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Open AccessFeature PaperReview

Multiconfiguration Dirac-Hartree-Fock Calculations with Spectroscopic Accuracy: Applications to Astrophysics

1
Materials Science and Applied Mathematics, Malmö University, SE-205 06 Malmö, Sweden
2
Institute of Theoretical Physics and Astronomy, Vilnius University, Saulėtekio av. 3, LT-10222 Vilnius, Lithuania
3
Chimie Quantique et Photophysique, Université libre de Bruxelles, B-1050 Brussels, Belgium
4
Department of Computer Science, University of British Columbia, Vancouver, BC V6T 1Z4, Canada
5
Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, UK
6
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
7
Division of Mathematical Physics, Department of Physics, Lund University, 221-00 Lund, Sweden
*
Author to whom correspondence should be addressed.
Academic Editor: Joseph Reader
Atoms 2017, 5(2), 16; https://doi.org/10.3390/atoms5020016
Received: 31 January 2017 / Revised: 5 April 2017 / Accepted: 7 April 2017 / Published: 14 April 2017
(This article belongs to the Special Issue Spectra of Ionized Atoms: From Laboratory to Space)

Abstract

Atomic data, such as wavelengths, spectroscopic labels, broadening parameters and transition rates, are necessary for many applications, especially in plasma diagnostics, and for interpreting the spectra of distant astrophysical objects. The experiment with its limited resources is unlikely to ever be able to provide a complete dataset on any atomic system. Instead, the bulk of the data must be calculated. Based on fundamental principles and well-justified approximations, theoretical atomic physics derives and implements algorithms and computational procedures that yield the desired data. We review progress and recent developments in fully-relativistic multiconfiguration Dirac–Hartree–Fock methods and show how large-scale calculations can give transition energies of spectroscopic accuracy, i.e., with an accuracy comparable to the one obtained from observations, as well as transition rates with estimated uncertainties of a few percent for a broad range of ions. Finally, we discuss further developments and challenges.
Keywords: transition energies; lifetimes; transition rates; multiconfiguration Dirac-Hartree-Fock; PACS; 31.15.am; 32.30.Jc; 32.70.Cs transition energies; lifetimes; transition rates; multiconfiguration Dirac-Hartree-Fock; PACS; 31.15.am; 32.30.Jc; 32.70.Cs

1. Introduction

Atomic data, such as wavelengths, spectroscopic labels, broadening parameters, excitation and transition rates, are necessary for many applications, especially in plasma diagnostics, and for interpreting laboratory and astrophysical spectra [1,2]. Plasma diagnostics are commonly applied to measure the physical state of the plasma, e.g., temperatures, densities, ion and chemical abundances. Atomic databases, such as CHIANTI [3,4], are widely used for such diagnostic purposes. Their accuracy relies on a range of atomic rates, the main ones being electron collision rates and transition rates. For the solar corona, lines from highly charged iron ions, emitted in the extreme ultraviolet (EUV) and soft X-ray region, are commonly used for diagnostics, together with those from all other abundant elements. Atomic data and line identifications involving states of the lowest configurations of an ion are now relatively well known and observed. However, much less data are available for lines from higher configurations; one example is the lack of line identifications and rates for transitions from n = 4 iron ions in the soft X-rays [5].
Line identification from observed spectra is a very difficult and challenging task. Different methods such as isoelectronic interpolation and extrapolation, perfected by Edlén [6], can be used, but the work is nowadays mostly done with the aid of calculated transition energies and simulated spectra. For calculated transition energies, or wavelengths, to be of practical use, they need to be very accurate with uncertainties of just a few mÅ, placing high demands on computational methodologies.
Transition rates and line ratios are needed for diagnostic purposes. Due to the almost complete lack of accurate experimental data for atoms a few times ionized or more, the bulk of the transition rates must be calculated. Not only the rates themselves should be provided, but also uncertainty estimates that can be propagated in plasma models for sensitivity analysis. Both accurate rates and uncertainty estimates pose a challenge, calling for methods for which computed properties can be monitored as the wave functions are systematically improved.
This review summarizes the results from recent accurate relativistic multiconfiguration calculations for lowly charged ions or more of astrophysical importance. Focus is on the transition energies and their uncertainties, but transition rates and the associated uncertainty estimates are also discussed. The astrophysical background is provided in the individual papers covered by the review. Neutral atoms and ions in the lowest charge states are not covered in the review.

2. Multiconfiguration Methods

Multiconfiguration methods are versatile and can, in principle, be applied to any atomic or ionic system [7]. Multiconfiguration methods generate approximate energies and wave functions for the each of the targeted states in a system. The wave functions can then be used to compute measurable quantities, such as transition rates, hyperfine structures or Landé g-factors [8]. Looking at strengths and weaknesses, multiconfiguration methods capture near degeneracies and valence-valence electron correlation very efficiently. They are however less good at accounting for core-core correlation, and here, perturbative methods relying on a complete orbital basis have advantages. Work has been done to combine multiconfiguration and perturbative methods in different ways [9,10,11,12], a development that will open up accurate results also for more complex systems [13].
The relativistic multiconfiguration method, to be described below, is implemented in the GRASP2K program package [14]. The package is generally available and utilizes a message passing interface (MPI) for the most time-consuming programs, allowing for large-scale computing on parallel computers.

2.1. Multiconfiguration Dirac-Hartree-Fock

Atomic calculations are based on a Hamiltonian. In the relativistic multiconfiguration Dirac- Hartree-Fock (RMCDHF) method [7,15], as implemented in the GRASP2K package, the Hamiltonian is taken as the Dirac-Coulomb Hamiltonian:
H DC = i = 1 N c α i · p i + ( β i 1 ) c 2 + V nuc ( r i ) + i > j N 1 r i j ,
where V nuc ( r i ) is the nuclear potential modelled from an extended nuclear charge distribution, r i j is the distance between electrons i and j and α and β are the Dirac matrices. Wave functions Ψ ( γ P J M J ) for fine-structure states labelled by parity, P, and angular quantum numbers, J M J , are expanded in antisymmetrized and coupled configuration state functions (CSFs):
Ψ ( γ P J M J ) = j = 1 N CSF c j Φ ( γ j P J M J ) .
The labels { γ j } denote the information of the CSFs, such as orbital occupancy and subshell quantum numbers in the angular momentum coupling tree. The CSFs are built from products of one-electron orbitals, having the general form:
ψ n κ , m ( r ) = 1 r P n κ ( r ) χ κ , m ( θ , φ ) ı Q n κ ( r ) χ κ , m ( θ , φ ) ,
where χ ± κ , m ( θ , φ ) are two-component spin-orbit functions and where the radial functions are numerically represented on a logarithmic grid. The selection of the CSFs depends on the atomic system at hand and is described in Section 3.
In applications, one often seeks to determine energies and wave functions for a number, sometimes up to a few hundred, of targeted states. This is most conveniently done in the extended optimal level (EOL) scheme [16]. Given initial estimates of the radial functions, the energies E and expansion coefficients c = ( c 1 , , c M ) t for the targeted states are obtained as solutions to the relativistic configuration interaction (RCI) problem:
H c = E c ,
where H is the RCI matrix of dimension M × M with elements:
H i j = Φ ( γ i P J M J ) | H DC | Φ ( γ j P J M J ) .
Once the expansion coefficients have been determined, the radial functions { P n κ ( r ) , Q n κ ( r ) } are improved by solving a set of differential equations that results from applying the variational principle on a weighted energy functional of the targeted states together with additional terms needed to preserve the orthonormality of the orbitals. Appropriate boundary conditions for the radial orbitals exclude undesired negative-energy solutions [15]. The RCI problem and the solution of the differential equations are iterated until the radial orbitals and the energy are converged to a specified tolerance.

2.2. Configuration Interaction

The RMCDHF calculations are used to generate an orbital basis. Given this basis, the final wave functions for the targeted states are obtained in RCI calculations based on the frequency dependent Dirac-Coulomb-Breit Hamiltonian:
H DCB = H DC i < j N α i · α j cos ( ω i j r i j / c ) r i j + ( α i · ) ( α j · ) cos ( ω i j r i j / c ) 1 ω i j 2 r i j / c 2 ,
where is the gradient operator involving differentiation with respect to r i j = r i r j and r i j = | r i j | [17]. In the RCI calculations leading quantum electrodynamic (QED) effects, vacuum polarization and self-energy are also taken into account. RCI calculations require less computational effort than do RMCDHF calculations, and currently, expansions with millions of CSFs can be handled. The relativistic multiconfiguration and configuration interaction calculations go together and are referred to as RMCDHF/RCI calculations.

2.3. Managing Large Expansions

To manage large expansions, CSFs can a priori be divided into two groups, referred to as a zero- and first-order partitioning. The first group, P, with m elements ( m M ) contains CSFs that account for the major parts of the wave functions. The second group, Q, with M m elements contains CSFs that represent minor corrections. Allowing interaction between CSFs in group P, interaction between CSFs in groups P and Q and diagonal interactions between CSFs in Q gives a matrix:
H ( P P ) H ( P Q ) H ( Q P ) H ( Q Q ) ,
where H i j ( Q Q ) = δ i j E i Q . The restriction of H ( Q Q ) to diagonal elements results in a huge reduction in the total number of matrix elements and the corresponding time for RCI calculations [12]. A similar reduction in computational time is obtained when constructing and solving the differential equations obtained from the weighted energy functional. Different computational strategies apply: RMCDHF calculations with limited interactions followed by RCI calculations with full interactions or RMCDHF calculations with limited interactions followed by RCI calculations with limited interaction, possibly with more CSFs in group P.

2.4. Labelling

In fully-relativistic calculations, quantum labels for the targeted states are obtained in j j -coupling. Most often, this wave function representation is not pure, i.e., there is no dominant CSF whose quantum numbers can be used to label a state in a proper way. Using the methods developed by Gaigalas and co-workers [18], the wave function representation in j j -coupling is transformed to an approximate representation in L S J -coupling. This representation is normally more pure and better suited for labelling. One should be aware of the fact that even in L S J -coupling, the labelling is not straight forward, and several components in the L S J -coupling representation must be used in a recursive way to find unique labels [19,20]. Programs for transforming wave functions and assigning unique labels are important parts of the GRASP2K package [21].

2.5. Transition Properties

Given wave functions from RMCDHF/RCI calculations, transition properties, such as rates, A, line strengths, S, and weighted oscillator strengths, g f , between two states γ P J and γ P J are computed in terms of reduced matrix elements:
Ψ ( γ P J ) T ( EMK ) Ψ ( γ P J ) ,
where the operator T ( EMK ) depends on the multipolarity, E1, M1, E2, M2, etc., of the transition. By including Bessel functions in the definition of the operator, GRASP2K accounts for more high-order effects than the usual transition operator used in non-relativistic calculations with Breit–Pauli corrections [15]. Inserting the CSF expansions for the wave functions, the reduced matrix element reduces to a sum over reduced matrix elements between CSFs. Using Racah algebra techniques, these matrix elements are finally obtained as sums over radial integrals [22,23]. The above procedure assumes that the two states γ P J and γ P J are built from the same set of orbitals. When this is not the case, e.g., when separate calculations have been done for the even and odd parity states, the representation of the wave functions are changed in such a way that the orbitals become biorthonormal [24,25], in which case the calculation continues along the lines above. For electric transitions, parameters can be computed in both length and velocity gauge [26], where the results in the length gauge are the preferred.

3. General Computational Methodology: The SD-MR Approach

Systematic calculations using multiconfiguration methods follow a determined scheme as described below. Details of the scheme are determined by the shell structure of the atom, the number of targeted states, the desired accuracy of the final results and the available computational resources. The atomic Hamiltonian is invariant with respect to space inversions, and there are no interactions between odd and even parity states. The odd and even parity states are thus often treated in separate sets of calculations. After validation for selected ions and states, computed transition energies and rates can be used to aid the analysis of unknown spectra.

3.1. Multireference and Gross Features of the Wave Functions

For highly ionized systems, a natural starting point is the multireference set (MR). In this review, we define the MR as the set of configurations associated with the targeted states of a given parity together with important closely degenerate configurations. Applying rules for the coupling of angular momenta, the configurations in the MR give rise to a set of CSFs that account for the most important gross features of the wave functions. The expansion coefficients of the CSFs and the orbitals are determined in an initial RMCDHF calculation. The orbitals for the initial calculation are called spectroscopic orbitals. They are required to have the same node structure as hydrogenic orbitals, i.e., the node structure is determined by the principal quantum number. The spectroscopic orbitals are kept frozen in all subsequent calculations.

3.2. Including Electron Correlation and Determining an Orbital Set

The initial approximation of the wave functions is improved by adding CSFs that account for electron correlation. Guided by a perturbative analysis, the CSFs are generated by the single (S) and double (D) multireference (SD-MR) active space method in which a number of configurations is obtained by SD substitutions of orbitals in the configurations of the MR with orbitals in an active set [7,8]. Again, applying rules for the coupling of angular momenta, the generated configurations give rise to the CSFs. Not all of these CSFs are important, and the CSFs are further required to be such that they interact (have non-zero Hamiltonian matrix elements) with the CSFs of the MR. The expansion coefficients of the CSFs and the radial parts of the orbitals in the active set are determined in RMCDHF calculations where, for large expansions, limited interactions are used.
The active set, often denoted by the number of orbitals with a specified symmetry, so that { 4 s 3 p 2 d 1 f } is a set with four s orbitals, three p orbitals, two d orbitals and one f orbital, is systematically enlarged one orbital layer at the time until the computed excitation energies and transition rates have converged to within some predetermined tolerance. For small systems, SD substitutions are done from all subshells of the configurations in the MR, and the generated CSFs account for valence-valence, core-valence and core-core electron correlation. For larger systems, it becomes necessary to define a core for which restrictions on the substitutions apply. In many cases, the SD-MR substitutions are restricted in such a way that there are only S substitutions from subshells that define a so-called active core. There may also be subshells deep down in the core for which there are no substitutions at all. CSFs obtained from S-MR substitutions from the active core together with SD-MR substitutions from the valence subshells account for valence-valence and core-valence correlation.

3.3. Final Configuration Interaction Calculations Including the Breit Interaction and QED Effects

The frequency dependent Breit (transverse photon) interaction and leading QED effects are included in final RCI calculations. To account for higher order correlation effects, the MR is sometimes enlarged at this final step leading to larger expansions. Full interaction is normally used, although limited interactions have been shown effective for including core-valence and core-core effects in larger systems [12,27].

4. Excitation Energies

In this section, RMCDHF/RCI excitation energies are compared with observations for a range of systems in order to illustrate the predictive power of highly accurate calculations. Generally, there are enough observations to validate computational methodologies and to distinguish between different approaches.

4.1. Energies for 2 s 2 2 p n , 2 s 2 p n + 1 and 2 p n + 2 States in the B-, C-, N-, O- and F-Like Sequences

Excitation energies and E1, M1, E2, M2 transition rates between 2 s 2 2 p n , 2 s 2 p n + 1 and 2 p n + 2 states of ions in the B-, C-, N-, O- and F-like sequences were calculated using the RMCDHF/RCI and SD-MR method [28,29,30,31,32]. The range of ions, as well as the details of the calculations are summarized in Table 1. Calculations of Landé g J factors, hyperfine structures and isotope shifts were done separately for ions in the Be-, B-, C- and N-like sequences [33,34].
A trend for all atomic structure calculations, including RMCDHF/RCI, is that the accuracy of the excitation energies is, relatively speaking, lower for lowly charged ions and that the accuracy then increases as the effects of electron correlation diminish. For the highly charged ions, the situation is less clear. Often experimental excitation energies are associated with large uncertainties or missing altogether. The situation is illustrated in Table 2 and Table 3 for the O-like sequence [31].
In Table 2, excitation energies in Ne III and Fe XIX from different calculations are compared with energies from observations. The most accurate calculations are the RMCDHF/RCI calculation [31] and the multireference second-order Möller-Plesset calculation (MRMP). For Ne III, the relative differences with observation for these two calculations are in the range of 0.2–0.4% (slightly worse for MRMP). For Fe XIX the relative errors go down by an order of magnitude, and now, the calculated energies are accurate enough to detect misidentifications or errors in observational data, but also to serve as a valuable tool for identifying new lines. The usefulness of computed energies is illustrated in Table 3 for Br XXVIII, where the RMCDHF/RCI and MRMP calculations clearly discriminate between observed energies [35] and energies from semiempirical fits [36], being in better agreement with the latter. This suggests that there may be some calibration problems in relation to the observed energies [35].
Summarizing the mean relative errors in the excitation energies for the 2 s 2 2 p n , 2 s 2 p n + 1 and 2 p n + 2 states of B-, C-, N-, O- and F-like Fe from RCI calculations [28,29,30,31,32], we have 0.022% for B-like, 0.022% for C-like, 0.050% for N-like, 0.042% for O-like and 0.011% for F-like Fe.

4.2. Energies of the 2 s 2 2 p 6 and 2 s 2 2 p 5 3 l States in the Ne-Like Sequence

The transitions connecting the 2 s 2 2 p 5 3 l , l = 0 , 1 , 2 configurations in Ne-like ions give rise to prominent lines in the spectra of many high temperature light sources. Some of these lines are considered for diagnostics of fusion plasmas. Excitation energies and E1, M1, E2, M2 transition rates between states of the above configurations in Ne-like Mg III and Kr XXVII sequences were calculated using the RMCDHF/RCI and SD-MR method [43]. The calculations were done based on expansions from SD substitutions from the 2 s 2 2 p 6 and 2 s 2 2 p 5 3 l configurations to active sets { 7 s 6 p 5 d 4 f 3 g 2 h 1 i } . The 1 s 2 was kept as a closed core. Some triple substitutions were allowed to capture higher order electron correlation effects. In Table 4, the RMCDHF/RCI excitation energies are displayed for Ca XI and Fe XVII. In the same table, the energies are compared with energies from NIST, as well as from MRMP calculations by Ishikawa et al. [44]. Again, the table illustrates the situation when it comes to experiments. For many ions, the excitation energies of the lower states are known from experiments. For other ions, such as Ca XI, energies are only known for a few states. The correlation model from the RMCDHF/RCI calculations predicts the excitation energies extremely well for all of the calculated ions. For Fe XVII, the relative differences with observations are around 0.005%. Calculated energies with this accuracy aid line identification in spectra and can be used to validate previous observations. As can be seen from the table, the RMCDHF/RCI and MRMP calculations both do very well, but the latter lose some of the accuracy at the neutral end of the sequence.
In Table 4, also the L S J composition is shown for each state. There are many states that are heavily mixed, with terms of almost the same weight. In these cases, labelling becomes difficult, and for many ions in the sequence, there are states that have the same leading term. Labeling is a general problem that needs considerable attention [21].

4.3. Energies for Higher States in the B-, C-, N-, O-, F- and Ne-Like Sequences

In plasma modelling and diagnostics, it is important to provide atomic data for more than just the states of the lowest configurations. To meet this demand, the RMCDHF/RCI and SD-MR calculations for the B-, C-, N-, O-, F- and Ne-like sequences have been extended to hundreds of states in what we refer to as spectrum calculations [45,46,47,48,49,50,51,52]. The range of ions, the targeted configurations and the number of studied states for each sequence are summarized in Table 5. Calculations were done by parity, i.e., odd and even parity states were treated in separate sets of calculations. The targeted configurations define the MR, and the expansions were obtained by SD-MR substitutions from all subshells to increasing active sets of orbitals. In addition to excitation energies, E1, M1, E2 and M2 transition rates were calculated.
Spectrum calculations are challenging for different reasons. The active sets of orbitals often have to be large, since many states with different charge distributions should be represented. The large active sets lead to large CSF expansions, and typically, the number of CSFs are a few millions for each parity. Another challenge is to handle the labelling. With closely degenerate configurations, the states are often not pure, but need to be described by the leading L S J composition. However, the L S J composition depends on the details of the calculation and different calculation may lead to different compositions. Thus, it is not unusual that there are inconsistencies in labelling, making comparisons between different sets of calculations, as well as with observations difficult and time consuming.
For many ions, excitation energies for lower lying states are known from observations. Going higher, comparatively less data are available, and these are often associated with large uncertainties. The situation is well illustrated for C-like Fe, and in Table 6, the RMCDHF/RCI excitation energies by Ekman et al. [47] are compared with observations. Due to near degeneracies, many states have the same leading L S J term. In these cases, labelling can be done either by giving the leading terms in the composition or, more simply, introducing an additional index A and B to separate the states. For the 20 first states belonging to the n = 2 configurations, observations are available from the NIST [37] and CHIANTI databases [3,4]. There is an agreement between the RMCDHF/RCI and relativistic many body calculations (RMBPT) by Gu [53] and observations at the 0.028–0.032% level (slightly worse for RMBPT). The RCI calculation using the Flexible Atomic Code (FAC) [42] is less accurate. For the higher lying states, experimental data are sparse. In many cases, there is excellent agreement between observations and calculations also for these states, but in some cases, there are obvious disagreements. For State Number 36, the excitation energy from NIST and CHIANTI disagree, and the calculations by Ekman et al. and Gu support the energy from the CHIANTI database. For State 54, all calculations agree, but differ markedly from the energies given by NIST and CHIANTI.
One should note the excellent agreement between the energies from RMCDHF/RCI and RMBPT, the mean difference being less than 0.013%. Although the energies are in very good agreement, there seems to be a small systematic shift for the higher states, as depicted in Figure 1. The reason for this shift is not known, and further research is needed to shed light on this. To further access the accuracy of the excitation energies, calculations for the N-, O-, F- and Ne-like sequences [49,50,51,52] were done using both the RMCDHF/RCI and RMBPT methods. Cross-validations show that the mean energy differences for N-like, O-like, F-like and Ne-like Fe are 0.023%, 0.011%, 0.01% and 0.029%, respectively. The energy differences increase for the ions closer to the neutral end, where the RMBPT method is less efficient in capturing correlation effects.
Obviously, calculations with high accuracy that also give the leading L S J compositions are indispensable tools for analysing astrophysical observations.

4.4. Energies for Higher Lying States in the Mg-, Al- and Si-Like Sequences

For larger atomic systems, one needs to think in terms of a core and a number of valence electrons. In many calculations, only valence-valence (VV) correlation is included. More accurate results are obtained when accounting for the interactions with the core through the inclusion of core-valence correlation (VV + CV). The final step is to include core-core correlation (VV + CV + CC). The situation has been analysed by Gustafsson et al. [12] for 3 l 3 l , 3 l 4 l , 3 s 5 l states in Mg-like Fe where 1 s 2 2 s 2 2 p 6 is taken as the core. The results of the analysis can be inferred from Figure 2 that shows the difference between the computed excitation energies and the observed energies from the NIST database as a function of the excitation energies for the three computational models: VV, VV + CV and VV + CV + CC.
From the figure, we see that the differences between the RMCDHF/RCI energies and observed energies are quite large, of the order of several thousand cm−1, for the VV model. For many of the low lying states, calculated energies are too high, whereas for the more highly lying states, calculated energies are too low. Adding core-valence correlation (VV + CV) substantially improves the calculated energies. To explain the difference in behaviour, as shown in the figure, between the low lying states and the more high lying states when core-valence correlation is added, we note that core-valence correlation is a combination of core polarization, an electrostatic long range rearrangement and an electron-electron cusp correcting effect [7,8]. The cusp correcting effect lowers all energies with an amount that depends on the overlap of the valence electron charge distribution and the core. The charge distributions of the low lying states from the 3 l 3 l configurations are to a larger extent overlapping the core region compared to the charge distributions from the higher states of the 3 l 4 l and 3 l 5 l configurations, leading to a more pronounced energy lowering for the former states. The core polarization, in turn, lowers all energies except the 3 s 2 1 S 0 ground state for which the valence electron charge density is spherically symmetric. In total, these two effects explain the observed behaviour. Whereas the low lying states are now in very good agreement with observations, the high lying states are still a little high compared to observations. The effect of the core-core correlation (VV + CV + CC) is small for the low lying states, but brings down the more highly states that are now in perfect agreement with observations.
The increased accuracy comes with a price. For an orbital set { 8 s 7 p 6 d 5 f 4 g 3 h 2 i } , the valence-valence (VV) expansions sizes are less than 3000 CSFs for each parity. Including the core-valence correlation (VV + CV) increases the expansions sizes to around 650,000 CSFs for each parity. Finally, including also core-core (VV + CV + CC) make the expansion sizes grow to around 6,000,000 CSFs for each parity. For these large expansions, it becomes necessary to use a zero- and first-order partition of the CSFs and include part of the interactions perturbatively as described in Section 2.3.
Based on the valence-valence and core-valence model (VV + CV), RMCDHF/RCI and SD-MR calculations have been done for the Mg-, Al- and Si-like sequences [54,55,56]. The range of ions, the targeted configurations and the number of studied states for each sequence are summarized in Table 7. Calculations were done by parity, i.e., odd and even parity states were treated in separate sets of calculations. The targeted configurations define the MR, and the expansions were obtained by SD-MR substitutions to increasing active sets of orbitals with the restriction that only one substitution is allowed from the 2 s 2 2 p 6 core. 1 s 2 is treated as an inactive core and is always closed.
To illustrate the accuracy of the RMCDHF/RCI calculation accounting for valence-valence and core-valence effects, we look at Si-like Fe [56]. In Table 8, the computed excitation RMCDHF/RCI energies are compared with observed energies from Del Zanna [57], as well as with energies by Vilkas and Ishikawa [58] using the MRMP method. The mean deviation is 0.076% for RMCDHF/RCI and only 0.034% for MRMP. The expansions for the even and odd states contained 1,500,000 and 4,500,000 CSFs, respectively.
The mean energy deviations for Mg-like, Al-like and Si-like iron from RMCDHF/RCI calculation accounting for valence-valence and core-valence effects are 0.051%, 0.039% and 0.076%, respectively. To improve the energies for the RMCDHF/RCI calculations, core-core correlation effects can be included as perturbative corrections, and work is in progress to develop tractable computational methods. For systems with five and more valence electrons, the expansions grow rapidly, and it may be necessary to start with valence-valence correlation and include core-valence effects as perturbative corrections.

5. Transition Probabilities

Whereas there are enough observations to validate calculated excitation energies, the situation is very different for transition rates. For highly charged ions, there are few experimental methods available to determine transition rates. Lifetimes for long-lived states of the ground configuration or the lowest excited configurations have been determined in accurate storage-ring and trapping experiments (see for example, the review by Träbert [59]) and are used for benchmarking. Lifetimes for a large range of short-lived states have been determined using beam-foil spectroscopy [60]. However, even if these beam-foil data are very valuable, they are in general not accurate enough to discriminate between different computational approaches. In addition, lifetimes are dominated by the strong decay channels down to the lower configurations, and the lack of experimental transition rates, including weak transitions, between states of the excited configurations is of a major concern.

5.1. Internal Validation and Uncertainty Estimates

Due to the almost complete lack of experimental transition rates for highly charged ions, internal validation becomes important. For RMCDHF/RCI calculations, the convergence of the transition rates should be monitored as the active set is increased. Then, based on the same logic, the convergence of the transition rates should be monitored as the more involved correlation models are used, e.g., VV, VV + CV and VV + CV + CC. Considering the fact that there often are tens of thousands of transitions for extended spectrum calculations, this validation method is impractical, and only smaller numbers of selected transitions can be monitored. Another internal validation method is based on the accuracy of the transition energy and the agreement between the computed line strength S in the length and velocity gauge. Along these lines, Froese-Fischer [61] has suggested that the uncertainties δ A of the calculated transition rates for L S allowed transitions can be estimated according to:
δ A = ( δ E + δ S ) A ,
where A is the energy-scaled transition rate computed from the observed transition energy ( E o b s ), δ E = | E c a l c E o b s | / E o b s is the relative error in the transition energy and δ S = | S l e n S v e l | / max ( S l e n , S v e l ) is the relative discrepancy between the length and velocity forms of the line strengths. In cases where the transition energies are not known, the expression reduces to:
δ A = ( δ S ) A .
Based on a statistical analysis of large datasets of accurate E1 transition rates from many independent calculations, Ekman et al. [62] found that the estimated errors from Equation (10) are correlated with and very close to the presumed actual errors. A validation of the method extended to intercombination lines reveals a smaller correlation in the statistical analysis and suggests that the uncertainty estimate in this case should only be used if averaging over a larger sample. The analysis further confirms the well-known fact that the uncertainty is large for weaker transitions, the general explanation being cancellations between the contributions to the matrix elements from different pairs of CSFs [63] or cancellations in the integrands of the transition integrals.

5.2. Transition Rates for the B- to Si-Like Sequences

The RMCDHF/RCI and SD-MR method has been used to compute tens of thousands of E1, M1, E2, M2 transitions rates for the B- to Si-like sequences [28,29,30,31,32,43,45,46,47,48,49,50,51,52,54,55,56]. The E1 and E2 rates are internally validated by giving δ A / A along with A. The results for C-like Fe [29], shown in Table 9, illustrate the typical uncertainties. The table displays computed transition energies along with relative uncertainties obtained by comparing with observations from NIST. The uncertainties for the transition energies are all well below 1%, and many of them are around 0.1%, which is highly satisfactory. The transition rates in the length form are given together with the uncertainty estimate δ A / A . The uncertainties for the transition rates are a few percent or less for the strong transitions, but go up to around 20% for some of the weak intercombination transitions. To further shed light on the situation, we compare the RMCDHF/RCI rates for Ne-like S [43] with rates from accurate MCHF-BP calculations [19] and with CI calculations using CIV3 [64] in Table 10. From the table, we see that there is in general a very good agreement between the rates from the different calculations. It is clear that the largest differences are for the weak transitions.

5.3. Systematic Comparisons between Methods

Wang and co-workers have systematically compared large sets of transition rates from accurate RMCDHF/RCI and RMBPT calculations [49,50,51,52]. These comparisons show that the rates from the two methods agree within a few percent for the strong transitions and that the agreement gets slightly worse for the weak intercombination and the two-electron, one photon transitions 1. The comparisons also show that the differences between the methods are large for transitions for which there are large differences between the rates in the length and velocity form, thus confirming the usefulness of δ A / A as an uncertainty estimate. In Figure 3, we show the results of a comparison between methods for O-like Fe [31]. The figure clearly shows the consistency of the RMCDHF/RCI and RMBPT transitions rates, but also the comparatively large differences with rates from the CHIANTI database. These types of comparisons point to the fact that transition rates can be computed with high accuracy, but that much effort remains in order to make data practically available for astronomers and astrophysicist in updated databases.

6. Conclusions

Current computational methodologies make it possible to compute excitation and transition energies to almost spectroscopic accuracy for many ionized systems. In an astrophysical context, this means that calculated transition energies can be used to unambiguously identify new lines from spectra or correct old identifications. Transition data are lacking for many ions, and calculated values fill this gap. Whereas many of the calculations have been done for systems with relatively few electrons with a full RCI matrix, zero- and first-order methods, allowing for parts of the interactions to be treated perturbatively, have extended the range of applicability, and many calculations with high accuracy are in progress for isoelectronic sequences starting from the third and fourth row of the periodic table.
Accurate and consistent transition rates are essential for collisional and radiative plasma modelling and for diagnostic purposes. Very few experimental data are available for the rates, and thus, the bulk of the data must be computed. The lack of experimental data means that internal validation of computed data becomes important. For accurate calculations that predict the energy structure at the per millelevel, the differences between E1 rates in the length and velocity forms can be used to estimate the uncertainties. Internal validation based on convergence analysis and agreement between rates in length and velocity, as well as systematic comparisons of rates from RMCDHF/RCI and RMBPT calculations show that the uncertainties of the E1 rates are at the level of a few percent for the strong transitions. For the weakest transitions, the uncertainties are higher and come with a more irregular pattern.

7. Further Developments and Outlook

The time for angular integration is a limiting factor for very large RCI calculations. This time can be cut down by regrouping CSFs from SD-MR expansions in blocks that can be represented symbolically. For example, in non-relativistic notation, 1 s 2 n s m p 3 P and 1 s 2 n p m d 3 P with n = 2 , , 12 and m = 2 , , 12 represent two blocks where the angular integration between CSFs in the blocks, as well as between CSFs in the different blocks is independent of the principal quantum numbers or can be reduced to only a few cases. For large n and m, the reduction in computing time is substantial. Discussed already decades ago [65], it seems essential that these ideas are now broadly implemented in the generally available computer codes.
With angular integration being a negligible part of the computation comes the possibility to extend the orbital set to higher n. Currently, the orbitals are variationally determined on a grid in RMCDHF calculations. The variational determination is computationally costly, and it would be valuable to augment the variationally-determined orbitals with analytical orbitals or orbitals determined in simplified and fast procedures. Work along these lines is in progress.
Among the targeted systems for improved computer codes are the α -elements, including Mg, Si, Ca and the iron group elements Sc, Ti, Cr, Mn, Fe at lower ionization states. These elements are of key importance for stellar and galactic evolution studies [66].

Acknowledgments

Per Jönsson, Henrik Hartman, Jörgen Ekman and Tomas Brage acknowledge support from the Swedish Research Council under Contract 2015-04842. Laima Radziute is thankful for the high performance computing resources provided by the Information Technology Open Access Center of Vilnius University. Michel R. Godefroid acknowledges support from the Belgian F.R.S.-FNRS Fonds de la Recherche Scientifique under CDR J.0047.16.

Author Contributions

All authors contributed equally to the work presented here.

Conflicts of Interest

The authors declare no conflict of interest.

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1.
Transitions between two states for which the configurations differ by more than one electron. These transitions are zero in the lowest approximation and are induced by CSFs that enter the calculation to correct for electron correlation effects.
Figure 1. Difference between RMCDHF/RCI and MBPT excitation energies in percent for C-like Fe as a function of the excitation energy in kcm−1. The dashed lines show the 0.02 % levels.
Figure 1. Difference between RMCDHF/RCI and MBPT excitation energies in percent for C-like Fe as a function of the excitation energy in kcm−1. The dashed lines show the 0.02 % levels.
Atoms 05 00016 g001
Figure 2. Difference between observed and RMCDHF/RCI excitation energies in cm−1 for Mg-like Fe [12] as a function of the excitation energy in kcm−1. valence-valence (VV) (blue squares), VV + core-valence (CV) (red circles) and VV + CV + CC (black+).
Figure 2. Difference between observed and RMCDHF/RCI excitation energies in cm−1 for Mg-like Fe [12] as a function of the excitation energy in kcm−1. valence-valence (VV) (blue squares), VV + core-valence (CV) (red circles) and VV + CV + CC (black+).
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Figure 3. Results of a comparison between methods for O-like Fe [31]. Deviation in percent between RMCDHF/RCI and RMBPT transition rates as a function of the transition rate in s−1. Deviations from the values of the CHIANTI database [3,4] are given in red. The dashed lines give the 10% levels.
Figure 3. Results of a comparison between methods for O-like Fe [31]. Deviation in percent between RMCDHF/RCI and RMBPT transition rates as a function of the transition rate in s−1. Deviations from the values of the CHIANTI database [3,4] are given in red. The dashed lines give the 10% levels.
Atoms 05 00016 g003
Table 1. Multireference (MR), active set, number of generated configuration state functions (CSFs) ( N CSFs ) and the range of ions for the relativistic multiconfiguration Dirac-Hartree-Fock (RMCDHF) and relativistic configuration interaction (RCI) calculations of the boron-, carbon-, nitrogen-, oxygen- and fluorine-like sequences.
Table 1. Multireference (MR), active set, number of generated configuration state functions (CSFs) ( N CSFs ) and the range of ions for the relativistic multiconfiguration Dirac-Hartree-Fock (RMCDHF) and relativistic configuration interaction (RCI) calculations of the boron-, carbon-, nitrogen-, oxygen- and fluorine-like sequences.
ConfigurationMR for RMCDHFMR for RCIActive Set N CSFs
boron-like, N III to Zn XXVI
1 s 2 2 s 2 2 p 1 s 2 { 2 s 2 2 p , 2 p 3 } 1 s 2 { 2 s 2 2 p , 2 p 3 , 2 s 2 p 3 d , 2 p 3 d 2 } { 9 s 8 p 7 d 6 f 5 g 3 h 1 i } 200 100
1 s 2 2 p 3 1 s 2 { 2 s 2 2 p , 2 p 3 } 1 s 2 { 2 s 2 2 p , 2 p 3 , 2 s 2 p 3 d , 2 p 3 d 2 } { 9 s 8 p 7 d 6 f 5 g 3 h 1 i } 360 100
1 s 2 2 s 2 p 2 1 s 2 2 s 2 p 2 1 s 2 { 2 s 2 p 2 , 2 p 2 3 d , 2 s 2 3 d , 2 s 3 d 2 } { 9 s 8 p 7 d 6 f 5 g 3 h 1 i } 300 100
carbon-like, F IV to Ni XXIII
1 s 2 2 s 2 2 p 2 1 s 2 { 2 s 2 2 p 2 , 2 p 4 } 1 s 2 { 2 s 2 2 p 2 , 2 p 4 , 2 s 2 p 2 3 d , 2 s 2 3 d 2 } { 8 s 7 p 6 d 5 f 4 g 2 h } 340 100
1 s 2 2 p 4 1 s 2 { 2 s 2 2 p 2 , 2 p 4 } 1 s 2 { 2 s 2 2 p 2 , 2 p 4 , 2 s 2 p 2 3 d , 2 s 2 3 d 2 } { 8 s 7 p 6 d 5 f 4 g 2 h } 340 100
1 s 2 2 s 2 p 3 1 s 2 2 s 2 p 3 1 s 2 { 2 s 2 p 3 , 2 p 3 3 d , 2 s 2 2 p 3 d , 2 s 2 p 3 d 2 } { 8 s 7 p 6 d 5 f 4 g 2 h } 1 000 100
nitrogen-like, F III to Kr XXX
1 s 2 2 s 2 2 p 3 1 s 2 { 2 s 2 2 p 3 , 2 p 5 } 1 s 2 { 2 s 2 2 p 3 , 2 p 5 , 2 s 2 p 3 3 d , 2 s 2 2 p 3 d 2 } { 8 s 7 p 6 d 5 f 4 g 1 h } 698 100
1 s 2 2 p 5 1 s 2 { 2 s 2 2 p 3 , 2 p 5 } 1 s 2 { 2 s 2 2 p 3 , 2 p 5 , 2 s 2 p 3 3 d , 2 s 2 2 p 3 d 2 } { 8 s 7 p 6 d 5 f 4 g 1 h } 382 100
1 s 2 2 s 2 p 4 1 s 2 2 s 2 p 4 1 s 2 { 2 s 2 p 4 , 2 p 4 3 d , 2 s 2 2 p 2 3 d , 2 s 2 p 2 3 d 2 } { 8 s 7 p 6 d 5 f 4 g 1 h } 680 100
oxygen-like, F II to Kr XXIX
1 s 2 2 s 2 2 p 4 1 s 2 { 2 s 2 2 p 4 , 2 p 6 } 1 s 2 { 2 s 2 2 p 4 , 2 p 6 , 2 s 2 p 4 3 d } { 8 s 7 p 6 d 5 f 4 g 3 h } 709 690
1 s 2 2 p 6 1 s 2 { 2 s 2 2 p 4 , 2 p 6 } 1 s 2 { 2 s 2 2 p 4 , 2 p 6 , 2 s 2 p 4 3 d } { 8 s 7 p 6 d 5 f 4 g 3 h } 67 375
1 s 2 2 s 2 p 5 1 s 2 2 s 2 p 5 1 s 2 { 2 s 2 p 5 , 2 p 5 3 d , 2 s 2 2 p 3 3 d } { 8 s 7 p 6 d 5 f 4 g 3 h } 702 892
fluorine-like, Si VI to WLXVI
1 s 2 2 s 2 2 p 5 1 s 2 2 s 2 2 p 5 1 s 2 2 s 2 2 p 5 { 8 s 7 p 6 d 5 f 4 g 3 h 2 i } 73 000
1 s 2 2 s 2 p 6 1 s 2 2 s 2 p 6 1 s 2 2 s 2 p 6 { 8 s 7 p 6 d 5 f 4 g 3 h 2 i } 15 000
Table 2. Excitation energies in cm−1 for O-like Ne and Fe from observations and different calculations. Relative errors in % for the calculated energies are shown in parenthesis. E o b s observation NIST [37], E R C I energies from RMCDHF/RCI [31], E M R M P energies from Möller-Plesset calculation (MRMP) [38], E M B P T energies from many-body perturbation theory [39], E B P energies from multiconfiguration Hartree-Fock-Breit-Pauli [19], E S S energies from super structure [40], E M C D F energies from RMCDHF [41] and E F A C  energies from RCI with the FAC code [42].
Table 2. Excitation energies in cm−1 for O-like Ne and Fe from observations and different calculations. Relative errors in % for the calculated energies are shown in parenthesis. E o b s observation NIST [37], E R C I energies from RMCDHF/RCI [31], E M R M P energies from Möller-Plesset calculation (MRMP) [38], E M B P T energies from many-body perturbation theory [39], E B P energies from multiconfiguration Hartree-Fock-Breit-Pauli [19], E S S energies from super structure [40], E M C D F energies from RMCDHF [41] and E F A C  energies from RCI with the FAC code [42].
Ne III
Level J E obs E RCI E MRMP E MBPT E BP E SS
2 s 2 2 p 4 3 P 200  (0.00)0  (0.00)0  (0.00)0  (0.00)0  (0.00)
1643645  (0.31)638  (0.77)645  (0.31)628  (2.33)744  (15.70)
0921923  (0.21)912  (0.97)926  (0.54)899  (2.38)1 069  (16.06)
2 s 2 2 p 4 1 D 225 84125 954  (0.43)26 097  (0.99)25 573  (1.03)25 759  (0.31)29 219  (13.07)
2 s 2 2 p 4 1 S 055 75356 058  (0.54)55 772  (0.03)55 459  (0.52)55 382  (0.66)72 484  (30.00)
2 s 2 p 5 3 P o 2204 290204 608  (0.15)204 718  (0.20)200 686  (1.76)204 635  (0.16)215 348  (5.41)
1204 873205 200  (0.15)205 297  (0.20)201 276  (1.75)205 236  (0.17)216 008  (5.43)
0205 194205 603  (0.19)205 617  (0.20)201 598  (1.75)205 539  (0.16)216 367  (5.44)
2 s 2 p 5 1 P o 1289 479290 315  (0.28)290 703  (0.42)288 219  (0.43)291 659  (0.75)315 511  (8.99)
Fe XIX
Level J E obs E RCI E MRMP E MBPT E MCDF E FAC
2 s 2 2 p 4 3 P 200  (0.000)0  (0.000)0  (0.00)0  (0.00)0  (0.00)
075 25075 313  (0.083)75 218  (0.042)74 742  (0.67)75 446  (0.26)75 198  (0.06)
189 44189 434  (0.007)89 251  (0.212)87 559  (2.10)88 791  (0.72)88 821  (0.69)
2 s 2 2 p 4 1 D 2168 852168 985  (0.078)168 792  (0.035)167 881  (0.57)170 847  (1.18)170 578  (1.02)
2 s 2 2 p 4 1 S 0325 140325 417  (0.085)324 949  (0.058)321 124  (1.23)326 536  (0.42)325 421  (0.08)
2 s 2 p 5 3 P o 2922 890923 044  (0.016)922 855  (0.003)917 435  (0.59)933 081  (1.10)929 231  (0.68)
1984 740984 920  (0.018)984 791  (0.005)978 242  (0.65)995 006  (1.04)991 246  (0.66)
01030 0201030 199  (0.017)1029 992  (0.002)1022 753  (0.70)1039 692  (0.93)1036 058  (0.58)
2 s 2 p 5 1 P o 11267 6001268 093  (0.038)1267 771  (0.013)1258 927  (0.68)1287 773  (1.59)1282 914  (1.20)
2 p 6 1 S 02134 1802134 958  (0.036)2132 810  (0.064)2120 211  (0.65)2175 645  (1.94)2160 701  (1.24)
Table 3. Excitation energies in cm−1 for O-like Br. Comparison between calculations, observations and semiempirical estimates. E o b s observation NIST [37] with original data from Kelly [35], E S E  semiempirical fit [36] and E R C I energies from RMCDHF/RCI [31], E M R M P energies from MRMP [38]; Δ E 1 , difference between calculated energies and E o b s ; Δ E 2 , difference between calculated energies and E S E . The calculations support energies from the semiempirical fit.
Table 3. Excitation energies in cm−1 for O-like Br. Comparison between calculations, observations and semiempirical estimates. E o b s observation NIST [37] with original data from Kelly [35], E S E  semiempirical fit [36] and E R C I energies from RMCDHF/RCI [31], E M R M P energies from MRMP [38]; Δ E 1 , difference between calculated energies and E o b s ; Δ E 2 , difference between calculated energies and E S E . The calculations support energies from the semiempirical fit.
LevelJ E obs E SE E RCI Δ E 1 Δ E 2 E MRMP Δ E 1 Δ E 2
2 s 2 2 p 4 3 P 200000000
0218 800153 478151 954−66 846−152415 2035−66 765−1443
1379 800371 663371 606−8 194−5737 1858−7 942195
2 s 2 2 p 4 1 D 2483 040470 699470 643−12 397−5647 0804−12 236105
2 s 2 2 p 4 1 S 0944 150912 501911 968−32 182-53391 2282−31 868−219
2 s 2 p 5 3 P o 2 1 579 9031 579 537 −3661 580 945 1042
1 1 755 0281 755 196 1681 756 684 1656
0 1 986 2741 985 784 −4901 987 396 1122
2 s 2 p 5 1 P o 1 2 229 3582 230 149 7912 231 636 2278
2 p 6 1 S 0 3 573 4163 575 415 19993 579 486 6070
Table 4. Excitation energies in cm−1 for Ne-like Ca and Fe from observations and different calculations. Relative errors in % for the calculated energies are shown in parenthesis. E o b s observation NIST [37], E R C I energies from RMCDHF/RCI [43] and E M R M P energies from MRMP [44].
Table 4. Excitation energies in cm−1 for Ne-like Ca and Fe from observations and different calculations. Relative errors in % for the calculated energies are shown in parenthesis. E o b s observation NIST [37], E R C I energies from RMCDHF/RCI [43] and E M R M P energies from MRMP [44].
Level LSJ Composition E obs E RCI E MRMP
Ca XI
2 p 6 1 S 0 1.00000
2 p 5 3 s 3 P 2 o 0.99 2 801 9892 801 819
2 p 5 3 s 3 P 1 o 0.62 + 0.38 1 P 1 o 2 810 9002 810 834 (0.0023)2 810 588 (0.011)
2 p 5 3 s 3 P 0 o 0.99 2 831 8002 831 670
2 p 5 3 s 1 P 1 o 0.62 + 0.38 3 P 1 o 2 839 9002 839 662 (0.0084)2 839 386 (0.018)
2 p 5 3 p 3 S 1 0.92 2 953 7912 953 594
2 p 5 3 p 3 D 2 0.68 + 0.24 1 D 2 2 978 4102 977 968
2 p 5 3 p 3 D 3 1.00 2 978 6502 978 276
2 p 5 3 p 3 D 1 0.43 + 0.36 1 P 1 + 0.20 3 P 1 2 986 9082 986 513
2 p 5 3 p 3 P 2 0.65 + 0.34 1 D 2 2 993 7602 993 336
2 p 5 3 p 3 D 1 0.54 + 0.40 1 P 1 3 007 3013 006 932
2 p 5 3 p 3 P 0 0.98 3 009 3453 009 000
2 p 5 3 p 1 D 2 0.41 + 0.31 3 D 2 + 0.27 3 P 2 3 016 7493 016 378
2 p 5 3 p 3 P 1 0.68 + 0.24 1 P 1 3 017 1753 016 845
2 p 5 3 p 1 S 0 0.98 3 101 1663 098 308
2 p 5 3 d 3 P 0 o 0.99 3 196 0753 195 830
2 p 5 3 d 3 P 1 o 0.953 199 3003 199 045 (0.0080)3 198 902 (0.012)
2 p 5 3 d 3 P 2 o 0.85 3 205 2783 205 169
2 p 5 3 d 3 F 4 o 1.00 3 208 3513 208 165
2 p 5 3 d 3 F 3 o 0.72 + 0.23 1 F 2 o 3 212 3923 212 144
2 p 5 3 d 3 F 2 o 0.53 + 0.29 1 D 2 o + 0.18 3 D 2 o 3 219 6553 219 428
2 p 5 3 d 3 D 3 o 0.55 + 0.41 1 F 3 o 3 224 3943 224 078
2 p 5 3 d 3 D 1 o 0.893 239 7003 239 502 (0.0061)3 239 308 (0.012)
2 p 5 3 d 3 F 2 o 0.47 + 0.38 1 D 2 o + 0.14 3 D 2 o 3 244 3483 244 161
2 p 5 3 d 3 D 2 o 0.58 + 0.27 1 D 2 o + 0.14 3 P 2 o 3 248 0173 247 805
2 p 5 3 d 3 D 3 o 0.40 + 0.35 1 F 3 o + 0.24 3 F 3 o 3 248 3453 248 099
2 p 5 3 d 1 P 1 o 0.913 284 3003 284 444 (0.0044)3 283 473 (0.025)
Fe XVII
2 p 6 1 S 0 1.00000
2 p 5 3 s 3 P 2 o 1.005 849 4905 849 108  (0.0065)5 848 891  (0.0102)
2 p 5 3 s 1 P 1 o 0.54 + 0.45 3 P 1 5 864 7605 864 469  (0.0049)5 864 138  (0.0106)
2 p 5 3 s 3 P 0 o 1.005 951 4785 951 003  (0.0079)5 950 877  (0.0100)
2 p 5 3 s 3 P 1 o 0.54 + 0.45 1 P 1 5 961 0225 960 633  (0.0065)5 960 410  (0.0102)
2 p 5 3 p 3 S 1 0.80 + 0.17 3 P 1 6 093 5686 093 573  (0.0000)6 093 209  (0.0058)
2 p 5 3 p 3 D 2 0.58 + 0.30 1 D 2 + 0.12 3 P 2 6 121 7566 121 769  (0.0002)6 121 253  (0.0082)
2 p 5 3 p 3 D 3 1.006 134 8156 134 794  (0.0003)6 134 360  (0.0074)
2 p 5 3 p 1 P 1 0.51 + 0.25 3 D 1 + 0.19 3 P 1 6 143 8976 143 898  (0.0000)6 143 431  (0.0075)
2 p 5 3 p 3 P 2 0.67 + 0.32 1 D 2 6 158 5406 158 481  (0.0009)6 158 010  (0.0086)
2 p 5 3 p 3 P 0 0.946 202 6206 202 542  (0.0012)6 202 238  (0.0061)
2 p 5 3 p 3 D 1 0.67 + 0.31 1 P 1 6 219 2666 219 185  (0.0013)6 218 795  (0.0075)
2 p 5 3 p 3 P 1 0.63 + 0.17 1 P 1 + 0.13 3 S 1 6 245 4906 245 346  (0.0023)6 245 018  (0.0075)
2 p 5 3 p 3 D 2 0.41 + 0.38 1 D 2 + 0.21 3 P 2 6 248 5306 248 390  (0.0022)6 248 024  (0.0080)
2 p 5 3 p 1 S 0 0.936 353 3566 353 605  (0.0039)6 351 136  (0.0349)
2 p 5 3 d 3 P 0 o 0.996 464 0956 463 913  (0.0028)6 463 611  (0.0074)
2 p 5 3 d 3 P 1 o 0.916 471 2336 471 519  (0.0044)6 471 317  (0.0012)
2 p 5 3 d 3 P 2 o 0.72 + 0.18 3 D 2 o 6 486 4406 486 166  (0.0042)6 485 977  (0.0071)
2 p 5 3 d 3 F 4 o 1.006 487 0006 486 745  (0.0039)6 486 514  (0.0074)
2 p 5 3 d 3 F 3 o 0.65 + 0.29 1 F 3 o 6 492 9246 492 689  (0.0036)6 492 387  (0.0082)
2 p 5 3 d 1 D 2 o 0.41 + 0.35 3 F 2 o + 0.24 3 D 2 o 6 506 8086 506 561  (0.0037)6 506 276  (0.0081)
2 p 5 3 d 3 D 3 o 0.64 + 0.34 1 F 3 o 6 515 4796 515 276  (0.0031)6 514 936  (0.0083)
2 p 5 3 d 3 D 1 o 0.74 + 0.20 1 P 1 o 6 552 2216 552 697  (0.0072)6 552 491  (0.0041)
2 p 5 3 d 3 F 2 o 0.63 + 0.29 1 D 2 o 6 594 6176 594 260  (0.0054)6 594 099  (0.0078)
2 p 5 3 d 3 D 2 o 0.50 + 0.27 3 P 2 o + 0.21 1 D 2 o 6 601 2106 600 855  (0.0053)6 600 688  (0.0079)
2 p 5 3 d 1 F 3 o 0.37 + 0.33 3 F 3 o + 0.30 3 D 3 o 6 605 4696 605 078  (0.0059)6 604 858  (0.0092)
2 p 5 3 d 1 P 1 o 0.78 + 0.18 3 D 1 o 6 660 8946 661 101  (0.0031)6 660 232  (0.0099)
Table 5. Sequence, ions and targeted configurations for the RMCDHF/RCI calculations. N is the number of studied states for each ion. In the table, l = 0 , 1 , 2 , l = 0 , 1 , 2 , 3 , l = 0 , , n 1 .
Table 5. Sequence, ions and targeted configurations for the RMCDHF/RCI calculations. N is the number of studied states for each ion. In the table, l = 0 , 1 , 2 , l = 0 , 1 , 2 , 3 , l = 0 , , n 1 .
SequenceIonsConfigurationsNRef.
B-likeSi, Ti-Cu 2 s 2 2 p , 2 s 2 p 2 , 2 p 3 , 2 s 2 3 l , 2 s 2 p 3 l , 2 p 2 3 l , 2 s 2 4 l , 2 s 2 p 4 l , 2 p 2 4 l 291[45]
B-likeNa 2 s 2 2 p , 2 s 2 p 2 , 2 p 3 , 2 s 2 3 l , 2 s 2 p 3 l , 2 p 2 3 l , 2 s 2 4 l , 2 s 2 p 4 s 133[46]
C-likeAr-Zn 2 s 2 2 p 2 , 2 s 2 p 3 , 2 p 4 , 2 s 2 2 p 3 l , 2 s 2 p 2 3 l , 2 p 3 3 l , 2 s 2 2 p 4 l 262[47]
N-likeCr, Fe, Ni, Zn 2 s 2 2 p 3 , 2 s 2 p 4 , 2 p 5 , 2 s 2 2 p 2 3 l , 2 s 2 p 3 3 l , 2 p 4 3 l 272[48]
N-likeAr-Zn 2 s 2 2 p 3 , 2 s 2 p 4 , 2 p 5 , 2 s 2 2 p 2 3 l , 2 s 2 p 3 3 l , 2 p 4 3 l , 2 s 2 2 p 2 4 l 359[49]
O-likeCr-Zn 2 s 2 2 p 4 , 2 s 2 p 5 , 2 p 6 , 2 s 2 2 p 3 3 l , 2 s 2 p 4 3 l 200[50]
F-likeCr-Zn 2 s 2 2 p 5 , 2 s 2 p 6 , 2 s 2 2 p 4 3 l , 2 s 2 p 5 3 l , 2 p 6 3 l , 2 s 2 2 p 4 4 l 200[51]
Ne-likeCr-Kr 2 s 2 2 p 6 , 2 s 2 p 6 3 l , 2 s 2 2 p 5 4 l , 2 s 2 p 6 4 l , 2 s 2 2 p 5 5 l , 2 s 2 2 p 5 6 l 201[52]
Table 6. Energies in cm−1 for levels in Fe XXI. E R C I energies from RMCDHF/RCI calculations [47], E R M B P T energies from RMBPT [53], E F A C energies from RCI calculations with FAC [42], E N I S T NIST recommended values [37] and E C H I observed energies from the CHIANTI database [3,4].
Table 6. Energies in cm−1 for levels in Fe XXI. E R C I energies from RMCDHF/RCI calculations [47], E R M B P T energies from RMBPT [53], E F A C energies from RCI calculations with FAC [42], E N I S T NIST recommended values [37] and E C H I observed energies from the CHIANTI database [3,4].
No.Level E RCI E RMBPT E FAC E NIST E CHI
1 2 s 2 2 p 2 3 P 0 00000
2 2 s 2 2 p 2 3 P 1 73 86473 86773 04173 85173 851
3 2 s 2 2 p 2 3 P 2 117 417117 372117 146117 354117 367
4 2 s 2 2 p 2 1 D 2 244 751244 581245 710244 561244 568
5 2 s 2 2 p 2 1 S 0 372 137372 261373 060371 980371 744
6 2 s 2 p 3 5 S 2 486 584487 683479 658486 950486 991
7 2 s 2 p 3 3 D 1 776 775777 005779 724776 690776 685
8 2 s 2 p 3 3 D 2 777 404777 655779 963777 340777 367
9 2 s 2 p 3 3 D 3 803 618803 869805 768803 540803 553
10 2 s 2 p 3 3 P 0 916 444916 773920 272916 330916 333
11 2 s 2 p 3 3 P 1 925 074925 408928 822924 920924 920
12 2 s 2 p 3 3 P 2 942 621942 986946 135942 430942 364
13 2 s 2 p 3 3 S 1 1 096 0191 095 8201 105 5781 095 6701 095 679
14 2 s 2 p 3 1 D 2 1 127 6721 127 4601 137 5331 127 2401 127 250
15 2 s 2 p 3 1 P 1 1 261 5771 261 2401 272 6271 261 1401 260 902
16 2 p 4 3 P 2 1 646 4371 646 4671 657 4111 646 3001 646 409
17 2 p 4 3 P 0 1 735 8231 735 8131 747 3011 735 7001 735 715
18 2 p 4 3 P 1 1 740 6231 740 7071 750 8481 740 5001 740 453
19 2 p 4 1 D 2 1 817 7861 817 3621 832 1021 817 1001 817 041
20 2 p 4 1 S 0 2 048 5122 047 8502 066 4632 048 2002 048 056
21 2 s 2 2 p 3 s 3 P 0 7 663 2837 664 0547 654 119
22 2 s 2 2 p 3 s 3 P 1 7 671 9717 672 7037 663 398 7 661 883
23 2 s 2 2 p 3 s 3 P 2 7 780 2987 781 1477 770 895
24 2 s 2 2 p 3 s 1 P 1 7 803 7647 804 4197 796 397
25 2 s 2 2 p 3 p 3 D 1 7 841 9037 842 9227 834 847
26 2 s 2 2 p 3 p 3 P 1 : A 7 898 1547 898 9747 891 978
27 2 s 2 2 p 3 p 3 D 2 7 901 5537 902 3787 895 497
28 2 s 2 2 p 3 p 3 P 0 7 914 8497 915 8117 909 434 7 915 463
29 2 s 2 2 p 3 p 3 P 1 : B 7 983 4467 984 3507 977 011
30 2 s 2 2 p 3 p 3 D 3 7 994 5887 995 3887 987 318
31 2 s 2 2 p 3 p 3 S 1 8 004 9878 005 7937 998 341
32 2 s 2 2 p 3 p 3 P 2 8 007 3268 008 3198 002 052
33 2 s 2 2 p 3 p 1 D 2 8 068 5378 069 0718 065 382
34 2 s 2 2 p 3 d 3 F 2 8 078 5408 079 1198 072 911 8 074 160
35 2 s 2 p 2 ( 4 P ) 3 s 5 P 1 8 080 5518 082 0018 070 805
36 2 s 2 2 p 3 d 3 F 3 8 116 0488 116 4808 111 3368 101 4008 118 008
37 2 s 2 2 p 3 d 3 P 2 : A 8 121 9228 122 5298 118 025 8 124 085
38 2 s 2 2 p 3 p 1 S 0 8 128 6458 129 3968 126 192 8 143 710
39 2 s 2 p 2 ( 4 P ) 3 s 5 P 2 8 131 9738 133 4608 121 258
40 2 s 2 2 p 3 d 3 D 1 8 139 2908 140 7358 135 9928 140 0008 141 785
41 2 s 2 p 2 ( 4 P ) 3 s 5 P 3 8 181 3318 182 5998 170 876
42 2 s 2 p 2 ( 4 P ) 3 s 3 P 0 8 182 1728 182 8448 179 292 8 180 254
43 2 s 2 2 p 3 d 3 F 4 8 202 0738 202 6708 195 771
44 2 s 2 2 p 3 d 1 D 2 8 208 7058 209 5978 204 329
45 2 s 2 p 2 ( 4 P ) 3 s 3 P 1 8 222 1568 222 9488 217 390
46 2 s 2 2 p 3 d 3 D 3 8 230 9188 231 8688 227 144(8 195 000)8 229 642
47 2 s 2 2 p 3 d 3 P 2 : B 8 245 4538 246 4288 241 4368 230 9008 229 642
48 2 s 2 2 p 3 d 3 P 1 8 245 7378 247 0758 241 557
49 2 s 2 2 p 3 d 3 P 0 8 247 7328 249 1648 243 033
50 2 s 2 p 2 ( 4 P ) 3 p 5 D 0 8 267 9638 269 2208 259 742
51 2 s 2 p 2 ( 4 P ) 3 p 5 D 1 8 270 5588 272 0888 262 373
52 2 s 2 p 2 ( 4 P ) 3 s 3 P 2 8 274 7048 275 4278 270 235
53 2 s 2 2 p 3 d 1 F 3 8 300 6188 301 1288 301 3798 313 600
54 2 s 2 2 p 3 d 1 P 1 8 303 7308 307 4288 305 3768 293 9008 293 791
55 2 s 2 p 2 ( 4 P ) 3 p 5 D 2 8 305 9178 309 1628 297 457
56 2 s 2 p 2 ( 4 P ) 3 p 3 S 1 8 312 4998 309 3598 300 070
57 2 s 2 p 2 ( 4 P ) 3 p 5 P 1 8 349 4568 350 8578 342 324 8 350 731
58 2 s 2 p 2 ( 4 P ) 3 p 5 D 3 8 351 7758 353 2778 342 522
59 2 s 2 p 2 ( 4 P ) 3 p 5 P 2 8 352 1178 353 7318 343 801
60 2 s 2 p 2 ( 4 P ) 3 p 3 D 1 8 379 9678 380 8908 373 689 8 376 741
61 2 s 2 p 2 ( 4 P ) 3 p 5 P 3 8 388 6348 390 2928 380 281
62 2 s 2 p 2 ( 4 P ) 3 p 5 D 4 8 399 5578 401 0398 390 478
63 2 s 2 p 2 ( 4 P ) 3 p 3 D 2 8 410 0778 411 1828 404 386
64 2 s 2 p 2 ( 2 D ) 3 s 3 D 1 8 420 5888 421 4268 420 569
65 2 s 2 p 2 ( 2 D ) 3 s 3 D 2 8 428 4058 429 2488 428 172
66 2 s 2 p 2 ( 2 D ) 3 s 3 D 3 8 440 9268 441 7588 437 725
67 2 s 2 p 2 ( 4 P ) 3 p 3 P 0 8 442 8138 443 7768 438 241
68 2 s 2 p 2 ( 4 P ) 3 p 5 S 2 8 443 6468 445 0378 440 027
69 2 s 2 p 2 ( 4 P ) 3 p 3 D 3 8 462 3658 463 5108 456 502
70 2 s 2 p 2 ( 4 P ) 3 p 3 P 1 8 467 6908 468 6808 462 413
71 2 s 2 p 2 ( 4 P ) 3 p 3 P 2 8 470 8718 471 9138 466 158
72 2 s 2 p 2 ( 4 P ) 3 d 5 F 1 8 480 6208 481 7358 471 575
73 2 s 2 p 2 ( 4 P ) 3 d 5 F 2 8 488 7828 489 9718 479 898 8 486 331
74 2 s 2 p 2 ( 2 D ) 3 s 1 D 2 8 496 9908 497 5128 498 660
75 2 s 2 p 2 ( 4 P ) 3 d 5 F 3 : A 8 506 1118 507 3438 497 311 8 511 385
76 2 s 2 p 2 ( 4 P ) 3 d 5 F 4 8 544 5758 545 9288 534 831
77 2 s 2 p 2 ( 2 P ) 3 s 3 P 1 8 545 4858 546 5078 544 603
78 2 s 2 p 2 ( 2 P ) 3 s 3 P 0 8 553 8858 554 7168 545 420
79 2 s 2 p 2 ( 4 P ) 3 d 5 D 0 8 554 7988 555 9188 546 365
80 2 s 2 p 2 ( 4 P ) 3 d 5 D 1 8 555 2978 556 5348 547 553
81 2 s 2 p 2 ( 4 P ) 3 d 5 D 2 8 555 4918 556 8508 558 611
82 2 s 2 p 2 ( 4 P ) 3 d 5 F 3 : B 8 561 6628 562 9698 552 789 8 564 535
83 2 s 2 p 2 ( 4 P ) 3 d 3 P 2 8 581 2748 582 7558 576 965 8 575 780
84 2 s 2 p 2 ( 4 P ) 3 d 5 F 5 8 586 6368 588 0148 577 151
85 2 s 2 p 2 ( 4 P ) 3 d 5 D 4 8 597 7358 599 1068 588 388
86 2 s 2 p 2 ( 2 D ) 3 p 3 F 2 8 606 1108 606 6548 607 148 8 605 427
87 2 s 2 p 2 ( 4 P ) 3 d 3 F 2 8 611 4328 611 8438 608 283
88 2 s 2 p 2 ( 4 P ) 3 d 5 P 3 8 619 3128 620 8478 611 148
228 2 p 3 ( 2 P ) 3 d 3 F 4 9 735 4809 736 1119 746 771
229 2 p 3 ( 2 P ) 3 d 3 P 1 9 740 6459 742 7199 754 847
230 2 p 3 ( 2 P ) 3 d 3 P 0 9 748 1849 749 7749 758 859
231 2 p 3 ( 2 P ) 3 d 3 P 2 : B 9 757 8909 758 1079 770 963
232 2 p 3 ( 2 P ) 3 d 3 D 1 9 765 6639 766 0569 781 499
233 2 p 3 ( 2 P ) 3 d 3 D 3 9 780 7389 781 0449 794 389
234 2 p 3 ( 2 P ) 3 d 1 F 3 9 800 3689 800 7429 819 206
235 2 p 3 ( 2 P ) 3 d 3 D 2 9 800 8529 801 7389 819 939
236 2 p 3 ( 2 P ) 3 d 1 P 1 9 879 4719 879 6559 902 175
237 2 s 2 2 p 4 s 3 P 0 10 368 077 10 362 393
238 2 s 2 2 p 4 s 3 P 1 10 371 121 10 365 58510 380 000
239 2 s 2 2 p 4 p 3 D 1 10 442 616 10 437 633
240 2 s 2 2 p 4 p 3 P 1 10 466 102 10 460 676
241 2 s 2 2 p 4 p 3 D 2 10 468 322 10 462 978
242 2 s 2 2 p 4 p 3 P 0 10 470 990 10 465 993
243 2 s 2 2 p 4 s 3 P 2 10 485 597 10 479 693
244 2 s 2 2 p 4 s 1 P 1 10 492 966 10 487 317
245 2 s 2 2 p 4 d 3 F 2 10 532 099 10 526 459
246 2 s 2 2 p 4 d 3 P 2 : A 10 548 542 10 542 323(10 547 000)10 547 249
247 2 s 2 2 p 4 d 3 F 3 10 549 480 10 543 48810 548 00010 548 160
248 2 s 2 2 p 4 d 3 D 1 10 554 447 10 548 34510 553 00010 553 955
249 2 s 2 2 p 4 p 1 P 1 10 568 810 10 563 327
250 2 s 2 2 p 4 p 3 D 3 10 574 912 10 569 41210 664 000
251 2 s 2 2 p 4 p 3 P 2 10 575 111 10 569 433
252 2 s 2 2 p 4 p 3 S 1 10 578 203 10 572 657
253 2 s 2 2 p 4 p 1 D 2 10 597 862
254 2 s 2 2 p 4 p 1 S 0 10 619 563
255 2 s 2 2 p 4 d 3 F 4 10 652 979
256 2 s 2 2 p 4 d 1 D 2 10 653 631 10 675 000
257 2 s 2 2 p 4 d 3 D 3 10 660 593
258 2 s 2 2 p 4 d 3 P 2 : B 10 666 807
259 2 s 2 2 p 4 d 3 P 1 10 666 946 10 688 000
260 2 s 2 2 p 4 d 3 P 0 10 667 948
261 2 s 2 2 p 4 d 1 F 3 10 683 984 10 681 000
262 2 s 2 2 p 4 d 1 P 1 10 687 400
Table 7. Sequence, ions and targeted configurations for the calculations. N is the number of studied states for each ion. In the table, l = 0 , , n 1 , l = 0 , , n 1 .
Table 7. Sequence, ions and targeted configurations for the calculations. N is the number of studied states for each ion. In the table, l = 0 , , n 1 , l = 0 , , n 1 .
SequenceIonsConfigurationsNRef.
Mg-likeCa-As, Kr 3 l 3 l , 3 l 4 l , 3 s 5 l 146[54]
Al-likeTi-Kr, Xe, W 3 s 2 { 3 l ; 4 l ; 5 l } , 3 p 2 { 3 d ; 4 l } , 3 s { 3 p 2 ; 3 d 2 } ,360[55]
3 s { 3 p 3 d ; 3 p 4 l ; 3 p 5 s ; 3 d 4 l } , 3 p 3 d 2 , 3 p 3 , 3 d 3
Si-likeTi-Ge, Sr, Zr, Mo 3 s 2 3 p 2 , 3 s 3 p 3 , 3 s 2 3 p 3 d 27[56]
Table 8. Comparison of calculated and observed excitation energies in cm−1. E R C I RMCDHF/RCI energies from [56], E M R M P MRMP energies from [58] and E D Z observed energies from [57]. Relative errors in % for the calculated energies are shown in parenthesis.
Table 8. Comparison of calculated and observed excitation energies in cm−1. E R C I RMCDHF/RCI energies from [56], E M R M P MRMP energies from [58] and E D Z observed energies from [57]. Relative errors in % for the calculated energies are shown in parenthesis.
Fe XIII
Level E RCI E MRMP E DZ
3 s 2 3 p 2 3 P 0  000
3 s 2 3 p 2 3 P 1  9281 (0.237)9295 (0.086)9303.1
3 s 2 3 p 2 3 P 2  18553 (0.048)18576 (0.075)18561.7
3 s 2 3 p 2 1 D 2  48236 (0.344)47985 (1.077)48069.7
3 s 2 3 p 2 1 S 0  91839 (0.357)91508 (0.003)91511.0
3 s 3 p 3 5 S 2 o  214152 (0.220)214540 (0.039)214624.0
3 s 3 p 3 3 D 1 o  287123 (0.028)287199 (0.002)287205.0
3 s 3 p 3 3 D 2 o  287270 (0.029)287348 (0.002)287356.0
3 s 3 p 3 3 D 3 o  290095 (0.029)290179 (0.000)290180.0
3 s 3 p 3 3 P 0 o  328974 (0.014)328980 (0.016)328927.0
3 s 3 p 3 3 P 1 o  329689 (0.015)329702 (0.019)329637.0
3 s 3 p 3 3 P 2 o  330323 (0.012)330334 (0.015)330282.0
3 s 3 p 3 1 D 2 o  362482 (0.020)362416 (0.002)362407.0
3 s 3 p 3 3 S 1 o  415577 (0.027)415519 (0.013)415462.0
3 s 2 3 p 3 d 3 F 2 o  430277 (0.035)430129 (0.001)430124.0
3 s 2 3 p 3 d 3 F 3 o  437064 (0.033)436905 (0.003)436919.0
3 s 3 p 3 1 P 1 o  438365 (0.063)438005 (0.018)438086.0
3 s 2 3 p 3 d 3 F 4 o  447134 (0.029)446959 (0.009)447001.0
3 s 2 3 p 3 d 3 P 2 o  486542 (0.037)486403 (0.009)486358.0
3 s 2 3 p 3 d 3 P 1 o  495102 (0.032)495242 (0.060)494942.0
3 s 2 3 p 3 d 1 D 2 o  499060 (0.038)498925 (0.011)498870.0
3 s 2 3 p 3 d 3 P 0 o  501676 (0.032)501667 (0.030)501514.0
3 s 2 3 p 3 d 3 D 1 o  506661 (0.030)506681 (0.034)506505.0
3 s 2 3 p 3 d 3 D 3 o  509303 (0.024)509479 (0.059)509176.0
3 s 2 3 p 3 d 3 D 2 o  509394 (0.028)509441 (0.037)509250.0
3 s 2 3 p 3 d 1 F 3 o  557432 (0.093)557303 (0.070)556911.0
3 s 2 3 p 3 d 1 P 1 o  571376 (0.110)571187 (0.077)570743.0
Table 9. Transition energies in cm−1 and E1 rates A in s−1 in the length gauge for Fe XXI from RMCDHF/RCI calculations [29]. Relative errors in % for the calculated transition energies and rates are shown in parenthesis. For the transition energies, the relative errors were obtained by comparison with observations from NIST. For the transition rates, the relative errors are estimated from Equation (10).
Table 9. Transition energies in cm−1 and E1 rates A in s−1 in the length gauge for Fe XXI from RMCDHF/RCI calculations [29]. Relative errors in % for the calculated transition energies and rates are shown in parenthesis. For the transition energies, the relative errors were obtained by comparison with observations from NIST. For the transition rates, the relative errors are estimated from Equation (10).
UpperLower Δ E calc A
Fe XXI
2 s 2 p 3 3 D 1 o 2 s 2 2 p 2 3 P 0 776750 (0.049)1.156E+10 (0.00)
2 s 2 p 3 3 P 1 o 2 s 2 2 p 2 3 P 0 925023 (0.120)4.213E+09 (0.07)
2 s 2 p 3 3 S 1 o 2 s 2 2 p 2 3 P 0 1096012 (0.089)9.460E+09 (0.16)
2 s 2 p 3 1 P 1 o 2 s 2 2 p 2 3 P 0 1261529 (0.205)2.850E+07 (1.82)
2 s 2 p 3 5 S 2 o 2 s 2 2 p 2 3 P 1 412701 (0.293)3.597E+07 (8.72)
2 s 2 p 3 3 D 1 o 2 s 2 2 p 2 3 P 1 702930 (0.049)7.606E+08 (1.64)
2 s 2 p 3 3 D 2 o 2 s 2 2 p 2 3 P 1 703550 (0.048)9.240E+09 (0.66)
2 s 2 p 3 3 P 0 o 2 s 2 2 p 2 3 P 1 842581 (0.122)2.200E+10 (0.31)
2 s 2 p 3 3 P 1 o 2 s 2 2 p 2 3 P 1 851203 (0.119)1.602E+10 (0.12)
2 s 2 p 3 3 P 2 o 2 s 2 2 p 2 3 P 1 868735 (0.120)3.820E+08 (1.85)
2 s 2 p 3 3 S 1 o 2 s 2 2 p 2 3 P 1 1022191 (0.133)2.533E+10 (0.11)
2 s 2 p 3 1 D 2 o 2 s 2 2 p 2 3 P 1 1053811 (0.089)3.907E+08 (1.68)
2 s 2 p 3 1 P 1 o 2 s 2 2 p 2 3 P 1 1187709 (0.205)4.909E+09 (0.34)
2 s 2 p 3 5 S 2 o 2 s 2 2 p 2 3 P 2 369157 (0.293)3.272E+07 (11.61)
2 s 2 p 3 3 D 1 o 2 s 2 2 p 2 3 P 2 659387 (0.050)8.335E+07 (6.39)
2 s 2 p 3 3 D 2 o 2 s 2 2 p 2 3 P 2 660006 (0.050)4.535E+06 (22.51)
2 s 2 p 3 3 D 3 o 2 s 2 2 p 2 3 P 2 686197 (0.050)6.105E+09 (0.80)
2 s 2 p 3 3 P 1 o 2 s 2 2 p 2 3 P 2 807659 (0.120)2.681E+09 (1.30)
2 s 2 p 3 3 P 2 o 2 s 2 2 p 2 3 P 2 825192 (0.121)2.038E+10 (0.04)
2 s 2 p 3 3 S 1 o 2 s 2 2 p 2 3 P 2 978647 (0.134)6.104E+10 (0.24)
2 s 2 p 3 1 D 2 o 2 s 2 2 p 2 3 P 2 1010267 (0.090)8.020E+09 (0.39)
2 s 2 p 3 1 P 1 o 2 s 2 2 p 2 3 P 2 1144165 (0.206)2.491E+08 (0.16)
2 s 2 p 3 5 S 2 o 2 s 2 2 p 2 1 D 2 241853 (0.714)1.307E+06 (19.51)
2 s 2 p 3 3 D 1 o 2 s 2 2 p 2 1 D 2 532083 (0.048)1.808E+08 (6.63)
2 s 2 p 3 3 D 2 o 2 s 2 2 p 2 1 D 2 532703 (0.048)3.827E+07 (6.27)
2 s 2 p 3 3 D 3 o 2 s 2 2 p 2 1 D 2 558894 (0.049)9.767E+08 (3.61)
2 s 2 p 3 3 P 1 o 2 s 2 2 p 2 1 D 2 680356 (0.051)2.577E+08 (2.44)
2 s 2 p 3 3 P 2 o 2 s 2 2 p 2 1 D 2 697888 (0.052)1.435E+08 (4.80)
2 s 2 p 3 3 S 1 o 2 s 2 2 p 2 1 D 2 851344 (0.084)3.607E+08 (3.16)
2 s 2 p 3 1 D 2 o 2 s 2 2 p 2 1 D 2 882963 (0.037)4.485E+10 (0.37)
2 s 2 p 3 1 P 1 o 2 s 2 2 p 2 1 D 2 1016862 (0.170)6.583E+10 (0.15)
2 s 2 p 3 3 D 1 o 2 s 2 2 p 2 1 S 0 404698 (0.165)4.070E+07 (2.97)
2 s 2 p 3 3 P 1 o 2 s 2 2 p 2 1 S 0 552971 (0.014)1.492E+08 (7.10)
2 s 2 p 3 3 S 1 o 2 s 2 2 p 2 1 S 0 723959 (0.008)6.511E+08 (2.13)
2 s 2 p 3 1 P 1 o 2 s 2 2 p 2 1 S 0 889477 (0.147)1.727E+10 (0.52)
2 p 4 3 P 2 2 s 2 p 3 5 S 2 o 1159940 (0.180)1.422E+09 (2.39)
2 p 4 3 P 1 2 s 2 p 3 5 S 2 o 1254085 (0.181)2.381E+08 (3.65)
2 p 4 1 D 2 2 s 2 p 3 5 S 2 o 1331255 (0.201)5.282E+07 (2.48)
2 p 4 3 P 2 2 s 2 p 3 3 D 1 o 869710 (0.102)3.455E+09 (0.75)
2 p 4 3 P 0 2 s 2 p 3 3 D 1 o 959080 (0.137)3.474E+10 (0.43)
2 p 4 3 P 1 2 s 2 p 3 3 D 1 o 963855 (0.102)1.434E+10 (0.27)
2 p 4 1 D 2 2 s 2 p 3 3 D 1 o 1041025 (0.103)4.033E+06 (20.67)
2 p 4 1 S 0 2 s 2 p 3 3 D 1 o 1271738 (0.137)2.331E+08 (1.32)
2 p 4 3 P 2 2 s 2 p 3 3 D 2 o 869090 (0.103)1.335E+10 (0.00)
2 p 4 3 P 1 2 s 2 p 3 3 D 2 o 963235 (0.104)2.124E+10 (0.28)
2 p 4 1 D 2 2 s 2 p 3 3 D 2 o 1040406 (0.106)7.878E+08 (0.20)
2 p 4 3 P 2 2 s 2 p 3 3 D 3 o 842899 (0.137)2.822E+10 (0.53)
2 p 4 1 D 2 2 s 2 p 3 3 D 3 o 1014215 (0.260)5.622E+09 (0.85)
2 p 4 3 P 1 2 s 2 p 3 3 P 0 o 824204 (0.038)4.813E+09 (0.02)
2 p 4 3 P 2 2 s 2 p 3 3 P 1 o 721437 (0.039)3.678E+09 (0.32)
2 p 4 3 P 0 2 s 2 p 3 3 P 1 o 810807 (0.040)1.827E+10 (0.93)
2 p 4 3 P 1 2 s 2 p 3 3 P 1 o 815582 (0.043)1.273E+08 (3.29)
2 p 4 1 D 2 2 s 2 p 3 3 P 1 o 892752 (0.085)1.241E+09 (1.85)
2 p 4 1 S 0 2 s 2 p 3 3 P 1 o 1123465 (0.233)4.222E+09 (1.30)
2 p 4 3 P 2 2 s 2 p 3 3 P 2 o 703905 (0.039)3.629E+09 (0.88)
2 p 4 3 P 1 2 s 2 p 3 3 P 2 o 798050 (0.040)1.598E+10 (0.75)
2 p 4 1 D 2 2 s 2 p 3 3 P 2 o 875220 (0.085)2.512E+09 (1.79)
2 p 4 3 P 2 2 s 2 p 3 3 S 1 o 550450 (0.021)6.005E+09 (0.76)
2 p 4 3 P 0 2 s 2 p 3 3 S 1 o 639819 (0.020)1.733E+10 (0.28)
2 p 4 3 P 1 2 s 2 p 3 3 S 1 o 644594 (0.049)1.290E+10 (0.00)
2 p 4 1 D 2 2 s 2 p 3 3 S 1 o 721764 (0.059)7.002E+06 (0.68)
2 p 4 1 S 0 2 s 2 p 3 3 S 1 o 952477 (0.060)5.111E+09 (0.25)
2 p 4 3 P 2 2 s 2 p 3 1 D 2 o 518830 (0.065)7.538E+08 (1.85)
2 p 4 3 P 1 2 s 2 p 3 1 D 2 o 612975 (0.124)3.767E+08 (3.31)
2 p 4 1 D 2 2 s 2 p 3 1 D 2 o 690145 (0.311)3.182E+10 (0.28)
2 p 4 3 P 2 2 s 2 p 3 1 P 1 o 384932 (0.274)8.409E+07 (3.44)
2 p 4 3 P 0 2 s 2 p 3 1 P 1 o 474302 (0.271)9.885E+06 (20.99)
2 p 4 3 P 1 2 s 2 p 3 1 P 1 o 479076 (0.265)6.387E+08 (1.58)
2 p 4 1 D 2 2 s 2 p 3 1 P 1 o 556247 (0.139)4.499E+09 (1.15)
2 p 4 1 S 0 2 s 2 p 3 1 P 1 o 786959 (0.156)7.549E+10 (0.09)
Table 10. Transition rates for Ne-like S. A R C I transition rates from RMCDHF/RCI [43], A B P transition rates from multiconfiguration Hartree-Fock-Breit-Pauli [19] and A C I V 3 transition rates from CI calculations using CIV3 [64].
Table 10. Transition rates for Ne-like S. A R C I transition rates from RMCDHF/RCI [43], A B P transition rates from multiconfiguration Hartree-Fock-Breit-Pauli [19] and A C I V 3 transition rates from CI calculations using CIV3 [64].
States Δ E (cm−1)Type A RCI A BP A CIV 3
UpperLower
S VII
2 p 5 3 s 3 P 2 o 2 p 6 1 S 0 1371667M27.638E+027.617E+02
2 p 5 3 s 3 P 1 o 0.81 + 0.18 1 P 1 o 2 p 6 1 S 0 1376084E11.855E+101.816E+101.989E+10
2 p 5 3 s 1 P 1 o 0.81 + 0.18 3 P 1 o 2 p 6 1 S 0 1388242E18.421E+108.507E+108.777E+10
2 p 5 3 p 3 D 2 0.79 + 0.16 1 D 2 2 p 6 1 S 0 1484530E23.021E+062.964E+06
2 p 5 3 p 3 P 2 0.56 + 0.42 1 D 2 2 p 6 1 S 0 1492576E28.028E+068.008E+06
2 p 5 3 d 3 P 1 o 2 p 6 1 S 0 1624769E12.160E+092.182E+092.312E+09
2 p 5 3 d 3 P 2 o 2 p 6 1 S 0 1627240M21.710E+041.728E+04
2 p 5 3 d 3 D 1 o 2 p 6 1 S 0 1644545E16.122E+106.206E+106.230E+10
2 p 5 3 d 1 P 1 o 2 p 6 1 S 0 1662346E19.452E+119.448E+119.087E+11
2 p 5 3 s 3 P 1 o 0.81 + 0.18 1 P 1 o 2 p 5 3 s 3 P 2 o 4417M11.587E+001.601E+00
2 p 5 3 s 1 P 1 o 0.81 + 0.18 3 P 1 o 2 p 5 3 s 3 P 2 o 16575M11.849E+011.867E+01
2 p 5 3 p 3 S 1 2 p 5 3 s 3 P 2 o 95373E16.393E+086.504E+086.480E+08
2 p 5 3 p 3 D 3 2 p 5 3 s 3 P 2 o 111609E11.566E+091.608E+091.594E+09
2 p 5 3 p 3 D 2 0.79 + 0.16 1 D 2 2 p 5 3 s 3 P 2 o 112863E16.439E+086.627E+086.418E+08
2 p 5 3 p 3 D 1 0.72 + 0.17 1 P 1 2 p 5 3 s 3 P 2 o 116432E11.994E+082.060E+082.023E+08
2 p 5 3 p 3 P 2 0.56 + 0.42 1 D 2 2 p 5 3 s 3 P 2 o 120909E11.000E+091.031E+091.006E+09
2 p 5 3 p 1 P 1 0.55 + 0.27 3 D 1 2 p 5 3 s 3 P 2 o 124214E17.124E+077.315E+076.807E+07
2 p 5 3 p 1 D 2 0.42 + 0.38 3 P 2 2 p 5 3 s 3 P 2 o 127448E12.464E+082.501E+082.832E+08
2 p 5 3 p 3 S 1 2 p 5 3 s 1 P 1 o 0.81 + 0.18 3 P 1 o 78797E11.188E+071.206E+071.253E+07
2 p 5 3 p 3 D 2 0.79 + 0.16 1 D 2 2 p 5 3 s 1 P 1 o 0.81 + 0.18 3 P 1 o 96287E16.945E+067.197E+065.508E+06
2 p 5 3 p 3 D 1 0.72 + 0.17 1 P 1 2 p 5 3 s 1 P 1 o 0.81 + 0.18 3 P 1 o 99856E13.825E+064.290E+064.536E+06
2 p 5 3 p 3 P 2 0.56 + 0.42 1 D 2 2 p 5 3 s 1 P 1 o 0.81 + 0.18 3 P 1 o 104333E12.932E+082.977E+083.152E+08
2 p 5 3 p 1 P 1 0.55 + 0.27 3 D 1 2 p 5 3 s 1 P 1 o 0.81 + 0.18 3 P 1 o 107638E17.415E+087.625E+087.557E+08
2 p 5 3 p 3 P 0 2 p 5 3 s 1 P 1 o 0.81 + 0.18 3 P 1 o 110460E11.908E+081.974E+082.006E+08
2 p 5 3 p 1 D 2 0.42 + 0.38 3 P 2 2 p 5 3 s 1 P 1 o 0.81 + 0.18 3 P 1 o 110872E11.206E+091.243E+091.199E+09
2 p 5 3 p 3 P 1 0.70 + 0.27 1 P 1 2 p 5 3 s 1 P 1 o 0.81 + 0.18 3 P 1 o 112070E17.082E+087.307E+087.000E+08
2 p 5 3 p 3 P 1 0.70 + 0.27 1 P 1 2 p 5 3 s 1 P 1 o 0.81 + 0.18 3 P 1 o 112070M21.139E−011.191E−01
2 p 5 3 p 1 S 0 2 p 5 3 s 1 P 1 o 0.81 + 0.18 3 P 1 o 165149E15.082E+095.118E+095.073E+09
2 p 5 3 d 3 P 1 o 2 p 5 3 p 1 D 2 0.42 + 0.38 3 P 2 125654E11.229E+081.235E+081.323E+08
2 p 5 3 d 3 P 2 o 2 p 5 3 p 1 D 2 0.42 + 0.38 3 P 2 128125E13.010E+083.017E+083.358E+08
2 p 5 3 d 3 P 2 o 2 p 5 3 p 1 D 2 0.42 + 0.38 3 P 2 128125M21.161E−011.175E−01
2 p 5 3 d 3 F 3 o 0.77 + 0.20 1 F 3 o 2 p 5 3 p 1 D 2 0.42 + 0.38 3 P 2 132832E11.471E+061.472E+061.078E+06
2 p 5 3 d 3 F 2 o 0.71 + 0.17 1 D 2 o 2 p 5 3 p 1 D 2 0.42 + 0.38 3 P 2 136157E11.190E+061.370E+061.643E+06
2 p 5 3 d 1 F 3 o 0.53 + 0.40 3 D 3 o 2 p 5 3 p 1 D 2 0.42 + 0.38 3 P 2 138763E11.511E+081.519E+081.131E+08
2 p 5 3 d 3 D 1 o 2 p 5 3 p 1 D 2 0.42 + 0.38 3 P 2 145430E13.071E+063.159E+061.717E+06
2 p 5 3 d 1 D 2 o 0.52 + 0.28 3 F 2 o 2 p 5 3 p 1 D 2 0.42 + 0.38 3 P 2 145447E16.172E+086.286E+086.276E+08
2 p 5 3 d 3 D 3 o 0.57 + 0.27 1 F 3 o 2 p 5 3 p 1 D 2 0.42 + 0.38 3 P 2 146716E14.030E+094.093E+094.183E+09
2 p 5 3 d 3 D 3 o 0.57 + 0.27 1 F 3 o 2 p 5 3 p 1 D 2 0.42 + 0.38 3 P 2 146716M21.029E+001.054E+00
2 p 5 3 d 3 D 2 o 0.65 + 0.27 1 D 2 o 2 p 5 3 p 1 D 2 0.42 + 0.38 3 P 2 147358E11.521E+081.531E+081.489E+08
2 p 5 3 d 1 P 1 o 2 p 5 3 p 1 D 2 0.42 + 0.38 3 P 2 163231E14.854E+074.851E+075.550E+07
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