Breit and QED contributions in atomic structure calculations of tungsten ions
aa r X i v : . [ phy s i c s . a t o m - ph ] N ov Breit and QED contributions in atomic structure calculations of tungsten ions
Karol Kozioł
Narodowe Centrum Bada´n J ˛adrowych (NCBJ), Andrzeja Sołtana 7, 05-400 Otwock- ´Swierk, Poland
Abstract
The F ac , G rasp k , and M cdfgme codes are compared in three case studies of the radiative transitions occurring in tungsten ions:(i) Ni1 and Ni2 lines in Ni-like tungsten, (ii) 3 p / − p / fine splitting in Cl-like tungsten, and (iii) K α and K α lines in W VIII.Various approaches to including the Breit interaction term and QED corrections in atomic calculations are examined. Electroncorrelation e ff ects are also investigated and compared to the Breit and QED contributions. The data presented here may be used toestimate theoretical uncertainties relevant to interpretation of high-resolution spectroscopic data. Keywords:
X-ray spectra; Multiconfiguration Dirac-Fock calculations; Line energies; Breit interaction; QED corrections
1. Introduction
The investigation of tungsten ions is of great importance intheoretical and applied atomic physics. Firstly, high- Z atomssuch as tungsten, are used to probe relativistic and quantum-electrodynamics (QED) e ff ects [1] and have been suggested aspotential candidates for testing the time variation of the finestructure constant [2]. Secondly, tungsten is chosen as a plasmafacing material in modern large tokamaks, such as JET (JointEuropean Torus) and ITER (International Thermonuclear Ex-perimental Reactor), and thus constitutes the majority of theimpurity ions in the tokamak plasma. Therefore, spectroscopicstudies of tungsten ions provide a diagnostic tool relevant toa wide range of electron temperatures [3, 4]. High-precisionatomic calculations are required to interpretation of complexspectra of high-charged tungsten ions and they constitute thesolid base for the farther benchmark modelling of the radiationemission from tokamak plasmas. Several codes are used to pre-dict the atomic structure and transition probabilities of ions thatare of interest in plasma research, such as R elac [5], the Cowancode [6], H ullac [7], G rasp [8] and G rasp k [9, 10], M cdfgme [11, 12], R mbpt [13], and F ac [14]. Recently, two of these,F ac and G rasp k , have become the most widely used codes.When comparing results obtained from F ac and G rasp k cal-culations, it is crucial to know which theoretical contributionsare included in the calculations and what may potentially pro-duce discrepancies. Hence, the aim of this article is to discussthe di ff erences in theoretical contributions taken into accountby the F ac , G rasp k , and M cdfgme codes. The relative rolesof the electron correlation contribution and the Breit and QEDcontributions is also examined.Both the G rasp k and the M cdfgme codes use the Multi-Configuration Dirac-Hartree-Fock (MCDHF) approach. The Email address:
[email protected] (Karol Kozioł) methodology of MCDHF calculations has been presented inmany papers; see, e.g., [15]. The G rasp (General-Purpose Rel-ativistic Atomic Structure Program) code was developed by theGrant group at University of Oxford and recently improved byFroese Fischer, Jönsson, and collaborators in order to performlarge-scale Configuration Interaction (CI) calculations. The M cd - fgme (Multi Configuration Dirac Fock and General Matrix Ele-ment) code was developed by Desclaux and Indelicato in France,and takes into account the Breit and QED corrections in a de-tailed way. The F ac (Flexible Atomic Code), utilising the modi-fied multiconfigurational Dirac-Hartree-Fock-Slater (DHFS) method,was developed by M. F. Gu at Stanford University for speed,multi-utility, and collisional-radiative modelling. The main dif-ference between the DHF and DHFS methods is that DHFSapproximates the non-local DHF exchange potential by a lo-cal potential. Since the DHFS method uses an approximateform of the electron-electron interaction potential, it is com-monly considered to be less accurate than the more sophisti-cated MCDHF method. This assumption is examined in thepresent work. The main aim of the research presented here is toestimate the theoretical uncertainties relevant to the interpreta-tion of high-resolution spectroscopic data.
2. Theoretical background
The methodology of MCDHF calculations performed in thepresent studies is similar to the one published earlier, in severalpapers (see, e.g., [11, 12, 15–18]). The e ff ective Hamiltonianfor an N -electron system is expressed byˆ H = N X i = ˆ h D ( i ) + N X j > i = V i j (1) Preprint submitted to Journal of Quantitative Spectroscopy & Radiative Transfer November 14, 2019 able 1: Various theoretical contributions to the energy of Ni1 and Ni2 transitions in Ni-like tungsten ion (eV). (cid:12)(cid:12)(cid:12) [ Mg ]3 p d E J = ( S ) (cid:12)(cid:12)(cid:12) [ Mg ]3 p / p / d d / E J = ( P ) (cid:12)(cid:12)(cid:12) [ Mg ]3 p / p / d d / E J = ( P )G rasp k G rasp k M cdfgme F ac G rasp k G rasp k M cdfgme F ac G rasp k G rasp k M cdfgme F ac Contribution + Q edmod + Q edmod + Q edmod Dirac-Fock -399290.21 -399287.25 -399266.93 -396923.31 -396920.38 -396900.97 -396898.12 -396895.20 -396875.68Breit( ω = + Rec. 444.25 443.17 444.66 440.02 438.89 440.73 439.30 438.20 440.02Mag. 493.73 488.60 487.82Ret.( ω =
0) -50.66 -49.81 -49.73Recoil 0.10 0.10 0.10Breit( ω >
0) -8.84 -8.80 -8.44 -8.40 -8.45 -8.40VPVP11 -71.10 -70.84 -71.76 -71.12 -70.86 -71.79 -71.12 -70.86 -71.79VP11 +
21 -71.66 -71.39 -71.69 -71.41 -71.69 -71.41VP11 + +
13 -68.80 -68.79 -68.83 -68.81 -68.83 -68.81SEWelt. 374.64 374.60 372.94 374.22 374.10 372.54 374.26 374.12 372.55dens. 380.15 379.73 379.76mod. 374.91 374.46 374.51QED h.o. -1.15 -1.15 -1.15Total(Welt.) -398551.82 -398548.20 -398521.09 -396189.20 -396185.75 -396159.49 -396164.70 -396161.24 -396134.90Total(dens.) -398546.31 -396183.69 -396159.20Total(mod.) -398548.69 -396186.11 -396161.59Ni1 ( P → S ) Ni2 ( P → S )G rasp k G rasp k M cdfgme F ac G rasp k G rasp k M cdfgme F ac Contribution + Q edmod + Q edmod Dirac-Fock 2392.09 2392.05 2391.25 2366.90 2366.87 2365.96Breit( ω = + Rec. -4.95 -4.97 -4.64 -4.23 -4.28 -3.93Mag. -5.91 -5.14Ret.( ω =
0) 0.93 0.85Recoil 0.00 0.00Breit( ω >
0) 0.38 0.39 0.40 0.40VPVP11 -0.02 -0.02 -0.03 -0.02 -0.02 -0.03VP11 +
21 -0.02 -0.02 -0.02 -0.02VP11 + +
13 -0.02 -0.02 -0.02 -0.02SEWelt. -0.38 -0.48 -0.39 -0.41 -0.51 -0.40dens. -0.39 -0.42mod. -0.40 -0.46QED h.o. 0.00 0.00Total(Welt.) 2387.12 2386.96 2386.19 2362.62 2362.45 2361.61Total(dens.) 2387.11 2362.62Total(mod.) 2387.10 2362.58 a b l e : V a r i ou s t h e o r e ti ca l c on t r i bu ti on s t o t h ee n e r gyo f p / − p / fi n e s p litti ng i n C l - li k e t ung s t e n i on ( e V ) . p / p / p / − p / G r a s p k G r a s p k M c d f g m e F a c G r a s p k G r a s p k M c d f g m e F a c G r a s p k G r a s p k M c d f g m e F a c C on t r i bu ti on + Q e d m o d + Q e d m o d + Q e d m o d D i r ac - F o c k - . - . - . - . - . - . . . . B r e it ( ω = ) + R ec . . . . . . . - . - . - . M a g . . . - . R e t . ( ω = )- . - . - . R ec o il . . . B r e it ( ω > )- . - . - . - . - . - . V P V P - . - . - . - . - . - . . . . V P + - . - . - . - . . . V P + + - . - . - . - . . . S E W e lt . . . . . . . . . .
22 d e n s . . . . m od . . . . Q E D h . o . - . - . . T o t a l ( W e lt . )- . - . - . - . - . - . . . . T o t a l ( d e n s . )- . - . . T o t a l ( m od . )- . - . . where ˆ h D ( i ) is the Dirac one-particle operator for i -th electronand the terms ˆ V i j account for the e ff ective electron-electron in-teractions.An atomic state function (ASF) with the total angular mo-mentum J , its z -projection M , and parity p is assumed in theform Ψ s ( JM p ) = X m c m ( s ) Φ ( γ m JM p ) (2)where Φ ( γ m JM p ) are configuration state functions (CSF), c m ( s )are the configuration mixing coe ffi cients for state s , γ m rep-resents all information required to uniquely define a certainCSF. The CSF is a Slater determinant of Dirac 4-componentbispinors: Φ ( γ m JM p ) = X i d i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ (1) · · · ψ ( N ) ... . .. ...ψ N (1) · · · ψ N ( N ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3)where the ψ i is the one-electron wavefunctions and the d i coe ffi -cients are determined by requiring that the CSF is an eigenstateof ˆ J and ˆ J z . The one-electron wavefunction is defined as ψ = r P n ,κ ( r ) · Ω m j κ, j ( θ, φ ) iQ n ,κ ( r ) · Ω m j − κ, j ( θ, φ ) (4)where Ω m j κ, j ( θ, φ ) is a angular 2-component spinor and P n ,κ ( r ) and Q n ,κ ( r ) are large and small radial part of the wavefunction, re-spectively.On the whole, the multiconfiguration DHS method is sim-ilar to the MCDF method, referring to e ff ective Hamiltonianand multiconfigurational ASF. The main di ff erence between theDirac–Hartree–Fock method and the Dirac–Hartree–Fock–Slatermethod is that, in the (Dirac–)Hartree–Fock–Slater approachthe nonlocal (Dirac–)Hartree–Fock exchange potential is ap-proximated by a local potential. The FAC code uses an im-proved form of the local exchange potential (see [14] for de-tails). The electron-electron interaction term is a sum of the Cou-lomb interaction ˆ V Ci j operator and the transverse Breit ˆ V Bi j oper-ator [19–21]: ˆ V i j = ˆ V Ci j + ˆ V Bi j (5)where the Coulomb interaction operator is ˆ V Ci j = / r i j , and theBreit operator in the Coulomb gauge isˆ V Bi j = − α i · α j e i ω ij r ij r i j − ( α i · ∇ i )( α j · ∇ j ) e i ω ij r ij − ω i j r i j (6)where ω i j = ( ε i − ε j ) / c is the frequency of one virtual photonexchanged and ε i and ε j are orbital energies of interacting elec-trons.3he unretarded (instantaneous) parts are obtained making ω i j →
0. Then the Breit terms are given asˆ V Bi j = − α i · α j r i j − ( α i · r i j )( α j · r i j )2 r i j = − α i · α j r i j | {z } V mag + α i · α j r i j − ( α i · r i j )( α j · r i j )2 r i j | {z } V ret (7)where V mag is called magnetic (Gaunt) [22] part and V ret iscalled retardation part.The zero-frequency approximation to the full transverse Breitinteraction, i.e. Eq. (7), is well suited for most computationsof many-electron atomic systems since the explicit frequency-dependent form, because of remedying the lack of covariance ofDirac-Coulomb-Breit Hamiltonian and the di ff erences of stateenergy by using frequency-independent and frequency-dependentBreit operator are usually small [11, 23, 24]. The Breit interac-tion can be included in two general ways: in the self-consistentfield process, such as in M cdfgme code [11, 25–27], or in per-turbational approach, such as in G rasp / G rasp k codes [8, 9]. The bound-state vacuum polarization (VP) contribution isrelated to the creation and annihilation of virtual electron-positronpairs in the field of the nucleus. It is a correction to the photonpropagator. The first term of order α ( Z α ) can be calculated asthe expectation value of the Uehling potential. The Uehling po-tential in the case of finite nuclear size and spherical symmetricnuclear charge distribution ρ ( ~ r ) can be expressed as [28]: U ( ~ r ) = − Z α ~ mr Z ∞ d r ′ r ′ ρ ( r ′ ) × " K mc ~ | r − r ′ | ! − K mc ~ | r + r ′ | ! (8)where the function K ( x ) is defined as: K ( x ) = Z ∞ dt e − xt t + t ! √ t − α ( Z α ) and by Wichmann and Kroll [30, 31] fororder α ( Z α ) .Self-energy (SE) contribution arises from the interaction ofthe electron with its own radiation field. It is a correction to theelectron propagator. For one-electron systems the most impor-tant (one-loop) self-energy term has been calculated exactly byMohr [32–34] and expressed as: ∆ E n κ = απ ( Z α ) n F n κ ( Z α ) m e c (10)where F n κ ( Z α ) is a slowly varying function of Z α . For many-electron atomic systems the self-energy correction to the energyis changed by the electron screening. There are three generalways to estimate self-energy screening for atoms. The major di ff erences between these approaches are for results of SE cor-rection to the energy of s subshells.In the ’Welton picture’ approach [12, 35, 36] the self-energycorrection for s -type Dirac-Fock orbitals is scaled from exacthydrogenic results from the following relation:( ∆ E ns ) DF = h ns |∇ V nucl ( r ) | ns i DF h ns |∇ V nucl ( r ) | ns i Hyd ( ∆ E ns ) Hyd (11)where V nucl ( r ) is a nuclear potential. This approach is imple-mented in M cdfgme code. Lowe et al. [37] created extension ofG rasp k package, that implements Welton picture approach toestimate SE screening into G rasp k suite. The G rasp k codenatively approximates the screening coe ffi cient by taking theratio of the Dirac-Fock wavefunction density in a small regionaround the nucleus ( r ≤ r ′ , r ′ = . a , a – Bohr’s radius)to the equivalent density for a hydrogenic orbital, i.e. [37]( ∆ E nl ) DF = h nl r ≤ r ′ | nl r ≤ r ′ i DF h nl r ≤ r ′ | nl r ≤ r ′ i Hyd ( ∆ E nl ) Hyd (12)This approach is called ’density approach’ further in the manuscript.Last years some modern approaches for the estimation of hy-drogenic SE data to many-electron atoms have been presented,such as the model Lamb-shift operator [38–40] and the spectralrepresentation (projection operator) of the Lamb shift [41]. Re-cently Shabaev et al. [40, 42] published Q edmod , a program forcalculating the model Lamb-shift operator basing on numericalradial wavefunctions. In this paper the G rasp k wavefunctionsare used as a Q edmod input.
3. Results and discussion
In present work the F ac , G rasp k , and M cdfgme codes arecompared in three case studies of radiative transitions occurringin tungsten ions. The first case study is focused on radiativetransitions among outer orbitals in highly ionised tungsten. Thestudy of characteristic x-ray radiation emitted by highly ionizedW atoms is of great importance for both theoretical and appliedatomic physics, including fusion applications [1, 4, 43]. In therecent works of Rzadkiewicz et al. [44] and Kozioł and Rzad-kiewicz [45], the energy levels of the ground and excited statesof W + and W + ions and the wavelengths and transition prob-abilities of the 4 d → p transitions were calculated using theMCDHF Configuration Interaction (CI) method, but no deeperanalysis of the Breit and QED contributions to the energy levelswas performed. Hence, the Ni1 and Ni2 transitions in Ni-like(W + ) tungsten were selected as the first case study. The ini-tial levels of these transitions are (cid:12)(cid:12)(cid:12)(cid:12) [ Mg ]3 p / p / d d / E J = ( P ) and (cid:12)(cid:12)(cid:12)(cid:12) [ Mg ]3 p / p / d d / E J = ( P ) and the final levelis (cid:12)(cid:12)(cid:12) [ Mg ]3 p d E J = ( S ). The Ni1 and Ni2 transitions are P → S and P → S , respectively.The second case study concerns the 3 p / − p / fine split-ting in Cl-like (W + ) tungsten. Recently, a Cl-like isoelec-tronic sequence was proposed as an electronic configuration4 able 3: Various theoretical contributions to the energy of 1 s − , 2 p − / , and 2 p − / hole states of W + (eV) and energy of K α , transitions. s − p − / p − / G rasp k G rasp k M cdfgme F ac G rasp k G rasp k M cdfgme F ac G rasp k G rasp k M cdfgme F ac Contribution + Q edmod + Q edmod + Q edmod Dirac-Fock -369478.91 -369476.90 -369453.39 -427763.39 -427760.56 -427729.68 -429115.93 -429112.94 -429082.02Breit( ω = + Rec. 234.57 234.29 234.08 425.12 424.64 425.46 440.54 439.40 441.22Mag. 270.66 474.72 489.49Ret.( ω =
0) -36.43 -50.18 -50.20Recoil 0.06 0.10 0.10Breit( ω >
0) -4.95 -5.00 -9.05 -9.08 -7.24 -7.26VPVP11 -43.31 -42.27 -42.59 -72.34 -71.11 -71.99 -72.69 -71.42 -72.36VP11 +
21 -42.61 -42.56 -71.91 -71.66 -72.26 -71.98VP11 + +
13 -40.90 -41.01 -69.05 -69.06 -69.38 -69.35SEWelt. 232.51 231.23 228.50 379.95 377.50 375.58 378.85 376.07 374.49dens. 232.56 382.62 381.52mod. 230.35 377.68 376.39QED h.o. -1.05 -1.26 -1.25Total(Welt.) -369059.38 -369058.44 -369033.40 -427039.28 -427037.80 -427000.63 -428376.04 -428375.32 -428338.66Total(dens.) -369059.33 -427036.62 -428373.37Total(mod.) -369059.84 -427038.69 -428375.63 K α K α G rasp k G rasp k M cdfgme F ac G rasp k G rasp k M cdfgme F ac Contribution + Q edmod + Q edmod Dirac-Fock 59637.02 59636.04 59628.63 58284.48 58283.65 58276.29Breit( ω = + Rec. -205.97 -205.11 -207.14 -190.54 -190.35 -191.38Mag. -218.84 -204.07Ret.( ω =
0) 13.77 13.76Recoil -0.04 -0.04Breit( ω >
0) 2.29 2.26 4.10 4.08VPVP11 29.42 29.19 29.77 29.07 28.88 29.40VP11 +
21 29.66 29.42 29.30 29.11VP11 + +
13 28.48 28.35 28.14 28.05SEWelt. -146.34 -144.85 -145.99 -147.44 -146.27 -147.08dens. -148.96 -150.06mod. -146.04 -147.33QED h.o. 0.20 0.21Total(Welt.) 59316.66 59316.88 59305.26 57979.90 57979.36 57967.23Total(dens.) 59314.04 57977.28Total(mod.) 59315.79 57978.85 hat could be used to accurately test current methods of com-puting the Breit and QED e ff ects [46]. The 3 p / − p / finesplitting in Cl-like tungsten has recently been investigated bothexperimentally [47] and theoretically [48–50].The third case study is focused on core radiative transitionsin stripped tungsten. The energy shifts of the K α , , K β , and K β lines of stripped high- Z atoms have been suggested as be-ing potentially relevant to diagnostics of high-energy-densitylaser-produced plasmas [51]. Hence, the K α (1 s − → p − / )and K α (1 s − → p − / ) transitions in W + were selected asthe third case study. The W + ion was selected because it hassimple closed-shell valence electronic configurations and, as aresult, there are only two transitions between initial and finalstates to be studied. The electronic configurations for the 1 s − ,2 p − / , and 2 p − / hole states are then 1 s / s p M N s p ,1 s s p / p / M N s p , and 1 s s p / p / M N s p ,respectively.Table 1 presents various theoretical contributions to the en-ergies of the [ Mg ]3 p d d ( J =
1) and [ Mg ]3 p d ( J = p / − p / fine splitting in a Cl-like tungsten ion. Table 3 presents various theoretical contri-butions to the energies of 1 s − , 2 p − / , and 2 p − / hole states ofW + and the K α and K α transitions. Relativistic and radia-tive e ff ects are treated slightly di ff erently in the di ff erent codes.The term ’Dirac-Fock’ in the tables indicates that the energyof the state was obtained using the self-consistent multicon-figurational Dirac-Hartree-Fock (G rasp k and M cdfgme codes)or the Dirac-Hartree-Fock-Slater (F ac code) procedure, withouthigher order corrections. ’Breit( ω = + Rec.’ refers to theBreit correction in the zero-frequency approximation (see, e.g.,[52] and references therein for details) plus the recoil correction.Using the M cdfgme code, it is also possible to print the partic-ular contributions included in this correction; they are: ’Mag.’– magnetic (Gaunt) part of the Breit interaction, ’Ret.( ω = ω > ac calcula-tions, which is common in many atomic computational codes.Each of studied codes has a di ff erent treatment of VP correction.The M cdfgme code takes into account three VP potentials: V11(Uehling potential), V21 (Källén and Sabry potential), and V13(Wichmann and Kroll potential). In the F ac code only V11 isincluded while the G rasp k code takes into account the sum ofV11 and V21. The di ff erent methods for including the SE cor-rection in many-electron atomic systems that have been usedare the density approach (’dens.’ in Tables), the Welton pictureapproach (’Welt.’ in Tables), and the model Lamb-shift opera-tor (’mod.’ in Tables).As seen from Tables 1, 2, and 3, the absolute level energiescalculated by the F ac code di ff er significantly from the energiescalculated by the G rasp k and M cdfgme codes, by about 20–30 eV. However, the energies of the radiative transitions di ff er much less, about 1 eV in the case of the Ni1 and Ni2 lines andbelow 1 eV for the 3 p / − p / fine splitting in W + . For the K α and K α transitions in W + , the F ac calculated numbersare smaller by about 10 eV than those obtained by the MCDHFcodes. This di ff erence is too large to estimate properly the outer-shell ionization level from K -shell x-ray lines shift [51]. It canbe concluded that the F ac code is su ffi ciently accurate in caseswhere radiative transitions are linked to an electron jump withinthe valence shells (if high accuracy is not required), but is lessaccurate for transitions that are linked to inner-shell hole states.Comparing the Breit contributions obtained from the G rasp k code (where the Breit term is treated perturbatively) and fromthe M cdfgme code (where Breit term is included in a variationalSCF process) allows the so-called ”variational e ff ect” to be es-timated. The magnitude of this e ff ect is about 1 eV (0.1–0.3%)in the cases studied here. However, it has been found that thevariational e ff ect is significantly reduced when active space isexpanding [53, 54]. The frequency-dependent Breit term isabout 2% of the frequency-independent one (having the oppo-site sign).As mentioned above, three di ff erent approximations to es-timate the SE corrections have been used: the Welton picture,the density approach, and the model Lamb-shift operator. ForG rasp k calculations, it is possible to compare these models byusing these same wavefunctions. In the case of W + , the ’den-sity’ approach gives SE contributions to the energy levels thatare significantly larger, by about 5 eV, than those of the othertwo approaches. However, this di ff erence vanishes for the Ni1and Ni2 transition energies. The case of the W + ion is similar.For the 1 s − , 2 p − / , and 2 p − / hole states of W + the ’modeloperator’ method gives SE contributions to the energy levelsthat are significantly smaller than those of the ’Welton picture’,which in turn are smaller than those of the ’density’ approach. It is interesting to compare the electron correlation contribu-tions to the Breit and QED contributions in selected cases. Forthe Ni1 and Ni2 lines, the correlation contribution was studiedextensively by Rzadkiewicz et al. [44] and Kozioł and Rzad-kiewicz [45] using a MCDHF Configuration Interaction (CI)calculation. They pointed out that electron correlation e ff ectranges from -1.87 eV to -2.87 eV for the Ni1 line energy andfrom -1.05 eV to -2.45 eV for the Ni2 line energy, dependingon the CI model used. The correlation e ff ect is then larger byan order of magnitude than the frequency-dependent Breit term(omitted in the calculations in [44, 45] due to the inclusion ofvirtual orbitals within the CI procedure) and larger by more thanan order of magnitude than the di ff erences arising from the useof di ff erent QED models.In the case of the 3 p / − p / fine splitting in W + theMCDHF-CI calculations were performed with the G rasp k code(see e.g. [44, 45] for details). The 1 s , 2 s , and 2 p subshells areinactive orbitals. All single (S) and double (D) substitutionsfrom the 3 s and 3 p orbitals to the active spaces (AS) of vir-tual orbitals are allowed. The virtual orbital sets used were:AS1 = {3d,4s,4p,4d,4f}, AS2 = AS1 + {5s,5p,5d,5f,5g}, AS36 able 4: The 3 p / − p / fine splitting calculated for various CI active spaces. Active space Energy (eV) Wavelength (Å)AS0 349.11 35.515AS1 347.26 35.703AS2 347.84 35.644AS3 347.62 35.666AS4 347.50 35.678AS5 347.49(1) 35.680(2)Experiment:Lennartsson et al. [47] 35.668(4)Other theory:Quinet [48] 35.633Singh and Puri [50] 35.632Aggarwal and Keenan [49] 35.686Bilal et al. [46] 35.765 = AS2 + {6s,6p,6d,6f,6g}, AS4 = AS3 + {7s,7p,7d,7f,7g}, andAS5 = AS4 + {8s,8p,8d,8f,8g}. Table 4 collects the resultsof the 3 p / − p / fine splitting calculated for various CI ac-tive spaces. The AS0 value is a number related to G rasp k calculations with the ’Welton picture’ approach for estimatingthe SE contribution. For the final active space, AS5, the theo-retical uncertainties are presented. They are related to conver-gence with the size of a basis set and estimated as an absolutevalue of di ff erence between energy(wavelengths) calculated forAS4 and AS5 stages. The wavelengths for the AS3-AS5 ap-proaches agree well with the experimental values from the workof Lennartsson et al. [47]. The correlation e ff ect is about1.62 eV, which is about three times larger than the frequency-dependent Breit term and the total QED e ff ect. Table 5: The K α and K α transitions energy in W + calculated for various CIactive spaces. Theoretical uncertainties for ’AS3 + Auger shift’ are related toconvergence with the size of a basis set and to error of interpolation Auger shiftcorrections.
Active space Energy (eV) K α K α AS0 59316.66 57979.90AS1 59315.82 57979.06AS2 59316.43 57979.60AS3 59316.63 57979.80AS3 + Auger shift 59318.77(30) 57983.13(30)Experiment (neutral W):Bearden [55] 59318.24 57981.7Deslattes et al. [56] 59318.847(50) 57981.77(14)Other theory (neutral W):Deslattes et al. [56] 59318.8(17) 57981.9(19)For the K α and K α transition energies in W + , MCDHF-CI calculations were performed to check correlation e ff ects. Theactive space of occupied orbitals contains orbitals involved inradiative transitions (1 s and 2 p ) and two the most outer sub-shells: 4 f and 5 p . All other occupied subshells are inactive core. All SD substitutions from the active space of occupiedorbitals to the active spaces of virtual orbitals are allowed. Thevirtual orbital active spaces used were: AS1 = {5d,5f,5g}, AS2 = AS1 + {6s,6p,6d,6f,6g}, and AS3 = AS2 + {7s,7p,7d,7f,7g}.The results of the K α and K α transition energies in W + , cal-culated with various CI active spaces, are shown in Table 5. Itis evident that the correlation e ff ects are small in this case. Sim-ilar order of magnitude for correlation e ff ects were found forthe K α , lines of Al and Si [57] and of Kr, Pb, U, Pu, Fm[58, 59]. Because inner-hole states are autoionising, the levelenergy shift due to the coupling between the two hole statesand one excited electron, called Auger shift, must be included[56]. The Auger shift contribution to the K α and K α transi-tion energies can be approximated through interpolation fromthe numbers given by Indelicato, Lindroth et al. [58–60]. Theinterpolated values for the Auger shift contribution are 2.14 eVand 3.33 eV for the K α and K α lines, respectively. Addingthe Auger shift contributions to the MCDHF-CI values reducesthe substantially discrepancy between theory and experimentfor K α line, however overestimate value for K α line.
4. Conclusions
The interpretation of atomic observations by theory and thetesting of computational predictions by experiment are inter-active processes. In this paper the F ac , G rasp k , and M cd - fgme codes have been compared in selected case studies in-volving radiative transitions occurring in the tungsten ions W + ,W + , and W + . Transitions involving electron jumps betweenouter or inner orbitals are both considered. Various approachesto including the Breit interaction term and QED correctionsin atomic calculations have been examined and their contribu-tions compared to those of electron correlations. In the casewhen transitions involve electron jumps between outer shells(the first and second cases in the present work) the frequency-dependent Breit contribution to transition energy is smaller fewtimes than the electron correlation contribution. Then, ommit-ing the frequency-dependent Breit term is not a big mistake.In the case when transitions involve electron jumps betweeninner and outer shells (the third case in the present work) thefrequency-dependent Breit contribution dominates over the elec-tron correlation contribution. In this case also the di ff erencesbetween QED models may be bigger than correlation contribu-tion. The presented data may be used to estimate theoreticaluncertainties relevant to interpretation of high-resolution spec-troscopic data. Acknowledgments
The work was partly supported by the Polish Ministry ofScience and Higher Education within the framework of the sci-entific financial resources in the years 2016–2019 allocated forthe realization of the international co-financed project. Thiswork has been carried out within the framework of the EURO-fusion Consortium and has received funding from the Euratom7esearch and Training Programme 2014–2019 under Grant Agree-ment No. 633053. The views and opinions expressed herein donot necessarily reflect those of the European Commission.
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