© ESO, 2016

1. Introduction

In relation to their high cosmic abundance, accurate and reliable atomic data for the iron-peak elements are crucial in astrophysics. The advent of high-resolution astrophysical spectroscopy has led to observing these elements in low-ionization stages in various astronomical objects. Emission lines of doubly ionized Fe-peak species have been observed in several nebular environments. Recent Hubble Space Telescope/Space Telescope Imaging Spectograph (HST/STIS) observations from the Weigelt blobs of Eta Carinae (ηCar) have revealed several forbidden lines of Fe III and Ni III (Zethson et al. 2012). Lines of doubly ionized species have also been detected in various galactic sources, such as Herbig-Haro objects in the Orion Nebula (Mesa-Delgado et al. 2009) and extragalactic objects, including active galactic nuclei (Vestergaard & Wilkes 2001). Reliable radiative data are therefore essential for interpreting these observations and obtaining a diagnostic of the physical conditions in the astrophysical plasma. However, our knowledge of doubly charged iron-peak ions is still incomplete, in particular when it comes to forbidden transitions between the low-lying metastable states.

Atomic data calculations for iron-peak elements are very challenging owing to the complexity of these systems due to the open 3d subshell. In particular, transition probabilities for magnetic dipole (M1) and electric quadrupole (E2) lines are very difficult to compute due to their extreme sensitivity to configuration interaction and level mixing. Forbidden radiative rates were only available for selected transitions in five ions of the doubly ionized Fe peak. As the simplest atomic structure considered in this work, doubly ionized scandium has been investigated extensively by Ali & Kim (1988) using the multiconfigurational Dirac-Fock (MCDF) method, by Zeippen (1990) with the SUPERSTRUCTURE code, and more recently by Sahoo et al. (2008) and Nandy et al. (2011), both using an all-order, perturbative, relativistic many-body approach, i.e. the relativistic coupled-cluster (RCC) method.

Biémont et al. (1992) published a list of ab initio transition probabilities of M1 and E2 transitions within the 3d² configuration of Ti III using the Relativistic Hatree-Fock (HFR) approach and the SUPERSTRUCTURE code. Raassen & Uylings (1997) also performed fully-relativistic multiconfiguration Dirac-Fock (MCDF) calculations for all the metastable levels of this ion. Irimia (2007) published theoretical lifetimes for the 33 levels belonging to the low-lying metastable terms of V III using the multiconfiguration Hartree-Fock (MCHF) method with Breit-Pauli (BP) corrections to a non-relativistic Hamiltonian. Selected transition probabilities were also presented in this paper. Radiative rates have been computed for the astrophysically important Fe III ion by Quinet (1996) using the HFR approach, by Deb & Hibbert (2009) with the CIV3 code, and more recently by Bautista et al. (2010) using the same theoretical methods as those presented in this work. Transition rates have also been published for forbidden lines in the 3d⁷ ground configuration of Co III by Hansen et al. (1984) using a parametric approach.

When computing forbidden radiative rates, it is common to assess the quality of the results by comparing them with a few metastable lifetime measurements performed with a storage ring (see, e.g., Lundin et al. 2007). When experimental data are missing, information on the accuracy of the radiative rates can be obtained by comparing calculations using different independent theoretical approaches. The agreement observed between the sets of results allows us to perform consistency checks and estimate the uncertainties affecting the data. This is the approach adopted in the present work for computing E2 and M1 transition probabilities where we compare the results of two different theoretical methods with each other, together with previous results when available. Since the odd levels can be de-excited by E1 transitions (several orders of magnitude stronger than E2 and M1 transitions) to the even states, odd-odd forbidden transitions are of little or no interest because they are very unlikely to be observed in an experimental and/or astrophysical spectrum. Therefore, we chose to limit our work to the even metastable states.

2. Theoretical models

The first theoretical approach used in this work is the pseudo-relativistic Hartree-Fock (HFR) method implemented in Cowan’s chain of computer codes (Cowan 1981). In our calculations, configuration interactions were considered by including the configurations of the type 3d^k, 3d^k−14s, 3d^k−15s, 3d^k−14d, 3d^k−24s², 3d^k−24p², 3d^k−24d², 3d^k−24s4d, 3d^k−24s5s, 3s3p⁶3d^k+1, 3s3p⁶3d^k4s, and 3s3p⁶3d^k−14s² with k = 1 (Sc III), k = 2 (Ti III), k = 3 (V III), k = 4 (Cr III), k = 5 (Mn III), k = 6 (Fe III), k = 7 (Co III), and k = 8 (Ni III). This method was then combined with a least-squares optimization routine that minimizes the differences between the calculated and available experimental energy levels belonging to the low-lying even configurations 3d^k and 3d^k−14s. For Sc III, Ti III, V III, Mn III, Co III, and Ni III, the experimental data used in this semi-empirical process were taken from the NIST compilation (Kramida et al. 2013), which is exclusively based on the previous compilation by Sugar & Corliss (1985).

For Cr III and Fe III, we used more recent data from Ekberg (1997) and Ekberg (1993), respectively. We also calculated radiative transition rates for M1 and E2 forbidden transitions using the atomic structure code AUTOSTRUCTURE (Badnell 1988). This code is based on the program SUPERSTRUCTURE originally developed by Eissner et al. (1974). In this approach the wavefunctions are written as a configuration interaction expansion of the type (1)where the coefficients c_ji are chosen so as to diagonalize ⟨ ψ_j | H | ψ_i ⟩, where H is the Breit-Pauli Hamiltonian and the basic functions φ_j are constructed from one-electron orbitals generated using the Thomas-Fermi-Dirac-Amaldi model potential (Eissner & Nussbaumer 1969).

The Breit-Pauli Hamiltonian for an N-electron system is given by (2)where H_nr is the usual non-relativistic Hamiltonian, and H_1b and H_2b are the one-body and two-body operators. The one-body relativistic operator (3)represents the spin-orbit interaction f_n(SO), the non-fine structure mass variation f_n(mass), and the one-body Darwin f_n(d) corrections. The two-body corrections (4)usually referred to as the Breit interaction, include, on one hand, the fine-structure terms g_nm(SO) (spin-other-orbit and mutual spin-orbit) and g_nm(SS) (spin-spin). On the other hand, they inclue the non-fine structure terms: g_nm(CSS) (spin-spin contact), g_nm(d) (Darwin), and g_nm(OO) (orbit-orbit). The scaling parameters λ_nl for each nl orbital are optimized by minimizing a weighted sum of the energies for all the metastable terms belonging to the 3d^k and 3d^k−14s configurations. Instead of optimizing each scaling parameter individually, we chose to optimize the core orbitals 1s, 2s, 2p, 3s, and 3p together to simulate the effect of missing open-core configurations in our model. Table 1 gives the values of the λ_nl for all the ions considered in this work.

The set of configurations used in the AUTOSTRUCTURE model is the same as the one used for the HFR calculations with the addition of 3d^k−14p and 3d^k−24s4p to ensure a better representation of the 4p orbital. Semi-empirical corrections take the form of term energy corrections (TEC). By considering the relativistic wavefunction, in a perturbation expansion of the non-relativistic functions (5)where H_1b and H_2b are, respectively, the one- and two-body parts of both fine-structure and non-fine-structure Hamiltonians. A modified non-relativistic Hamiltonian is constructed with improved estimates of the differences so as to adjust the centers of gravity of the spectroscopic terms to the available experimental values. Term energy corrections (TEC) have been applied to all the metastable terms considered in this work. Tables A.1 to A.8 compare, respectively, the level energies (in cm^-1), which were obtained before applying the TEC, for all the metastable levels of Sc III, Ti III, VIII, Cr III, Mn III, Fe III, Co III, and Ni III along with the corresponding TEC. The average TEC along the sequence are about 10% or less of the calculated energies. For the manganese ion, we were not able to apply TEC to all the metastable terms because this resulted in a switch in the energies and an incorrect representation of the ground state. Therefore, we kept the ab initio term energies for the 3d⁵ c ²D1, 3d⁴(³D)4s c ⁴D, and 3d⁴(¹S2)4s b ²S.

3. Forbidden M1 and E2 transition probabilities

In this section, we discuss the radiative data calculations for each ion considered in this work. Transition probabilities can be found in Tables A.9 to A.17 for all the forbidden lines depopulating the metastable levels belonging to the 3d^k and the 3d^k−14s configurations. The lack of space means that only the total A-values (M1+E2) contributing more than 10% to the total de-excitation of each level are presented here. The weakest transition probabilities are available upon request to the authors.

3.1. Scandium (Z = 21)

Only three metastable levels arise from 3d and 4s configurations in Sc III. This gives three spectral lines for which we computed the magnetic dipole (M1) and electric quadrupole (E2) contributions. In Table A.9, our HFR and AUTOSTRUCTURE results are compared to the calculations previously published by Ali & Kim (1988), Zeippen (1990), Sahoo et al. (2008), and Nandy et al. (2011). As seen from this table, the agreement between all sets of data is excellent (within 10%).

3.2. Titanium (Z = 22)

In Table A.10 and Fig. 1, we compare our present HFR transition probabilities for M1 and E2 lines involving the levels of 3d² and 3d4s configurations in Ti III with our AUTOSTRUCTURE results and the previous data published by Biémont et al. (1992) and Raassen & Uylings (1997). Very good agreement (within 5%) is observed between our HFR radiative rates and the MCDF results from Raassen & Uylings (1997). We also note that our present AUTOSTRUCTURE calculations agree within 15% with both our HFR and Raassen & Uylings’ results for transitions with A values greater than 10^-2 s^-1, such as 3d²³F₃–3d²¹G₄ (λ = 703.55 nm), 3d²³F₄–3d²¹G₄ (λ = 715.33 nm), 3d²³F₂–3d²¹D₂ (λ = 1180.65 nm), and 3d²³F₃–3d²¹D₂ (λ = 1206.31 nm). Despite a systematic discrepancy between the SUPERSTRUCTURE transition probabilities of Biémont et al. (1992) and all the other sets of results, the overall agreement is still reasonable (within 25%).

3.3. Vanadium (Z = 23)

Computed transition probabilities as obtained in the present work for forbidden lines arising from 3d³ and 3d²4s configurations in V III are reported in Table A.11 and compared in Fig. 2. We can see that our HFR and AUTOSTRUCTURE data are in excellent agreement (within 5%) for the strongest transitions, while larger discrepancies are observed for some weak lines, in particular for those depopulating the 3d³²G, ²P, ²D2, and ²H terms. When comparing the lifetimes of the eight levels corresponding to the last terms (see Table A.12), we find an average dispersion of about 30% between our two sets of results. This disagreement is even greater (up to several orders of magnitude) when comparing our new theoretical data with those obtained by Irimia (2007) using a multiconfiguration Hartree-Fock approach that includes Breit-Pauli corrections. Such weak transitions are extremely sensitive and dependent upon the physical model and the configuration expansion considered in the calculations, but in view of the satisfactory agreement between our HFR and AUTOSTRUCTURE results, we can reasonably expect the A values obtained in the present work for those lines to be more reliable than the data published by Irimia (2007).

3.4. Chromium (Z = 24)

The two lowest configurations of doubly ionized chromium (Cr III) are 3d⁴ and 3d³4s. Transition probabilities obtained in the present work for forbidden lines involving levels of these last configurations are reported in Table A.13 and compared in Fig. 3. It is clearly seen that both our HFR and AUTOSTRUCTURE models give results in very good agreement (within 15–20%), if we except the E2 transition 3d⁴¹G₄–3d³4s ¹D₂ located at 201.41 nm. For this transition the A value computed with AUTOSTRUCTURE (A = 1.29 × 10² s^-1) is 65% higher than the result obtained with HFR (A = 7.78 × 10¹ s^-1). This discrepancy could be explained by the sensitive mixing of the lower level at 25 137.91 cm^-1 composed of 64% 3d⁴ a ¹G₄ and 35% 3d⁴ b ¹G₄.

Fig. 1 Comparison between our present HFR and AUTOS calculations with the previous results of Biémont et al. (1992) and Raassen & Uylings (1997) for forbidden transitions in Ti III. The straight line of equality has been drawn, and the two dashed lines represent a 10% deviation from equality.

Fig. 2 Comparison between our present HFR and AUTOS calculations for V III. The straight line of equality has been drawn, and the two dashed lines represent a 10% deviation from equality.

Fig. 3 Comparison between our present HFR and AUTOS calculations in Cr III. The straight line of equality has been drawn, and the two dashed lines represent a 10% deviation from equality.

Fig. 4 Comparison between our present HFR and AUTOS calculations in Mn III. The straight line of equality has been drawn, and the two dashed lines represent a 10% deviation from equality.

3.5. Manganese (Z = 25)

Mn III is characterized by the ground configuration with a half-filled 3d subshell (3d⁵), which is well known to be rather complicated to deal with theoretically. This complexity affects the calculations of forbidden transition probabilities performed in the present work and reported in Table A.14. Although an overall agreement of 20% is observed when comparing the results obtained with the HFR and AUTOSTRUCTURE approaches for the strongest lines, some rather large discrepancies appear for transitions depopulating highly excited levels. This is illustrated in Fig. 4, which shows a slightly wider scatter in the results than observed in the other ions Sc III, Ti III, V III, and Cr III. This seems to indicate that the A values obtained in this work for forbidden lines in Mn III are probably affected by larger uncertainties in the range of 20–30% for the strongest transitions.

3.6. Iron (Z = 26)

Extensive calculations were carried out recently for Fe III forbidden transitions by Bautista et al. (2010) using similar theoretical approaches to those employed in this work. If their HFR model was the same as the one considered in the present study, their AUTOSTRUCTURE multiconfiguration expansions would include a total of 40 configurations. Figure 5 compares our present HFR and AUTOSTRUCTURE results to the data of Bautista et al. (2010). Although a slight systematic discrepancy is observed with the HFR A values (15 to 20%), the agreement between the two different AUTOSTRUCTURE calculations is really good (often within a few percentage points), indicating that our configuration expansion is sufficient for an accurate calculation of the A- values. The new A values obtained in this work are compared in Table A.15.

3.7. Cobalt (Z = 27)

In Fig. 6, we compare our HFR and AUTOSTRUCTURE results with the calculations published by Hansen et al. (1984) for forbidden lines in Co III. Only A values that correspond to transitions within the 3d⁷ ground configuration are shown in this figure since Hansen et al. (1984) only used a single-configuration model in their computations. Overall good agreement is observed between the three sets of data, but in view of the much larger multiconfiguration bases used in our models and the very good agreement (within 10%) reached between Co III forbidden transition probabilities obtained with these two models, the new data reported in Table A.16 are expected to be more accurate than those of Hansen et al. (1984).

Fig. 5 Comparison between our present HFR and AUTOS calculations in Fe III. The straight line of equality has been drawn, and the two dashed lines represent a 10% deviation from equality.

Fig. 6 Comparison between our present HFR and AUTOS calculations in Co III and the results of Hansen et al. (1984). The straight line of equality has been drawn, and the two dashed lines represent a 10% deviation from equality.

3.8. Nickel (Z = 28)

Transition probabilities for forbidden lines involving the 3d⁸ and 3d⁷4s configurations of Ni III are listed in Table A.17, while a comparison between HFR and AUTOSTRUCTURE results is illustrated in Fig. 7. Even if the AUTOSTRUCTURE A values seem to be systematically smaller than the HFR ones, we observe an overall satisfactory agreement on the order of 20% for the most intense transitions (A ≥ 10² s^-1) if we except the E2 line at 127.71 nm (3d⁸³F₄–3d⁷4s ³P₂), for which the HFR approach gives a transition probability (A = 3.76 × 10² s^-1) that is a factor of 1.50 greater than the AUTOSTRUCTURE result (A = 2.55 × 10² s^-1).

We noticed a slight systematic shift in the A values for several ions considered in this work (Mn III, Fe III, Co III, and Ni III) and found the AUTOSTRUCTURE calculations to be very sensitive to the configuration expansion, to the optimization procedure of the scaling parameters λ_nl, and to the TEC applied to the metastable states. To assess the sensitivity of our AUTOSTRUCTURE results to the optimization of the scaling parameters, we performed a second calculation in Ni III where we optimized the λ_nl on the terms belonging to the 3d⁸, 3d⁷4s and 3d⁷4p configurations instead of limiting ourselves to the terms of 3d⁸ and 3d⁷4s metastable configurations, While the scaling parameters for the 4s and 4p orbitals only varied by about 10%, we observed a general shift in the A values of about 15% between the two AUTOSTRUCTURE calculations, bringing the disagreement between the HFR and the new AUTOSTRUCTURE calculations to 30% instead of 18%. Therefore, the HFR results are expected to carry a much smaller uncertainty than the AUTOSTRUCTURE A values.

Fig. 7 Comparison between our present HFR and AUTOS calculations in Ni III. The straight line of equality has been drawn, and the two dashed lines represent a 10% deviation from equality.

4. Conclusions

Detailed and systematic calculations were carried out for magnetic dipole and electric quadrupole transitions in doubly-ionized iron peak elements from Sc III to Ni III. Using two independent methods based on the pseudo-relativistic Hartree-Fock (HFR) approach and the Thomas-Fermi-Dirac-Amaldi potential approximation implemented in the AUTOSTRUCTURE code allowed us to estimate the uncertainties on the radiative transition probabilities obtained in the present work. For most of

the strongest lines, we observed a general agreement of 20% or better between both sets of data. This is consistent with the usual uncertainty expected when considering radiative parameters for forbidden lines. Transition probabilities for some of the weakest lines were found to be affected by larger uncertainties because of their higher sensitivity to level mixing and configuration interaction. These faint lines can also be affected by cancellation effects in the line strength calculation. However, in most cases those lines do not contribute much to the total de-excitation of a level and are, therefore, not listed in the tables reported in this paper.

The overall good agreement obtained in the present work between transition probabilities computed with two different methods indicates that the new results should be reliable. They represent the most comprehensive and consistent study of forbidden lines available to date for doubly-charged ions belonging to the iron group. It is expected that these new data will help astrophysicists with interpreting numerous stellar spectra in which such lines are detected.

Acknowledgments

We acknowledge financial support from the NASA Astronomy and Physics Research and Analysis Program (award NNX10AU60G), the National Science Foundation (award 1313265), and the Belgian F.R.S.-FNRS where P.Q. is a Research Director. V.F. is currently a post-doctoral researcher of the Return Grant program of the Belgian Scientific Policy (BELSPO).

References

Appendix A: Additional tables

All Tables

All Figures

	Fig. 1 Comparison between our present HFR and AUTOS calculations with the previous results of Biémont et al. (1992) and Raassen & Uylings (1997) for forbidden transitions in Ti III. The straight line of equality has been drawn, and the two dashed lines represent a 10% deviation from equality.
In the text

	Fig. 2 Comparison between our present HFR and AUTOS calculations for V III. The straight line of equality has been drawn, and the two dashed lines represent a 10% deviation from equality.
In the text

	Fig. 3 Comparison between our present HFR and AUTOS calculations in Cr III. The straight line of equality has been drawn, and the two dashed lines represent a 10% deviation from equality.
In the text

	Fig. 4 Comparison between our present HFR and AUTOS calculations in Mn III. The straight line of equality has been drawn, and the two dashed lines represent a 10% deviation from equality.
In the text

	Fig. 5 Comparison between our present HFR and AUTOS calculations in Fe III. The straight line of equality has been drawn, and the two dashed lines represent a 10% deviation from equality.
In the text

	Fig. 6 Comparison between our present HFR and AUTOS calculations in Co III and the results of Hansen et al. (1984). The straight line of equality has been drawn, and the two dashed lines represent a 10% deviation from equality.
In the text

	Fig. 7 Comparison between our present HFR and AUTOS calculations in Ni III. The straight line of equality has been drawn, and the two dashed lines represent a 10% deviation from equality.
In the text