Configuration Interaction Calculations of Energy Levels and Radiative Parameters of Calcium-Like Ions, 22 ≤ Z ≤ 30

The energy levels, oscillator strengths, and E1 transition probabilities have been calculated for calcium-like ions with Z = 22 − 30 (excluding iron). The calculations have been executed using the CIV3 and LANL codes for a set of configuration arrays including 63 fine structure levels (in this paper we mentioned for examples about 40 energy levels) belonging to 3p 6 3d 2 and 3p 6 3d4l configurations, where l = s, p, d, f. The correlations up to 6l orbitals are included to optimize the wave functions generated by the CIV3 code. In spite of the complexity of the Ca sequence, the present results are in fairly good agreement with the experimental and theoretical data available in the literature. The present study provides calculations of various atomic structure data which are necessary for many fields of researches and applications, especially in the astrophysics and plasma diagnostics.


Introduction
Members of the calcium isoelectronic sequence such as vanadium, iron, cobalt, and nickel are of importance for astrophysics as well as for plasma modeling. The importance of calcium-like ions (vanadium through zinc) for astrophysics lies within its abundance in the solar system, so, transitions arise from these atomic systems possible to appear in the solar spectra and in stellar corona. The spectra of calcium isoelectronic sequence had been studied early by many authors [1][2][3][4], these works had provided results of observed transition parameters and energies of Ca-like manganese, scandium, cobalt, iron, nickel, copper, and zinc. Fawcett and Cowan [5] have studied the spectral lines arise from Ca-like iron (Fe VII), the atomic self-consistent field calculations have been applied to identify the spectra belonging to the 3d 2 -3p 5 3d 3 transition arrays. The spectrum of Fe VII has been observed by Ekberg [6] using low-and high-voltage sliding spark, as well as the Fe VII spectra has been studied in many other different works such as [7][8][9][10][11][12][13][14][15][16][17][18][19][20]. A measured values of lifetimes for some transitions of the 3d 2 → 3d4p arrays in Ti III have been reported [21,22] using the beam-foil technique. Abbott [23] has used the non-relativistic SUPERSTRUCTURE program to calculate the transition probabilities of calcium-like ionized Cr, Mn, Fe, and Ni, the calculations have been carried out for transitions belonging to 3d 2 , 3d4s, 3d4p, 4s4p, 3d4d, and 3d4f. Many lines in Ca I, Ti III, Cr V, Mn VI, Fe VII, and Ni IX observed in the solar spectra have been identified in works such as [24,25]. Resonance transitions in Co VIII, Ni IX and Cu X have been analyzed as well as the energies and weighted oscillator strengths have been calculated for Co VIII by Fawcett et al. [26]. The E1, M1, and M2 transition probabilities of some transitions in Ti III and V IV have been calculated using the orthogonal operator method by Raassen and Uylings [27]. Safronova et al. [28,29] have used the relativistic many-body perturbation theory (RMBPT) to calculate the excitation energies and radiative rates of Ca-like ions. the calculated Parameters of Calcium-Like Ions, 22 ≤ Z ≤ 30 spectra are arising from E2 and M1 transitions. Observed wavelengths of the 3d4p → 3d4d and 3d4d → 3d4f transitions in the vacuum ultraviolet spectra of calcium-like Mn VI, Fe VII and Co VIII have been identified [30], the MCHF method and isoelectronic comparison have been used for the classification.
The forbidden transitions E2, M1 have been studied using three different methods (MCDF, HFR, and SST), transition probabilities and oscillator strengths in Ca-like Ti III have been computed [31]. Oscillator strengths have been calculated for transitions between the 3d 2 , 3p 5 3d 3 , 3d4p, 3d4f levels in Ca-like iron, cobalt and nickel using the pseudorelativistic Hartree-Fock (HFR), the calculated results are reported in Ref. [20]. Recently, observed values of oscillator strengths of Ca I have been reported by Haq et al. [32] using the thermionic diode ion detector in conjunction with a Nd:YAG pumped dye laser system. The spectrum of Cu X has been studied using high-resolution vacuum spectrographs, and the excited energies of the 3d 2 , 3p 5 3d 3 , 3d4p, 3d4f levels have been determined using the HF method [33]. Zhang et al. [34] calculated oscillator strengths and transition probabilities of Ti III using the weakest bound electron potential model (WBEPM). The calcium like titanium has been studied again [35] where, the oscillator strengths and radiative rates have been computed using the CIV3 program. The energy levels and collision strengths of Sc II have been computed using R-matrix package, transitions arising from the 3d4s, 3d 2 , 4s 2 , 3d4p, 4s4p, 3d5s, 3d4d, 3d5p, 4p 2 , and 3d4f configurations are included [36].
The early compilation of atomic data of Ca-like ions performed by Wiese et al. [37], and Smith and Wise [38]. Compiled results including ions belonging to calcium isoelectronic sequence have been presented by Morton [39], the transition probabilities and oscillator strengths of several lines in Ca I, Sc II, and Ti III are reported. Another compilations performed by Corlis and Sugar [40], and Sugar et al. [41][42][43] have reported energy levels of calcium through zinc. Fuhr et al. [44,45] have compiled experimental and theoretical data for transition probabilities for iron through nickel. Another compilations of transition probabilities for scandium through manganese ions have been performed by Martin et al. [46], and for energy levels of all ionization stages of scandium by Kaufman and Sugar [47]. The compilation processes still go on where, Shirai et al. [48,49] have collected experimental and theoretical results of energy levels, oscillator strengths, and transition probabilities of Cr V and Co VIII, and for Ca-like Ti through Cu in the published monograph [50]. Wavelengths, energy levels, level classifications, oscillator strengths, and radiative transition probabilities for the nickel ions Ni IX to Ni XXVIII have been collected by Shirai et al. [51].
From the previous review we can figure out that the calcium isoelectronic sequence has been widely studied, despite this a limited works have provided an extensive calculations of energies and radiative parameters of many ionic species in this sequence, but for most of ions we still lack abundance of atomic data, especially for nickel and zinc. In the present study, the configuration interaction calculations for calcium isoelectronic sequence with atomic charge Z=22 to 30 have been performed using the non-relativistic CIV3 code [52,53], and Los Alamos atomic physics code (LANL) based on Cowan's method which publicly available via website (http://aphysics2.lanl.gov/cgi-bin/ION/runlanl08d.pl).
The iron ion was excluded from the study. The configuration groups 3d 2 , 3d4s, 3d4p, 3d4d, and 3d4f with different angular momenta (J) and parities are included in the configuration state list.

The Choice of Wave Functions
The present calculations have been performed using the configuration interaction method (CIV3 code) including relativistic effects of Hibbert [52,53]  ( The single configuration wave functions ϕ i are consist of one electron functions. α i,LS defines the coupling of angular momenta to form a total L and S common to all configurations in Eq 1. The CIV3 method theory has been described in details in many previous works, for example Refs. [52][53][54][55][56][57][58][59][60][61][62]. In the present calculations the 1s, 2s, 2p, 3s, 3p and 3d radial functions are taken as the Hartree-Fock orbitals of the ground state 3p 6 3d 2 of Ca-like ions as given by Clementi and Roetti [63]. The other radial functions for 4l are chosen as a spectroscopic type and are optimized using the CIV3 program [52]. The 5s, 5p, 5d, and 5f, 6s-6f orbitals are chosen as a correlation orbitals (contributions from higher order levels with n + 2 to improve the wave functions of the spectroscopic orbitals). The oscillator strengths in length and velocity gauge forms are given by: The LANL atomic code based on Cowan method is described in details in Ref. [64]. Extensive configurationinteraction (CI) wave functions in intermediate coupling scheme have been generated. Despite that, the produced results are in LSJ-coupling, the intermediate LS coupling configuration state list is successfully effective in the evaluation of the Hamiltonian matrix elements [65,66]. These wave functions are used to calculate the excitation energies, oscillator strengths and transition probabilities for allowed electric-dipole and intercombination transitions among the states of Ca-like ions with 22 ≤ Z ≤ 30.

Energy Levels
The calculated values of energies of 40 fine structure levels of Ca-like ions (Z = 22 − 30) are listed in Tables 1-3 the data of iron ion have been excluded because we believe that we will not provide any significant results more than hitherto published. The present energies from CIV3 and LANL codes have been arranged in ascending order and compared with the energy values taken from NIST atomic database [67]. The comparisons show good agreement with those in the literature, where the deviations from the values of NIST are better than 1.0% for most calculated levels. The cited values by NIST [67] are primarily taken from the compiled experimental and theoretical results by Sugar and Corliss [41] for Ca-like Ti III− Ni IX ions, from [42] for Cu X, and from [43] for Zn XI.  Cr V: In the Cr IV ion the 3d 2 3 F 3 level deviates from the corresponding NIST value by 4.73% for the CIV3 and the value of 5.6% is recorded as a deviation of the 3d 2 1 D 2 level from NIST value for the LANL calculations, the value of energy levels have been listed in Table 2.
Mn VI: The same as Cr IV was figured out for the Mn ion where, the 3d 2 3 F 3 level deviates from the corresponding NIST value by 4.65% for the CIV3 and the value of 5.2% is recorded as a deviation of the 3d 2 1 D 2 level from NIST value for the LANL calculations, as shown in Table 2.
Co VIII: For most of levels calculated using CIV3 the energies agree well with those from NIST and the percent differences are less than 1.0%, except the 3d 2 1 G 4 where the deviation rises to 6.0%. The maximum deviations for the results produced by LANL are 4.9, 4.8, 4.8% for 3d 2 3 P 0,1,2 , respectively, as shown in Table 2.
Ni IX: The ground state and excitation energies of Ca-like Ni IX are shown in Table 3, the data have been compared with the minor ones available by NIST atomic spectra database [67], and it is in a reasonable agreement with the literature [67]. But in some cases this agreement disappears for example, the levels 3d 2 1 D 2 and 3d 2 3 F 3,4 which showing differences from NIST by 4.0, 3.1, 3.7%, respectively.   Cu X: The energy levels of Cu X have been listed in Table  3 and have been compared with the corresponding values of NIST. For the few available levels by NIST the comparisons show good agreement and the maximum deviation is found for the 3d 2 3 F 4 where the error value is about 5.38%.
Zn XI: The Zn XI data are tabulated in Table 3, it is clear that the computed energies using the CIV3 code are much more better than those computed using LANL, we can figure out this from the results deviations from NIST values. The deviation of the 3d 2 Where, figures (1a-2b) show that, the NIST and CIV3 values of E 2 /Z decrease systematically vs 1/Z over all ions except for zinc ion, while the calculations of LANL fit well with the equation (4) over all ions including Zn.    The fine structure parameter ζi depend on the angular momentum l > 0 and is given from: . Figure (3a) show that the CIV3 calculations of hyper fine structure parameter vary systematically with Z, except at Zn value which decreases rapidly. The ζ2/Z value was adjusted at 0.808 for the Zn ion to match the calculated energies by CIV3 with NIST, from the previous argument we think that the abnormal value of LANL calculations of the 3d 2 3 F 3 energy level in Zn XI are accurate and may be an error has been occurred in the compilation process for the zinc data in the literature [43].

Oscillator Strengths and Radiative Rates
The configuration interaction calculations of radiative parameters (wavelengths, oscillator strengths, and transition probabilities) are given in Tables 4-7. In the present work, we have used two approximations to get better values for oscillator strengths and transition probabilities. First, the calculations using the non-relativistic configuration interaction method CIV3 including Breit-Pauli Hamiltonian. Second, the ab-initio calculations using the LANL atomic code based on Cowan's method [64]. It obvious that for the listed data of oscillator strengths, the data produced from CIV3 calculations are so close to those from LANL calculations for the majority of transitions.
The accuracy of the results could be estimated by many ways such as, the comparison with the previous published theoretical calculations and experimental data, the use of different methods in the calculations, and/or the convergence between the length and velocity gauge forms [69]. In the present work we have used two different methods in the calculations, and compared the calculated data of oscillator strengths from the two methods with the available sources, and checked the agreement between length and velocity gauge values. The comparisons between CIV3 and LANL calculations of oscillator strengths and other literature have been accomplished in Table 8. Where the present calculations have been compared with the values of NIST [67] as well as with the cited values in the literature [20, 26, 34, 44-46, 48, 49]. For most cited transitions in Table 8 our data either from CIV3 or from LANL are in a reasonable agreement with those of NIST and the literature, the deviation for some transitions reaches large values such as 3d4p( 3 D 3 ) -3d4d( 3 D 3 ) in V IV ion, which deviates from NIST by as much as 99% but the average deviation for this ion lies within 10%. To avoid worse accuracies like that we should include a large number of correlations within the used configuration state list using CIV3 by adding the 7l and 8l correlations. Procedure like this needs high speed computer containing large memory (not available for us at the present time).
An important point should be mentioned here, that is the accuracy (D, E) of most of NIST-values for oscillator strengths are about 40% ≤ D ≤ 50% and E > 50% which means that our calculated results of most oscillator strengths are in the uncertainty range of NIST. Parameters of Calcium-Like Ions, 22 ≤ Z ≤ 30 For ions from nickel to zinc there are no data available at NIST online database. The present f-values of the Ni IX ion have been compared with those in reference [20] (see Table  8), the comparison shows that the ab-initio results from LANL are close to those in Ref. [20] than the data calculated by CIV3. The behavior of the oscillator strength of a given transition along an isoelectronic sequence can be illustrated using the formula [68] = & + % () ) + ⋯ The Z-dependence of oscillator strengths may be shown in the plot of oscillator strengths for the transition 3d 2 ( 3 P 2 )-3d4p( 3 P 2 ) versus the nuclear charge in figure (3b). The present calculations decrease with increasing Z, while the fvalues by NIST [67] follow non-specific behavior with Z.
Another criterion is used to determine the accuracy of oscillator strengths, the precision of the theoretical oscillator strengths can be judged by the convergence between length and velocity gauge values. If exact wave functions are used then = " [70,71]. For most of the present transitions the length and velocity gauge values are in a fairly good agreement, where the ratio L/V for most transitions are about one. The good agreement between and " gives some indications (but not sufficient) for the present calculations accuracy [72]. Note: a: The oscillator strengths from Ref. [20] b: The oscillator strengths from Ref. [45] c: The oscillator strengths from Ref. [48] d: The oscillator strengths from Ref. [26] e: The oscillator strengths from Ref. [44]

Conclusion
In the present study, configuration interaction calculations using CIV3 code and ab-initio calculations using the LANL code have been performed for Ca-like ions. An extensive calculations of energy levels, oscillator strengths, and radiative rates have been evaluated for configuration arrays including the 3d 2 , 3d4s, 3d4p, 3d4d, and 3d4f levels with different angular momenta and parity. The present calculations of energies and oscillator strengths have been compared with the available experimental and theoretical sources, and it show a reasonable agreement with the literature. The atomic structure data are useful for many applications such as the identification of the solar spectra, the plasma diagnostics, and the thermonuclear fusion researches. In this paper we mentioned some data we obtained, not all the data we calculated, Such as tables include energy levels for Ca like ions (Z=22-30) except Fe, ions and tables include oscillator strength for Co VIII, Ni IX, Cu X, Zn XI ions, also the table includes comparison our data with the other published literature.