Combining Multiconfiguration and Perturbation Methods: Perturbative Estimates of Core–core Electron Correlation Contributions to Excitation Energies in Mg-like Iron

Large configuration interaction (CI) calculations can be performed if part of the interaction is treated perturbatively. To evaluate the combined CI and perturbative method, we compute excitation energies for the 3l3l , 3l4l and 3s5l states in Mg-like iron. Starting from a CI calculation including valence and core–valence correlation effects, it is found that the perturbative inclusion of core–core electron correlation halves the mean relative differences between calculated and observed excitation energies. The effect of the core–core electron correlation is largest for the more excited states. The final relative differences between calculated and observed excitation energies is 0.023%, which is small enough for the calculated energies to be of direct use in line identifications in astrophysical and laboratory spectra.


Introduction
Transitions from highly charged ions are observed in the spectra of astrophysical sources as well as in Tokamak and laser-produced plasmas, and they are routinely used for diagnostic purposes [1].Often, transitions between configurations in the same complex are used, but transitions from higher lying configurations are also important (see, e.g., [2] for a discussion of the higher lying states in the case of Mg-like iron).Transition energies are available from experiments for many ions and collected in various data bases [3], but large amounts of data are still lacking.Although experimental work is aided by a new generation of light sources such as EBITs [4], spectral identifications are still a difficult and time-consuming task.A way forward is provided by theoretical transition energies that support line identification and render consistency checks for experimental level designations.
Much work has been done to improve both multiconfiguration methods and perturbative methods, each with their strengths and weaknesses, in order to provide theoretical transition energies of spectroscopic accuracy, i.e., transition energies with uncertainties of the same order as the ones obtained from experiments and observations using Chandra, Hinode or other space based missions in the X-ray and EUV spectral ranges [5][6][7][8].Further advancements for complex systems with several electrons outside a closed atomic core calls for a combination of multiconfiguration and perturbative methods [9] and also for methods based on new principles [10,11].
In this paper, we describe how the multiconfiguration Dirac-Hartree-Fock (MCDHF) and relativistic configuration interaction (CI) methods can be modified to include perturbative corrections that account for core-core electron correlation.Taking Mg-like iron as an example, we show how the corrections improve excitation energies for the more highly excited states.

Multiconfiguration Dirac-Hartree-Fock and Configuration Interaction
In the MCDHF method [12,13], as implemented in the GRASP2K program package [14], the wave function Ψ(γPJ M J ) for a state labeled γPJ M J , where J and M J are the angular quantum numbers and P is the parity, is expanded in antisymmetrized and coupled configuration state functions (CSFs) The labels {γ j } denote other appropriate information of the configuration state functions, such as orbital occupancy and coupling scheme.The CSFs are built from products of one-electron orbitals, having the general form where χ ±κ,m (θ, ϕ) are 2-component spin-orbit functions.The radial functions {P nκ (r), Q nκ (r)} are numerically represented on a grid.Wave functions for a number of targeted states are determined simultaneously in the extended optimal level (EOL) scheme.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 configuration interaction (CI) problem where H is the CI matrix of dimension M × M with elements In relativistic calculations, the Hamiltonian H is often taken as the Dirac-Coulomb Hamiltonian.Once the expansion coefficients have been determined, the radial functions are improved by solving a set of differential equations results from applying the variational principle on a weighted energy functional of the targeted states together with additional terms needed to preserve orthonormality of the orbitals.The CI problem and the solution of the differential equations are iterated until the radial orbitals and the energy are converged to a specified tolerance.
The MCDHF calculations are often followed by CI calculations where terms representing the transverse photon interaction are added to the Dirac-Coulomb Hamiltonian and the vacuum polarization effects are taken into account by including the Uehling potential.Electron self-energies are calculated with the screened hydrogenic formula [12,15].Due to the relative simplicity of the CI method, often much larger expansions are included in the final CI calculations compared to the MCDHF calculations.

Large Expansions and Perturbative Corrections
The number of CSFs in the wave function expansions depend on the shell structure of the ionic system as well as the model for electron correlation (to be discussed in Section 3).For accurate calculations, a large number of CSFs are required, leading to very large matrices.To handle these large matrices, the CSFs can a priori be divided into two groups.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 group 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 corresponding computational time.The assumptions of the approximation and the connections to the method of deflation in numerical analysis are discussed in [13].This form of the CI matrix, which has been available in the non-relativistic and relativistic multiconfiguration codes for a long time [16,17], yields energies that are similar to the ones obtained by applying second-order perturbation theory (PT) corrections to the energies of the smaller m × m matrix.The method is therefore referred to here as CI combined with second-order Brillouin-Wigner perturbation theory [18].Note, however, that the CI method with restrictions on the interactions gives, in contrast to ordinary perturbative methods, wave functions that can be directly used to evaluate expectation values such as transition rates.

Calculations
Calculations were performed for states belonging to the 3s 2 , 3p 2 , 3s3d, 3d 2 , 3s4s, 3s4d, 3p4p, 3p4 f , 3d4s, 3d4d, 3s5s, 3s5d, 3s5g even configurations and the 3s3p, 3p3d, 3s4p, 3p4s, 3s4 f , 3p4d, 3d4p, 3d4 f , 3s5p, 3s5 f odd configurations of Mg-like iron.For 3d4 f , only states below the 3p5s configuration were included.The above configurations define the multireference (MR) for the even and odd parities, respectively.Following the procedure in [19], an initial MCDHF calculation for all even and odd reference states was done in the EOL scheme.The initial calculation was followed by separate calculations in the EOL scheme for the even and odd parity states.The MCDHF calculations for the even states were based on CSF expansions obtained by allowing single (S) and double (D) substitutions of orbitals in the even MR configurations to an increasing active set of orbitals.In a similar way, the calculations for the odd states were based on CSF expansions obtained by allowing single (S) and double (D) substitutions of orbitals in the odd MR configurations to an increasing active set of orbitals.To prevent the CSF expansions from growing unmanageably large and in order to obtain orbitals that are spatially localized in the valence and core-valence region, at most, single substitutions were allowed from the 2s 2 2p 6 core.The 1s 2 shell was always closed.The active sets of orbitals for the even and odd parity states were extended by layers to include orbitals with quantum numbers up to n = 8 and l = 6, at which point the excitation energies are well converged.
To investigate the effects of electron correlation, three sets of CI calculations were done.In the first set of CI calculations, one calculation was done for the even states and one calculation for the odd states, the SD substitutions were only allowed from the valence shells of the MR, and the CSFs account for valence-valence correlation.In the second set of calculations, SD substitutions were such that there was at most one substitution from the 2s 2 2p 6 core, and the CSFs account for valence-valence and core-valence correlation.In the final set of calculations, all SD substitutions were allowed, and the CSFs account for valence-valence, core-valence and core-core correlation.When all substitutions are allowed, the number of CSFs grows very large.For this reason, we apply CI with second-order perturbation corrections.The CSFs describing valence-valence and core-valence effects (SD substitutions with at most one substitution from the 2s 2 2p 6 core) were included in group P, whereas the CSFs accounting for core-core correlation (D substitutions from 2s 2 2p 6 ) were included in group Q and treated in second-order perturbation theory.The number of CSFs for the different CI calculations are given in Table 1.
Table 1.Number of CSFs for the even and odd parity expansions for the different sets of CI calculations.VV are the expansions accounting for valence-valence correlation, VV+CV are the expansions accounting for valence-valence and core-valence correlation and VV+CV+CC are the expansions accounting for valence-valence, core-valence and core-core correlation.

Results
The excitation energies from the different CI calculations, along with observed energies from the NIST database [3], are displayed in Table 2. From the table, we see that states belonging to 3l3l , with the exception of 3s3p 3 P 0,1,2 , are too high for the valence-valence correlation calculation.The states belonging to 3l4l and 3s5l, on the other hand, are too low.When including also the core-valence correlation, the states belonging to 3l3l go down in energy and approach the observed excitation energies.The states belonging to 3l4l and 3s5l go up and are now too high.Including also the core-core correlation results in a rather small energy change for the states belonging to 3l3l .The main effect of the core-core correlation is to lower the energies of the states belonging to 3l4l and 3s5l, bringing them in very good agreement with observations.The labeling of levels is normally done by looking at the quantum designation of the leading component in the CSF expansion [20].There are two levels (67 and 69) with 3p4d 3 D 3 as the leading component in the corresponding CSF expansion.To distinguish these levels, we added subscripts A and B to the labels of the dominant component.In a similar way, subscripts A and B were added to distinguish levels 78 and 80, both with 3p4 f 3 F 3 as the leading component.
Table 2 indicates that there are a few states that are either misidentified or assigned with a label that is inconsistent with the labels of the current calculation.The observed energy for 3p4 f 3 D 2 (level 84) is 2417 cm −1 too low compared to the calculated value and the observed energy for 3s5s 3 S 1 (level 92) is 33,948 cm −1 too high.There seem to be no other computed energy levels that match the observed energies.The observed energy for 3s5p 1 P o 1 (level 100) is 3733 cm −1 too low.The observed energy matches the computed energy of 3s5p 3 P o 1 (level 97), and, thus, it seems like an inconsistency in the labeling.Finally, 3s5 f 1 F o 3 (level 117) is 101,545 cm −1 too high and there is no other computed energy level that matches.Removing the energy outliers above, the mean relative energy differences are, respectively, 0.217%, 0.051%, 0.023% for the valence, the valence and core-valence and the valence, core-valence and core-core calculations.The energy differences are mainly due to higher-order electron correlation effects that have not been accounted for in the calculations.At the same time, one should bear in mind that the observed excitation energies are also associated with uncertainties as reflected in the limited number of valid digits displayed in the NIST tables.
In Table 3, the excitation energies obtained by including core-core correlation in the CI calculations are compared with energies from calculations by Landi [2] using the FAC code and with energies by Aggarwal et al. [21] using CIV3 in the Breit-Pauli approximation.The uncertainties of the excitation energies for the latter calculations are substantially larger.The calculations by Landi support the conclusion that some of the levels in the NIST database are misidentified.One may note that Landi gives levels 78 and 80 the labels 3p4 f 3 F 3 and 3p4 f 1 F 3 , respectively, whereas Aggarwal et al. reverse the labels.This illustrates that labeling is dependent on the calculation and that the labeling process is far from straightforward [20].
Table 2. Comparison of calculated and observed excitation energies in Mg-like iron (Fe XV).E VV are energies from CI calculations that account for valence-valence correlation.E VV+CV are energies from CI calculations that account for valence-valence and core-valence electron correlation.E VV+CV+CC are energies that account for valence-valence and core-valence electron correlation and where core-core electron correlation effects have been included perturbatively.E N IST are observed energies from the NIST database ( [3]).∆E are energy differences with respect to E N IST .All energies are in cm −1 .

No.
Level

Conclusions
CI with restrictions on the interactions (CI combined with second-order Brillouin-Wigner perturbation theory) makes it possible to handle large CSF expansions.The calculations including core-core correlation take around 20 h with 10 nodes on a cluster and bring the computed and observed excitation energies into very good agreement.To improve the computed excitation energies, the orbital set would need to be further extended leading to even larger matrices.The combined CI and perturbation method can be applied to include core-valence correlation in systems with many valence electrons and calculations.Calculations including valence-valence correlation and where core-valence correlation is treated perturbatively are in progress for P-, S-, and Cl-like systems.

Table 3 .
[3]]arison of calculated and observed excitation energies in Mg-like iron (Fe XV).E VV+CV+CC are energies that account for valence-valence and core-valence electron correlation and where core-core electron correlation effects have been included perturbatively.E FAC are energies by Landi[2]using the FAC code.E CIV3 are energies by Aggarwal et al.[21]using the CIV3 code.E N IST are observed energies from the NIST database ([3]).∆E are energy differences with respect to E N IST .All energies are in cm −1 .