2021 Compiled using MNRAS L A TEX style file v3.0
Extended theoretical transition data in C i – iv
W. Li, , ★ A. M. Amarsi, A. Papoulia , J. Ekman and P. Jönsson Department of Materials Science and Applied Mathematics, Malmö University, SE-205 06, Malmö, Sweden Theoretical Astrophysics, Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala, Sweden Division of Mathematical Physics, Lund University, Post Office Box 118, SE-221 00 Lund, Sweden
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Accurate atomic data are essential for opacity calculations and for abundance analyses of the Sun and other stars. The aim ofthis work is to provide accurate and extensive results of energy levels and transition data for C i – iv.The Multiconfiguration Dirac–Hartree–Fock and relativistic configuration interaction methods were used in the present work.To improve the quality of the wave functions and reduce the relative differences between length and velocity forms for transitiondata involving high Rydberg states, alternative computational strategies were employed by imposing restrictions on the electronsubstitutions when constructing the orbital basis for each atom and ion.Transition data, e.g., weighted oscillator strengths and transition probabilities, are given for radiative electric dipole (E1)transitions involving levels up to 1s
7f for C ii, up to 1s
8g for C iv. Usingthe difference between the transition rates in length and velocity gauges as an internal validation, the average uncertainties of allpresented E1 transitions are estimated to be 8.05%, 7.20%, 1.77%, and 0.28%, respectively, for C i – iv. Extensive comparisonswith available experimental and theoretical results are performed and good agreement is observed for most of the transitions. Inaddition, the C i data were employed in a reanalysis of the solar carbon abundance. The new transition data give a line-by-linedispersion similar to the one obtained when using transition data that are typically used in stellar spectroscopic applicationstoday.
Key words:
Atomic data — Atomic processes — Line: formation — Radiative transfer — Sun: abundances — Methods:numerical
Accurate atomic data are of fundamental importance to many differ-ent fields of astronomy and astrophysics. This is particularly true forcarbon. As the fourth-most abundant metal in the cosmos (Asplundet al. 2009), carbon is a major source of opacity in the atmospheresand interiors of stars. Complete and reliable sets of atomic data forcarbon are essential for stellar opacity calculations, because of theirsignificant impact on stellar structure and evolution (e.g. VandenBerget al. 2012; Chen et al. 2020).Accurate atomic data for carbon are also important in the context ofspectroscopic abundance analyses and Galactic Archaeology. Carbonabundances measured in late-type stars help us to understand thenucleosynthesis of massive stars and AGB stars, and thus the Galacticchemical evolution (e.g. Franchini et al. 2020; Jofré et al. 2020;Stonkut˙e et al. 2020). In early-type stars, carbon abundances helpconstrain the present-day Cosmic Abundance Standard (e.g. Nieva& Przybilla 2008, 2012; Alexeeva et al. 2019). In the Sun, the carbonabundance is precisely measured in order to put different cosmicobjects onto a common scale (e.g. Caffau et al. 2010; Amarsi et al.2019). In all of these cases, oscillator strengths for C i (cool stars)and for C i – iv (hot stars) underpin the spectroscopic analyses; thisis especially the case for studies that relax the assumption of local ★ E-mail: [email protected] thermodynamic equilibrium (LTE; e.g. Przybilla et al. 2001; Nieva& Przybilla 2006), in which case much larger sets of reliable atomicdata are needed.On the experimental side, a number of studies of transition datahave been presented in the literature. Neutral C i transition probabili-ties for the 2p4p → → → → P → P o at a heavy-ion storage ring, and the total measured radiative decay rates to theground term were 125.8 ± − for P / , 9.61 ± − for P / ,and 45.35 ± − for P / . The aforementioned results are, how-ever, not in agreement with the values measured by Fang et al. (1993)using a radio-frequency ion trap, i.e., 146.4(+8.3, -9.2) s − for P / , © a r X i v : . [ phy s i c s . a t o m - ph ] J a n W. Li et al. − for P / , and 51.2(+2.6, -3.5) s − for P / .Goly & Weniger (1982) measured the transition probabilities from ahelium-carbon arc for some multiplets of { , } → and2s →
3p with estimated relative uncertainty of 50%. Using anelectric shock tube, Roberts & Eckerle (1967) provided the relativeoscillator strengths of some C ii multiplets with relative uncertaintiesof 7%. Reistad et al. (1986) gave lifetimes for 11 C ii levels using thebeam-foil excitation technique and extensive cascade analyses.For C iii, the IC decay rate of the 2s2p P o1 → S transitionwas measured to be 121.0 ± − by Kwong et al. (1993) using aradio-frequency ion trap and 102.94 ± − by Doerfert et al.(1997) using a heavy-ion storage ring. The discrepancy between thevalues obtained from the two different methods is quite large, i.e., ofthe order of 15%. The result given by the latter measurement is closerto earlier 𝑎𝑏 𝑖𝑛𝑖𝑡𝑖𝑜 calculations ranging between 100 and 104 s − (Fleming et al. 1994; Fischer 1994; Ynnerman & Fischer 1995).Several measurements have also been performed for the lifetimesof the low-lying levels of C iii (Reistad & Martinson 1986; Mickey1970; Nandi et al. 1996; Buchet-Poulizac & Buchet 1973a).For the system of Li-like C iv, the transition probabilities ofthe 1s P o1 / , / → S / transitions were measured byBerkner et al. (1965) using the foil-excitation technique and byKnystautas et al. (1971) using the beam-foil technique, respectively.There are also a number of measurements of lifetimes in C iv usingthe beam-foil technique (Donnelly et al. 1978; Buchet-Poulizac &Buchet 1973b; Jacques et al. 1980).On the theoretical side, Froese Fischer et al. have performed de-tailed studies of C i – iv, focusing on the low-lying levels. They car-ried out Multiconfiguration Hartree-Fock (MCHF) calculations andused the Breit-Pauli (MCHF-BP) approximation for computing en-ergy levels and transition properties, e.g., transition probabilities, os-cillator strengths, and lifetimes, in C i (Tachiev & Fischer 2001; Fis-cher 2006; Fischer & Tachiev 2004), C ii (Tachiev & Fischer 2000),C iii (Tachiev & Fischer 1999; Fischer 2000), and C iv (Godefroidet al. 2001; Fischer et al. 1998).Hibbert et al. have presented extensive calculations for opticaltransitions. They used the CIV3 code (Hibbert 1975) to calculateoscillator strengths and transition probabilities in C i (Hibbert et al.1993), C ii (Corrégé & Hibbert 2004), and C iii (Kingston & Hibbert2000). In the calculations of Hibbert et al. (1993); Corrégé & Hibbert(2004), empirical adjustments were introduced to the diagonal matrixelements in order to accurately reproduce energy splittings. Their C ioscillator strengths are frequently used in the abundance analyses ofcool stars (Sect. 5).A number of other authors have also presented theoretical transi-tion data for carbon. Zatsarinny & Fischer (2002) calculated the oscil-lator strengths for transitions to high-lying excited states of C i usinga spline frozen-cores method. Nussbaumer & Storey (1984) providedthe radiative transition probabilities using the 𝐿𝑆 -coupling approx-imation and intermediate coupling approximation, respectively, forthe six energetically lowest configurations of C i. Nussbaumer &Storey (1981) calculated the transition probabilities for C ii, fromterms up to 2s F o , using the 𝐿𝑆 -coupling and close coupling(CC) approximation, respectively.In view of the great astrophysical interest for large sets of homo-geneous atomic data, extensive spectrum calculations of transitiondata in the carbon atom and carbon ions were carried out under theumbrella of the Opacity Project using the CC approximation of theR-matrix theory, and the results are available in the Opacity Projectonline database (TOPbase; Cunto & Mendoza (1992); Cunto et al.(1993)). The latest compilation of C i transition probabilities was made available by Haris & Kramida (2017), and those of C ii-iv canbe found in earlier compilations by Wiese & Fuhr (2007b,a) and Fuhr(2006).In this context, the General-purpose Relativistic Atomic Struc-ture Package (Grasp) has, more recently, been used by Aggarwal &Keenan (2015) to predict the radiative decay rates and lifetimes of166 levels belonging to the n (cid:54) In the Multiconfiguration Dirac-Hartree-Fock (MCDHF) method(Grant 2007; Fischer et al. 2016), wave functions for atomic states 𝛾 ( 𝑗 ) 𝑃𝐽 𝑀 , 𝑗 = , , . . . , 𝑁 with angular momentum quantum num-bers 𝐽 𝑀 and parity 𝑃 are expanded over 𝑁 CSFs configuration statefunctions Ψ ( 𝛾 ( 𝑗 ) 𝑃𝐽 𝑀 ) = 𝑁 CSFs ∑︁ 𝑖 𝑐 ( 𝑗 ) 𝑖 Φ ( 𝛾 𝑖 𝑃𝐽 𝑀 ) . (1)The configuration state functions (CSFs) are 𝑗 𝑗 -coupled many-electron functions, recursively built from products of one-electronDirac orbitals. As for the notation, 𝛾 𝑖 specifies the occupied subshellsof the CSF with their complete angular coupling tree information.The radial large and small components of the one-electron orbitalsand the expansion coefficients { 𝑐 ( 𝑗 ) 𝑖 } of the CSFs are obtained, fora number of targeted states, by solving the Dirac-Hartree-Fock ra-dial equations and the configuration interaction eigenvalue problemresulting from applying the variational principle on the statisticallyweighted energy functional of the targeted states with terms addedfor preserving the orthonormality of the one-electron orbitals. Theenergy functional is based on the Dirac-Coulomb (DC) Hamiltonianand accounts for relativistic kinematic effects.Once the radial components of the one-electron orbitals are de-termined, higher-order interactions, such as the transverse photoninteraction and quantum electrodynamic (QED) effects (vacuum po-larization and self-energy), are added to the Dirac-Coulomb Hamilto-nian. Keeping the radial components fixed, the expansion coefficients{ 𝑐 ( 𝑗 ) 𝑖 } of the CSFs for the targeted states are obtained by solving theconfiguration interaction eigenvalue problem.The evaluation of radiative E1 transition data (transition probabili-ties, oscillator strengths) between two states: 𝛾 (cid:48) 𝑃 (cid:48) 𝐽 (cid:48) 𝑀 (cid:48) and 𝛾𝑃𝐽 𝑀 is MNRAS , 1–21 (2020) xtended theoretical transition data in C i – iv non-trivial. The transition data can be expressed in terms of reducedmatrix elements of the transition operator T ( ) : (cid:104) Ψ ( 𝛾𝑃𝐽 ) (cid:107) T ( ) (cid:107) Ψ ( 𝛾 (cid:48) 𝑃 (cid:48) 𝐽 (cid:48) ) (cid:105) = ∑︁ 𝑗,𝑘 𝑐 𝑗 𝑐 (cid:48) 𝑘 (cid:104) Φ ( 𝛾 𝑗 𝑃𝐽 ) (cid:107) T ( ) (cid:107) Φ ( 𝛾 (cid:48) 𝑘 𝑃 (cid:48) 𝐽 (cid:48) ) (cid:105) , (2)where 𝑐 𝑗 and 𝑐 (cid:48) 𝑘 are, respectively, the expansion coefficients of theCSFs for the lower and upper states, and the summation occurs overall the CSFs for the lower and upper states. The reduced matrix ele-ments are expressed via spin-angular coefficients 𝑑 ( ) 𝑎𝑏 and operatorstrengths as: (cid:104) Φ ( 𝛾 𝑗 𝑃𝐽 ) (cid:107) T ( ) (cid:107) Φ ( 𝛾 (cid:48) 𝑘 𝑃 (cid:48) 𝐽 (cid:48) ) (cid:105) = ∑︁ 𝑎,𝑏 𝑑 ( ) 𝑎𝑏 (cid:104) 𝑛 𝑎 𝑙 𝑎 𝑗 𝑎 (cid:107) T ( ) (cid:107) 𝑛 𝑏 𝑙 𝑏 𝑗 𝑏 (cid:105) . (3)Allowing for the fact that we are now using Brink-and-Satchler typereduced matrix elements, we have (cid:104) 𝑛 𝑎 𝑙 𝑎 𝑗 𝑎 (cid:107) T ( ) (cid:107) 𝑛 𝑏 𝑙 𝑏 𝑗 𝑏 (cid:105) = (cid:18) ( 𝑗 𝑏 + ) 𝜔𝜋𝑐 (cid:19) / (− ) 𝑗 𝑎 − / (cid:18) 𝑗 𝑎 𝑗 𝑏 − (cid:19) 𝑀 𝑎𝑏 , (4)where 𝑀 𝑎𝑏 is the radiative transition integral defined by Grant(1974). The factor in front of 𝑀 𝑎𝑏 is the Wigner 3-j symbol thatgives the angular part of the matrix element. The 𝑀 𝑎𝑏 integral canbe written 𝑀 𝑎𝑏 = 𝑀 𝑒𝑎𝑏 + 𝐺 𝑀 𝑙𝑎𝑏 , where 𝐺 is the gauge parameter.When 𝐺 = 𝐺 = √ 𝑑𝑇 , defined as (Froese Fischer 2009; Ekman et al. 2014) 𝑑𝑇 = | 𝐴 𝑙 − 𝐴 𝑣 | max ( 𝐴 𝑙 , 𝐴 𝑣 ) , (5)where 𝐴 𝑙 and 𝐴 𝑣 are transition rates in length and velocity form, canbe used as an estimation of the uncertainty of the computed rate. Calculations were performed in the extended optimal level (EOL)scheme (Dyall et al. 1989) for the weighted average of the even andodd parity states. The CSF expansions were determined using themultireference-single-double (MR-SD) method, allowing single anddouble (SD) substitutions from a set of important configurations,referred to as the MR, to orbitals in an active set (AS) (Olsen et al.1988; Sturesson et al. 2007; Fischer et al. 2016). The orbitals in the ASare divided into spectroscopic orbitals, which build the configurationsin the MR, and correlation orbitals, which are introduced to correctthe initially obtained wave functions. During the different steps ofthe calculations for C i – iv, the CSF expansions were systematicallyenlarged by adding layers of correlation orbitals.MCDHF calculations aim to generate an orbital set. The orbitalset is then used in RCI calculations based on CSF expansions thatcan be enlarged to capture additional electron correlation effects. For the same CSF expansion, different orbital sets give different resultsfor both energy levels and transition data. Conventionally, MCDHFcalculations are performed for CSF expansions obtained by allowingsubstitutions not only from the valence subshells, but also from thesubshells deeper in the core, accounting for valence-valence (VV),core-valence (CV), and core-core (CC) electron correlation effects.Using orbital sets from such calculations, Pehlivan Rhodin et al.(2017) predicted large 𝑑𝑇 values for transitions between low-lyingstates and high Rydberg states, indicating substantial uncertaintiesin the corresponding transition data. For transitions involving highRydberg states, it was shown that the velocity gauge gave the moreaccurate results, which is contradictory to the general belief that thelength gauge is the preferred one (Hibbert 1974). Analyzing the sit-uation more carefully, Papoulia et al. (2019) found that correlationorbitals resulting from MCDHF calculations based on CSF expan-sions obtained by allowing substitutions from deeper subshells arevery contracted in comparison with the outer Rydberg orbitals. As aconsequence, the outer parts of the wave functions for the Rydbergstates are not accurately described. Thus, the length form that probesthe outer part of the wave functions does not produce trustworthyresults, while the velocity form that probes the inner part of the wavefunctions yields more reliable transition rates. In the same work, theauthors showed how transition rates that are only weakly sensitive tothe choice of gauge can be obtained, by paying close attention to theCSF generation strategies for the MCDHF calculations.In the present work, following the suggestion by Papoulia et al.(2019), the MCDHF calculations were based on CSF expansions forwhich we impose restrictions on the substitutions from the inner sub-shells and obtain, as a consequence, correlation orbitals that overlapmore with the spectroscopic orbitals of the higher Rydberg states,adding to a better representation of the outer parts of the correspond-ing wave functions. The MR and orbital sets for each atom and ionare presented in Table 1. The computational scheme, including CSFgeneration strategies, for each atom and ion is discussed in detailbelow. The MCDHF calculations were followed by RCI calculations,including the Breit interaction and leading QED effects. As seen in Table 1, in the computations of neutral carbon, con-figurations with 𝑛 = ( 𝑙 = s ) ; 6 ( 𝑙 = p , d ) , which are not of directrelevance, were included in the MR set to obtain orbitals that arespatially extended, improving the quality of the outer parts of thewave functions of the higher Rydberg states. The MCDHF calcu-lations were performed using CSF expansions that were producedby SD substitutions from the valence orbitals of the configurationsin the MR to the active set of orbitals, with the restriction of al-lowing maximum one substitution from orbitals with 𝑛 =
2. The1s core was kept closed and, at this point, the expansions of theatomic states accounted for VV electron correlation. As a final step,an RCI calculation was performed for the largest SD valence expan-sion augmented by a CV expansion. The CV expansion was obtainedby allowing SD substitutions from the valence orbitals and the 1s core of the configurations in the MR, with the restriction that thereshould be at most one substitution from 1s . The numbers of CSFsin the final even and odd state expansions are, respectively, 14 941842 and 15 572 953, distributed over the different 𝐽 symmetries. Similarly to the computations in C i, in the computations of the singly-ionized carbon, the configurations 2s { , , , } , which are not MNRAS , 1–21 (2020)
W. Li et al.
Table 1.
Summary of the computational schemes for C i – iv. The first column displays the configurations of the targeted states. MR and AS, respectively, denotethe multireference sets and the active sets of orbitals used in the MCDHF and RCI calculations, and 𝑁 CSFs are the numbers of generated CSFs in the final RCIcalculations, for the even (e) and the odd (o) parity states.Targeted configurations MR AS 𝑁 CSFs
C i, N levels = {11s,10p,10d,9f, e: 14 941 8422s { 𝑛 s , 𝑛 p , 𝑛 d , } { 𝑛 s , 𝑛 p , 𝑛 d , } (cid:54) 𝑛 (cid:54)
6, 2 (cid:54) 𝑛 (cid:54)
5, 3 (cid:54) 𝑛 (cid:54)
5) (3 (cid:54) 𝑛 (cid:54)
6, 2 (cid:54) 𝑛 (cid:54)
6, 3 (cid:54) 𝑛 (cid:54) { 𝑛 s , 𝑛 p , 𝑛 d } (3 (cid:54) 𝑛 (cid:54)
6, 3 (cid:54) 𝑛 (cid:54)
5, 3 (cid:54) 𝑛 (cid:54) { , , , , , } { 𝑛 s , 𝑛 p , 𝑛 d , } (cid:54) 𝑛 (cid:54)
6, 3 (cid:54) 𝑛 (cid:54)
5, 3 (cid:54) 𝑛 (cid:54) levels = 𝑛𝑙 ( 𝑛 (cid:54) , 𝑙 (cid:54)
4) 2s2p , 2s { 𝑛 s , 𝑛 p , 𝑛 d , 𝑛 f , 𝑛 g } {14s , , , , e: 6 415 7982s 𝑙 (cid:54) ( (cid:54) 𝑛 (cid:54) , (cid:54) 𝑛 (cid:54) , (cid:54) 𝑛 (cid:54)
7, 10g , , 2p , 4 (cid:54) 𝑛 (cid:54) , (cid:54) 𝑛 (cid:54) ) , 2p { 𝑛 s , 𝑛 p , 𝑛 d , 𝑛 f , 𝑛 g }( (cid:54) 𝑛 (cid:54) , (cid:54) 𝑛 (cid:54) , (cid:54) 𝑛 (cid:54) (cid:54) 𝑛 (cid:54) , (cid:54) 𝑛 (cid:54) ) levels = 𝑛𝑙 ( 𝑛 (cid:54) , 𝑙 (cid:54)
4) 2s 𝑛𝑙 ( 𝑛 (cid:54) , 𝑙 (cid:54)
4) {12s , , , , e: 1 578 6202p , 2p { , , } , 2p { , , } , levels = 𝑛𝑙 ( 𝑛 (cid:54) , 𝑙 (cid:54)
4) 1s 𝑛𝑙 ( 𝑛 (cid:54) , 𝑙 (cid:54)
4) {14s , , , , , e: 1 077 8721s
6h 1s
6h 8h , our prime targets, were included in the MR set (see also Table 1).In this manner, we generated orbitals that are localized farther fromthe atomic core. The MCDHF calculations were performed usingCSF expansions obtained by allowing SD substitutions from thevalence orbitals of the MR configurations. During this stage, the1s core remained frozen and the CSF expansions accounted forVV correlation. The final wave functions of the targeted states weredetermined in an RCI calculation, which included CSF expansionsthat were formed by allowing SD substitution from all subshells ofthe MR configurations, with the restriction that there should be atmost one substitution from the 1s core. The numbers of CSFs in thefinal even and odd state expansions are, respectively, 6 415 798 and4 988 973, distributed over the different 𝐽 symmetries. In the computations of beryllium-like carbon, the MR simply con-sisted of the targeted configurations (see also Table 1). The CSF ex-pansions used in the MCDHF calculations were obtained by allowingSD substitutions from the valence orbitals, accounting for VV cor-relation effects. The final wave functions of the targeted states weredetermined in subsequent RCI calculations, which included CSFsthat were formed by allowing single, double, and triple (SDT) substi-tutions from all orbitals of the MR configurations, with the limitationof leaving no more than one hole in the 1s atomic core. The finaleven and odd state expansions, respectively, contained 1 578 620 and1 274 147 CSFs, distributed over the different 𝐽 symmetries. Likewise the computations in C iii, the MR in the computationsof lithium-like carbon was solely represented by the targeted con-figurations (see also Table 1). In the MCDHF calculations, the CSFexpansions were acquired by implementing SD electron substitutionsfrom the configurations in the MR, with the restriction of allowingmaximum one hole in the 1s core. In this case, the shape of thecorrelation orbitals was established by CSFs accounting for valence(V) and CV correlation effects. In the subsequent RCI calculations,the CSF expansions were enlarged by enabling all SDT substitutionsfrom the orbitals in the MR to the active set of orbitals. The finalexpansions of the atomic states gave rise to 1 077 872 CSFs witheven parity and 1 287 706 CSFs with odd parity, respectively, sharedamong the different 𝐽 symmetry blocks. The energy spectra and wave function composition in 𝐿𝑆 -couplingfor the 100, 69, 114, and 53 lowest states, respectively, for C i – ivare given in Table A1. In the tables, the states are given with uniquelabels (Gaigalas et al. 2017), and the labelling is determined by theCSFs with the largest coefficient in the expansion of Eq. (1). We firstsummarise the results here, before discussing the individual ions indetail in Sects. 4.1 – 4.4, below.The accuracy of the wave functions from the present calculationswas evaluated by comparing the calculated energy levels with ex-perimental data provided via the National Institute of Standards andTechnology (NIST) Atomic Spectra Database (Kramida et al. 2019).In the left panel of Fig. 1, energy levels computed in this work are MNRAS , 1–21 (2020) xtended theoretical transition data in C i – iv No. -2.0-1.5-1.0-0.50.00.51.01.52.0 ( E RC I - E N I S T ) / E N I S T ( % ) C IC IIC IIIC IV
No. -3 -2 -1 | l - v | / m a x ( l , v ) ( % ) C IC IIC IIIC IV
Figure 1.
Left panel: Comparison of computed energy levels in the present work with data from the NIST database, for C i – iv. The dashed lines indicate the − compared with the NIST data. A closer inspection of the figure re-veals that the relative discrepancies between the experimental and thecomputed in this work energies are, in most cases, about − − configuration in C i, the disagreements are largerthan 1.0%. The average difference of the computed energy levelsrelative to the energies from the NIST database is 0.41%, 0.081%,0.041%, and 0.0044%, respectively, for C i – iv. In Table A1, life-times in length and velocity gauges are also presented. The rightpanel of Fig. 1 presents the relative differences between the lifetimesin length and velocity forms for C i – iv. Except for a few long-livedstates that can decay to the ground state only through IC transitions,the relative differences are well below 5%.The accuracy of calculated transition rates can be estimated eitherby comparisons with other theoretical works and experimental re-sults, when available, or by the quantity 𝑑𝑇 , which is defined in Eq. (5)as the agreement between the values in length and velocity gauges(Froese Fischer 2009; Ekman et al. 2014). The latter is particularlyuseful when no experimental measurements are available. Transitiondata, e.g., wavenumbers; wavelengths; line strengths; weighted os-cillator strengths; transition probabilities of E1 transitions; and theaccuracy indicators 𝑑𝑇 , are given in Tables A2 – A5, respectively,for C i – iv. Note that the wavenumbers and wavelengths are adjustedto match the level energy values in the NIST database, which arecritically evaluated by Haris & Kramida (2017) for C i and Moore &Gallagher (1993) for C ii-iv. When no NIST values are available, thewavenumbers and wavelengths are from the present MCDHF/RCIcalculations and marked with * in the tables.To better display the uncertainties 𝑑𝑇 of the computed transitionsrates and their distribution in relation to the magnitude of the transi-tion rate values 𝐴 , the transitions are organized in five groups basedon the magnitude of the 𝐴 values. A statistical analysis of the uncer-tainties 𝑑𝑇 of the transitions is performed for the 1553, 806, 1805,and 386 E1 transitions, respectively, for C i – iv. In Table 2, the meanvalue of the uncertainties (cid:104) 𝑑𝑇 (cid:105) and standard deviations 𝜎 are givenfor each group of transitions. As seen in Table 2, most of the estimateduncertainties 𝑑𝑇 are well below 10%. Most of the strong transitionswith 𝐴 > 10 s − are associated with small uncertainties 𝑑𝑇 , lessthan 2%, especially for C iii and C iv, for which (cid:104) 𝑑𝑇 (cid:105) is 0.297% ( 𝜎 =0.01) and 0.205% ( 𝜎 = 0.0041), respectively. It is worth noting that, by employing the alternative optimization scheme of the radial or-bitals in the present calculations, the uncertainties 𝑑𝑇 for transitionsinvolving high Rydberg states are significantly reduced.Contrary to the strong transitions, the weaker transitions are asso-ciated with relatively large 𝑑𝑇 values. This is even more pronouncedfor the first two groups of transitions in C i and C ii, where 𝐴 isless than 10 s − . These weak E1 transitions are either IC or two-electron one-photon transitions. The rates of the former transitions,in relativistic calculations, are small due to the strong cancellationcontributions to the transition moment (Ynnerman & Fischer 1995),whereas the rates of the latter transitions are identically zero in thesimplest approximation of the wave function and only induced bycorrelation effects (Bogdanovich et al. 2007; Li et al. 2010). Thesetypes of transitions are extremely challenging, and therefore interest-ing from a theoretical point of view, and improved methodology isneeded to further decrease the uncertainties of the respective transi-tion data.Fortunately, the weak transitions tend to be of lesser astrophysi-cal importance, either for opacity calculations, or for spectroscopicabundance analyses. Thus, only the transitions with 𝐴 (cid:62) s − forC i and C ii, and 𝐴 (cid:62) s − for C iii and C iv, are discussed in thepaper; although the complete transition data tables, for all computedE1 transitions in C i – iv, are available online. The scatterplots of 𝑑𝑇 versus 𝐴 are given in Fig. 2. The mean 𝑑𝑇 for all presented E1transitions shown in Fig. 2 is 8.05% ( 𝜎 = 0.12), 7.20% ( 𝜎 = 0.13),1.77% ( 𝜎 = 0.05), and 0.28% ( 𝜎 = 0.0059), respectively, for C i – iv.A statistical analysis of the proportions of the transitions with 𝑑𝑇 less than 20%, 10%, and 5% in all the presented E1 transitions isalso performed and shown in the last three rows of Table 2.Finally, the present work can be compared with other theoreticalcalculations. In Fig. 3, log 𝑔 𝑓 values from the present work are com-pared with results from MCHF-BP (Fischer 2006; Tachiev & Fischer2000, 1999; Fischer et al. 1998), CIV3 (Hibbert et al. 1993; Corrégé& Hibbert 2004), and TOPbase data (Cunto & Mendoza 1992), whenavailable. As shown in the figure, the differences between the log 𝑔 𝑓 values computed in the present work and respective results from othersources are rather small for most of the transitions. Comparing theMCDHF/RCI results with those from CIV3 calculations by Hibbertet al. (1993), which are frequently used in the abundance analyses,292(228) out of 378 transitions are in agreement within 20% (10%) MNRAS , 1–21 (2020)
W. Li et al.
Table 2.
Distribution of the uncertainties 𝑑𝑇 (in %) of the computed transition rates in C i – iv depending on the magnitude of the rates. The transition ratesare arranged in five groups based on the magnitude of the 𝐴 values (in s − ). The number of transitions, No., the mean 𝑑𝑇 , (cid:104) 𝑑𝑇 (cid:105) , (in %), and the standarddeviations, 𝜎 , are given for each group of transitions, in C i – iv, respectively. The last three rows show the proportions of the transitions with 𝑑𝑇 less than 20%,10%, and 5% in all the transitions with 𝐴 (cid:62) s − for C i and C ii and 𝐴 (cid:62) s − for C iii and C iv, respectively.C i C ii C iii C ivGroup No. (cid:104) 𝑑𝑇 (cid:105)( % ) 𝜎 No. (cid:104) 𝑑𝑇 (cid:105)( % ) 𝜎 No. (cid:104) 𝑑𝑇 (cid:105)( % ) 𝜎 No. (cid:104) 𝑑𝑇 (cid:105)( % ) 𝜎<
62 52.6 0.34 80 29.6 0.32 137 10.8 0.18 20 5.92 0.06110 −
156 34.0 0.25 134 17.1 0.24 239 5.57 0.096 10 2.38 0.01710 −
451 13.2 0.15 128 14.4 0.19 354 2.48 0.050 6 0.667 0.004710 −
600 7.20 0.11 167 11.8 0.15 360 1.44 0.034 43 0.267 0.0035 >
284 1.68 0.020 297 1.53 0.023 715 0.297 0.010 307 0.205 0.0041 𝑑𝑇 < 20% 87.4% 89.5% 98.4% 100% 𝑑𝑇 < 10% 77.3% 80.7% 95.7% 100% 𝑑𝑇 < 5% 62.0% 68.7% 91.7% 99.4% for C i, and 78(66) out of 87 transitions are within the same range forC ii. The results from the MCDHF/RCI and MCHF-BP calculationsare found to be in very good agreement for C iii–iv, with the relativedifferences being less than 5% for all the computed transitions. Moredetails about the comparisons with other theoretical calculations, aswell as with experimental results, are given in Sects 4.1 – 4.4. The computed excitation energies, given in Table A1, are comparedwith results from NIST (Kramida et al. 2019). With the exception ofthe levels belonging to the 2s2p configuration, for which the averagerelative difference between theory and experiment is 1.22%, the meanrelative difference for the rest of the states is 0.35%. The completetransition data, for all computed E1 transitions in C i, can be foundin Table A2. Based on the statistical analysis of the uncertainties 𝑑𝑇 shown in table 2, out of the 1335 transitions with 𝐴 (cid:62) s − , theproportions of the transitions with 𝑑𝑇 less than 20%, 10%, and 5%are, respectively, 87.4%, 77.3%, and 62.0%.In C i, experimental transition data are available for the2p3p → → → 𝐿𝑆 configurationin the interaction matrix in the determination of the wavefunctions.The estimated uncertainties 𝑑𝑇 of the MCDHF/RCI line strengthsare given as percentages in parentheses. In most cases, the theoreticalvalues fall into, or only slightly outside, the range of the estimateduncertainties of the experimental values.Comparing the MCDHF/RCI results with the results from theCIV3 calculations by Hibbert et al. (1993), we see that 41 outof the 50 transitions in common are in good agreement, with therelative differences being less than 10% (see Table A6). For the2p4s P o → P transitions and the 2p4s P o2 → D tran-sition, the 𝑆 values deduced from the present MCDHF/RCI calcula-tions differ substantially from the experimental values, i.e., by morethan 20%, while the values from the CIV3 calculations appear to be inbetter agreement with the corresponding experimental values. Basedon the agreement between the length and velocity forms, the esti-mated uncertainties 𝑑𝑇 of the present MCDHF/RCI calculations forthe above-mentioned transitions are of the order of 8.5% and 1.4%,respectively. For the 2p3d P o2 → P , 2p4s P o2 → D , and 2p3d D o2 → D transitions, both theoretical results areoutside the range of the estimated uncertainties of the experimen-tal values. For the 2p3d D o → P transitions, the evaluatedrelative line strengths by Golly et al. (2003) slightly differ from theobservations by Bacawski et al. (2001). The latter seem to be in betteroverall agreement with the transition rates predicted by the presentcalculations.In Table A7, the computed line strengths and transition rates arecompared with values from the spline frozen-cores (FCS) methodby Zatsarinny & Fischer (2002) and the MCHF-BP calculations byFischer (2006). Zatsarinny & Fischer (2002) presented oscillatorstrengths for transitions from the 2p
P term to high-lying excitedstates, while Fischer (2006) considered only transitions from 2p P, D, and S to odd levels up to 2p3d P o . As seen in the table, thepresent MCDHF/RCI results seem to be in better agreement withthe values from spline FCS calculations. 76 out of 98 transitionsfrom Zatsarinny & Fischer (2002) agree with present values within10%, while only 38 out of 78 transitions from Fischer (2006) arewithin the same range. The relatively large differences with Fischer(2006) may be due to the fact that limited electron correlations wereincluded in their calculations. In the MCHF-BP calculations, twotypes of correlation, i.e., VV, CV, have been accounted for; however,the CC correlation has not been considered. Additionally, CSF ex-pansions obtained from SD substitutions are not as large as the CSFexpansions used in the present calculations. For the majority of thestrong transitions with 𝐴 > 10 s − , there is a very good agreementbetween the MCDHF/RCI results and the spline FCS values, withthe relative difference being less than 5%. On the other hand, for the2p3d F → P and 2p4s P → P transitions, the observeddiscrepancies between these three methods, i.e., MCDHF/RCI, splineFCS, and MCHF-BP, are quite large. These transitions are all 𝐿𝑆 -forbidden transitions, the former is with Δ 𝐿 = 2 and the latter isspin-forbidden transition; these types of transitions are challengingfor computations and are always with large uncertainties. For ex-ample, for the 2p3d F → P transition, the 𝐴 values fromMCDHF/RCI, spline FCS, and MCHF-BP calculations are, respec-tively, 7.92E+06, 6.24E+06, and 1.14E+07 s − , with the relativedifference between each two of them being greater than 20%. Ex-perimental data are, therefore, needed for validating these theoreticalresults. On the contrary, based on the agreement between the lengthand velocity forms displayed in the parentheses, the estimated un-certainties of the MCDHF/RCI calculations for the above-mentionedtransitions are all less than 0.5%. MNRAS , 1–21 (2020) xtended theoretical transition data in C i – iv A (s -1 ) -3 -2 -1 d T ( % ) d T vs. A for C I, d T = 8.05%, = 0.12 10 A (s -1 ) -3 -2 -1 d T ( % ) d T vs. A for C II, d T = 7.20%, = 0.1310 A (s -1 ) -4 -3 -2 -1 d T ( % ) d T vs. A for C III, d T = 1.77%, = 0.050 A (s -1 ) -4 -3 -2 -1 d T ( % ) d T vs. A for C IV, d T = 0.284%, = 0.0059 Figure 2.
Scatterplot of d 𝑇 values vs. transition rates 𝐴 of E1 transitions, for C i – iv. The solid lines indicate the 10% relative agreement between the lengthand velocity gauges. The relative differences between theory and experiment for all theenergy levels of 2s2p are 0.16%, while the mean relative differencefor the rest of the states is 0.071% (see Table A1). The completetransition data, for all computed E1 transitions in C ii, can be foundin Table A3. Out of the presented 592 E1 transitions with 𝐴 (cid:62) s − , the proportions of the transitions with 𝑑𝑇 less than 20%, 10%,and 5% are, respectively, 89.5%, 80.7%, and 68.7%.In Table A6, the lifetimes from the present MCDHF/RCI calcu-lations are compared with available results from the MCHF-BP cal-culations by Tachiev & Fischer (2000) and observations by Reistadet al. (1986) and Träbert et al. (1999). Träbert et al. (1999) measuredlifetimes for the three fine-structure components of the 2s2p P termin an ion storage ring. For the measured lifetimes by Reistad et al.(1986) of the doublets terms using the beam-foil technique, a singlevalue for the two fine-structure levels is provided. It can be seen that,in all cases, the MCDHF/RCI computed lifetimes agree with the ex-perimental values by Reistad et al. (1986) within the experimentalerrors. For the 2s2p P / , / , / states, as discussed in Sect. 1, thediscrepancies between the measured transition rates by Fang et al.(1993) and by Träbert et al. (1999) are quite large. It is found that theMCDHF/RCI values are in better agreement with the results given bythe latter measurements, with a relative difference less than 3%. For these long-lived states, the measured lifetimes are better representedby the MCDHF/RCI results than by the MCHF-BP values.The computed line strengths and transition rates are compared withvalues from the MCHF-BP calculations by Tachiev & Fischer (2000)and the CIV3 calculations by Corrégé & Hibbert (2004) in TableA8. We note that the agreement between the present MCDHF/RCIand the MCHF-BP transition rates exhibits a broad variation. In theearlier MCHF-BP and our MCDHF/RCI calculations, the same cor-relation effects, i.e., VV and CV, have been accounted for. However,the CSF expansions obtained from SD substitutions in the MCHF-BP calculations are not as large as the CSF expansions used in thepresent calculations, and as a consequence, the 𝐿𝑆 -composition ofthe configurations might not be predicted as accurately in the formercalculations. The MCDHF/RCI results seem to be in better over-all agreement with the values from the CIV3 calculations, exceptfor transitions from 2p P o to 2s2p { P , S } and to 2s D. Forthese transitions, involving 2p P o as the upper level, the transitionrates 𝐴 are of the order of 10 – 10 s − . The 𝑑𝑇 values are relativelylarge in the present calculations. This is due to the strong cancella-tion effects caused by, e.g., the strong mixing between the 2p P o and 2s2p3s P o levels for 2p P o → S, and the mixing be-tween the 2p P o and 2s P o levels for 2p P o → D.Large discrepancies are also observed between the MCDHF/RCI andMCHF-BP results, as well as between the MCHF-BP and CIV3 re-
MNRAS000
MNRAS000 , 1–21 (2020)
W. Li et al. -4 -3 -2 -1 0 1 log gf -3.0-2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.53.0 ( l og g f ) (1) C I MCHF-BPCIV3TOPbase -4 -3 -2 -1 0 1 log gf -2.0-1.5-1.0-0.50.00.51.01.52.0 ( l og g f ) (2) C II MCHF-BPCIV3TOPbase -4 -3 -2 -1 0 1 log gf -4.0-3.0-2.0-1.00.01.02.03.04.0 ( l og g f ) (3) C III MCHF-BPTOPbase -4 -3 -2 -1 0 1 log gf -0.10-0.08-0.06-0.04-0.020.000.020.040.060.080.10 ( l og g f ) (4) C IV MCHF-BPTOPbase
Figure 3.
Differences between the calculated log 𝑔 𝑓 values in this work and results from other theoretical calculations: MCHF-BP (red asterisk), CIV3 (blueplus sign), and TOPbase (black point), for C i – iv. sults for these transitions. Experimental data are, therefore, crucialfor validating the aforementioned theoretical results. On the contrary,for the majority of the strong transitions with 𝐴 > 10 s − , there isa very good agreement between the MCDHF/RCI results and thosefrom the two previous calculations, with the relative differences beingless than 5%. The average relative discrepancy between the computed excitationenergies, shown in Table A1, and the NIST recommended values is0.041%. The complete transition data, for all computed E1 transitionsin C iii, can be found in Table A4. Out of the 1668 transitions with 𝐴 (cid:62) s − , 91.7% (98.4%) of them have 𝑑𝑇 values less than 5%(20%). Further, the mean 𝑑𝑇 for all transitions with 𝐴 (cid:62) s − is1.8% with 𝜎 = 0.05.The lifetimes of the 2s2p P o1 , 2p { S , D } , and 2s3s S stateswere measured by Reistad et al. (1986) using the beam-foil tech-nique, and the oscillator strengths for the 2s2p P o1 → S andthe 2p { S , D } → P o1 transitions were also provided. Ta-ble A6 gives the comparisons between the observed and computedoscillator strengths and lifetimes in C iii. Looking at the table, we seean excellent agreement between the present calculations and thosefrom the MCHF-BP calculations (Tachiev & Fischer 1999) with the relative difference being less than 0.7%. In all cases, the computedoscillator strengths and lifetimes agree with experiment within theexperimental errors. The exceptions are the oscillator strength of the2p S → P o1 transition and the lifetime of the 2p S state,for which the computed values slightly differ from the observations.In Table A9, the computed line strengths and transition rates arecompared with values from the MCHF-BP calculations by Tachiev& Fischer (1999) and the Grasp calculations by Aggarwal & Keenan(2015). For the majority of the transitions, there is an excellent agree-ment between the MCDHF/RCI and MCHF-BP values with the rel-ative differences being less than 1%. Only 4 out of 60 transitionsdisplay discrepancies that are greater than 20%. These large discrep-ancies are observed for the IC transitions, e.g., 2s3d D → P o1 and 2s3d D → P o1 , for which the 𝑑𝑇 is relatively large. Thediscrepancies between the MCDHF/RCI and Grasp values are over-all large; this is due to the fact that limited electron correlationswere included in their calculations. Based on the excellent agree-ment between the MCDHF/RCI and MCHF-BP results as well aswith experiment, we believe that the present transition rates togetherwith the MCHF-BP transition data are more reliable than the onesprovided by Aggarwal & Keenan (2015). MNRAS , 1–21 (2020) xtended theoretical transition data in C i – iv The mean relative discrepancy between the computed excitation en-ergies, given in Table A1, and the NIST values is 0.0044%. Out ofthe presented 366 transitions with 𝐴 (cid:62) s − shown in Table A5,only two of them have 𝑑𝑇 values greater than 5%; 94.0% of themwith 𝑑𝑇 being less than 1%. The mean 𝑑𝑇 for all transitions with 𝐴 (cid:62) s − is 0 .
28% with 𝜎 = 0.0059.For C iv, there are a number of measurements of transition proper-ties. The transition rates of the 2p P o1 / , / → S / transitionswere measured by Knystautas et al. (1971) using the beam-foil tech-nique. By using the same technique, the lifetimes for a number ofexcited states were measured in four different experimental works(Donnelly et al. 1978; Buchet-Poulizac & Buchet 1973b; Jacqueset al. 1980; Peach et al. 1988). In Table A6, we compare the theoret-ical results, from present calculations and MCHF-BP calculations,with the NIST recommended values and observed values. The transi-tion rates of the 2p P o1 / , / → S / transitions from the presentwork agree perfectly with the values from the MCHF-BP calcula-tions by Fischer et al. (1998), while they are slightly smaller than theNIST data and the values by Knystautas et al. (1971). A comparisonof the lifetimes of the { , , , , , , , } states is madewith other theoretical results, i.e., from the MCHF-BP calculationsand the Model Potential method. The agreements between thesedifferent theoretical results are better than 1% for all these states.Furthermore, the agreement between the computed values and thosefrom observations is also very good except for the 3s S / level, forwhich the MCDHF/RCI calculations give a slightly smaller lifetimeof 0.2350 ns than the observed value of 0.25 ± One can also attempt to verify the present atomic data empirically, inan astrophysical context. To demonstrate this, a solar carbon abun-dance analysis was carried out, based on permitted C i lines. Largererrors in the atomic data usually impart a larger dispersion in theline-by-line abundance results, as well as trends in the results withrespect to the line parameters.The solar carbon abundance analysis recently presented in Amarsiet al. (2019) was taken as the starting point. Their analysis is based onequivalent widths measured in the solar disk-centre intensity, for 14permitted C i lines in the optical and near-infrared, as well as a singleforbidden [C i] line at 872 . ± .
01 dex.Here, we post-correct the solar carbon abundances inferred inAmarsi et al. (2019) from the 14 permitted C i lines, using the newatomic data derived in the present study (see Table 3). To first-order, for a given spectral line, the change in the inferred abundancesare related to the difference in the adopted transition probabilities
500 1000 1500 2000 2500 3000 35008.308.408.508.60 500 1000 1500 2000 2500 3000 3500 λ air / nm8.308.408.508.60 l og ε C A19: µ =8.44 σ =0.06L20: µ =8.50 σ =0.07 low / eV8.308.408.508.60 l og ε C A19: µ =8.44 σ =0.06L20: µ =8.50 σ =0.07 −5.8 −5.6 −5.4 −5.2 −5.08.308.408.508.60 −5.8 −5.6 −5.4 −5.2 −5.0log W / λ l og ε C A19: µ =8.44 σ =0.06L20: µ =8.50 σ =0.07 Figure 4.
Inferred solar carbon abundances. Black points (A19) are the 3Dnon-LTE results of Amarsi et al. (2019) for 14 permitted C i lines. Blue points(L20) are these same results but post-corrected using the new log 𝑔 𝑓 data.Error bars reflect ±
5% uncertainties in the measured equivalent widths asstipulated by those authors. The four lines between 1254 nm and 1259 nmdiscussed in the text have been highlighted in red. The unweighted means 𝜇 (including all 14 lines) and the standard deviations of the samples 𝜎 arestated in each panel. MNRAS000
5% uncertainties in the measured equivalent widths asstipulated by those authors. The four lines between 1254 nm and 1259 nmdiscussed in the text have been highlighted in red. The unweighted means 𝜇 (including all 14 lines) and the standard deviations of the samples 𝜎 arestated in each panel. MNRAS000 , 1–21 (2020) W. Li et al.
Table 3.
The 14 permitted C i lines used as abundance diagnostics in Amarsi et al. (2019). Shown are the upper and lower configurations, oscillator strengthsobtained from the present calculations, and oscillator strengths from NIST; the latter being based on the calculations from CIV3 (Hibbert et al. 1993). Theestimated uncertainties 𝑑𝑇 of the oscillator strengths are given as percentages in parentheses. The final two columns show the abundances derived in Amarsiet al. (2019), and the post-corrected values derived here based on the formula Δ log ε Cline = − Δ log 𝑔 𝑓 line .log 𝑔 𝑓 Upper Lower 𝜆 air (nm) NIST MCDHF/RCI( 𝑑𝑇 ) log ε A19C log ε L20C D P o1 P P o1 P o1 P F o2 D F o4 D D P o2 F o2 D P o1 P P o0 P P o1 P P o2 P P o1 S D F o3 D o2 P simply as Δ log ε lineC = − Δ log 𝑔 𝑓 line . We briefly note that second-order effects on the inferred abundances, propagated forward fromchanges to the non-LTE statistical equilibrium when adopting thefull set of new log 𝑔 𝑓 data in the non-LTE model atom, were alsotested; these were found to be negligible.The results of this post-correction are illustrated in Fig. 4. We findthat the dispersion in the line-by-line abundance results are similarwhen using the new and the old sets of log 𝑔 𝑓 data. We also find thatthe trends in the results with respect to the line parameters are ofsimilar gradients. This is consistent with the finding in Sect. 4.1, thatthe precision of this new, much larger atomic data set is comparableto that of Hibbert et al. (1993).This new analysis implies a solar carbon abundance of 8 .
50 dex,which is 0 .
06 dex larger than that inferred in Amarsi et al. (2019)from C i lines, and 0 .
07 dex larger than the current standard valuefrom Asplund et al. (2009) that is based on C i lines as well as onmolecular diagnostics. This increase in the mean abundance is dueto 12 of the 14 permitted C i lines having lower oscillator strengthsin the present calculations, compared to the NIST data set. Six ofthe lines give results that are larger than the mean (log ε (cid:62) . .
53 and 8 .
60 dex.These four lines have the same upper level configuration, 2p3d P o ,and a closer inspection of the 𝐿𝑆 -composition reveals that thesestates are strongly mixed (of the order of 26%) with 2s2p P o states,which are less accurately described in the present calculations. Asa consequence, as shown in Table 3, these transitions appear to beassociated with slightly larger uncertainties 𝑑𝑇 than most of the otherlines. Omitting these four lines, or adopting NIST oscillator strengthsfor them, would reduce the mean abundance from 8 .
50 to 8 .
47 dex.Given that the scatter and trends in the results do not support oneset of data over the other, we refrain from advocating a higher so-lar carbon abundance at this point. Nevertheless, this quite drasticchange in the resulting solar carbon abundance highlights the impor-tance of having accurate atomic data for abundance analyses. Thisis especially relevant in the context of the solar modelling problem,wherein standard models of the solar interior, adopting the solarchemical composition of Asplund et al. (2009), fail to reproduce keyempirical constraints, including the depth of the convection zone and interior sound speed that are precisely inferred from helioseismicobservations (Basu & Antia 2008; Zhang et al. 2019). Extra opacityin the solar interior near the boundary of the convection zone wouldresolve the problem (Bailey et al. 2015). Carbon contributes about5% of the opacity in this region (Blancard et al. 2012), so a highercarbon abundance would help alleviate the problem, albeit only veryslightly.
In the present work, energy levels and transition data of E1 transitionsare computed for C i – iv using the MCDHF and RCI methods. Spe-cial attention is paid to the computation of transition data involvinghigh Rydberg states by employing an alternative orbital optimizationapproach.The accuracy of the predicted excitation energies is evaluated bycomparing with experimental data provided by the NIST database.The average relative differences of the computed energy levels com-pared with the NIST data are 0.41%, 0.081%, 0.041%, and 0.0044%,respectively, for C i – iv. The accuracy of the transition data is eval-uated based on the relative differences of the computed transitionrates in the length and velocity gauges, which is given by the quan-tity 𝑑𝑇 , and by extensive comparisons with previous theoretical andexperimental results. For most of the strong transitions in C i – iv,the 𝑑𝑇 values are less than 5%. The mean 𝑑𝑇 for all presented E1transitions are 8.05% ( 𝜎 = 0.12), 7.20% ( 𝜎 = 0.13), 1.77% ( 𝜎 =0.050), and 0.28% ( 𝜎 = 0.0059), respectively, for C i – iv. Particu-larly, for strong transitions with 𝐴 > s − , the mean 𝑑𝑇 is 1.68%( 𝜎 = 0.020), 1.53% ( 𝜎 = 0.023), 0.297% ( 𝜎 = 0.010), and 0.205%( 𝜎 = 0.0041), respectively, for C i – iv. By employing alternativeoptimization schemes of the radial orbitals, the uncertainties 𝑑𝑇 ofthe computed transition data for transitions involving high Rydbergstates are significantly reduced. The agreement between computedtransition properties, e.g., line strengths, transition rates, and life-times, and experimental values is overall good. The exception is theweak transitions, e.g., the IC transitions, for which the strong cancel-lation effects are important; however, these effects cannot be properlyconsidered in the present calculations. The present calculations are MNRAS , 1–21 (2020) xtended theoretical transition data in C i – iv extended to high Rydberg states that are not covered by previousaccurate calculations and this is of special importance in variousastrophysical applications.The accurate and extensive sets of atomic data for C i – iv arepublicly available for use by the astronomy community. These datashould be useful for opacity calculations and for models of stel-lar structures and interiors. They should also be useful to non-LTEspectroscopic analyses of both early-type and late-type stars. ACKNOWLEDGEMENTS
This work is supported by the Swedish research council under con-tracts 2015-04842, 2016-04185, 2016-03765, and 2020-03940, andby the Knut and Alice Wallenberg Foundation under the projectgrant KAW 2013.0052. Some of the computations were enabledby resources provided by the Swedish National Infrastructure forComputing (SNIC) at the Multidisciplinary Center for AdvancedComputational Science (UPPMAX) and at the High PerformanceComputing Center North (HPC2N) partially funded by the SwedishResearch Council through grant agreement no. 2018-05973. Thiswork was also supported by computational resources provided by theAustralian Government through the National Computational Infras-tructure (NCI) under the National Computational Merit AllocationScheme (NCMAS), under project y89. We thank Nicolas Grevessefor insightful comments on an earlier version of this manuscript. Wewould also like to thank the anonymous referee for her/his usefulcomments that helped improve the original manuscript.
DATA AVAILABILITY
The full tables of energy levels (Table A1) and transition data (TablesA2 – A5) are available in the online Supplementary Material.
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APPENDIX A: ADDITIONAL TABLES
MNRAS , 1–21 (2020) xtended theoretical transition data in C i – iv Table A1.
Wave function composition (up to three 𝐿𝑆 components with a contribution > 𝐿𝑆 -coupling, energy levels (incm − ), and lifetimes (in s; given in length ( 𝜏 𝑙 ) and velocity ( 𝜏 𝑣 ) gauges) for C i – iv. Energy levels are given relative to the ground state and compared withNIST data (Kramida et al. 2019). The full table is available online.Species No. State 𝐿𝑆 -composition 𝐸 𝑅𝐶𝐼 𝐸 𝑁 𝐼𝑆𝑇 𝜏 𝑙 𝜏 𝑣 C I 1 2s ( P ) P P 7p P 0 0C I 2 2s ( P ) P P 7p P 16 16C I 3 2s ( P ) P P 7p P 43 43C I 4 2s ( D ) D P 7p D + 0.03 2s P 3p D 10 275 10 193C I 5 2s ( S ) S P 7p S + 0.06 2p ( S ) S 21 775 21 648C I 6 2s S 2p ( S ) S ◦ S 2p ( P ) P 7p S ◦
33 859 33 735 3.00E-02 1.26E-02C I 7 2s P 3s P ◦ ( P ) P 3s P ◦
60 114 60 333 3.00E-09 3.04E-09C I 8 2s P 3s P ◦ ( P ) P 3s P ◦
60 133 60 353 3.00E-09 3.04E-09C I 9 2s P 3s P ◦ ( P ) P 3s P ◦
60 174 60 393 3.00E-09 3.04E-09C I 10 2s P 3s P ◦ ( P ) P 3s P ◦
61 750 61 982 2.78E-09 2.83E-09– – – – – – – –
Table A2.
Electric dipole transition data for C i from present calculations. Upper and lower states, wavenumber, Δ 𝐸 , wavelength, 𝜆 , line strength, 𝑆 , weightedoscillator strength, 𝑔 𝑓 , transition probability, 𝐴 , together with the relative difference between two gauges of 𝐴 values, 𝑑𝑇 , provided by the present MCDHF/RCIcalculations are shown in the table. Wavelength and wavenumber values are from the NIST database (Kramida et al. 2019) when available. Wavelengths andwavenumbers marked with * are from the present calculations. Only the first ten rows are shown; the full table is available online.Upper Lower Δ 𝐸 (cm − ) 𝜆 (Å) 𝑆 (a.u. of a e ) 𝑔 𝑓 𝐴 (s − ) 𝑑𝑇 D o2 P D o1 P D o3 P P o2 P D o1 P D o2 P P o1 P P o2 P D o1 P P o1 P Table A3.
Electric dipole transition data for C ii from present calculations. Upper and lower states, wavenumber, Δ 𝐸 , wavelength, 𝜆 , line strength, 𝑆 , weightedoscillator strength, 𝑔 𝑓 , transition probability, 𝐴 , together with the relative difference between two gauges of 𝐴 values, 𝑑𝑇 , provided by the present MCDHF/RCIcalculations are shown in the table. Wavelength and wavenumber values are from the NIST database (Kramida et al. 2019) when available. Only the first tenrows are shown; the full table is available online.Upper Lower Δ 𝐸 (cm − ) 𝜆 (Å) 𝑆 (a.u. of a e ) 𝑔 𝑓 𝐴 (s − ) 𝑑𝑇 D / P o1 / D / P o3 / D / P o3 / D / P o1 / D / P o3 / D / P o3 / P / P o1 / P / P o1 / P / P o3 / P / P o3 /000
Electric dipole transition data for C ii from present calculations. Upper and lower states, wavenumber, Δ 𝐸 , wavelength, 𝜆 , line strength, 𝑆 , weightedoscillator strength, 𝑔 𝑓 , transition probability, 𝐴 , together with the relative difference between two gauges of 𝐴 values, 𝑑𝑇 , provided by the present MCDHF/RCIcalculations are shown in the table. Wavelength and wavenumber values are from the NIST database (Kramida et al. 2019) when available. Only the first tenrows are shown; the full table is available online.Upper Lower Δ 𝐸 (cm − ) 𝜆 (Å) 𝑆 (a.u. of a e ) 𝑔 𝑓 𝐴 (s − ) 𝑑𝑇 D / P o1 / D / P o3 / D / P o3 / D / P o1 / D / P o3 / D / P o3 / P / P o1 / P / P o1 / P / P o3 / P / P o3 /000 , 1–21 (2020) W. Li et al.
Table A4.
Electric dipole transition data for C iii from present calculations. Upper and lower states, wavenumber, Δ 𝐸 , wavelength, 𝜆 , line strength, 𝑆 , weightedoscillator strength, 𝑔 𝑓 , transition probability, 𝐴 , together with the relative difference between two gauges of 𝐴 values, 𝑑𝑇 , provided by the present MCDHF/RCIcalculations are shown in the table. Wavelength and wavenumber values are from the NIST database (Kramida et al. 2019) when available. Wavelengths andwavenumbers marked with * are from the present calculations. Only the first ten rows are shown; the full table is available online.Upper Lower Δ 𝐸 (cm − ) 𝜆 (Å) 𝑆 (a.u. of a e ) 𝑔 𝑓 𝐴 (s − ) 𝑑𝑇 P o1 S P o1 S P o1 S P o1 S P o1 S P o1 S P o1 S P o1 S D o1 S P o1 S Table A5.
Electric dipole transition data for C iv from present calculations. Upper and lower states, wavenumber, Δ 𝐸 , wavelength, 𝜆 , line strength, 𝑆 , weightedoscillator strength, 𝑔 𝑓 , transition probability, 𝐴 , together with the relative difference between two gauges of 𝐴 values, 𝑑𝑇 , provided by the present MCDHF/RCIcalculations are shown in the table. Wavelength and wavenumber values are from the NIST database (Kramida et al. 2019). Only the first ten rows are shown;the full table is available online.Upper Lower Δ 𝐸 (cm − ) 𝜆 (Å) 𝑆 (a.u. of a e ) 𝑔 𝑓 𝐴 (s − ) 𝑑𝑇 P o3 / S / P o1 / S / P o3 / S / P o1 / S / P o3 / S / P o1 / S / P o3 / S / P o1 / S / D / P o1 / D / P o3 / , 1–21 (2020) xtended theoretical transition data in C i – iv Table A6.
Comparison of relative line strengths ( 𝑆 ), weighted oscillator strengths ( 𝑔 𝑓 ), and lifetimes ( 𝜏 ), or transition probabilities ( 𝐴 ),with other theoretical work and experimental results for C i – iv. The present values from the MCDHF/RCI calculations are given in theBabushkin(length) gauge. The values in the parentheses are the relative differences between the length and velocity gauges. The references forthe experiments are shown in the last column. Note that the sums of the line strengths 𝑆 have been normalized to 100 for each multiplet in C i.C iTransition array Mult. 𝐽 𝑢 − 𝐽 𝑙 𝑆 (a.u. of a e )MCDHF/RCI CIV3 ( 𝑎 ) Expt. Expt.2s − D − P o ± ( 𝑏 ) ± ( 𝑏 ) ± ( 𝑏 ) ± ( 𝑏 ) ± ( 𝑏 ) − P − P o ± ( 𝑏 ) ( 𝑑 ) ± ( 𝑏 ) ( 𝑑 ) ± ( 𝑏 ) ( 𝑑 ) ± ( 𝑏 ) ( 𝑑 ) ± ( 𝑏 ) ( 𝑑 ) ± ( 𝑏 ) ( 𝑑 ) − S − P o ± ( 𝑏 ) ± ( 𝑏 ) ± ( 𝑏 ) − P o − S 2 - 1 57.27(3.0%) 56.85 59.0 ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) − P o − P 2 - 2 42.67(3.3%) 42.63 43.5 ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) − P o − P 2 - 2 62.05(5.7%) 50.76 51.2 ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) − D o − P 3 - 2 47.89(1.2%) 46.50 45.5 ± ( 𝑐 ) ( 𝑑 ) ± ( 𝑐 ) ( 𝑑 ) ± ( 𝑐 ) ( 𝑑 ) ± ( 𝑐 ) ( 𝑑 ) ± ( 𝑐 ) ( 𝑑 ) − P o − D 2 - 3 44.42(3.2%) 44.77 44.5 ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) − D o − D 3 - 3 50.31(1.5%) 49.19 50.0 ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) − F o − D 4 - 3 43.41(0.4%) 43.53 44.9 ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) MNRAS000
Comparison of relative line strengths ( 𝑆 ), weighted oscillator strengths ( 𝑔 𝑓 ), and lifetimes ( 𝜏 ), or transition probabilities ( 𝐴 ),with other theoretical work and experimental results for C i – iv. The present values from the MCDHF/RCI calculations are given in theBabushkin(length) gauge. The values in the parentheses are the relative differences between the length and velocity gauges. The references forthe experiments are shown in the last column. Note that the sums of the line strengths 𝑆 have been normalized to 100 for each multiplet in C i.C iTransition array Mult. 𝐽 𝑢 − 𝐽 𝑙 𝑆 (a.u. of a e )MCDHF/RCI CIV3 ( 𝑎 ) Expt. Expt.2s − D − P o ± ( 𝑏 ) ± ( 𝑏 ) ± ( 𝑏 ) ± ( 𝑏 ) ± ( 𝑏 ) − P − P o ± ( 𝑏 ) ( 𝑑 ) ± ( 𝑏 ) ( 𝑑 ) ± ( 𝑏 ) ( 𝑑 ) ± ( 𝑏 ) ( 𝑑 ) ± ( 𝑏 ) ( 𝑑 ) ± ( 𝑏 ) ( 𝑑 ) − S − P o ± ( 𝑏 ) ± ( 𝑏 ) ± ( 𝑏 ) − P o − S 2 - 1 57.27(3.0%) 56.85 59.0 ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) − P o − P 2 - 2 42.67(3.3%) 42.63 43.5 ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) − P o − P 2 - 2 62.05(5.7%) 50.76 51.2 ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) − D o − P 3 - 2 47.89(1.2%) 46.50 45.5 ± ( 𝑐 ) ( 𝑑 ) ± ( 𝑐 ) ( 𝑑 ) ± ( 𝑐 ) ( 𝑑 ) ± ( 𝑐 ) ( 𝑑 ) ± ( 𝑐 ) ( 𝑑 ) − P o − D 2 - 3 44.42(3.2%) 44.77 44.5 ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) − D o − D 3 - 3 50.31(1.5%) 49.19 50.0 ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) − F o − D 4 - 3 43.41(0.4%) 43.53 44.9 ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) ± ( 𝑐 ) MNRAS000 , 1–21 (2020) W. Li et al. ± ( 𝑐 ) C iiConfiguration Term
𝐽 𝜏 (ns) Ref.MCDHF/RCI MCHF-BP ( 𝑒 ) Expt.2s2p
S 1/2 0.4497 (0.7%) 0.4523 0.44 ± S 1/2 2.292 (0.6%) 2.266 2.4 ± S 1/2 2.017 (0.1%) 1.9 ± S 1/2 3.774 (0.1%) 3.7 ± P o ± ± P o ± ± P o ± ± P o ± ± P o ± ± D 3/2 0.3490(0.2%) 0.3493 0.34 ± ± D 3/2 0.7299(0.2%) 0.75 ± ± 𝐽 𝜏 (ms) Ref.MCDHF/RCI MCHF-BP ( 𝑒 ) Expt.2s2p
P 1/2 8.151 (47.6%) 7.654 7.95 ± ± ± 𝐽 𝑢 − 𝐽 𝑙 𝑔 𝑓 Ref.MCDHF/RCI MCHF-BP ( ℎ ) Expt.2s2p - 2s P o − S 1-0 0.7592(0.1%) 0.7583 0.75 ± - 2s2p S − P o < ± - 2s2p D − P o ± 𝐽 𝜏 (ns) Ref.MCDHF/RCI MCHF-BP ( ℎ ) Expt.2s2p P o ± S 0 0.4766( < ± D 2 7.240(0.5%) ns 7.191 7.2 ± S 0 1.164( < ± 𝐽 𝑢 − 𝐽 𝑙 𝐴 ( s − ) Ref.MCDHF/RCI MCHF-BP ( 𝑗 ) NIST Expt.2p − P o − S 1/2 - 1/2 2.632( < ± − P o − S 3/2 - 1/2 2.646( < ± 𝐽 𝜏 (ns) Ref.MCDHF/RCI MCHF-BP ( 𝑗 ) Model Potential ( 𝑜 ) Expt.3s S 1/2 0.2350( < ± S 1/2 0.3755( < ± P o < ± < P o < ± MNRAS , 1–21 (2020) xtended theoretical transition data in C i – iv < P o < ± < D 3/2 0.05717( < ± < D 3/2 0.1312 ( < ± < D 3/2 0.2511 ( < ± < ( 𝑎 ) Hibbert et al. (1993); ( 𝑏 ) Musielok et al. (1997); ( 𝑐 ) Bacawski et al. (2001); ( 𝑑 ) Golly et al. (2003); ( 𝑒 ) Tachiev & Fischer (2000); ( 𝑓 ) Reistadet al. (1986); ( 𝑔 ) Träbert et al. (1999); ( ℎ ) Tachiev & Fischer (1999); ( 𝑗 ) Fischer et al. (1998); ( 𝑘 ) Knystautas et al. (1971); ( 𝑙 ) Donnelly et al.(1978); ( 𝑚 ) Buchet-Poulizac & Buchet (1973b); ( 𝑛 ) Jacques et al. (1980); ( 𝑜 ) Peach et al. (1988).
Table A7.
Comparison of line strengths ( 𝑆 ) and transition rates ( 𝐴 ) with other theoretical results for C i. The present values from theMCDHF/RCI calculations are given in the Babushkin(length) gauge. The wavenumber Δ 𝐸 and wavelength 𝜆 values are taken from the NISTdatabase. The estimated uncertainties 𝑑𝑇 of the transition rates are given as percentages in parentheses. Transition array Mult. 𝐽 𝑢 − 𝐽 𝑙 Δ 𝐸 𝜆
MCDHF/RCI Spline FCS ( 𝑎 ) MCHF-BP ( 𝑏 ) ( cm − ) (Å) 𝑆 (a.u. of a e ) 𝐴 (s − ) 𝑆 (a.u. of a e ) 𝐴 (s − ) 𝑆 (a.u. of a e ) 𝐴 (s − ) − F o − P − F o − P − F o − P − P o − P − P o − P − D o − P − P o − P − D o − P − P o − P − D o − P < − P o − P − F o − P − P o − P − P o − P MNRAS000
MCDHF/RCI Spline FCS ( 𝑎 ) MCHF-BP ( 𝑏 ) ( cm − ) (Å) 𝑆 (a.u. of a e ) 𝐴 (s − ) 𝑆 (a.u. of a e ) 𝐴 (s − ) 𝑆 (a.u. of a e ) 𝐴 (s − ) − F o − P − F o − P − F o − P − P o − P − P o − P − D o − P − P o − P − D o − P − P o − P − D o − P < − P o − P − F o − P − P o − P − P o − P MNRAS000 , 1–21 (2020) W. Li et al.
Table A7.
Continued.
Transition array Mult. 𝐽 𝑢 − 𝐽 𝑙 Δ 𝐸 𝜆
MCDHF/RCI Spline FCS ( 𝑎 ) MCHF-BP ( 𝑏 ) ( cm − ) (Å) 𝑆 (a.u. of a e ) 𝐴 (s − ) 𝑆 (a.u. of a e ) 𝐴 (s − ) 𝑆 (a.u. of a e ) 𝐴 (s − )1 - 2 79275 1261.425 2.74E-01 9.26E+07(2.1%) 2.72E-01 9.31E+07 2.23E-01 7.55E+072 - 1 79294 1261.122 2.54E-01 5.15E+07(2.1%) 2.52E-01 5.19E+07 1.98E-01 4.02E+071 - 1 79302 1260.996 1.69E-01 5.72E+07(2.1%) 1.68E-01 5.75E+07 1.39E-01 4.70E+070 - 1 79306 1260.926 2.20E-01 2.24E+08(2.1%) 2.18E-01 2.24E+08 1.79E-01 1.82E+081 - 0 79318 1260.735 2.12E-01 7.19E+07(2.1%) 2.11E-01 7.23E+07 1.69E-01 5.73E+07 − F o − D − P o − D − P o − D − D o − D − P o − D − D o − D − P o − D − D o − D − P o − D − F o − D − P o − D − P o − D − P o − S − P o − S − P o − S − P o − S − D o − S − P o − S − P o − S < − P o − S − D o − P − P o − P − D o − P < < − P o − P − F o − P − P o − P < − P o − P − D o − P < − P o − P − D o − P < ( 𝑎 ) Zatsarinny & Fischer (2002); ( 𝑏 ) Fischer (2006).
MNRAS , 1–21 (2020) xtended theoretical transition data in C i – iv Table A8.
Comparison of line strengths ( 𝑆 ) and transition rates ( 𝐴 ) with other theoretical results for C ii. The present values from theMCDHF/RCI calculations are given in the Babushkin(length) gauge. The wavenumber Δ 𝐸 and wavelength 𝜆 values are taken from the NISTdatabase. The estimated uncertainties 𝑑𝑇 of the transition rates are given as percentages in parentheses. Transition array Mult. 𝐽 𝑢 − 𝐽 𝑙 Δ 𝐸 𝜆
MCDHF/RCI MCHF-BP ( 𝑎 ) CIV3 ( 𝑏 ) ( cm − ) (Å) 𝑆 (a.u. of a e ) 𝐴 (s − ) 𝑆 (a.u. of a e ) 𝐴 (s − ) 𝐴 (s − ) − D − P o < − S − P o − P − P o − S − P o − D − P o − P o − P − S o − P − D o − P − P o − P − P o − P − P o − D − D o − D < − P o − D − P o − D − P o − S − D o − S − P o − S − P o − P − D o − P − P o − P − P o − P − P o − S − P o − S − D − P o − P o − D MNRAS000
MCDHF/RCI MCHF-BP ( 𝑎 ) CIV3 ( 𝑏 ) ( cm − ) (Å) 𝑆 (a.u. of a e ) 𝐴 (s − ) 𝑆 (a.u. of a e ) 𝐴 (s − ) 𝐴 (s − ) − D − P o < − S − P o − P − P o − S − P o − D − P o − P o − P − S o − P − D o − P − P o − P − P o − P − P o − D − D o − D < − P o − D − P o − D − P o − S − D o − S − P o − S − P o − P − D o − P − P o − P − P o − P − P o − S − P o − S − D − P o − P o − D MNRAS000 , 1–21 (2020) W. Li et al.
Table A8.
Continued.
Transition array Mult. 𝐽 𝑢 − 𝐽 𝑙 Δ 𝐸 𝜆
MCDHF/RCI MCHF-BP ( 𝑎 ) CIV3 ( 𝑏 ) ( cm − ) (Å) 𝑆 (a.u. of a e ) 𝐴 (s − ) 𝑆 (a.u. of a e ) 𝐴 (s − ) 𝐴 (s − )3/2 - 5/2 23427 4268.462 1.15E-04 7.49E+02(32.7%) 5.57E-01 3.58E+06 3.33E+04 ( 𝑎 ) Tachiev & Fischer (2000); ( 𝑏 ) Corrégé & Hibbert (2004).
Table A9.
Comparison of line strengths ( 𝑆 ) and transition rates ( 𝐴 ) with other theoretical results for C iii. The present values from theMCDHF/RCI calculations are given in the Babushkin(length) gauge. The wavenumber Δ 𝐸 and wavelength 𝜆 values are taken from the NISTdatabase. The estimated uncertainties 𝑑𝑇 of the transition rates are given as percentages in parentheses. Transition array Mult. 𝐽 𝑢 − 𝐽 𝑙 Δ 𝐸 𝜆
MCDHF/RCI MCHF-BP ( 𝑎 ) Grasp ( 𝑏 ) ( cm − ) (Å) 𝑆 (a.u. of a e ) 𝐴 (s − ) 𝑆 (a.u. of a e ) 𝐴 (s − ) 𝑆 (a.u. of a e ) 𝐴 (s − ) − P o − S − P o − S < − P o − S − P − P o < − D − P o − S − P o − S − P o − S − P o − D − P o < < < < < < − D − P o − P − P o − D − P o − S − P o < − S − P o − S − P o − D − P o − D − P o < − P o − P − P o − P − P o − D < − P o − D − P o − S − P o − S − P o − S − P o − S < < < − P o − S < − P o − S − D − P o − D − P o < − D − P o < < < < < < − D − P o MNRAS , 1–21 (2020) xtended theoretical transition data in C i – iv ( 𝑎 ) Tachiev & Fischer (1999); ( 𝑏 ) Aggarwal & Keenan (2015).
Table A10.
Comparison of line strengths ( 𝑆 ) and transition rates ( 𝐴 ) with other theoretical results for C iv. The present values from theMCDHF/RCI calculations are given in the Babushkin(length) gauge. The wavenumber Δ 𝐸 and wavelength 𝜆 values are taken from the NISTdatabase. The estimated uncertainties 𝑑𝑇 of the transition rates are given as percentages in parentheses.Transition array Mult. 𝐽 𝑢 − 𝐽 𝑙 Δ 𝐸 𝜆
MCDHF/RCI MCHF-BP ( 𝑎 ) (cm − ) (Å) 𝑆 (a.u. of a e ) 𝐴 (s − ) 𝑆 (a.u. of a e ) 𝐴 (s − )2p − P o − S 1/2 - 1/2 64484 1550.772 9.68E-01 2.63E+08( < < − P o − S 1/2 - 1/2 320050 312.451 1.39E-01 4.63E+09( < < − S − P o < < − D − P o < < < − S − P o < < − P o − S 1/2 - 1/2 17201 5813.582 6.10E+00 3.15E+07( < < − D − P o < < < − S − P o < < ( 𝑎 ) Fischer et al. (1998).
This paper has been typeset from a TEX/L A TEX file prepared by the author. MNRAS000