Extended Calculations of Atomic Structure Parameters for Na-like Ar, Kr and Xe Ions Using Relativistic MCDHF and MBPT Methods

Abstract

In this study, comprehensive calculations of energies, hyperfine structure constants, Landé $g_J$ factors and isotope shifts have been performed for the lowest 71 states of Na-like Ar${}^{7+}$, Kr${}^{25+}$ and Xe${}^{43+}$ ions. Radiative parameters viz., wavelengths, transition rates, oscillator strengths and lifetimes are estimated for the electric dipole E1 transitions among these levels. The states under consideration include $1s^{2}2s^{2}2p^{6}nl$ for n = 3–9, l = 0–6, and the fully relativistic multiconfiguration Dirac–Hartree–Fock (MCDHF) method integrated in the latest version of the general-purpose relativistic atomic structure package (GRASP2018) is used for the calculations. The additional corrections, such as the Breit interaction and quantum electrodynamics effects are included in the relativistic configuration interaction calculations, and their effects on energies and other parameters are analysed. We examined the impact of including the core–core and core–valence correlations on level energies. Furthermore, to inspect the reliability of our MCDHF results, we performed another set of calculations using the many-body perturbation theory built into the Flexible Atomic Code (FAC). Moreover, we estimated the uncertainties in the computed lifetimes and transition parameters and assigned their accuracy class. A thorough comparison between the two obtained calculations and with the previous theoretical and experimental results, wherever available, is carried out and a good agreement is observed.

1. Introduction

The chemical inertness of Ar, Kr and Xe makes them befitting candidates for use in tokamaks, fusion plasma research and diagnostics, where all charge states of Ar and highly charged ions (HCIs) of Kr and Xe can exist [1,2,3,4,5]. Their HCIs are of particular interest in plasma research [2,6,7], astrophysics [8,9,10,11] and extreme ultraviolet lithography [12]. Hence, spectroscopic data of HCIs of Ar, Kr and Xe are essential to interpret the spectra correctly and to model the conditions in plasmas containing these species. Therefore, we carried out detailed relativistic calculations of atomic parameters for sodium-like Ar${}^{7+}$, Kr${}^{25+}$ and Xe${}^{43+}$ ions.

Many experimental and theoretical studies have been performed in the past five decades to investigate the level energies and transition rates of these ions. Here we discuss only those previous works with which we have compared our present calculations. Complete lists of references to previous studies on Na-like ions can be obtained from the National Institute of Standards and Technology (NIST) bibliographic databases for level energies [13] and transition parameters [14]. On the theoretical front, the energies of $4s,5s,3p,4p,5p,3d,4d,5d,4f,5f$ and $6f$ states of Na-like ions with Z = 25–80 were calculated by Ivanov and Ivanova [15] with a relativistic model potential method including quantum electrodynamics (QED) effects. To ease the reading, we omit the closed-shell, i.e., $1s^{2}2s^{2}2p^6$, for representing the states of Na-like ions. Theodosiou and Curtis [16] calculated the lifetime of the $3p$ and $3d$ levels of Na-like ions for Z = 11–54, 74, 79, 90 and 92 using the experimental data and multiconfiguration Dirac–Fock (MCDF) method. Siegel et al. [17] applied the single-configuration Dirac–Fock theory to calculate oscillator strength (f) values for the transitions between states with n = 3–5 and $l=s,p,d,f$ of the sodium iso-electronic sequence from Na to Ca. Semi-empirical energies of Na-like Zn XX to Nd L ions are reported by Matsushima et al. [18] for $2p^{6}nl$ states with n = 3–6 and l = 0–6. Applying the Configuration Interaction, Version 3 (CIV3) code of Hibbert, Younis et al. [19] determined the energy levels and electric dipole (E1) transition oscillator strengths of transitions between the n = 3–5 and l = 0–3 levels of Na-like ions up to krypton. Froese Fischer et al. [20] reported the energy levels, transition probabilities and lifetimes of Ar${}^{7+}$ using the MCDF method for n = 3–4 and l = 0–3. Calculations of energies, E1 transitions rates (A) and lifetimes for the five lowest levels of Na-like Xe${}^{43+}$ have been perfomed by Vilkas et al. [21] using the relativistic many-body Møller–Plesset perturbation theory. Liang et al. [22] computed the level energies, A, and weighted oscillator strengths ($gf$) among $nl$–$n^{\prime }l(n,n^{\prime }$ = 3–6 and l = 0–5) levels using the AUTOSTRUCTURE [23] program for the Na-like Mg${}^{+}$ to Kr${}^{25+}$ ions. Sampson et al. [24] used the Dirac–Slater central potential to determine the electronic radial functions, transition energies, f and further employed the relativistic distorted wave theory to calculate collision strengths for the $3l$–$nl$, n = 4–5 transitions of Na-like ions with Z = 22–92. Furthermore, neglecting the relativistic effects, large-scale calculations for allowed transitions between $n\le 10$ and $l\le 4$ levels of Ar${}^{7+}$ were performed under the Opacity Project [25]. However, these results are available only for transitions between $LS$ terms with unresolved fine structure, since they were obtained with a non-relativistic approach.

On the experimental side, Reistad et al. [26] measured the f values for $3s$–$3p$ transition in Ar${}^{7+}$ and also evaluated the lifetimes for $3p{}^2{P}_{1/2,3/2}$. Kink et al. [27] carried out a lifetime measurement of $3p{}^2{P}_{3/2}$ level of Kr${}^{25+}$ using beam-foil excitation and carrying out a cascade-corrected analysis. The lifetimes of $3p$ and $3d$ levels of Xe${}^{43+}$ have been measured by Träbert et al. [28] using the time-resolved spectroscopy of foil-excited ion beams.

Studies on hyperfine structures (HFS), isotope shifts (IS) and Landé $g_J$ factors are scarce, particularly for higher excited levels. Dutta and Majumder [29] used the relativistic coupled-cluster (RCC) theory and calculated the HFS constants $A_J$ and $B_J$ of states with configuration $nl(n$ = 3 and l = 0–2) for Na-like Si to V ions. They also studied the electron correlation effects on HFS. Using the relativistic many-body perturbation theory (RMBPT), Safronova and Johnson [30] derived the field shift (FS) and specific mass shift (SMS) constants for $3s,3p$ and $3d$ levels in sodium-like ions with Z = 12–18 and 26. Tupitsyn et al. [31] reported the FS and SMS for $3p_{1/2}$–$3s_{1/2}$ transition of Na-like Ar${}^{7+}$ and Xe${}^{43+}$ ions by performing large-scale configuration interaction Dirac–Fock (CIDF) calculations. Using the electron beam ion trap (EBIT) facility at NIST, Silwal et al. [32,33] measured the mass shifts (MS) and FS for D1 and D2 transitions between sodium-like ${}^{124}$Xe and ${}^{136}$Xe ions.

We find from the above discussion that, for Ar${}^{7+}$, fine structure energies and radiative parameters of a few higher levels, viz., $7f,7g,7i,8p,8f,8g,8i$ are absent from the literature. For Kr${}^{25+}$, these parameters are available for n = 3–6, $l=s,p,d,f$ levels only. The situation is even worse for Xe${}^{43+}$. Although its level energies are available up to n = 3–6, $l=s,p,d,f$, transition probabilities have been published for only a few transitions among the low-lying states. Thus, an evident scarcity of the atomic structures data for higher excited states of the three Na-like ions can be pictured from the studies mentioned above, especially for Kr${}^{25+}$ and Xe${}^{43+}$.

Therefore, to fulfil the growing demand for electronic structure data of inert gas ions in the current fusion research projects [2], we carry out fully relativistic calculations for energies, E1 transition parameters and lifetimes for the lowest 71 levels of Na-like Ar${}^{7+}$, Kr${}^{25+}$ and Xe${}^{43+}$ ions in this work. Furthermore, we also determine the HFS constants $A_J$ and $B_J$, Landé $g_J$ and IS factors of these levels. For this purpose, we use the MCDHF procedure incorporated in the latest version of the GRASP2018 [34]. The Breit interaction, QED effects, normal and specific mass shift corrections are also taken into consideration to enhance the accuracy of the calculated parameters. The configurations of our interest include $1s^{2}2s^{2}2p^{6}nl$, where n = 3–9 and l = 0–6. The present study adds new results for n = 7, 8, 9 levels of Kr${}^{25+}$ and Xe${}^{43+}$ and remarkably improves the amount of atomic data for these ions. Due to lack of existing data for $n\ge 7$ and further, to examine the reliability of our results, we have performed separate calculations using the many-body perturbation theory (MBPT) inbuilt in the FAC [35,36]. A good agreement is achieved by comparing the two results and also with the available previous results.

2. Computational Procedures

2.1. The MCDHF−RCI Method

A detailed explanation of the MCDHF method is available in Grant’s book [37]. In this method, the relativistic atomic state functions (ASFs) are given as linear combinations of configuration state functions (CSFs). These CSFs are built from the anti-symmetric products of Dirac’s one-electron orbitals.

The states of interest are grouped according to their parity into the even and odd sets. Independent calculations in the extended optimal level (EOL) scheme are performed for these sets. The multi-reference (MR) set for odd states includes 34 levels of the configurations $1s^{2}2s^{2}2p^{6}nl$ for n = 3–9, $l=p,f,h$. Similarly, the even parity states contain 37 levels pertaining to the configurations $1s^{2}2s^{2}2p^{6}nl$ for n = 3–9, $l=s,d,g,i$, which constitute the even MR set.

The relativistic self-consistent field (RSCF) approach optimizes the radial part of the Dirac orbitals and the expansion coefficients to self-consistency. These calculations are first performed on the MR sets. In the next step, expansion of CSFs is completed in a restrictive active space approach, by single and double substitutions of electrons from the MR set orbitals to higher orbitals upto $n=11$ and $l=s-i$ to form the active space (AS). The $1s$ orbital is considered as inactive core, whereas the $2s$ and $2p$ orbitals are taken as the active core. The core–valence (CV) correlations are considered by allowing only single excitations from $2s^{2}2p^6$ orbitals of the MR set to the AS for all the three ions. To monitor the convergence of the computed electronic structure parameters, the active space is divided into layers of the same n and all the corresponding orbital quantum numbers up to $l=6$. So, we define the AS’s as,

$$\begin{array}{c}\hfill AS\left\{9\right\}\left(MR\right)=\left(nl\right)\forall n=3–9\mathrm{and}l=0–\min (n-1,6)\\ \hfill AS\left\{10\right\}=AS\left\{9\right\}+(10s,10p,10d,10f,10g,10h,10i)\\ \hfill AS\left\{11\right\}=AS\left\{10\right\}+(11s,11p,11d,11f,11g,11h,11i)\end{array}$$

For each AS, only the outer orbitals are optimized, while the inner ones are kept frozen. The numbers of CSFs generated are given in

Table 1. The number of CSFs generated for each AS considering the CV correlations for Ar${}^{7+}$, Kr${}^{25+}$ and Xe${}^{43+}$.

Active Space	Number of CSFs
	Even	Odd
AS{10}	53,352	47,191
AS{11}	71,820	63,339

.

Furthermore, we incorporate the leading-order corrections, such as transverse photon exchange, QED effects and nuclear recoil corrections (specific and normal mass corrections) [38] while performing the relativistic configuration integration (RCI) calculations. Moreover, due to the non-orthogonality of initial and final states, biorthogonal transformations are performed on the odd and even sets to remove the complexities in estimating transition probabilities.

To see the correlation effects, another set of calculations is performed only for Xe${}^{43+}$ by considering the core–core (CC) correlations. In this approach, all possible excitations are allowed from the $2s$ and $2p$ orbitals to AS orbitals with a restriction that correlation orbitals will always have at least two electrons. The number of CSFs generated are given in

Table 2. The number of CSFs generated for each AS considering the CC correlations for Xe${}^{43+}$.

Active Space	Number of CSFs
	Even	Odd
AS{10}	726,101	853,388
AS{11}	933,241	1,099,439

. It can be seen that the numbers of CSFs in this case are very large, making the computations time-consuming and constraining the computational resources. To mitigate such a situation, we rearranged the CSFs into the zero- and first-order spaces. The zero-order space consists of the most important CSFs, which primarily constitute the CSFs formed from the MR sets. The first-order space includes almost all the CSFs generated by excitation to virtual orbitals and are less important. We consider the full interaction of the CSFs in the zero-order space, while only the diagonal interaction due to the CSFs in the first-order space are included.

Furthermore, the specific mass shift, normal mass shift and field shift are calculated using the relativistic isotope shift (RIS4) code [39]. Though it is a subprogram written for GRASP2K [40], we performed the required modifications to use it in alliance with the GRASP2018 code.

2.2. The MBPT Approach

We performed the MBPT calculations as an additional accuracy test of our results. This method is available in the FAC package [35,36] and has successfully provided atomic data of high accuracy [41,42,43]. The details of MBPT can be found in [44,45,46,47]. In this method, the Hilbert space of the Hamiltonian is divided into the M and N orthogonal spaces. Here, M represents the model space containing the non-Hermitian effective Hamiltonian, and N includes the perturbation expansion. The electron correlation effects are accounted for within the M space, while the perturbation method is applied to work out the interaction between the M and N model spaces. In the present work, the $2s^{2}2p^{6}nl$ configurations with $n=3$ to 9 and the corresponding orbital quantum numbers (up to $l=6$) are considered in the model space, M. All the possible configurations originated by single–double $\left(SD\right)$ substitutions from the M space in the N space are considered. The N space includes virtual orbitals with $n\le 125$ and $l\le$ min$(n-1,25)$. The inner and outer electrons are substituted up to $n=50$ and $n=125$, respectively. Furthermore, the leading-order QED corrections, such as vacuum polarization and self-energy are also taken into account in addition to the Dirac–Coulomb Hamiltonian.

3. Results

3.1. Energy Levels

We carried out systematic MCDHF—RCI calculations for AS{9} to AS{11} to determine the energies of the lowest 71 levels of Na-like Ar${}^{7+}$, Kr${}^{25+}$ and Xe${}^{43+}$. The corresponding AS label defines the calculated energies for each active space. Further, to examine the importance of the Breit and QED corrections, we performed another set of calculations for AS{11} without incorporating these corrections (labeled as RSCF). For simplicity, AS{9} and AS{11} are referred to as MR and RCI. Furthermore, to ascertain the accuracy of our results, independent calculations were performed using the MBPT approach. Both results (RCI and MBPT) are presented in

Table 3. The present RCI and MBPT energies (in cm${}^{-1}$) for the lowest 71 levels of Ar${}^{7+}$, Kr${}^{25+}$ and Xe${}^{43+}$.

Level Index	Level	Ar${}^{7+}$		Kr${}^{25+}$		Xe${}^{43+}$
		RCI	MBPT	RCI	MBPT	RCI	MBPT
0	$3s{}^2{S}_{1/2}$	0	0	0	0	0	0
1	$3p{}^2{P}_{1/2}$	139,953	140,274	454,134	455,455	801,367	808,237
2	$3p{}^2{P}_{3/2}$	142,666	142,891	558,960	559,481	1,496,482	1,502,292
3	$3d{}^2{D}_{3/2}$	333,132	332,432	1,163,042	1,165,044	2,516,981	2,523,955
4	$3d{}^2{D}_{5/2}$	333,243	332,517	1,183,462	1,184,507	2,672,886	2,678,868
5	$4s{}^2{S}_{1/2}$	576,777	574,152	4,491,857	4,490,602	12,252,891	12,256,273
6	$4p{}^2{P}_{1/2}$	628,927	626,667	4,677,700	4,676,217	12,588,459	12,590,454
7	$4p{}^2{P}_{3/2}$	629,938	627,633	4,719,395	4,717,747	12,870,945	128,72,672
8	$4d{}^2{D}_{3/2}$	698,340	695,384	4,945,424	4,943,817	13,257,895	13,259,163
9	$4d{}^2{D}_{5/2}$	698,396	695,430	4,954,338	4,952,453	13,325,520	13,326,499
10	$4f{}^2{F}_{5/2}$	717,886	715,163	5,066,255	5,064,743	13,531,106	13,532,278
11	$4f{}^2{F}_{7/2}$	717,915	715,189	5,069,743	5,068,164	13,559,901	13,560,874
12	$5s{}^2{S}_{1/2}$	808,230	804,477	6,456,747	6,453,990	17,699,409	17,699,727
13	$5p{}^2{P}_{1/2}$	833,159	829,603	6,549,823	6,546,785	17,868,528	17,867,898
14	$5p{}^2{P}_{3/2}$	833,643	830,065	6,570,566	6,567,459	18,010,338	18,009,552
15	$5d{}^2{D}_{3/2}$	866,223	862,258	6,680,518	6,677,411	18,199,690	18,198,589
16	$5d{}^2{D}_{5/2}$	866,252	862,284	6,685,604	6,681,897	18,234,629	18,233,351
17	$5f{}^2{F}_{5/2}$	876,343	872,354	6,740,695	6,737,534	18,336,031	18,334,769
18	$5f{}^2{F}_{7/2}$	876,358	872,367	6,742,477	6,739,282	18,350,816	18,349,449
19	$5g{}^2{G}_{7/2}$	877,077	873,415	6,749,310	6,746,304	18,365,271	18,363,833
20	$5g{}^2{G}_{9/2}$	877,087	873,424	6,750,267	6,747,365	18,374,078	18,372,607
21	$6s{}^2{S}_{1/2}$	924,460	920,093	7,490,604	7,485,851	20,586,459	20,581,083
22	$6p{}^2{P}_{1/2}$	938,248	934,033	7,543,326	7,539,380	20,683,308	20,681,117
23	$6p{}^2{P}_{3/2}$	938,517	934,288	7,555,064	7,551,140	20,764,316	20,762,103
24	$6d{}^2{D}_{3/2}$	956,611	952,137	7,616,898	7,613,010	20,871,265	20,868,869
25	$6d{}^2{D}_{5/2}$	956,627	952,154	7,620,748	7,615,620	20,891,528	20,889,039
26	$6f{}^2{F}_{5/2}$	962,479	957,976	7,651,212	7,647,270	20,949,042	20,946,584
27	$6f{}^2{F}_{7/2}$	962,488	957,982	7,652,255	7,648,282	20,957,618	20,955,092
28	$6g{}^2{G}_{7/2}$	962,952	958,542	7,656,625	7,652,685	20,966,905	20,964,212
29	$6g{}^2{G}_{9/2}$	962,957	958,547	7,657,180	7,653,299	20,972,000	20,969,289
30	$6h{}^2{H}_{9/2}$	963,006	958,829	7,657,613	7,653,794	20,972,634	20,970,072
31	$6h{}^2{H}_{11/2}$	963,009	958,833	7,658,023	7,654,204	20,976,027	20,973,456
32	$7s{}^2{S}_{1/2}$	991,046	986,382	8,100,962	8,095,864	22,298,896	22,295,426
33	$7p{}^2{P}_{1/2}$	999,464	994,886	8,133,773	8,128,925	22,359,317	22,356,355
34	$7p{}^2{P}_{3/2}$	999,629	995,042	8,140,625	8,136,226	22,409,872	22,406,788
35	$7d{}^2{D}_{3/2}$	1,010,721	1,005,973	8,179,357	8,174,518	22,476,203	22,473,056
36	$7d{}^2{D}_{5/2}$	1,010,734	1,005,984	8,180,519	8,176,163	22,488,972	22,485,744
37	$7f{}^2{F}_{5/2}$	1,014,418	1,009,649	8,200,303	8,195,889	22,524,753	22,521,566
38	$7f{}^2{F}_{7/2}$	1,014,424	1,009,654	8,200,911	8,196,526	22,530,156	22,526,928
39	$7g{}^2{G}_{7/2}$	1,014,733	1,009,973	8,203,697	8,199,412	22,536,330	22,532,942
40	$7g{}^2{G}_{9/2}$	1,014,737	1,009,976	8,204,182	8,199,798	22,539,536	22,536,139
41	$7h{}^2{H}_{9/2}$	1,014,771	1,010,151	8,204,464	8,200,126	22,540,016	22,536,692
42	$7h{}^2{H}_{11/2}$	1,014,773	1,010,153	8,204,780	8,200,384	22,542,152	22,538,824
43	$7i{}^2{I}_{11/2}$	1,014,781	1,010,287	8,204,719	8,200,534	22,542,218	22,538,981
44	$7i{}^2{I}_{13/2}$	1,014,783	1,010,289	8,204,902	8,200,719	22,543,742	22,540,502
45	$8s{}^2{S}_{1/2}$	1,032,739	1,027,889	8,490,600	8,485,550	23,397,105	23,393,244
46	$8p{}^2{P}_{1/2}$	1,038,247	1,033,457	8,512,288	8,507,495	23,437,259	23,433,755
47	$8p{}^2{P}_{3/2}$	1,038,355	1,033,559	8,517,176	8,512,341	23,470,900	23,467,310
48	$8d{}^2{D}_{3/2}$	1,045,646	1,040,740	8,542,410	8,537,688	23,514,880	23,511,238
49	$8d{}^2{D}_{5/2}$	1,045,657	1,040,747	8,543,389	8,538,790	23,523,433	23,519,737
50	$8f{}^2{F}_{5/2}$	1,048,124	1,043,203	8,556,678	8,551,915	23,547,210	23,543,548
51	$8f{}^2{F}_{7/2}$	1,048,127	1,043,206	8,556,963	8,552,341	23,550,832	23,547,141
52	$8g{}^2{G}_{7/2}$	1,048,342	1,043,400	8,558,975	8,554,321	23,555,103	23,551,282
53	$8g{}^2{G}_{9/2}$	1,048,344	1,043,402	85,59,249	8,554,580	23,557,251	23,553,423
54	$8h{}^2{H}_{9/2}$	1,048,368	1,043,512	8,559,399	8,554,796	23,557,605	23,553,811
55	$8h{}^2{H}_{11/2}$	1,048,370	1,043,513	8,559,704	8,554,970	23,559,036	23,555,240
56	$8i{}^2{I}_{11/2}$	1,048,375	1,043,608	8,559,628	8,555,078	23,559,089	23,555,355
57	$8i{}^2{I}_{13/2}$	1,048,376	1,043,609	8,559,724	8,555,202	23,560,110	23,556,374
58	$9s{}^2{S}_{1/2}$	1,072,059	1,055,602	8,754,502	8,749,676	24,143,209	24,139,044
59	$9p{}^2{P}_{1/2}$	1,064,363	1,059,445	8,769,806	8,764,978	24,171,188	24,167,329
60	$9p{}^2{P}_{3/2}$	1,064,438	1,059,515	8,773,532	8,768,357	24,194,694	24,190,767
61	$9d{}^2{D}_{3/2}$	1,069,484	1,064,486	8,790,833	8,786,005	24,225,352	24,221,384
62	$9d{}^2{D}_{5/2}$	1,069,493	1,064,491	8,794,390	8,786,779	24,231,371	24,227,351
63	$9f{}^2{F}_{5/2}$	1,071,225	1,066,215	8,800,845	8,795,952	24,247,965	24,243,983
64	$9f{}^2{F}_{7/2}$	1,071,227	1,066,218	8,800,977	8,796,251	24,250,379	24,246,506
65	$9g{}^2{G}_{7/2}$	1,071,382	1,066,342	8,802,584	8,797,664	24,253,578	24,249,467
66	$9g{}^2{G}_{9/2}$	1,071,383	1,066,344	8,802,626	8,797,845	24,255,086	24,250,970
67	$9h{}^2{H}_{9/2}$	1,071,403	1,066,414	8,802,844	8,797,992	24,255,353	24,251,247
68	$9h{}^2{H}_{11/2}$	1,071,404	1,066,415	8,803,049	8,798,114	24,256,358	24,252,250
69	$9i{}^2{I}_{11/2}$	1,071,407	1,066,484	8,802,971	8,798,192	24,256,401	24,252,334
70	$9i{}^2{I}_{13/2}$	1,071,408	1,066,484	8,803,056	8,798,279	24,257,118	24,253,050

for all the three ions. For Ar${}^{7+}$, except for the 9s state, the present RCI energies agree well with MBPT energies. The average deviations between the two energy results are 0.42%, 0.05% and 0.04% for Ar${}^{7+}$, Kr${}^{25+}$ and Xe${}^{43+}$, respectively. Thus, the two calculations are in excellent agreement. A more detailed analysis of the present results for each ion are presented in the following subsections.

3.1.1. Ar${}^{7+}$

For Ar${}^{7+}$, the variation in energies with AS is examined in Figure 1a by plotting the energy difference between the consecutive AS. The maximum difference reduced from 4000 cm${}^{-1}$ for AS{10}−MR to 2000 cm${}^{-1}$ for RCI−AS{10}. In terms of percentage variation, we found that mean relative deviation between RCI and AS{10} is only 0.22%, which indicates a good convergence of the results.

We further compared the present energy values for Ar${}^{7+}$ with the MCDHF results for 12 levels reported by Froese Fischer et al. [20], AUTOSTRUCTURE calculations of Liang et al. [22] for 32 levels, and the corresponding results available in the NIST Atomic Spectra Database (ASD) [48]. The deviation of the present (MR, AS{10}, RCI, RSCF and MBPT) and previous [20,22] results relative to the NIST values are also shown in Figure 1b (excluding the MCDHF−RCI calculated 9s level with level index 58). We find that increasing AS has a constructive effect on reducing the discrepancies in level energies with mean absolute deviations of 0.45%, 0.16%, 0.13% and 0.16% for MR, AS{10}, RCI and RSCF results, respectively. In the present study, the mean absolute deviation refers to the arithmetic mean of absolute values of the deviations. The present RCI values show a maximum difference of 1.17% for the 9s state but differ from the NIST reference values by less than 0.15% for all the other states. The results of Froese Fischer et al. [20] and Liang et al. [22] have average variations of 0.31% and 0.33%, respectively, compared to the NIST values. Furthermore, the present MBPT results deviate by 0.33% relative to the NIST data. The opacity database [25] also provides center-of-gravity energies for $LS$ terms. Therefore, we calculated centre of gravity of our RCI, MBPT and NIST fine structure energies. On comparing the RCI, MBPT and opacity [25] results with respect to the NIST values, we found that the RCI energies are in the best agreement with the NIST data, whereas energies from [25] show an absolute mean discrepancy of 0.74% and a maximum difference of about 2%. For the sake of brevity, we did not display graphical comparison of these results.

Thus, our results are more precise, as evident from Figure 1b, except for the 9s level. The present MCDHF−RCI energy for this level comes out to be greater than that for $9p$. Although we made efforts to obtain a correct energy for the 9s level, for example, by considering different integration methods, we have been unable to detect the cause of this error. Therefore, the RCI energy of 9s in Table 3 should be considered incorrect. Our results for almost all the levels are more accurate than those of [20,22,25]. Moreover, we also provide energies of a few fine structure levels that are absent in the NIST database.

3.1.2. Kr${}^{25+}$

For Kr${}^{25+}$, the impact of introducing several layers of AS in reducing the energy difference and improving the present results is evident from Figure 1c. The difference between AS{10} and MR goes up to 300,000 cm${}^{-1}$, while it is reduced to 8000 cm${}^{-1}$ for RCI−AS{10}. The average relative deviation between the RCI and AS{10} is only 0.06%. Hence, a remarkable convergence is achieved in the case of Kr${}^{25+}$. Furthermore, the relative difference in energies of the present and previous [19,22] results with respect to the NIST values is displayed pictorially in Figure 1d. To distinguish the different results, we did not include the MR values in Figure 1d, as the maximum deviation is found to be 6% for this set, although the mean deviation is 0.88% only. It is noticeable from Figure 1d that the present calculations converge well towards the NIST values with an increase in the active space and show the mean differences of 0.091% and 0.032% for AS{10} and RCI, respectively. The mean agreement with the NIST results improves from 0.15% to 0.032% as we move from RSCF to RCI calculations. Especially for the lowest four levels, the RCI calculations introduce a remarkable rectification in reducing the energy deviation from 0.7% to 0.05%. In our MBPT calculations, the mean discrepancy is only 0.079% relative to the NIST values. Younis et al. [19] and Liang et al. [22] have mean deviations of 0.43% and 0.14%, respectively, from the NIST ASD [48], which are greater than for our results. Moreover, energies reported in [19] for a few levels show exceptionally large differences. For example, the energy for the $3p{}^2{P}_{1/2}$ state strays by 3.8% in [19], whereas the corresponding values deviate by only 0.06%, 0.23% and 1.37% in the present RCI, present MBPT and AUTOSTRUCTURE [22] calculations, respectively. An excellent agreement of our calculated energies for Kr${}^{25+}$ ion with the NIST ASD [48] assists in establishing the reliability of the other electronic structure parameters determined in this work. Moreover, the results for $1s^{2}2s^{2}2p^{6}nl$ levels, where n = 7–9 and l = 0–6, are reported for the first time.

3.1.3. Xe${}^{43+}$

For Xe${}^{43+}$, the pattern of convergence with increasing AS is analysed in Figure 1e. The energy difference between the successive AS layers reduced considerably for all the levels. Its maximum value drops from 8000 cm${}^{-1}$ for AS{10}−MR to 800 cm${}^{-1}$ for RCI−AS{10}. In terms of relative deviation between RCI and AS{10}, the mean value is 0.004%. This greatly assists in proving the convergence of our calculations. Similar to Ar${}^{7+}$ and Kr${}^{25+}$, we show the relative percentage deviation of the present and previous works [15,21] with respect to the NIST values in Figure 1f. Clearly, the agreement with the NIST results improves with increasing AS except for the first four levels, where the deviation in the MR is the least. However, the absolute mean deviation is only 0.08% in the case of RCI calculations compared to those obtained for MR (0.1%), AS{10} (0.09%) and RSCF (0.22%). Therefore, we find that the quality of the wave functions and hence, the electronic parameters, improve by including the QED effects and other corrections.

The results of Ivanov and Ivanova [15] and Vilkas et al. [21] have a mean deviation of 0.20% and 0.03%, respectively, with respect to the NIST data [48]. The present RCI results deviate by 0.31% and 0.34% with respect to the energies obtained from these two theoretical works [15,21] and hence, are in good agreement. Overall, we can say that the present calculations provide fairly precise energies along with the new results for 43 levels for Xe${}^{43+}$.

We have also calculated the level energies for Xe${}^{45+}$ using the CC correlations. However, the computed energies have a mean deviation of 0.25% with respect to NIST ADS [48]. This shows that the CC energies become degraded compared to the CV correlations; therefore, these results are not reported here.

Furthermore, in Figure 2a, the difference between the RSCF and RCI energies is presented for Ar${}^{7+}$, Kr${}^{25+}$ and Xe${}^{43+}$ to demonstrate the influence of Breit corrections, QED effects and nuclear recoil correction with increasing nuclear charge. Their effects are clearly seen to be most significant for xenon. The energies are improved by a maximum of 33,797 cm${}^{-1}$ for Xe${}^{43+}$. The largest difference of around 8000 cm${}^{-1}$ between the RSCF and RCI energies is found in Kr${}^{25+}$, whereas for Ar${}^{7+}$, it is only 250 cm${}^{-1}$. Overall, the increasing role of RCI calculations in heavy ions is affirmed by this study.

Moreover, to illustrate the role of Breit interaction in obtaining accurate energies, we also performed another set of calculations for Xe${}^{43+}$ by excluding the Breit interaction and including all the other leading-order corrections (QED and nuclear recoil). In Figure 2b, we present the difference between the present RSCF, RCI and RCI without Breit interaction energies with respect to the NIST ASD [48]. The error bars show the uncertainty in the NIST values. Except for $3p{}^2{P}_{3/2},3d{}^2{D}_{3/2},4s{}^2{S}_{1/2},4p{}^2{P}_{1/2}$ levels, the RCI results are in better agreement with the NIST [48] data compared to those obtained without including the Breit effects. Hence, the significance of including the Breit interaction in improving the energies is perceptible from the present study.

For more elaborated analyses, we also investigated the individual contributions of the Breit interaction, QED effects (vacuum polarization (VP) and self energy (SE)) and nuclear recoil correction to the energies of Xe${}^{43+}$ and tabulated them in

Table 4. The contributions from QED (SE + VP ) effects, Breit interaction and nuclear recoil correction on energies (cm${}^{-1}$) of low lying states of Xe${}^{43+}$ and the comparison of total energy with the NIST [48] values. The level energies are given relative to the ground state.

Level	Energies (cm${}^{-1}$)
	RSCF	SE	VP	Nuclear Recoil	Breit	Total	NIST [48]
$3p{}^2{P}_{1/2}$	808,702	−19,696	1779	−64	10,643	801,364	806,985
$3p{}^2{P}_{3/2}$	1,512,126	−18,236	1855	−63	797	1,496,479	1,501,276
$3d{}^2{D}_{3/2}$	2,539,364	−20,144	1857	−86	−4016	2,516,975	2,523,660
$3d{}^2{D}_{4/2}$	2,700,953	−19,679	1855	−87	−10,160	2,672,882	2,679,380
$4s{}^2{S}_{1/2}$	12,277,698	−16,315	1083	−49	−9534	12,252,883	12,263,000
$4p{}^2{P}_{1/2}$	12,611,836	−19,865	1797	−74	−5238	12,588,456	12,596,000
$4p{}^2{P}_{3/2}$	12,897,971	−19,591	1829	−74	−9194	12,870,941	12,880,000

. Here, we show results for only seven low-lying levels as a similar pattern of contribution from the above corrections has been observed in the higher levels. The SE and Breit interaction play a significant role in improving the RSCF energies, whereas the influence of nuclear recoil correction is very small.

3.2. Transition Parameters

Using the MCDHF−RCI and MBPT methods, we calculated the wavelengths, transition rates, oscillator strengths and line strengths $\left(S\right)$ corresponding to the E1 transitions among the $nl$ levels with n = 3–9 and $l=s,p,d,f,g,h,i$ of Na-like Ar${}^{7+}$, Kr${}^{25+}$ and Xe${}^{43+}$ ions. The transition probabilities are computed in the Coulomb (velocity) and Babushkin (length) gauges.

3.2.1. Estimation of Uncertainty

In the present study, the accuracy of the E1 transition parameters is determined using two approaches. The first accuracy estimator is to check the discrepancy between the S values from the velocity ($S_V$) and length ($S_L$) gauges. Figure 3a,b show the comparison between the log${}_{10}\left(S_V\right)$ and log${}_{10}(S_L/S_V)$ values in Ar${}^{7+}$, Kr${}^{25+}$ and Xe${}^{43+}$. It is quite evident that the results from the two gauges are in good agreement for all the three ions.

The second way of determining uncertainties is to evaluate the discrepancy between the present MBPT ($S_{MBPT}$) and RCI ($S_{RCI})$ line strengths, which are taken in the length gauge. Figure 3c,d depict the relation between the log${}_{10}{S}_{MBPT}$ and log${}_{10}(S_{MBPT}/S_{RCI})$ values in the three ions. A remarkably good agreement between the two theories can be noticed here. Furthermore, for evaluating the percentage uncertainty, we have used the method that was suggested by Kramida [49] and further modified in the works of Attia and El-Sayed [50] and El-Sayed [51]. The step-wise procedure is as follows:

• The transitions are divided into five groups based on the different ranges of $S_{MBPT}$ values.
• The mean value of $S_{MBPT}$, i.e., $S_{av}$, and the root mean squares of ln$(S_{MBPT}/S_{RCI})$, i.e., rms${}_{ln\left(S\right)}$, are evaluated for each of the chosen intervals.
• The rms${}_{ln\left(S\right)}$ vs ln($S_{av}$) curve is fitted by a suitable function.
• The uncertainty $\delta S$ in S is chosen as the maximum of the two values obtained from the formulated function and the actual value of ln$(S_{MBPT}/S_{RCI})$.
• Finally, the uncertainty in A is evaluated from the expression as given below,

$$\delta A=\sqrt{(\delta S^2+{\left(3\delta W\right)}^2)},$$

(1)

where $\delta W$ is the percentage uncertainty in wavelength, which is given as,

$$\delta W=\frac{\sqrt{(\delta E_u^2+\delta E_l^2)}}{(E_u-E_l)}\times 100.$$

(2)

Here, $E_u$ and $E_l$ are the energies of the upper and lower levels and $\delta E_u$, and $\delta E_l$ are their corresponding uncertainties, which are estimated as the root mean square of the percentage deviations between the present RCI and the NIST ASD [48] energies. Furthermore, it can be noticed from Figure 2b for Xe${}^{43+}$, the deviations in the RCI energies from the NIST results are almost constant as a function of energy. Therefore, the present RCI excitation energies are adjusted to the NIST values by adding a factor of 5600 cm${}^{-1}$. This adjustment accommodates the possible error in the theoretically computed position of the ground state which may be occurring due to a large contribution from the QED corrections in the RCI calculations for this level. The RCI transition rates, weighted oscillator strengths, lifetimes and the uncertainty percentages for Xe${}^{43+}$ are calculated using these adjusted energy values and the obtained line strengths.

We converted $\delta A$ values into the terminology used by the NIST [48]. It varies between $A+$ and E, with $A+\le 2\%,A\le 3\%,B+\le 7\%,B\le 10\%,C+\le 18\%,C\le 25\%,$$D+\le 44\%,D\le 54\%$ and $E>54\%$.

Furthermore, in the present work, the amount of correlation effects that are accounted for are much smaller for levels with $n>7$ as compared to those with $n\le 7$. Therefore, we divided our results into two sets based on the transitions from the upper levels with $n\le 7$ and $n>7$. All the above steps ($1-5$) are performed independently to determine the uncertainty for these two sets.

Table 5. Percentage of E1 transitions with uncertainty ranges.

Accuracy	Percentage of E1 Transitions
	$\mathbf{n}\le \mathbf{7}$			$\mathbf{n}>\mathbf{7}$
	Ar${}^{\mathbf{7}+}$	Kr${}^{\mathbf{25}+}$	Xe${}^{\mathbf{43}+}$	Ar${}^{\mathbf{7}+}$	Kr${}^{\mathbf{25}+}$	Xe${}^{\mathbf{43}+}$
A+	0	7.48	27.31	0	0	0
A	0	13.72	4.83	0	0.42	0
B+	14.20	15.38	6.93	0	18.30	35.08
B	12.92	2.70	1.68	2.33	15.59	11.76
C+	14.19	2.49	2.10	6.78	14.55	6.09
C	1.91	0.83	0.42	9.53	4.37	1.47
D+	1.91	0.83	0.21	9.32	1.45	1.89
D	0	0.21	0	5.08	0	0
E	1.06	0	0	20.76	1.66	0.21

shows the percentage of E1 transitions falling in each uncertainty category for the three ions. The percentage is given separately for all the transitions with upper level $n\le 7$ and $n>7$. In the case of Ar${}^{7+}$, maximum transitions show uncertainty from $B+$ to $D+$, while 22% of the total transitions are in the E category. The majority of transitions in the E class are with $n>7$, and only 1.06% of such transitions belong to $n\le 7$ levels. For Kr${}^{25+}$ and Xe${}^{43+}$, most of the transitions have an accuracy class between $A+$ and $C+$.

We have also performed a detailed comparison of the E1 radiative parameters for each ion as described below.

3.2.2. Ar${}^{7+}$

For Ar${}^{7+}$, the present radiative parameters for the E1 transitions, computed using the RCI and MBPT approaches and the previous results of Siegel et al. [17], Froese Fischer et al. [20], Liang et al. [22] and NIST ASD [48], are presented in Table S1 (Supplementary Data). However, the transitions parameters with the upper level 9s are not reported here due to the unphysical MCDHF−RCI energy of this state as mentioned in Section 3.1.1. We observe that the $gf$ values from the present work and Siegel et al. [17] show an excellent agreement. Our oscillator strengths vary by a maximum of only 19% and 1.5% with respect to Liang et al. [22] and Froese Fischer et al. [20], respectively. Figure 3e shows a clear picture of the relation between the S values from the present and previous studies [20,22,48]. Our S results are fairly consistent with the NIST values except for $3d{}^2{D}_{5/2}$–$4p{}^2{P}_{3/2}$ and $3d{}^2{D}_{3/2}$–$4p{}^2{P}_{1/2}$ transitions. However, all the theoretical results match reasonably well for these two transitions as can be seen in Table S1. The estimated uncertainty for the NIST values lies within 18–25%. In fact, for a few transitions, the accuracy range of the NIST values extends up to 50%. Hence, our results are within the uncertainties of the recommended reference data and can be considered reliable. Overall, most of the present radiative parameters are consistent with previous values in the literature. Furthermore, for a few allowed transitions of Ar${}^{7+}$, the radiative parameters are reported here for the first time.

3.2.3. Kr${}^{25+}$

For Kr${}^{25+}$, the radiative parameters for the E1 transitions from the present RCI and MBPT methods and previous results of [22,24,48] are given in Table S2 of the Supplementary Data. The present wavelengths agree extremely well with the reference data for 26 transitions available in the NIST ASD [48] with an average deviation of 0.049%. Our weighted oscillator strengths have a mean deviation of only 2.71% relative to the theoretical results of Sampson et al. [24], with only six transitions exhibiting a variation above 5%. The relation between S values from the RCI calculations and the corresponding results of Liang et al. [22] is shown in Figure 3f. Our calculated line strengths lie primarily within 20% of those reported in [22] with a few exceptions. However, significant discrepancies are encountered when comparing the transition rates of Younis et al. [19] with both the present and the AUTOSTRUCTURE [22] calculations. Therefore, we do not report a complete comparison with [19]. A good agreement of our results with previous studies and the uncertainty estimations in Table 5 indicate the validity of our calculations. Moreover, new results are reported for the levels with $n\ge$ 7.

3.2.4. Xe${}^{43+}$

The transition parameters of the E1 transitions for Xe${}^{43+}$ computed from the RCI and MBPT methods in the present work are tabulated in Table S3 (Supplementary Data). The previous results [18,21,24,48,52] are also listed in Table S3 for comparison. Our results are in good agreement with the wavelengths of the four transitions available in the NIST ASD [48] and have a maximum difference of 0.81 Å for $3p{}^2{P}_{1/2}$–$3d{}^2{D}_{3/2}$ transitions. The wavelengths of 34 allowed E1 transitions from the work of Matsushima et al. [18] show an excellent match with the present results within a very marginal average absolute difference of 0.003 Å. The present $gf$ values also agree with the theoretical calculations of [24]. The largest difference of 0.058 between the two results occurs for $4f{}^2{F}_{7/2}$–$3d{}^2{D}_{5/2}$ transition, whereas the mean absolute difference is found to be only 0.007. Furthermore, the mean absolute deviation in our transition rates is 0.3% and 0.1% compared to the results reported for five and four transitions by Vilkas et al. [21] and Johnson et al. [52], respectively.

Overall, the agreement among all the results is very satisfactory. To our knowledge, no other comprehensive results exist for Xe${}^{43+}$; hence, most of the radiative parameters reported here are new.

3.3. Lifetime

We have calculated the lifetimes of the presently considered 71 levels in both the length and velocity forms for all the three Na-like ions.

Table 6. The lifetimes (unit: ${10}^{-10}$ s) from the present work and previous studies of Theodosiou and Curtis [16], Froese Fischer et al. [20], Vilkas et al. [21], Reistad et al. [26], Kink et al. [27] and Träbert et al. [28] for Na-like Ar${}^{7+}$, Kr${}^{25+}$ and Xe${}^{43+}$ ions.

Ion	Level	RCI	Other Theories	Measurements
Ar${}^{7+}$
	$3p{}^2{P}_{1/2}$	4.15	4.09 [16]	4.17 ± 0.1 [26] ${}^a$
			4.12 [20]
	$3p{}^2{P}_{3/2}$	3.91	3.86 [16]	3.89 ± 0.1 [26] ${}^a$
			3.87 [20]
	$3d{}^2{D}_{3/2}$	1.34	1.34 [16]	1.70 ± 0.1 [26] ${}^b$
			1.32 [20]
	$3d{}^2{D}_{5/2}$	1.37	1.38 [16]	1.66 ± 0.08 [26] ${}^b$
			1.36 [20]
	$4s{}^2{S}_{1/2}$	0.30	0.30 [20]
	$4p{}^2{P}_{1/2}$	0.51	0.51 [20]
	$4p{}^2{P}_{3/2}$	0.52	0.52 [20]
	$4d{}^2{D}_{3/2}$	0.55	0.54 [20]
	$4d{}^2{D}_{5/2}$	0.54	0.54 [20]
	$4f{}^2{F}_{5/2}$	0.16	0.16 [20]
	$4f{}^2{F}_{7/2}$	0.16	0.16 [20]
Kr${}^{25+}$
	$3p{}^2{P}_{1/2}$	0.87	0.87 [16]
	$3p{}^2{P}_{3/2}$	0.46	0.46 [16]	0.45 ± 0.02 [27]
	$3d{}^2{D}_{3/2}$	0.27	0.27 [16]
	$3d{}^2{D}_{5/2}$	0.37	0.36 [16]
Xe${}^{43+}$
	$3p{}^2{P}_{1/2}$	0.43	0.42 [16]	0.42 ± 0.04 [28]
			0.43 [21]
	$3p{}^2{P}_{3/2}$	0.06	0.06 [16]	0.07 ± 0.007 [28]
			0.06 [21]
	$3d{}^2{D}_{3/2}$	0.06	0.06 [16]	0.06 ± 0.015 [28]
			0.06 [21]
	$3d{}^2{D}_{5/2}$	0.15	0.15 [16]	0.14 ± 0.04 [28]
			0.15 [21]

^a—Reistad et al. [26] ANDC results, ^b—Reistad et al. [26] multiexponential fits.

compares the present calculations with the available theoretical [16,20,21] and experimental [26,27,28] results. Our results for $3p$${}^2{P}_{1/2,3/2}$ levels of Ar${}^{7+}$ lie within the error bars of the arbitrarily normalized decay curves (ANDC) results of Reistad et al. [26], but for $3d$${}^2{D}_{3/2,5/2}$ levels, the match is not as good. This could be because the quoted uncertainties were obtained from the multi-exponential fits for the latter two levels by Reistad et al. [26], and additional sources of uncertainties are not added to the statistical uncertainties of the fit. Furthermore, Ar${}^{7+}$ and Xe${}^{43+}$ have an excellent match with the corresponding values of Froese Fischer et al. [20] and Vilkas et al. [21], respectively. Our calculated lifetimes are within 1.4%, 0.8% and 3%, for Ar${}^{7+}$, Kr${}^{25+}$ and Xe${}^{43+}$, respectively, in comparison with the results of Theodosiou and Curtis [16]. The present lifetimes for the levels of Kr${}^{25+}$ and Xe${}^{43+}$ also show a good agreement with the measurements of Kink et al. [27] and Träbert et al. [28] and lie within the experimental uncertainties.

The complete lifetime results in length and velocity forms, along with computed uncertainties for each level, are presented in Table S4 (Supplementary Data). The uncertainties in the present lifetimes are estimated by propagating the uncertainties in A values of each decay branch and using the following expression,

$$\frac{\delta {\tau }_u}{{\tau }_u}={\tau }_u{\left(\sum _l\delta A_{ul}^2\right)}^{1/2}.$$

(3)

Here ${\tau }_u$ is the lifetime of level u, $\delta {\tau }_u$ is uncertainty in ${\tau }_u$, and $\delta A_{ul}$ is the uncertainty in the $A_{ul}$ value; l refers to all states lower in energy than u. It is observed that the value of $\delta {\tau }_u$ varies between 1% to 14 %, while the average of $\delta {\tau }_u$ is only 3.8% for Kr${}^{25+}$ and 2.5% for Xe${}^{43+}$. However, for Ar${}^{7+}$, the lifetimes corresponding to the levels $n>7$ show large uncertainties. This is because most of the transition rates for $n>7$ are in the E class of uncertainty as seen from Table 5. Hence, the lifetimes for a few levels viz., $8s,8p,8d,9s,9p$ and $9d$ that have percentage error greater than 35% are not reported here for Ar${}^{7+}$. Excluding these levels, the average $\delta {\tau }_u$ is only 10.32%. This analysis helps in establishing the reliability of the present results. Moreover, for each ion, we calculated the ratio of lifetimes from the two gauges and found its value close to one for almost all levels. Thus, a decent consistency exists between the two sets of results. This is also expected from the good agreement between the length and velocity forms of the present line strengths as observed from Figure 3a,b. Thus, our study provides new lifetime results for a majority of the levels, particularly for Kr${}^{25+}$ and Xe${}^{43+}$.

3.4. Hyperfine Interaction Constants and Landé $g_J$ Factors

We have computed the hyperfine constants ( $A_J$ and $B_J$) and Landé $g_J$ factors for the 71 levels of the presently studied Na-like inert gas ions (except for the $9s$ level of Ar${}^{7+}$, for the reason mentioned in Section 3.1.1 ). To provide generalized results, we assumed the nuclear magnetic (${\mu }_I$) and quadrupole moments (Q) to equal one. Table S5 (Supplementary Data) represents our RCI results of $A_J/{\mu }_I$, $B_J/Q$ and Landé $g_J$ factors for the three ions. The comparison of the present values with the only available RCC calculations of Dutta and Majumder [29] for Ar${}^{7+}$ is shown in

Table 7. Comparison of the present hyperfine constants $A_J/{\mu }_I$ (MHz per units of ${\mu }_I$), $B_J/Q$ (MHz/barn) with the corresponding results of Dutta and Majumder [29] for Ar${}^{7+}$.

	${\mathit{A}}_{\mathit{J}}$/${\mathit{\mu }}_{\mathit{I}}$ (MHz/Units of ${\mathit{\mu }}_{\mathit{I}}$ )				${\mathit{B}}_{\mathit{J}}/\mathit{Q}$ (MHz/Barn)
Level	RCI	RCC [29]	RSCF_MR	DF [29]	RCI	RCC [29]	RSCF_MR	DF [29]
$3s{}^2{S}_{1/2}$	5638.00	5669.86	5200.00	5080.69	-	-	-	-
$3p{}^2{P}_{1/2}$	1529.00	1518.29	1379.00	1346.27	-	-	-	-
$3p{}^2{P}_{3/2}$	302.00	261.54	268.70	262.34	1953.00	1924.15	1744.00	1742.25
$3d{}^2{D}_{3/2}$	87.27	84.15	85.00	83.10	182.70	180.22	183.70	183.68
$3d{}^2{D}_{5/2}$	9.69	10.59	36.36	35.54	260.30	256.92	261.34	261.30

. $A_J/{\mu }_I$ and $B_J/Q$ have been reported for only $3s,3p$ and $3d$ states in [29]. The maximum absolute difference between the present and RCC values is observed for the $3p{}^2{P}_{3/2}$ state where $A_J/{\mu }_I$ and $B_J/Q$ stray by 40.5 (MHz/units of ${\mu }_I$) and 28.9 (MHz/barn), respectively. Furthermore, Dutta and Majumder [29] also calculated the HFS for the Dirac–Fock (DF) reference states without considering the correlations and Gaunt corrections. Hence, we computed the HFS for the MR set by using the wavefunction obtained by the MCDHF method and without including the Breit and QED corrections and excitation to virtual orbits (correlation effects). These HFS values are labeled as RSCF_MR. The comparison of these two results is also given in Table 7. An excellent agreement with a maximum relative deviation of 0.1% exists between the present RSCF_MR and previous DF $B_J/Q$ values. However, our RCI $A_J/{\mu }_I$ values exhibit a better match with the RCC calculations [29] compared to the agreement between the RSCF_MR and DF results [29]. Therefore, the slight variation between the RCI and RCC results could be due to the correlation and leading-order Breit and QED corrections considered in these two calculations.

3.5. Isotope Shifts

Using the RIS4 module, we computed the electronic factors corresponding to normal mass, specific mass and field shifts for the 71 levels of Na-like Ar${}^{7+}$, Kr${}^{25+}$ and Xe${}^{43+}$ ions and reported them in Table S6 (Supplementary Data). The $9s$ level is excluded for Ar${}^{7+}$ as we could not obtain its correct energy. Previous results of IS factors are available from the MBPT calculations of Safronova and Johnson [30] for Ar${}^{7+}$, CIDF results of Tupitsyn et al. [31] for Ar${}^{7+}$ and Xe${}^{43+}$, and RMBPT, GRASP and experimental values of Silwal et al. [32] for Xe${}^{43+}$.

Table 8. The normal mass shift (NMS), specific mass shift (SMS) and field shift (FS) values from the present work and the results of Safronova and Johnson [30], Tupitsyn et al. [31] and Silwal et al. [32].

Ion	Transition	NMS (GHz u)		SMS (GHz u)		FS (GHz/fm${}^2$)
		Present	Others	Present	Others	Present	Others
Ar${}^{7+}$	$3p{}^2{P}_{1/2}$–$3s{}^2{S}_{1/2}$	−2184	-	−6133	−6504 [31]	2418	2418 [31]
					−6363 [30]		2427 [30]
	$3p{}^2{P}_{3/2}$–$3s{}^2{S}_{1/2}$	−2215	-	−6117	−6328 [30]	2422	2429 [30]
	$3d{}^2{D}_{3/2}$–$3s{}^2{S}_{1/2}$	−5572	-	−6367	−7559 [30]	2334	2283 [30]
	$3d{}^2{D}_{5/2}$–$3s{}^2{S}_{1/2}$	−5553	-	−6423	−7537 [30]	2334	2283 [30]
Xe${}^{43+}$	$3p{}^2{P}_{1/2}$–$3s{}^2{S}_{1/2}$	−11,922		−167,581	−171,701 [31]	958,270	965,738 [31]
			−13,100 [32] ${}^a$		−171,000 [32] ${}^a$		959,000 [32] ${}^a$
			−13,100 [32] ${}^b$		−170,000 [32] ${}^b$		966,000 [32] ${}^b$
	$3p{}^2{P}_{3/2}$–$3s{}^2{S}_{1/2}$	−22,361	-	−166,657	-	988,260	-
	$3d{}^2{D}_{3/2}$–$3s{}^2{S}_{1/2}$	−40,549	-	−244,604	-	984,560	-
	$3d{}^2{D}_{5/2}$–$3s{}^2{S}_{1/2}$	−41,931	-	−250,235	-	983,250	-

^a—Silwal et al. [32] GRASP2K results, ^b—Silwal et al. [32] RMBPT results.

displays comparison of the present and aforementioned studies [30,31,32]. We observed that our calculated FS values show an excellent agreement with [31], but the SMS results have a relative error of 5.7% and 2.4% in Ar${}^{7+}$ and Xe${}^{43+}$, respectively. The minimum and maximum deviations of 0.3% and 2% are noticed in present FS results of Ar${}^{7+}$ when compared with values from Safranova and Johnson [30]. In contrast, the difference goes up to 15% for SMS parameters. Furthermore, the present total IS value of 75.9 fm agrees well with the Silwal et al. [32] measured total IS (${}^{136}$Xe−${}^{124}$Xe) value of 65.5 ± 20.6 fm. Moreover, the present result deviates only by 0.5% and 0.4% from the GRASP2K and RMBPT calculations of [32]. Altogether, a reasonably good agreement exists between the present and previous results. Most of the present results are the first to appear in the literature for all three ions. Our calculated IS parameters will be useful in plasma modeling, heavy-ion storage experiments and the investigations of nuclear charge radii.

4. Conclusions

We performed a systematic and comprehensive computation of level energies, lifetimes, E1 radiative rates, transition wavelengths, hyperfine constants, Landé $g_J$ and isotope shift factors for lowest 71 states of the $nl$$n\le 9;l\le 6$ configurations of Na-like Ar${}^{7+}$, Kr${}^{25+}$ and Xe${}^{43+}$ ions. We presented a detailed examination on the significance of electron correlation and QED and Breit effects on these atomic parameters with increasing nuclear charge. These calculations were made with the MCDHF -RCI method as implemented in the GRASP2018 code package and its module RIS4. A large section of the present results for the three ions is reported for the first time and provides benchmarks for future theoretical and experimental studies. The absence of previous theoretical or experimental results for higher excited states, in particular for Kr${}^{25+}$ and Xe${}^{43+}$, led us to carry out similar independent calculations using the MBPT method to compare with our MCDHF−RCI results. The average deviations between the present MCDHF−RCI and MBPT level energies are 0.43%, 0.05% and 0.06% for Ar${}^{7+}$, Kr${}^{25+}$ and Xe${}^{43+}$, respectively. The absolute mean difference of only 0.13%, 0.03% and 0.08% in the present MCDHF−RCI energies is observed with reference to the NIST values for Ar${}^{7+}$, Kr${}^{25+}$ and Xe${}^{43+}$, respectively. The transition parameters computed using the MCDHF−RCI and MBPT approaches are within 10–20% agreement for most of the transitions, which confirms the reliability of the present results. Our calculated lifetimes in the length and velocity gauges are highly consistent and agree well with the other results. Furthermore, we carried out a detailed uncertainly estimation in our calculated A values for E1 transitions and lifetimes of the 71 levels. We found that the accuracy is between 7–18% in most cases. This analysis confirms the reliability of our calculated results. There is a good match between our HFS and IS results with the earlier theoretical and experimental works. Thus, the present MCDHF−RCI calculations yield fairly accurate relativistic atomic structures for Na-like ions of Ar, Kr and Xe.