Energy Levels and Radiative Rates for Transitions in F-like Sc~XIII and Ne-like Sc~XII and Y~XXX

Energy levels, radiative rates and lifetimes are reported for F-like Sc~XIII and Ne-like Sc~XII and Y~XXX for which the general-purpose relativistic atomic structure package ({\sc grasp}) has been adopted. For all three ions limited data exist in the literature but comparisons have been made wherever possible to assess the accuracy of the calculations. In the present work the lowest 102, 125 and 139 levels have been considered for the respective ions. Additionally, calculations have also been performed with the flexible atomic code ({\sc fac}) to (particularly) confirm the accuracy of energy levels.

As with grasp, with fac too we have performed a series of calculations, but focus on only three, i.e. (i) FAC1, which includes 113 levels as in GRASP1, (ii) FAC2, which includes 501 levels as in GRASP3, and finally (iii) FAC3, which includes in total 38 089 levels arising from all possible combinations of the (2*5) 3*2, 4*2, 5*2, 3*1 4*1, 3*1 5*1, and 4*1 5*1 configurations, plus those of FAC2. Although calculations have also been performed with even larger CI, these are not discussed here because the calculated energy levels show no appreciable differences, either in magnitude or orderings, i.e. the results have fully converged in FAC3. For brevity, for fac calculations we have used a short notation here (and elsewhere in the text) for describing configurations. As an example, 3*2 means 3ℓ3ℓ ′ resulting in 3s3p, 3s3d, 3p3d, 3s 2 , 3p 2 , and 3d 2 .
Our calculated energies for the lowest 102 levels of Sc XIII are listed in Table 1. These levels mostly belong to the 2s 2 2p 5 , 2s2p 6 , 2s 2 2p 4 3ℓ, and 2s2p 5 3ℓ configurations, and beyond these from others intermix, such as 2s 2 2p 4 4ℓ. However, energies for higher levels can be obtained from the author on request. Our energies calculated with grasp, without and with the inclusion of Breit and QED (quantum electro-dynamic) effects, are listed in the table, along with all three calculations with the fac, mentioned above. Also the experimental energies, compiled by the NIST (National Institute for Standards and Technology) team and available at the website http://www.nist.gov/pml/data/asd.cfm, are listed here along with the theoretical results of Jupén et al. [2], obtained from the Hartree-Fock Relativistic (HFR) code of Cowan -see [19]. However, these theoretical results have been adjusted with least square fitting with the available measurements for a few levels, and that is the reason that there are no appreciable differences for the levels in common with the NIST. We also note that for two levels (40/41) the NIST and HFR energies are non differentiable, but not in any of our calculations with both codes.
The contributions of the Breit and QED effects on the energy levels of Sc XIII are not very significant, and are below 0.04 Ryd. However, these contributions have slightly lowered the energies and subsequently, discrepancies with those of NIST have increased, because comparatively there is a better match between the NIST and our energies obtained without them. Nevertheless, discrepancies between our results with grasp (including the contributions from Breit and QED) and those of NIST are within 0.1 Ryd, and hence are highly satisfactory. Furthermore, there are no discrepancies in the level orderings between theory and measurements, and neither there are any ambiguities in level designations for this ion.
For most of the levels, there are no significant differences between the FAC1 and FAC2 energies, although the latter calculations include CI larger by more than a factor of four. However, for a few levels (such as 83 and 100) the differences are up to 0.04 Ryd, and energies in FAC2 are (mostly) lower. The same are the differences between the GRASP and FAC2 energies which include the same CI, but the latter ones are higher. Such small differences in energies between calculations with different codes are not uncommon and mainly arise due to the differences in algorithms, methodologies and formulations. Our FAC3 calculations include much larger CI and as a result the energies obtained are lower, by up to ∼0.2 Ryd, in comparison to those from FAC2. This has resulted in a better agreement with the GRASP energies. Although the FAC3 energies should be comparatively more accurate, differences with our GRASP or NIST are up to 0.25 Ryd -see for example levels 45-49 and 56-58. Since the FAC3 energies are the lowest, we consider our results with grasp to be comparatively more accurate, with agreement within 0.1 Ryd (0.3%) with those of NIST, except for level 3 (2s2p 6 2 S 1/2 ) for which the discrepancy is 2%, or 0.14 Ryd. For this level the energy calculated by Jönsson et al. [10] is closer to that of NIST, because not only they have included a significantly larger CI but their methodology is also different. Similarly, combining CI with many-body perturbation theory (MBPT) approach, Gu [20] calculated the energy 6.945 Ryd, which is lower than that of NIST by only 0.014 Ryd, a tenth of the difference we have.    For most of the levels, there are no significant differences between the FAC1 and FAC2 energies, although the latter calculations include CI larger by more than a factor of four. However, for a few levels (such as 83 and 100) the differences are up to 0.04 Ryd, and energies in FAC2 are (mostly) lower. The same are the differences between the GRASP and FAC2 energies which include the same CI, but the latter ones are higher.
Such small differences in energies between calculations with different codes are not uncommon and mainly arise due to the differences in algorithms, methodologies and formulations. Our FAC3 calculations include much larger CI and as a result the energies obtained are lower, by up to ∼0.2 Ryd, in comparison to those from FAC2. This has resulted in a better agreement with the GRASP energies. Although the FAC3 energies should be comparatively more accurate, differences with our GRASP or NIST are up to 0.25 Ryd -see for example levels 45-49 and 56-58. Since the FAC3 energies are the lowest, we consider our results with grasp to be comparatively more accurate, with agreement within 0.1 Ryd (0.3%) with those of NIST, except for level 3 (2s2p 6 2 S 1/2 ) for which the discrepancy is 2%, or 0.14 Ryd. For this level the energy calculated by Jönsson et al. [10] is closer to that of NIST, because not only they have included a significantly larger CI but their methodology is also different. Similarly, combining CI with many-body perturbation theory (MBPT) approach, Gu [20] calculated the energy 6.945 Ryd, which is lower than that of NIST by only 0.014 Ryd, a tenth of the difference we have.
In Table 2, we list our final energies from both grasp and fac for the lowest 125 levels, because beyond these from other configurations intermix, particularly from 2s 2 2p 5 6ℓ. However, energies for higher levels can be obtained from the author on request. In general, energies obtained in FAC1 (not listed here but discussed below) are lower than of GRASP by ∼0.1 Ryd, and both calculations include the same CI. This observation is similar to that noted earlier for Sc XIII. However, the FAC3 energies listed in Table 2 are lower than of GRASP by ∼0.2 Ryd, i.e. the effect of additional CI (by more than a factor of 20) is about 0.1 Ryd. In addition, for a few levels, such as 79-84, there are some (minor) differences in energy orderings, but overall there are no (major) discrepancies between calculations with two different codes. This result was expected and has been noted earlier for several ions, including some Ne-like [14,15], although some authors, such as [21], have shown differences of up to ∼2 Ryd, but their calculations are incorrect as discussed in [14,15] and further explained in [22,23] -see also [24] for many other examples of discrepancies. Although a good agreement between the two calculations in our work confirms the accuracy of the calculated energies, we discuss these further below.
As stated in Section 1, the only other energies available in the literature, but only for the lowest 27 levels, are by Cogordan and Lunell [5] and Jönsson et al. [10], who have also used (the different versions of) the grasp code. Since experimental energies compiled by NIST are also available for a few levels of Sc XII, in Table 3 we compare different sets of energies for the lowest 37 levels, which belong to the 2s 2 2p 6 , 2s 2 2p 5 3ℓ, 2s2p 6 3ℓ, and 2s 2 2p 5 4s configurations. The FAC1 and FAC2 energies differ at most by 0.2 Ryd (see levels 29-33) which indicates a small effect of additional CI included in the latter. However, further inclusion of CI in FAC3 is not of any (great) advantage because differences with FAC2 are below 0.02 Ryd, i.e. the results have converged. However, energy differences between the FAC3 and NIST are the largest, and are up to 0.3 Ryd for several levels, and those from the former are invariably lower. Therefore as for Sc XIII, energies calculated with fac for Sc XII too are comparatively less accurate. On the other hand, our calculations (and those of Cogordan and Lunell [5]) with grasp compare well with those of NIST, because the differences are within ∼0.1 Ryd (0.3%), with the measurements being (slightly) on the higher side. A notable exception is the level 15 (2s 2 2p 5 3p 1 S 0 ) for which the energy calculated by Cogordan and Lunell is (unusually) lower than our calculation by 0.14 Ryd. In all our calculations (with increasing CI) with the grasp code the energy obtained for this level is invariably higher, and the contributions of Breit and QED effects are only 0.02 Ryd. Therefore, the reason for this (comparatively) large difference is neither in the inclusion of (much) larger CI in our calculations nor in the modified version of the code adopted, but is due to the fact that they have treated this level separately in a different manner. Anyway, their calculated energy for this level is as much lower than of NIST as ours is higher, and therefore the overall differences with measurements remain the same. Finally, the energies calculated by Jönsson et al. [10] are the most accurate because they have been able to produce results closer to those of NIST, for the same reasons as explained in Section 2.1 for Sc XIII.

Y XXX
For Y XXX we have performed similar calculations with grasp and fac as for Sc XII, described in Section 2.2.
In Table 4 we list our final results with both these codes for the lowest 139 levels, as beyond these is an intermix from other configurations, such as 2s 2 2p 5 6ℓ. However, energies for higher levels can be obtained from the author on request. As stated in Section 1, similar results for some levels are available in the literature by Zhang and and agree within 0.1 Ryd, which is highly satisfactory. However, as for other two ions, the corresponding results with fac are the lowest among those listed in Table 4, although differ by a maximum of 0.2 Ryd with GRASP1. Before drawing our conclusion we make a few other comparisons below.
In Table 5 we compare our GRASP results with those of Cogordan and Lunell [5] and Quinet et al.  Table 4. Since the FAC energies are the lowest, irrespective of the level of CI, these results are assessed to be comparatively less accurate, and therefore our

Radiative rates
Our results for the A-values calculated with the grasp code are listed in Tables 6-8  for the electric dipole (E1) transitions for the magnetic dipole (M1) transitions for the electric quadrupole (E2) transitions and for the magnetic quadrupole (M2) transitions We also note here that f -and A-values are related as where m and e are the electron mass and charge, respectively, c is the velocity of light, λ ji is the transition wavelength inÅ, and ω i and ω j are the statistical weights of the lower i and upper j levels, respectively. This relationship is the same irrespective of the type of a transition, and λ ji are also listed in Tables 6-8      YODA: Earlier results of Hagelstein and Jung [6] Assessing the accuracy of our calculated results for A-values (and other related parameters) is not straightforward. This is because no measurements are available for any transition of the ions concerned. However, limited theoretical results are available in the literature which will perhaps be helpful for some accuracy assessments. For Sc XIII, Jönsson et al. [3] have listed A-values for the 1-3 E1 (4.635×10 10 s −1 ), 2-3 E1 (1.968×10 10 s −1 ), 1-2 M1 (9.773×10 2 s −1 ), and 1-2 E2 (3.849×10 −2 s −1 ) transitions, which match very well (within 10%) with our corresponding results of 5.143×10 10 , 2.193×10 10 , 9.571×10 2 , and 3.849×10 −2 s −1 , respectively. However, this direct comparison of A-values is very limited. Some further assessments of accuracy can be made by comparing the length and velocity forms (i.e. the Babushkin and Coulomb gauges in the relativistic terms) of the A-values, and their ratio (R) for all E1 transitions are listed in Tables 6-8. Ideally R should be closer to unity but in practice it is not, particularly for the weak(er) transitions. For many strong transitions with f ≥ 0.1, R is within 10% of unity as may be noted for the 1-49/52/55/56/59 transitions in Table 6 [10] and therefore in Table 9 we make comparisons for the E1 and E2 transitions from the lowest 5 to higher excited levels. Generally, for all E1 transitions the agreement between the two calculations is within ∼20%, which is highly satisfactory. However, for three weak transitions, namely 2-11 (f = 1.2×10 −4 ), 3-13 (f = 1.2×10 −4 ) and 5-9 (f = 4.4×10 −5 ), discrepancies are up to a factor of two. As already stated above, accuracies for such weak transitions are often not reliable and hence any of the two calculations can be (in)correct. Similarly, for the comparatively weak E2 transitions the two calculations agree within 20% for most, but discrepancies are up to a factor of two for four (2-24, 3-24, 3-25, and 5-20), whereas it is factor of four for one, i.e. 5-21 (f = 3.2×10 −10 ). Similar comparisons for the M1 and M2 transitions are made in Table 10. There are no appreciable discrepancies for the M2 transitions (except for 1-25), but for a few M1 the differences are up to two orders of magnitude, see in particular 2-18 for which our f = 9.7×10 −13 . Such weak transitions (and discrepancies between different calculations) do not affect the modelling, or the subsequent calculations of lifetimes, τ = 1.0/Σ i A ji , which includes contributions from all types of transitions, i.e. E1, E2, M1, and M2. This is further confirmed by comparing our results for τ , included in Tables 6-8, for the lowest 27 levels of Sc XII in Table 11, with those of Hibbert et al. [9] and Jönsson et al. [10], for which the agreements are within 10% for most levels.  1 98 2.046+01 0.000+00 0.000+00 0.000+00 0.000+00 1.006+01 0.000+00 0.0+00 1 101 2.046+01 0.000+00 0.000+00 0.000+00 7.876+07 0.000+00 0.000+00 0.0+00 1 103 2.046+01 0.000+00 0.000+00 0.000+00 1.535+08 0.000+00 0.000+00 0.0+00 1 104 2.046+01 0.000+00 0.000+00 0.000+00 0.000+00 0.000+00 6.763−01 0.0+00 1 114 2.038+01 0.000+00 0.000+00 0.000+00 0.000+00 0.000+00 7.163+03 0.0+00 1 115 2.038+01 0.000+00 0.000+00 0.000+00 0.000+00 0.000+00 2.434+04 0.0+00 1 117 2.035+01 1.052+12 1.960−01 1.313−02 0.000+00 0.000+00 0.000+00 9.2−01 1 121 2.030+01 0.000+00 0.000+00 0.000+00 1.136+08 0.000+00 0.000+00 0.0+00 For Y XXX, the only results available in the literature for comparison purposes are the f -values of Zhang and Sampson [7] for E1 transitions from the ground level and these are compared in Table 12. For a few weak transitions the differences are large, in particular for 1-63 (2s 2 2p 6 1 S 0 -2s 2 2p 5 4s 3 P o 1 ), a spin changing intercombination transition, for which our f -value is very small (∼10 −6 ), and subsequently the discrepancy is of three orders of magnitude. However, such comparisons are very limited and hence cannot be confidently relied upon. In conclusion, on the basis of the (whatever possible) comparisons have been made for all three ions, our experience on a wide range of other ions, including F-like [4,13] and Ne-like [14,15], and considering that we have included a large CI as well as relativistic effects in generating wavefunctions, we assess the accuracy of our radiative rates to be about 20%, for a majority of strong transitions with f ≥ 0.1. Table 8: Transition wavelengths (λ ij inÅ), radiative rates (A ji in s −1 ), oscillator strengths (f ij , dimensionless), and line strengths (S, in atomic units) for electric dipole (E1), and A ji for E2, M1 and M2 transitions in Y XXX. The last column gives the ratio R of the velocity and length forms of A(E1). a±b ≡ a×10 ±b .

Conclusions
In this paper energy levels have been reported for three ions, namely F-like Sc XIII and Ne-like Sc XII and Y XXX. For the calculations the grasp code has been adopted and CI has been included among a large number of configurations. Additional calculations have also been performed with fac, by including even larger CI. This was necessary for accuracy assessments [24] because the existing data available for these ions are very limited. Energies have been listed for the lowest 102, 125 and 139 levels of the respective ions, although calculations have been performed for a much larger ranges. This is because beyond these levels is a mixing from other configurations. However, energies for higher levels can be obtained from the author on request. On the basis of a variety of comparisons, the listed energies (in general) are assessed to be accurate to better than 1% for most levels. However, this assessment of accuracy may change if laboratory measurements in future become available for a larger number of levels.