Ab-Initio Calculations of Level Energies, Oscillator Strengths and Radiative Rates for E1 Transitions in Beryllium-Like Iron

Ahmed Abou El-Maaref; Stefan Schippers; Alfred Müller

doi:10.3390/atoms3010002

,

and

¹

Physics Department, Faculty of Science, Al-Azhar University Assiut, 71524 Assiut, Egypt

²

Institut für Atom- und Molekülphysik, Justus-Liebig-Universität Giessen, D-35392 Giessen, Germany

^*

Author to whom correspondence should be addressed.

Atoms2015, 3(1), 2-52;https://doi.org/10.3390/atoms3010002

This article belongs to the Special Issue Atomic Data for Tungsten

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Abstract

In the present work, energy levels, oscillator strengths, radiative rates and wavelengths of Be-like iron (Fe²²⁺) from ab-initio calculations using the multiconfiguration Dirac-Hartree-Fock method are presented. These quantities have been calculated for a set of configurations in the general form 1s² nl n′l′, where n = 2, 3 and n′ = 2, 3, 4, 5 and l = s, p, d and l′ = s, p, d, f, g. In addition, excitations of up to four electrons, including core-electron excitations, have been considered to improve the quality of the wave functions. This study comprises an extensive set of E1 transition rates between states with different J. The present results are compared with the available experimental and theoretical data.

Keywords:

energy levels; oscillator strengths; radiative rates; be-like iron

1. Introduction

Accurate atomic data for iron ions are of interest in astrophysics, especially for the identification of solar spectra [1–4], as well as in the physics of controlled fusion [5] and plasma diagnostic [6]. From the astrophysical point of view, the importance of iron ions lies in the fact that iron is the cosmically most abundant heavy element beyond silicon [7]. The beryllium isoelectronic sequence including the Fe²²⁺ ion has been studied using different theoretical approaches [8–21]. Most of the earlier calculations have produced results for one- or two-electron excitations to low-lying levels (up to n = 2 or 3). Excitation to high-lying levels was studied by Moribayashi and Kato [22], including configurations up to 2pnl (n ≤ 20, l = s, p, d) using Cowan’s code [23].

Recently, many theoretical calculations have been carried out for Fe²²⁺ to meet the needs for accurate atomic structure data. Chidichimo et al. have calculated level energies for n ≤ 4, as well as wavelengths and weighted oscillator strengths using the Belfast R-matrix programs [24,25]. Del Zanna et al. [26] have compiled experimental observations of energy levels and wavelengths, and performed calculations of weighted oscillator strengths for Be-like iron using the non-relativistic SUPERSTRUCTURE program. Santos et al. [27] calculated probabilities for transitions from the 1s²2s3p ³P₀ level for selected beryllium-like ions, from Z = 5 to 92. They used the MCDF method including relativistic effects, QED (quantum-electrodynamics) effects, and correlations up to the 4f subshell, however, they neglected the Breit interaction. Jian-Hui et al. [28] have calculated energy levels, oscillator strengths, transition probabilities and wavelengths of Fe²²⁺ using the MCDF method with the inclusion of vacuum polarization and Breit interaction. Charro et al. [29,30] have calculated some oscillator strengths of Be-like iron up to n = 2, 3 using the relativistic quantum defect orbital (RQDO) method and the MCDF method, but there were no Breit corrections included in the calculations. By using the FAC code Landi and Gu [31] calculated energy levels, oscillator strengths and transition probabilities for 166 fine-structure levels of Fe²²⁺ belonging to the complexes 1s²2lnl′, n = 2 – 5, l = s, p, d, f, g. In the present work an extensive set of configuration state functions (CSF) including subshell populations up to the 5g subshell is used. In addition, the Breit interaction and QED effects are incorporated, which have been neglected in most of the previous calculations. The present comprehensive treatment of the Fe²²⁺ atomic structure aims at providing more accurate results than hitherto available.

For the present extensive atomic structure calculations of beryllium-like iron, Fe²²⁺, we have used the multiconfiguration Dirac-Fock (MCDF) method [32] as implemented in the GRASP2K code [33]. Excitations from n = 3 to n = 4, 5 (doubly-excited levels) are included, and EOL (extended optimal level) type calculations have been performed. Wavelengths, energy levels, and E1 transition parameters (oscillator strengths, transition probabilities, and line strengths) have been computed for 182 fine-structure levels. The calculations have been divided into two main groups, with even and odd parity. The odd-parity group contains 90 levels while the even-parity group has 92 levels. The present calculations of oscillator strengths and radiative rates are generally in a good agreement with corresponding values in the NIST atomic data compilation [34]. The good agreement between our length and velocity gauge values provides some indication (although not a sufficient one) for the accuracy of the wave functions used in the present study.

2. Method of Calculation

Details of the MCDF method as implemented in the GRASP2K code can be found in References [32,33]. For the nuclear charge distribution within the ⁵⁶Fe nucleus, we used the default Fermi distribution parameters suggested in GRASP2K. The initial estimate for the radial orbitals is generated by solving the Dirac equation in a Thomas-Fermi potential for a single reference configuration (i.e., the 2s² level for even levels and the 2s2p level for odd levels) by allowing the single, double, triple, and quadruple excitations to active orbital sets with n = 2, 3, 4, 5. The self-consistent procedure (RSCF) including EOL type calculations (extended optimal levels) is done layer by layer, at each stage the outer orbitals are optimized. The EOL type calculations construct orbitals from an average energy functional in which the fine-structure levels are given the weight (2J + 1) [30]. This procedure is performed for every J-value separately. The splitting of the atomic levels into different groups has been found to be a useful compromise between two basic requirements in the atomic calculations, first, to get accurate wave functions for the radiative-rate calculations, and second, to keep the procedures manageable even with a large number of CSFs [35]. The same computational method has been applied to the even parity levels. The RSCF calculations were followed by relativistic configuration interaction (RCI) calculations including the Breit interaction Hamiltonian [36]. For the Breit interaction, we used the default low-frequency-limit approximation of the first-order perturbation theory, as implemented in the GRASP2K code. The GRASP2K procedure JJ2LSJ was used for the transformation of ASFs (atomic state functions) from a jj-coupled CSF basis into a LSJ-coupled CSF basis [37].

The beryllium-like iron atomic system has four electrons. In the ground state two electrons reside in the closed K-shell and the other two in the closed 2s subshell. In the first step of building the ground state wave function of the beryllium-like ion, only interactions between the two outer shell electrons were considered. In a second step the interaction with the atomic core was additionally considered since it is very important for the calculation of the wave functions of the excited states. The ground level of the Fe²²⁺ ion is the 1s²2s^{2 1}S₀ level, and the excited levels under consideration in this work belong to 1s² nl n′l′ configurations, where n = 2, 3 and n′ = 2, 3, 4, 5 and l = s, p, d and l′ = s, p, d, f, g with different angular momenta and parities. The open K-shell states are already included in our calculations as admixing correlations, but not explicitly provided. We categorized these levels into groups having the same angular momentum and parity. For example the even parity states with J = 3 are represented by 57161 jj-coupled CSF. As shown in Table 1, the wave function expansions increase rapidly in size by increasing nl which means that we can get unpractically high numbers of CSFs for n > 5. The numbers of CSFs which are generated by quadruple excitations are shown in Table 1 which illustrates the degree of complexity of the present calculations.

Table 1. Number of configuration state functions (CSFs) used in the atomic state function expansion for the given angular momentum and parity (J^P) considering only quadruple excitations.

3. Results and Discussion

The calculated total energies (in a.u.) and energy levels (in eV) are shown in ascending order in Table 2 where also comparisons with literature values [26,34,38,39] are included. Our calculated level energies for Fe²²⁺ are in good agreement with the NIST levels [34]. The relative deviation is generally better than 1.0%, except for the levels (2); 1s²2s2p(³P₀) and (3); 1s²2s2p(³P₁), where the relative deviations from NIST energies are 1.16% and 1.21%, respectively. These deviations may be due to the limited numbers of CSFs in the calculations. However, including more correlations by adding 6l orbitals to the expansion would produce a number of CSFs greater than our computer memory can tolerate. As an illustration, the number of CSFs generated by quadruple (Q) excitation to 6l orbitals with J^P = 2⁺ is 260,702. The energy of level 33, 1s²2p3d(¹D₂), differs from the corresponding NIST [34] value by 0.13%, while the values by Del Zanna et al. [26] and the CHIANTI database [38] deviate from the NIST [34] value by 0.94% and 0.97%, respectively.

Table 2. Total energies E_total (in a.u.) and energy levels (in eV) of Be-like iron (Fe²²⁺). The sixth column lists the present energy levels, and the next 5-columns provide the energies from NIST database [34], the observed energies by Del Zanna et al. [26] and Gu et al. [39], and the calculated (Theor.) and observed (Exp.) values from the CHIANTI database [38].

One might wonder whether quadruple excitations are really necessary. Two examples illustrate the improvement of the results when quadruple excitations are included. The first example is level 14, 1s²2s3p(³P₀), for which the present calculation gives an energy of 1124.70 eV which agrees to within less than 0.2% with the value 1126.47 eV recorded in the CHIANTI database [38]. If only triple excitations are included in the calculations the level energy drops to 1109.07 eV with an increase of the deviation from the CHIANTI energy by as much as 1.5%. The second example, that we want to mention, is level 21, 1s²2p3s(³P₀), which has a calculated energy of 1158.54 eV. This corresponds reasonably well to the NIST energy [34] 1152.40 eV with a relative difference of less than 0.55%. Again, when only triple excitations are considered, the level energy drops drastically, in this case to 1143.12 eV producing a difference in the resulting level energy of 15.42 eV and a relative deviation from the NIST energy of about 0.9%. Obviously, the agreement of the calculations with the CHIANTI and NIST databases becomes considerably better with the use of 4-electron excitations.

Dirac-Fock wave functions with a minimum number of radial functions are not sufficient to represent the occupied orbitals. Extra configurations have to be added to adequately represent electron correlations (i.e., mixing coefficients). These extra configurations are represented by CSFs and must have the same angular momentum and parity as the occupied orbital [40]. For instance, the level

1 s^{2} 2 s 2 p (^{3} P_{0}^{-})

is represented by 0.996 of

1 s^{2} 2 s 2 p (^{3} P_{0}^{-})

and 0.0660 of

1 s^{2} 2 s 3 p (^{3} P_{0}^{-})

. The mixing coefficients for the wave functions of some calculated levels are shown in Table 3. The most important contributions to the total wave function of a given level are those from the same configuration. For example, the configuration-mixed wave function for the 1s²2p3p(³P₀) level is represented as

| 1 s^{2} 2 p 3 p (^{3} P_{0}) ⟩ = 0.9501 | 1 s^{2} 2 p 3 p (^{3} P_{0}) ⟩ + 0.3 | 1 s^{2} 2 p 3 p (^{1} S_{0}) ⟩ - 0.0697 | 1 s^{2} 2 s 3 s (^{1} S_{0}) ⟩

where 0.9501, 0.3, and −0.0697 are the configuration mixing coefficients. Coefficients less than 0.05 were calculated but are not explicitly given. Expansion coefficients for several levels by Bhatia and Mason [10] are listed in Table 3 for comparison. Clearly, the present and the previous [10] results are very close to one another in the description of the configuration-interaction wave functions.

Table 3. The configuration mixing coefficients (> 0.05) for some levels in Fe²²⁺. The number in the bra-kets refers to the level number (the key in Table 2).

A comparison between the calculated wavelengths and other published experimental and theoretical values [1,2,8,24,26,28,29,34,41] is shown in Table 4. The accuracy of calculated wavelengths (in Å) relative to measurements [24,26,34] can be assessed from Table 4, where the agreement is within <0.2% for all available transitions such as 1–5; 2s²(¹S₀)−2s2p(¹P₁) with a calculated wavelength λ = 132.939 Å which deviates from the measurements [24,26,34] by about ±0.0248 Å and from the calculated wavelength of Reference [42] by about ±0.091 Å. In a few cases the agreement with measurements is slightly worse such as for the transitions 2–7, 3–6, 7–21, and 10–25 with λ =147.922, 172.469, 12.025, and 12.371 Å. In these cases the deviations from the measurements amount to 0.463%, 0.485%, 0.579%, and 0.451%, respectively, which are much greater than the experimental uncertainties. The deviations actually reflect the estimated errors in the wavelengths. In Table 6 many more Fe²²⁺ transitions in the soft X-ray region are listed than in any of the previous studies. This will help with the identification of spectral lines of the solar corona [1,2]. In particular, our calculations presented in Table 6 provide comprehensive new data for Be-like Fe with λ in the range 8 to 17 Å and at 132.85 Å which are very important in the solar spectrum [3,4].

Table 4. Comparison between the present calculations of oscillator strengths f_L (calc) and wavelengths (λ_calc in Å) and other published data for some transitions in Be-like iron. LL. = lower level; UL. = upper level, λ_NIST denotes the Ritz wavelength from the NIST database spectra.

Table 6. Oscillator strengths f, wavelengths (in Å), transition probabilities A (in s⁻¹) and line strengths S (in a.u.) for the transitions in Be-like iron. UL. = upper level; LL. = lower level, V and L indicate velocity and length forms.

The calculated values of transition probabilities in s⁻¹, oscillator strengths f, and line strengths S in a.u. (in both velocity and length forms) are listed in Table 6. A comparison between the present calculations of oscillator strengths (f_L) and other published data [8,24,26,29,34,41] is presented in Table 4. Most of our calculations of oscillator strengths show a good agreement with the NIST [34] values, but in a few cases the relative differences reach tens of %. Actually, the NIST values are a compilation of experimental and theoretical data from previous works [8,41,43–47], and the estimated accuracies of most oscillator strengths and transition probabilities recorded in the NIST database [34] are quoted to be between 25% and 50%. For transitions 1–5, 2–7, and 3–6 the uncertainty of the NIST data is estimated to be ≤ 10%. The largest deviation in our calculations of oscillator strengths is found for the transition 3–23 where the deviation from NIST is 36%. Generally, this error is still acceptable for oscillator strengths as is shown in previous publications, where the f_L values for the transition 1–49 in References [24,26] differ from the corresponding NIST [34] value by 35% and 33.25%, respectively, and for the line 5–48 the difference is 20% [26]. Our deviations for the same lines are about 6.75%, 12%, respectively. The transition 1–5 has an experimental f_L-value of 0.15 [48] with an estimated accuracy of 10%. The agreement of our calculated value with this reference [48] is within 2%. In the study by Jian-Hui et al. [28], significant differences compared with the NIST database [34] are revealed for several f_L values, where the deviations of the transitions 5–23, 5–27, and 7–36 are more than one order of magnitude, while our calculations agree with NIST [34] within about 17%, 21.5% and 2.8%, respectively. The precision of the theoretical calculations can be judged by the convergence between length and velocity gauge values of oscillator strengths. If exact wave functions are used then f_L = f_V [49], which is convincingly fulfilled by the approximate wave functions that are used in our calculations. The average deviation between f_L and f_V values is about 30%. There are several transitions with much larger deviations such as, the transitions 2–34, 2–47, 2–97, 3–48, 4–154, 11–116, 18–143, 93–149, 134–156, and 140–152. Comparisons between our transition probabilities and values from the NIST database [34] are available for some transitions. For instance, transitions 1–5, 1–13, and 1–15 in Table 6 have A_L values of 1.91E+10, 4.99E+12 and 7.94E+12 s⁻¹ while the values from the NIST database for these transitions are 2.0E+10, 4.9E+12 and 7.9E+12 s⁻¹, respectively. A comparison between the present GRASP2K calculations of transition probabilities and other published data [28,41,42,50,51] is shown in Table 5.

Table 5. Comparison between the present calculations of transition probabilities (A_L in s⁻¹) and other references.

4. Conclusions

In the present study, fine structure energy levels, oscillator strengths, line strengths, transition probabilities and wavelengths for transitions among levels belonging to Be-like iron are presented. We have used extensive CI wave functions based on large CSF expansions to produce 182 LSJ-coupling levels with various J-values. The self-consistent field approximation and the Breit interaction Hamiltonian as well as QED effects have been included in the calculations to improve the generated wave functions. The calculated energy levels and weighted oscillator strengths show a good agreement with both theoretical and experimental data from the literature. In addition, we have obtained some new and previously unpublished energy levels for this ion. Our results are useful for many applications such as controlled thermonuclear fusion, laser and plasma physics as well as astrophysics.

Author Contributions

Ahmed Abou El-Maaref suggested the main idea of the paper, performed the calculations, and wrote the great part of the manuscript. Alfred Müller and Stefan Schippers were involved in the discussion, particularly, of the astrophysical data needs, provided the required computer hardware, searched the literature and wrote parts of the manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

Supplement

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Supplementary (XLSX, 127 KB)

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Table 1. Number of configuration state functions (CSFs) used in the atomic state function expansion for the given angular momentum and parity (J^P) considering only quadruple excitations.

**Table 1.** Number of configuration state functions (CSFs) used in the atomic state function expansion for the given angular momentum and parity (J^P) considering only quadruple excitations.
J⁺	3l	4l	5l	J⁻	3l	4l	5l
0	211	2149	13,592	0	180	2040	13,302
1	436	5384	36,634	1	460	5476	36,894
2	534	7250	52,481	2	516	7168	52,238
3	380	6930	57,161	3	392	6988	57,354
4	228	5588	53,512	4	222	5540	53,342
5	89	3650	43,358	5	90	3672	43,466

Table 2. Total energies E_total (in a.u.) and energy levels (in eV) of Be-like iron (Fe²²⁺). The sixth column lists the present energy levels, and the next 5-columns provide the energies from NIST database [34], the observed energies by Del Zanna et al. [26] and Gu et al. [39], and the calculated (Theor.) and observed (Exp.) values from the CHIANTI database [38].

**Table 2.** Total energies E_total (in a.u.) and energy levels (in eV) of Be-like iron (Fe²²⁺). The sixth column lists the present energy levels, and the next 5-columns provide the energies from NIST database [34], the observed energies by Del Zanna et al. [26] and Gu et al. [39], and the calculated (Theor.) and observed (Exp.) values from the CHIANTI database [38].
Key	Configuration	J	Parity	E_total	Present	NIST	Reference [26]	Reference [38] Theor.	Reference [38] Exp.	Reference [39]
1	1s²2s²(¹S)	0	+	−812.1123386	0.00	0.00	0.00	0.00	0.00	0.00
2	1s²2s2p(³P )	0	−	−810.5074865	43.67	43.1688	43.16	42.86	43.13	43.219
3	1s²2s2p(³P )	1	−	−810.4056986	46.44	47.0055	47	46.83	46.99	47.046
4	1s²2s2p(³P )	2	−	−809.9784327	58.07	58.4933	58.49	58.16	58.51	58.538
5	1s²2s2p(¹P )	1	−	−808.6849085	93.26	93.2869	93.28	93.78	93.31	93.308
6	1s²2p²(³P )	0	+	−807.7638371	118.33	118.541	118.54	118.27	118.44	118.599
7	1s²2p²(³P )	1	+	−807.4272328	127.49	127.357	127.35	126.86	127.26	127.412
8	1s²2p²(³P )	2	+	−807.2353664	132.71	132.874	132.87	132.87	132.84	132.937
9	1s²2p²(¹D)	2	+	−806.6250012	149.31	149.302	149.3	149.41	149.42	149.341
10	1s²2p²(¹S)	0	+	−805.6167768	176.75	176.38	176.38	176.93	176.42	176.383
11	1s²2s3s(³S)	1	+	−771.5817273	1102.87	1102.72	1105.07	1105.84	1105.10
12	1s²2s3s(¹S)	0	+	−771.2425609	1112.1		1112.77	1114.73	1112.80
13	1s²2s3p(³P )	1	−	−770.8241423	1123.48	1125.28	1125.26	1126.46	1125.20
14	1s²2s3p(³P )	0	−	−770.779389	1124.7			1126.47	1125.18
15	1s²2s3p(¹P )	1	−	−770.679708	1127.41	1129.12	1129.16	1130.22	1129.20
16	1s²2s3p(³P )	2	−	−770.6540603	1128.11		1129.72	1130.64	1129.69
17	1s²2s3d(³D)	1	+	−770.2600664	1138.83	1140.53	1140.43	1141.79	1140.33
18	1s²2s3d(³D)	2	+	−770.2535233	1139.01	1141.77	1140.63	1142.39	1140.96
19	1s²2s3d(³D)	3	+	−770.1730742	1141.2	1142.14	1142.03	1143.35	1141.97
20	1s²2s3d(¹D)	2	+	−769.9289828	1147.84	1149.71	1149.62	1151.31	1149.65
21	1s²2p3s(³P )	0	−	−769.5356786	1158.54	1152.43		1160.33	1159.01
22	1s²2p3s(³P )	1	−	−769.495422	1159.64			1162.56	1161.19
23	1s²2p3p(³D)	1	+	−769.0671809	1171.29	1172.27		1173.74	1172.67
24	1s²2p3s(³P )	2	−	−769.0006032	1173.1		1174.91	1175.29	1174.46
25	1s²2p3s(¹P )	1	−	−768.7864995	1178.93	1174.1	1180.15	1181.23	1180.19
26	1s²2p3p(³D)	2	+	−768.7789536	1179.13	1180.83	1180.8	1181.79	1180.88
27	1s²2p3p(¹P )	1	+	−768.7562887	1179.75	1181.07		1181.94	1181.03
28	1s²2p3p(³P )	0	+	−768.5650136	1184.96			1184.87	1183.43
29	1s²2p3d(³F )	2	−	−768.5068132	1186.54			1189.09	1188.16
30	1s²2p3p(³P )	1	+	−768.3334484	1191.26			1192.9	1192.22
31	1s²2p3d(³F )	3	−	−768.287093	1192.52	1193.35	1193.27	1194.45	1193.11
32	1s²2p3p(³D)	3	+	−768.2775959	1192.78	1193.22	1193.12	1193.59	1193.56
33	1s²2p3d(¹D)	2	−	−768.2540191	1193.42	1194.96	1206.11	1206.61	1195.29
34	1s²2p3p(³S)	1	+	−768.2240512	1194.23			1195.94	1195.17
35	1s²2p3p(³P )	2	+	−768.218499	1194.38	1195.7	1195.64	1196.42	1195.61
36	1s²2p3d(³D)	1	−	−768.1717335	1195.66	1194.84	1198.03	1198.49	1197.51
37	1s²2p3p(¹D)	2	+	−767.9295858	1202.25	1203.76	1203.61	1204.21	1203.62
38	1s²2p3d(³D)	2	−	−767.8457058	1204.53	1206.12	1194.87	1196.01	1204.91
39	1s²2p3d(³F )	4	−	−767.8033102	1205.68		1205.09	1205.33	1206.13
40	1s²2p3d(³D)	3	−	−767.7047396	1208.36	1208.72	1209.19	1209.8	1209.32
41	1s²2p3d(³P )	1	−	−767.6426327	1210.05			1212.27	1211.77
42	1s²2p3d(³P )	2	−	−767.641839	1210.08	1209.22	1211.63	1212.18	1211.80
43	1s²2p3d(³P )	0	−	−767.5905258	1211.47			1212.55	1212.07
44	1s²2p3p(¹S)	0	+	−767.5265103	1213.21			1214.11	1213.37
45	1s²2p3d(¹F )	3	−	−767.3347315	1218.43	1218.8	1218.61	1220.05	1218.89
46	1s²2p3d(¹P )	1	−	−767.3208321	1218.81	1218.52		1221.25	1220.56
47	1s²2s4s(³S)	1	+	−757.6906561	1480.86			1484.44	1482.28
48	1s²2s4s(¹S)	0	+	−757.5602444	1484.4	1485.45	1485.4	1487.3	1485.43
49	1s²2s4p(³P )	1	−	−757.3878963	1489.09	1490.79	1491.06	1492.46	1490.46
50	1s²2s4p(³P )	0	−	−757.370068	1489.58			1492.24	1490.73
51	1s²2s4p(³P )	2	−	−757.3205243	1490.93			1493.93	1492.42
52	1s²2s4p(¹P )	1	−	−757.3072762	1491.29	1493.27	1493.23	1494.55	1493.23
53	1s²2s4d(³D)	2	+	−757.1611615	1495.26	1497.11	1497.07	1498.77	1497.02
54	1s²2s4d(³D)	1	+	−757.1547887	1495.44	1496.86	1496.83	1498.57	1497.25
55	1s²2s4d(³D)	3	+	−757.1038753	1496.82	1497.85	1497.79	1499.15	1497.68
56	1s²2s4d(¹D)	2	+	−757.051647	1498.24	1499.96	1499.93	1501.61	1499.97
57	1s²2s4f(³F )	2	−	−757.029191	1498.85			1502.11	1500.45
58	1s²2s4f(³F )	3	−	−757.0029963	1499.57			1502.21	1500.55
59	1s²2s4f(¹F )	3	−	−756.9746835	1500.34			1503.03	1500.75
60	1s²2s4f(³F )	4	−	−756.9456257	1501.13			1502.39	1501.37
61	1s²2p4s(³P )	1	−	−755.8087174	1532.06			1535.31	1532.53
62	1s²2p4s(³P )	0	−	−755.8024205	1532.24			1534.72	1533.18
63	1s²2p4p(³D)	1	+	−755.6064384	1537.57			1540.47	1538.88
64	1s²2p4p(³D)	2	+	−755.482694	1540.94	1542.74		1543.87	1542.36
65	1s²2p4p(³P )	1	+	−755.4767887	1541.1			1543.64	1542.64
66	1s²2p4d(³F )	2	−	−755.3757501	1543.85			1546.68	1542.87
67	1s²2p4d(³P )	2	−	−755.3004331	1545.9		1547.36	1548.6	1545.15
68	1s²2p4d(³F )	3	−	−755.2586403	1547.03	1547.82	1547.7	1549.2	1547.27
69	1s²2p4d(³D)	1	−	−755.2580034	1547.05	1548.31	1548.22	1549.88	1547.79
70	1s²2p4f(³G)	4	+	−755.241964	1547.49			1551.14	1548.74
71	1s²2p4s(³P )	2	−	−755.229847	1547.82			1550.3	1548.90
72	1s²2p4f(³F )	2	+	−755.2296258	1547.82			1550.98	1549.08
73	1s²2p4f(³G)	3	+	−755.2292762	1547.83			1550.57	1549.50
74	1s²2p4f(³D)	3	+	−755.2093178	1548.37			1551.08	1549.62
75	1s²2p4s(¹P )	1	−	−755.2038121	1548.52			1551.12	1549.62
76	1s²2p4p(¹P )	1	+	−754.9800163	1554.61			1556.65	1549.66
77	1s²2p4p(³P )	2	+	−754.9598815	1555.16			1560.57	1555.67
78	1s²2p4p(³D)	3	+	−754.9258014	1556.09	1557.2	1557.12	1557.48	1556.78
79	1s²2p4p(³S)	1	+	−754.925425	1556.1			1558.15	1556.62
80	1s²2p4p(¹D)	2	+	−754.8394761	1558.44		1557.12	1557.52	1557.12
81	1s²2p4d(¹D)	2	−	−754.7748959	1560.2	1561.83	1561.79	1562.34	1559.90
82	1s²2p4p(¹S)	0	+	−754.7230568	1561.61			1564.05	1561.31
83	1s²2p4d(³D)	3	−	−754.7134465	1561.87	1562.57	1562.54	1563.33	1561.67
84	1s²2p4d(³P )	1	−	−754.7071109	1562.04	1564.06	1564.02	1564.28	1562.55
85	1s²2p4d(³P )	2	−	−754.7010038	1562.21	1563.94	1563.9	1564.28	1563.46
86	1s²2p4d(³F )	4	−	−754.6940971	1562.39			1562.34	1563.70
87	1s²2p4f(³F )	4	+	−754.6817078	1562.73			1565.57	1563.66
88	1s²2p4d(³P )	0	−	−754.6731993	1562.96			1564.42	1563.79
89	1s²2p4f(³F )	2	+	−754.6511138	1563.56
90	1s²2p4f(¹G)	4	+	−754.6473417	1563.67			1566.63	1565.69
91	1s²2p4f(¹F )	3	+	−754.6449993	1563.73			1565.24	1564.31
92	1s²2p4f(³D)	3	+	−754.6115814	1564.64
93	1s²2p4f(³D)	1	+	−754.603951	1564.85			1566.91	1566.04
94	1s²2p4f(¹D)	2	+	−754.5999827	1564.95			1566.07	1565.16
95	1s²2p4d(¹P )	1	−	−754.5893505	1565.24		1563.96	1567.36	1566.94
96	1s²2p4d(¹F )	3	−	−754.5841823	1565.38	1566.04	1565.88	1566.78	1566.04
97	1s²2s5s(³S)	1	+	−751.392181	1652.24			1653.77
98	1s²2s5s(¹S)	0	+	−751.3292255	1653.95		1655.94	1655.09
99	1s²2s5p(³P )	1	−	−751.246659	1656.2		1659.28	1657.83
100	1s²2s5p(³P )	0	−	−751.221321	1656.89			1657.73
101	1s²2s5p(³P )	2	−	−751.2099355	1657.2			1658.66
102	1s²2s5p(¹P )	1	−	−751.2011308	1657.44	1659.28	1659.28	1659.03
103	1s²2s5d(³D)	2	+	−751.1374152	1659.17	1661	1661.34	1661.04
104	1s²2s5d(³D)	1	+	−751.126486	1659.47	1657.54		1660.94
105	1s²2s5d(¹D)	2	+	−751.0866158	1660.56	1666.1	1666.05	1662.22
106	1s²2s5d(³D)	1	+	−751.126486	1659.47		1659.43	1660.94
107	1s²2s5d(³D)	3	+	−751.0860762	1660.57	1661.88	1661.77	1661.23
108	1s²2s5g(³G)	4	+	−751.0857587	1660.58			1662.95
109	1s²2s5g(¹G)	4	+	−751.0826556	1660.66			1663.04
110	1s²2s5f(³F )	2	−	−751.0680262	1661.06			1662.78
111	1s²2s5f(³F )	3	−	−751.041488	1661.78			1662.81
112	1s²2s5f(¹F )	3	−	−751.0238241	1662.26			1663.23
113	1s²2s5g(³G)	3	+	−751.0218288	1662.32			1662.95
114	1s²2s5g(³G)	5	+	−751.0212189	1662.34			1663.02
115	1s²2s5f(³F )	4	−	−750.9786648	1663.49			1662.9
116	1s²2p5s(³P )	1	−	−749.564771	1701.97			1703.76
117	1s²2p5s(³P )	0	−	−749.5389946	1702.67			1703.54
118	1s²2p5p(³D)	1	+	−749.4542569	1704.97			1706.64
119	1s²2p5p(³P )	0	+	−749.4006235	1706.43			1708.4
120	1s²2p5p(³D)	2	+	−749.3994227	1706.47			1708.39
121	1s²2p5p(³P )	1	+	−749.3905489	1706.71			1708.26
122	1s²2p5d(³F )	2	−	−749.3402395	1708.08			1709.85
123	1s²2p5g(³H)	5	−	−749.3026955	1709.1			1712.02
124	1s²2p5d(³P )	2	−	−749.2983472	1709.22			1710.88
125	1s²2p5d(³D)	1	−	−749.2867415	1709.53			1711.22
126	1s²2p5f(³G)	4	+	−749.2824452	1709.65			1712.13
127	1s²2p5f(³F )	2	+	−749.2733538	1709.9			1712.07
128	1s²2p5f(³G)	3	+	−749.2659755	1710.1			1711.84
129	1s²2p5d(³F )	3	−	−749.2653948	1710.11	1711.48		1711.07
130	1s²2p5f(³D)	3	+	−749.2555687	1710.38			1712.11
131	1s²2p5g(³G)	3	−	−749.2352951	1710.93			1711.97
132	1s²2p5g(³H)	4	−	−749.1684219	1712.75			1711.93
133	1s²2p5g(³F )	4	−	−749.1647046	1712.85			1712.05
134	1s²2p5s(³P )	2	−	−748.9871642	1717.68			1719.42
135	1s²2p5s(¹P )	1	−	−748.9732118	1718.06			1719.89
136	1s²2p5p(³D)	2	+	−748.8524328	1721.35
137	1s²2p5p(¹P )	1	+	−748.8511734	1721.38			1723.03
138	1s²2p5p(³S)	1	+	−748.8228875	1722.15			1723.76
139	1s²2p5p(³D)	3	+	−748.8108735	1722.48	1723.88		1723.55
140	1s²2p5p(¹D)	2	+	−748.7936606	1722.95			1723.32
141	1s²2p5d(¹D)	2	−	−748.7499592	1724.14	1726.11		1725.94
142	1s²2p5p(¹S)	0	+	−748.7228871	1724.88			1726.27
143	1s²2p5d(³P )	1	−	−748.7220533	1724.9			1726.75
144	1s²2p5d(³P )	2	−	−748.7152748	1725.08
145	1s²2p5f(³F )	4	+	−748.7106954	1725.21			1727.69
146	1s²2p5d(³D)	3	−	−748.7069321	1725.31	1726.98		1726.34
147	1s²2p5f(³F )	2	+	−748.6949316	1725.64
148	1s²2p5f(¹G)	4	+	−748.6906284	1725.75			1728.16
149	1s²2p5d(³P )	0	−	−748.6802854	1726.03			1726.82
150	1s²2p5f(¹F )	3	+	−748.6721365	1726.26			1727.54
151	1s²2p5f(¹D)	2	+	−748.6702054	1726.31			1727.89
152	1s²2p5d(¹P )	1	−	−748.6683712	1726.36			1728.05
153	1s²2p5g(³F )	2	−	−748.6669093	1726.4			1728.1
154	1s²2p5f(³D)	1	+	−748.6651871	1726.45			1728.25
155	1s²2p5d(³F )	4	−	−748.658763	1726.62			1725.99
156	1s²2p5f(³D)	3	+	−748.6564126	1726.68
157	1s²2p5g(³G)	3	−	−748.6544227	1726.74
158	1s²2p5g(³G)	5	−	−748.6511383	1726.83			1727.75
159	1s²2p5d(¹F )	3	−	−748.6435973	1727.03	1728.96		1727.74
160	1s²2p5g(¹H)	5	−	−748.6425123	1727.06			1727.95
161	1s²2p5g(³F )	3	−	−748.63994	1727.13			1728.18
162	1s²2p5f(³G)	5	+	−748.6322597	1727.34			1727.98
163	1s²2p5g(³G)	4	−	−748.599593	1728.23			1727.87
164	1s²2p5g(³F )	4	−	−748.5922492	1728.43
165	1s²2p4p(³P )	0	+	−746.0926537	1796.45
166	1s²3s²(¹S)	0	+	−729.7643403	2240.75
167	1s²3s3p(³P )	0	−	−729.3728951	2251.4
168	1s²3s3p(³P )	1	−	−729.3646407	2251.63
169	1s²3s3p(³P )	2	−	−729.2408485	2255
170	1s²3s3p(¹P )	1	−	−728.8209355	2266.42
171	1s²3s3d(¹D)	2	+	−728.7962086	2267.1
172	1s²3s3d(³D)	1	+	−728.6774571	2270.33
173	1s²3s3d(³D)	2	+	−728.667205	2270.61
174	1s²3s3d(³D)	3	+	−728.604182	2272.32
175	1s²3p²(³P )	0	+	−728.5677089	2273.31
176	1s²3p²(³P )	1	+	−728.5185534	2274.65
177	1s²3p²(³P )	2	+	−728.4270966	2277.14
178	1s²3p3d(³F )	2	−	−728.2811758	2281.11
179	1s²3p3d(³F )	3	−	−728.1545448	2284.56
180	1s²3p3d(¹D)	2	−	−728.1238826	2285.39
181	1s²3p²(¹D)	2	+	−728.0448765	2287.54
182	1s²3p²(¹S)	0	+	−728.0435435	2287.58

Table 3. The configuration mixing coefficients (> 0.05) for some levels in Fe²²⁺. The number in the bra-kets refers to the level number (the key in Table 2).

**Table 3.** The configuration mixing coefficients (> 0.05) for some levels in Fe²²⁺. The number in the bra-kets refers to the level number (the key in Table 2).
J^P Configuration	Present Work	Reference [10]
J = 0⁺
1s²2s²(¹S)	0.9772 \|1⟩ + 0.2058 \|10⟩	0.98\|1⟩ + 0.21 \|10⟩
1s²2p²(³P)	0.9663 \|6⟩ + 0.2431 \|10⟩ −0.075 \|1⟩	0.97 \|6⟩ + 0.24 \|10⟩ − 0.07 \|1⟩
1s²2p²(¹S)	0.9465 \|10⟩ − 0.2533 \|6⟩ − 0.19241 \|1⟩	0.95 \|10⟩ − 0.25 \|6⟩ − 0.19 \|1⟩
1s²2s3s(¹S)	0.9814 \|12⟩ + 0.1688 \|44⟩ − 0.0708\|48⟩	0.99 \|12⟩ + 0.17 \|44⟩
1s²2p3p(³P)	0.9501 \|28⟩ + 0.3 \|44⟩ − 0.0697 \|12⟩

J = 1⁺
1s²2p²(³P)	0.9992 \|7⟩	1.00 \|7⟩
1s²2s3s(³S)	0.9913 \|11⟩ + 0.1301 \|34⟩	0.99 \|11⟩ + 0.13 \|34⟩
1s²2s3d(³D)	0.9896 \|17⟩ + 0.1399 \|23⟩	0.99 \|17⟩ + 0.14 \|23⟩
1s²2p3p(³D)	0.8202 \|36⟩ + 0.5328 \|27 ⟩ − 0.1612 \|34⟩ + 0.1265 \|17⟩
1s²2p3p(¹P)	0.5302 \|27⟩ + 0.5204 \|34⟩ + 0.4903 \|30⟩ + 0.4449 \|23⟩

J = 2⁺
1s²2p²(³P)	0.8640 \|8⟩ − 0.5014 \|9⟩
1s²2p²(¹D)	0.8637 \|9⟩ + 0.5016 \|8⟩
1s²2s3d(³D)	0.9903 \|18⟩ − 0.1169 \|26⟩
1s²2s3d(¹D)	0.9895 \|20⟩ − 0.1180 \|37⟩
1s²2p3d(³D)	0.9169 \|26⟩ − 0.2744 \|37⟩ + 0.2595 \|35⟩ + 0.1143 \|18⟩

J =3⁺
1s²2s3d(³D)	0.9939 \|19⟩ − 0.0943 \|32⟩
1s²2p3p(³D)	0.9948 \|32⟩ + 0.0939 \|19⟩
1s²2s4d(³D)	0.9976 \|55⟩
1s²2p4f(³G)	0.8423 \|73⟩ + 0.4515 \|91⟩ + 0.2737 \|89⟩

J =4⁺
1s²2p4f(³G)	0.6431 \|70⟩ − 0.5492 \|90⟩ + 0.5300 \|87⟩
1s²2p4f(³F)	0.8133 \|87⟩ − 0.5656 \|70⟩ + 0.1220 \|90⟩
1s²2p4f(¹G)	0.8241 \|90⟩ + 0.5122 \|70⟩ + 0.2327 \|87⟩
1s²2s5g(³G)	0.8292 \|108⟩ +0.5561 \|109⟩

J =5⁺
1s²2s5g(³G)	0.9982 \|114⟩
1s²2p5f(³G)	0.9980 \|162⟩

J =0⁻
1s²2s2p(³P)	0.9960 \|2⟩ + 0.0660 \|14⟩	1.00\|2⟩
1s²2s3p(³P)	0.9908 \|14⟩ − 0.0821 \|43⟩
1s²2p3s(³P)	0.9927 \|21⟩ + 0.0714 \|43⟩
1s²2p3d(³P)	0.9918 \|43⟩ + 0.0860 \|14⟩

J =1⁻
1s²2s2p(³P)	0.9840 \|3⟩ + 0.1564 \|5⟩	0.99 \|3⟩ + 0.16 \|5⟩
1s²2s2p(¹P)	0.9833 \|5⟩ − 0.1568 \|3⟩	0.99 \|5⟩ − 0.16 \|3⟩
1s²2s3p(³P)	0.7772 \|13⟩ + 0.5935 \|15⟩ − 0.1644 \|25⟩	0.80 \|13⟩ + 0.58 \|15⟩ − 0.16 \|25⟩
1s²2s3p(¹P)	0.7411 \|15⟩ − 0.6127 \|13⟩ − 0.238 \|25⟩	0.76 \|15⟩ − 0.59 \|13⟩ − 0.24 \|25⟩

J =2⁻
1s²2s2p(³P)	0.9965 \|4⟩
1s²2s3p(³P)	0.9890 \|16⟩
1s²2p3s(³P)	0.9843 \|24⟩
1s²2p3d(³F)	0.8727 \|29⟩ + 0.4391 \|33⟩ + 0.1375 \|38⟩

J =3⁻
1s²2p3d(³F)	0.8829 \|31⟩ + 0.3724 \|40⟩ − 0.2776 \|45⟩
1s²2p3d(³D)	0.8737 \|40⟩ − 0.4342 \|31⟩ − 0.2094 \|45⟩
1s²2p3d(¹F)	0.9344 \|45⟩ + 0.3064 \|40⟩ + 0.1649 \|31⟩
1s²2s4f(³F)	0.9807 \|58⟩ + 0.1863 \|59⟩

J =4⁻
1s²23d(³F)	0.9970 \|39⟩
1s²2s4f(³F)	0.9983 \|60⟩
1s²2p4d(³F)	0.9973 \|86⟩

J =5⁻
1s²2p5g(³G)	0.8469 \|158⟩ − 0.4368 \|2p5g(³H)⟩ + 0.2964 \|160⟩
1s²3d4f (³H)	0.9974 \|268⟩

Table 4. Comparison between the present calculations of oscillator strengths f_L (calc) and wavelengths (λ_calc in Å) and other published data for some transitions in Be-like iron. LL. = lower level; UL. = upper level, λ_NIST denotes the Ritz wavelength from the NIST database spectra.

**Table 4.** Comparison between the present calculations of oscillator strengths f_L (calc) and wavelengths (λ_calc in Å) and other published data for some transitions in Be-like iron. LL. = lower level; UL. = upper level, λ_NIST denotes the Ritz wavelength from the NIST database spectra.
LL.	UL.	f_L (calc)	f_L Reference [34,38]	Other f_L Values	λ_calc	Other Values of λ	λ_obs	λ_NIST
1	5	1.52E−01	1.55E−011	1.50E−01a, 1.55E−01b, 1.54E−01c	132.939	132.906a, 132.830d	132.906	132.910
1	13	2.73E−01	2.70E−011	2.50E−01a, 2.50E−01b, 2.54E−01c	11.036	11.0063a, 11.018b, 11.017c	11.018	11.020
1	15	4.32E−01	4.30E−011	4.10E−01a, 4.10E−01b, 4.14E−01c	10.997	10.9697a, 10.980b, 10.980c	10.980	10.980
1	22	9.82E−03	1.27E−022	1.30E−02b, 2.11E−02c	10.691	10.6642, 10.6645b, 10.505c
1	36	5.97E−03	5.92E−032	5.00E−03b, 5.91E−03c	10.369	10.3492, 10.3448b, 10.349c		10.377
1	41	2.41E−04	2.32E−042	2.00E−04b, 2.32E−04c	10.246	10.2332, 10.2272b, 10.227c
1	46	1.94E−02	1.90E−022	1.70E−02b, 1.89E−02c	10.172	10.1522, 10.152b, 10.153c		10.175
1	49	3.73E−02	4.00E−021	2.60E−02a, 2.60E−02b, 3.60E−02d	8.326	8.3072a, 8.315b, 8.316d	8.317	8.320
1	52	1.55E−01	1.50E−011	1.40E−01a, 1.40E−01b, 1.54E−01d	8.314	8.2956a, 8.303b, 8.303d	8.305	8.300
1	75	5.55E−04	4.65E−042	3.00E−04b	8.007	7.9932, 7.9931b
1	84	6.79E−05	7.35E−052	6.30E−05b	7.937	7.9272, 7.9258b		7.927
1	95	5.44E−03	0.00E+002	4.60E−03b	7.921	7.9272, 7.9102b
1	102	8.18E−02	6.30E−021	6.30E−02d	7.480	7.480d	7.472	7.470
2	7	6.21E−02	6.43E−021	6.34E−02b, 6.43E−02e	147.922	147.254b, 146.900e	147.240	147.300
2	11	2.37E−02	2.70E−021	2.47E−02c, 1.85E−02f	11.705	11.676c		11.702
2	17	7.47E−01	7.50E−011	7.51E−01c, 7.7E−01d, 6.58E−01f	11.320	11.299c, 11.298d, 11.285g	11.298	11.300
2	54	1.38E−01	1.40E−011	1.30E−01a, 1.30E−01d	8.540	8.529a, 8.528d	8.529	8.530
3	6	1.80E−02	1.85E−021	1.83E−02a	172.469	173.318a	173.310	173.300
3	11	2.43E−02	2.70E−021	2.48E−02c, 1.89E−02f	11.740	11.718c, 11.702g	11.748	11.740
3	17	1.82E−01	1.80E−011	1.23E−01c, 1.65E−01f	11.350	11.338c, 11.366g		11.338
3	23	3.14E−02	4.90E−021	2.50E−01c	11.020	11.003c		11.018
3	54	3.37E−02	3.50E−021	3.23E−02a	8.560	8.551a		8.550
5	23	2.48E−02	3.00E−021	6.46E−06c	11.500	11.113c		11.491
5	27	2.59E−02	3.30E−021	1.06E−03c	11.410	11.033c		11.398
5	48	4.17E−03	3.70E−031	3.70E−03d	8.912	8.910d	8.906	8.910
6	36	1.27E+00	1.29E+001	1.27E+00c, 1.29E+00d	11.508	11.486c, 11.489d	11.519	11.520
6	69	2.34E−01	2.30E−011	2.40E−01a	8.678	8.672a	8.672	8.670
7	21	2.07E−02	1.70E−021		12.025		12.095	12.100
7	36	9.15E−02	8.90E−021	7.45E−04c	11.607	11.822c	11.614	11.610
7	84	5.02E−02	4.90E−021		8.643		8.630	8.630
10	25	6.44E−02	5.50E−021	2.11E−03c	12.371	12.579c	12.427	12.400
10	46	1.27E+00	1.29E+001	1.28E+00c, 1.30E+00d	11.898	11.873c, 11.840d	11.898	11.900
10	75	1.53E−02	1.77E−022	1.80E−02d	9.038	9.0222, 9.028d
10	95	2.37E−01	3.22E−022		8.929	8.9352, 8.920h

^1,2f -values and wavelengths from the NIST [34] and CHIANTI [38] databases, respectively;

^aExperimental observations (for λ) and benchmark calculations (for f_L) by Del Zanna et al. [26];

^bR-matrix calculations by Chidichimo et al. [24,25];

^cMCDF calculations by Jian-Hui et al. [28];

^dCompilation of experimental and theoretical data by Fuhr et al. [41];

^eMCDF calculations by Cheng et al. [8];

^fRQDO calculations by Charro et al. [29];

^gMeasured wavelengths by Brown et al. [1];

^hMeasured wavelengths by Chen et al. [2]; λ_obs Observed wavelengths from NIST database [34]. Note: For numbers in this paper, aEb is a × 10^b.

Table 5. Comparison between the present calculations of transition probabilities (A_L in s⁻¹) and other references.

**Table 5.** Comparison between the present calculations of transition probabilities (A_L in s⁻¹) and other references.
Trans.	Present	Fuhr et al.*	Theoretical	CHIANTI
1–5	1.91E + 10	1.95E+10	1.93E+10a	1.94E+10
1–13	4.99E+12	4.90E+12	4.65E+12b	4.57E+12
1–15	7.94E+12	7.90E+12	7.64E+12b	7.65E+12
2–7	6.31E+09	6.59E+09	6.43E+09c	6.49E+09
2–54	4.19E+12	4.30E+12	1.23E+13d	4.06E+12
3–6	1.21E+10	1.23E+10	1.22E+10c	1.22E+10
3–54	2.97E+12	3.20E+12	2.97E+12d	2.95E+12
3–76	7.03E+11	8.30E+11	1.86E+12d	4.87E+11
6–69	6.91E+12	6.80E+12	2.10E+13d	7.00E+12
7–84	4.42E+12	4.40E+12	4.43E+12d	4.41E+12

^*Compilation of experimental and theoretical data [41];

^aCalculated data by Cheng et al. [50];

^bCalculated data by Jian-Hui et al. [28];

^cCalculated data by Nussbaumer and Storey [42];

^dCalculated data by Wyart et al. [51]; CHIANTI: Compilation of experimental and theoretical data by the CHIANTI atomic database [38].