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

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.


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.

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.

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: where V nuc (r i ) is the nuclear potential modelled from an extended nuclear charge distribution, r ij is the distance between electrons i and j and α and β are the Dirac matrices.Wave functions Ψ(γPJ 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): 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: 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: where H is the RCI matrix of dimension M × M with elements: Once the expansion coefficients have been determined, the radial functions {P nκ (r), Q nκ (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.

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: where ∇ is the gradient operator involving differentiation with respect to r ij = r i − r j and r ij = |r ij | [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.

Managing Large Expansions
To manage large expansions, CSFs can a priori be divided into two groups, referred to as a zeroand 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: where The restriction of H (QQ) 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.

Labelling
In fully-relativistic calculations, quantum labels for the targeted states are obtained in jj-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 jj-coupling is transformed to an approximate representation in LSJ-coupling.This representation is normally more pure and better suited for labelling.One should be aware of the fact that even in LSJ-coupling, the labelling is not straight forward, and several components in the LSJ-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].

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 γPJ and γ P J are computed in terms of reduced matrix elements: 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 γPJ 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.

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.

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.

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 {4s3p2d1 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.

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].

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 2s 2 2p n , 2s2p n+1 and 2p n+2 States in the B-, C-, N-, O-and F-Like Sequences Excitation energies and E1, M1, E2, M2 transition rates between 2s 2 2p n , 2s2p n+1 and 2p 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 Tables 2 and 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].

Level
The transitions connecting the 2s 2 2p 5 3l, 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 2s 2 2p 6 and 2s 2 2p 5 3l configurations to active sets {7s6p5d4 f 3g2h1i}.The 1s 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 LSJ 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].

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 LSJ composition.However, the LSJ 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.Relative errors in % for the calculated energies are shown in parenthesis.E obs observation NIST [37], E RCI energies from RMCDHF/RCI [43] and E MRMP energies from MRMP [44].

Level
LSJ  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 LSJ 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.Table 6.Energies in cm −1 for levels in Fe XXI.E RCI energies from RMCDHF/RCI calculations [47], E RMBPT energies from RMBPT [53], E FAC energies from RCI calculations with FAC [42], E N IST NIST recommended values [37] and E CH I observed energies from the CHIANTI database [3,4].

No.
Level

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 3l3l , 3l4l , 3s5l states in Mg-like Fe where 1s 2 2s 2 2p 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 3l3l configurations are to a larger extent overlapping the core region compared to the charge distributions from the higher states of the 3l4l and 3l5l configurations, leading to a more pronounced energy lowering for the former states.The core polarization, in turn, lowers all energies except the 3s 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 {8s7p6d5 f 4g3h2i}, 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 2s 2 2p 6 core.1s 2 is treated as an inactive core and is always closed.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.

Sequence Ions
Configurations N Ref.
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.
Table 8.Comparison of calculated and observed excitation energies in cm −1 .E RCI RMCDHF/RCI energies from [56], E MRMP MRMP energies from [58] and E DZ observed energies from [57].Relative errors in % for the calculated energies are shown in parenthesis.

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.

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 δA of the calculated transition rates for LS allowed transitions can be estimated according to: where A is the energy-scaled transition rate computed from the observed transition energy (E obs ), δE = |E calc − E obs |/E obs is the relative error in the transition energy and δS = |S len − S vel |/ max(S len , S vel ) 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: 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.

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 δA/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 δA/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.
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).

Upper
Lower ∆E calc A Fe XXI 2s2p

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 δA/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.[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.

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 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.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.

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, 1s 2 nsmp 3 P and 1s 2 npmd 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].

Figure 1 .
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 3 .
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.

Table 4 .
Excitation energies in cm −1 for Ne-like Ca and Fe from observations and different calculations.