Dielectronic Recombination Lines of C+

The current paper presents atomic data generated to investigate the recombination lines of C II in the spectra of planetary nebulae. These data include energies of bound and autoionizing states, oscillator strengths and radiative transition probabilities, autoionization probabilities, and recombination coefficients. The R-matrix method of electron scattering theory was used to describe the C2+ plus electron system.

Many atomic processes can contribute to the production of carbon spectral lines, including radiative and collisional excitation, and radiative and dielectronic recombination. The current study is concerned with some of the lines produced by dielectronic recombination and subsequent radiative cascades.
Capture of an incoming electron to a target ion may occur through a non-resonant background continuum, which is radiative recombination (RR), or through a process involving doubly-excited states (resonances) known as dielectronic recombination (DR). The latter can lead either to autoionization, a radiationless transition to a lower state of the ion with the ejection of a free electron, or to stabilization by radiative decay to a lower true bound state.
The rate for RR rises as the free electron temperature falls and hence tends to be the dominant recombination process at low temperatures. It is particularly significant in the cold ionized plasmas found, for example, in some supernova remnants [20][21][22].
We may distinguish three types of DR mechanism with relevance at different temperatures 1. High-temperature DR (HTDR) which occurs through Rydberg series of autoionizing states, and in which the radiative stabilization is via a decay within the ion core.
2. Low-temperature DR (LTDR) which operates via a few near-threshold resonances with radiative stabilization usually through decay of the outer, captured electron. These resonances are usually within a few thousand wave numbers of threshold, so this process operates at thousands to tens of thousands of degrees Kelvin.
3. Fine-structure DR (FSDR) which is due to Rydberg series resonances converging on the fine-structure levels of the ground term of the recombining ion, and is necessarily stabilized by outer electron decays. This process can operate at very low temperatures, down to tens or hundreds of Kelvin.
In this paper we are concerned only with LTDR and the resulting spectral lines.
With regard to the recombination lines of C ii, there are many theoretical and observational studies mainly within astronomical contexts. Examples are [5, 6, 9-12, 15-18, 20, 23-34]. From the perspective of atomic physics, the most comprehensive of these studies and most relevant to the current investigation is that of Badnell [31] who calculated configuration mixing effective dielectronic recombination coefficients for the recombined C + ion at temperatures T = 10 3 − 6.3 × 10 4 K applicable to planetary nebulae, and Davey et al [16] who computed effective recombination coefficients for C ii transitions between doublet states for the temperature range 500-20000 K with an electron density of 10 4 cm −3 relevant to planetary nebulae. Badnell performed calculations in both LS and intermediate coupling schemes and over a wide temperature range, using the Autostructure code which treats the autoionizing states as bound and the interaction with continuum states by a perturbative approach. On the other hand, Davey et al performed their calculations using the R-matrix code which uses a unified treatment of bound and continuum states but worked in the LS-coupling scheme and hence their results were limited to doublet states.
The aim of the current study is to treat the near-threshold resonances and subsequent radiative decays using the unified approach of the R-matrix method in intermediate coupling. The investigation includes all the autoionizing resonance states above the threshold of C 2+ 1 S e with a principal quantum number n < 5 for the captured electron as an upper limit. This condition was adopted mainly due to computational limitations. In total, 61 autoionizing states (27 doublets and 34 quartets) with this condition have been theoretically found by the R-matrix method. Of these 61 resonances, 55 are experimentally observed according to the NIST database [35]. More details will follow in the forthcoming sections.

Autoionizing and Bound States Calculations
We use the R-matrix code [36], and Autostructure [37][38][39] to compute the properties of autoionizing and bound states of C + . Orbitals describing the target for the R-matrix scattering calculations were taken from Berrington et al [40], who used a target comprising the six terms 2s 2 1 S e , 2s2p 3 P o , 1 P o and 2p 2 3 P e , 1 D e , 1 S e , constructed from seven orthogonal orbitals; three physical and four pseudo orbitals. These orbitals are: 1s, 2s, 2p, 3s, 3p, 3d and 4f, where the bar denotes a pseudo orbital. The purpose of including pseudo orbitals is to represent electron correlation effects and to improve the target wavefunctions. The radial parts, P nl (r), of these orbitals are Slater Type Orbital (STO) generated by the CIV3 program of Hibbert [41]. Each STO is defined by where C i is a coefficient and P i and ξ i are indicial parameters in this specification, r is the radius, and i is a counting index that runs over the orbitals of interest. The values of these parameters are given in Table (A). In this work we construct a scattering target of 26 terms which includes the 6 terms of Berrington et al [40] plus all terms of the configurations 2s3l and 2p3l, l= 0, 1, 2. This includes the terms outside the n = 2 complex which make the largest contribution to the dipole polarizability of the 2s 2 1 S e and 2s2p 3 P o states of C 2+ .
The R-matrix calculations were carried out in the intermediate coupling (IC) scheme by including the spin-orbit interaction terms of the Breit-Pauli Hamiltonian. The requirement for the IC scheme arises from the fact that in LScoupling the conserved quantities are LSπ and hence only the doublet states that conserve these quantities, such as 2 S e and 2 P o , can autoionize. Therefore, in LS-coupling no autoionization is allowed for quartet terms or some doublet states, such as 2 S o and 2 P e .
The number of continuum basis orbitals used to express the wavefunction in the inner region (MAXC in STG1 of the R-matrix code) was varied between 6-41 and the results were analyzed. It was noticed that increasing the number of basis functions, with all ensuing computational costs, does not necessarily improve the results; moreover a convergence instability may occur in some cases. It was decided therefore to use MAXC = 16 in all calculations as a compromise between the computational resources requirement and accuracy. The effect of varying the size of the inner region radius in the R-matrix formulation was also investigated and a value of 10 atomic units was chosen on the basis of numerical stability and convergence.
Several methods exist for finding and analyzing resonances that arise during the recombination processes; these methods include the Time-Delay method of Stibbe and Tennyson [42] which is based on the use of lifetime matrix eigenvalues [43] to locate the resonance position and identify its width, and the QB method of Quigley et al [44,45] which applies a fitting procedure to the reactance matrix eigenphase near the resonance position using the analytic properties of the R-matrix theory.
For the low-lying resonances just above the 1 S e ionization threshold, the scattering matrix S has only one channel, and hence the reactance matrix, K, is a real scalar with a pole near the resonance position. According to the collision theory of Smith [43], the lifetime matrix M is related to the S-matrix by where i is the imaginary unit, (= h/2π) is the reduced Planck's constant, S * is the complex conjugate of S, and E is the energy. Now, a K-matrix with a pole at energy E 0 superimposed on a background K o can be approximated by where K i is the value of K-matrix at energy E i and g is a physical parameter with dimension of energy. It follows that in the case of single-channel scattering the M-matrix is real with a value given by Using the fact demonstrated by Smith [43] that the lifetime of the state is the expectation value of M , it can be shown from Equation (4) that the position of the resonance peak E r is given by while the full width at half maximum ∆ E is given by The two parameters of primary interest to our investigation are the resonance energy position E r , and the resonance width ∆ r which equals the full width at half maximum ∆ E . However, for an energy point E i with a K-matrix value K i , Equation (3) has three unknowns, K o , g and E 0 , which are needed to find E r and ∆ r . Hence, three energy points in the immediate neighborhood of E 0 are required to identify these unknowns. As the K-matrix changes sign at the pole, the neighborhood of E 0 is located by testing the K-matrix value at each point of the energy mesh for sign change.
Consequently, the three points are obtained and can be used to find E r and ∆ r .
To improve the performance of this approach, an interactive graphical technique was developed to read and plot the K-matrix data directly while searching for poles. In a later stage, more efficient non-graphical tools for pole searching were used. The purpose of these tools is to search for any sudden increase or decrease in the background of the K-matrix where poles do exist and where a search with a finer energy mesh would then be carried out.
With regard to sampling the three points for the K-matrix calculations, it was observed that sampling the points very close to the pole makes the energy position and width of resonances susceptible to fluctuations and instabilities. Therefore, a sampling scheme was adopted in which the points are selected from a broad range not too close to the pole. This approach was implemented by generating two meshes, coarse and fine, around the pole as soon as the pole is found.
To check the results, several different three-point combinations for each resonance were used to find the position and width of the resonance. In each case, the results from these different combinations were compared. In all cases they were identical within acceptable numerical errors. The results of QB confirm this conclusion as they agree with the K-matrix results as can be seen in Table (2). The results for all bound and autoionizing states are given in Tables (1-2). In total, 142 bound states belonging to 11 symmetries (2J = 1, 3, 5, 7, 9 even and 2J = 1, 3, 5, 7, 9, 11 odd) and 61 resonances belonging to 11 symmetries (2J = 1, 3, 5, 7, 9, 11 even and 2J = 1, 3, 5, 7, 9 odd) were identified. As seen in Tables (1-2), the theoretical results for both bound and resonance states agree very well with the available experimental data both in energy levels and in fine structure splitting. Experimental energies are not available for the very broad resonances as they are difficult to find experimentally. The maximum discrepancy between experiment and theory in the worst case does not exceed a few percent. Furthermore, the ordering of the energy levels is the same between the theoretical and experimental in most cases. Order reversal in some cases is indicated by a minus sign in the fine structure splitting.

Oscillator Strength Calculations
The oscillator strengths for free-free, free-bound and bound-bound transitions are required to find the radiative probabilities for these transitions. As there is no free-free stage in the available R-matrix code, the f -values for the free-free transitions could not be produced by R-matrix. Therefore, these values were generated by Autostructure in the intermediate coupling scheme where 60 electron configurations were included in the Autostructure input: 2s 2 nl (2p≤ nl ≤7s), 2s2p nl (2p≤ nl ≤7s), 2p 3 , and 2p 2 nl (3s≤ nl ≤7s). An iterative procedure was followed to find the orbital scaling parameters (λ's) required in Autostructure. These parameters are given in Table (B). The scaling parameters, which are obtained by Autostructure in an automated optimization variational process, are required to minimize the weighted energy sum of the included target states [37,46,47].
Regarding the free-bound transitions, the f -values for more than 2500 free-bound transitions were computed by integrating the photoionization cross sections (in mega barn) over the photon energy (in Rydberg). This was done for each bound state and for all resonances in the corresponding cross-section. The area under the cross-section curve comprises a background contribution, assumed linear with energy, and the contribution due to the resonance, which is directly related to the bound-free oscillator strength. The photoionization cross sections as a function of photon energy were generated by stage STGBF of the R-matrix code. Some representative cross-sections are shown in Figure   (

Emissivity Calculations
We are concerned with spectral lines formed by dielectronic capture followed by radiative decay. Including only these two processes and autoionization, the number density, N l of a state l is given by where c is the rate coefficient for dielectronic capture to state l and Γ a l is the autoionization probability of that state, related by where the subscript S refers to the value of the ratio given by the Saha equation, and N e and N i are the number density of electrons and ions respectively. If state l lies below the ionization limit, c = Γ a l = 0. Equation 7 can be solved for the populations l by a stepwise downward iteration using the 'Emissivity' code [48]. Note that all other processes have been neglected, so that the results obtained here are incomplete for states likely to be populated by radiative recombination or collisional excitation and de-excitation. The results for free-free and free-bound transitions can be used directly to predict line intensities from low-density astrophysical plasmas such as gaseous nebulae but those between bound states underestimate the line intensities in general and should only be used as part of a larger ion population model including all relevant processes.
The emissivity in transition u → l is given by where N u is the population of the upper state, Γ r ul is the radiative transition probability between the upper and lower states, h is the Planck's constant, and ν is the frequency of the transition line. The equivalent effective recombination coefficient f , which is linked to the emissivity by the following relation, can also be computed where N e and N i are the number density of electrons and ions respectively. In these calculations, all theoretical data for the energy of resonances and bound states were replaced with experimental data from NIST when such experimental data were available.
As part of this investigation, the C ii lines from several observational line lists found in the literature, such as that of Zhang et al [18] for the planetary nebula NGC 7027, were analyzed using our theoretical line list and all correctlyidentified C ii recombination lines in these observational lists were identified in our theoretical list apart from very few exceptions which are outside our wavelength range. The analysis also produced an electron temperature for the lineemitting regions of a number of astronomical objects in reasonably good agreement with the values obtained by other researchers using different data and employing other techniques. The details of these investigations will be the subject of a forthcoming paper.

Comparison to Previous Work
In this section we make a brief comparison of some of our results against a sample of similar results obtained by other researchers previously. These include radiative transition probabilities, given in Table (C), autoionization probabilities, given in Table (D), and dielectronic recombination coefficients, given in Table (E). Transition probabilities generally show good agreement between the various calculations for the strongest electric dipole transitions. There are some significant differences for intercombination transitions, indicative of the increased uncertainty of the results for these cases. There are also significant differences between the present results and those of Nussbaumer & Storey [6] for some of the allowed but two-electron transitions from the 2s 2 3p configuration, where we would expect our results to be superior, given the larger scattering target.
In Table (D) we compare our calculated autoionization probabilities with those of De Marco et al [49]. They combined the LS-coupled autoionization probabilities calculated in the close-coupling approximation by Davey et al [16] with oneand two-body fine structure interactions computed with SUPERSTRUCTURE [37] to obtain autoionization probabilities for four states that give rise to spectral lines seen in carbon-rich Wolf-Rayet stars. For the three larger probabilities, there is agreement within 25%. For the 4f 2 G e 9/2 state there is a factor of two difference but this state has a very small autoionization probability corresponding to an energy width of only 4.2 × 10 −8 Rydberg.
In Table (E) we compare our results with effective recombination coefficients from Table (1) of Badnell [31]. His results are tabulated between terms, having been summed over the J of the upper and lower terms of the transition, so results for individual lines cannot be compared. Badnell only tabulates results for one transition in which the upper state is allowed to autoionize in LS-coupling (2s2p( 3 P o )3d 2 F o ) and here the agreement is excellent as one might expect.
On the other hand, the two levels of 2s2p( 3 P o )3d 2 D o have very small autoionization widths and our coefficients are factor of 5× smaller than Badnell's for transitions from this term. We note that the fine structure splitting of this term is well represented in our calculation as is its separation from neighboring states with large autoionization widths, giving us confidence in our result.

Results
In this section, we present a sample of the data produced during this investigation. In Table (1) the theoretical results for the energies of the bound states of C + below the C 2+ 1 S e 0 threshold are given alongside the available experimental data from the NIST database [35]. Similarly, Table (2) presents the energy and autoionization width data for the resonances as obtained by the K-matrix and QB methods. In these tables, a negative fine structure splitting indicates that the theoretical levels are in reverse order compared to their experimental counterparts. It is noteworthy that due to limited precision of figures in these tables, some data may appear inconsistent, e.g. a zero fine structure splitting from two levels with different energies. Full-precision data in electronic format are available from the Centre de Données astronomiques de Strasbourg (CDS) database.
Regarding the bound states, all levels with effective quantum number between 0.1-13 for the outer electron and 0 ≤ l ≤ 5 (142 states) were found. The 8 uppermost bound states in Table ( Tables (3)(4)(5) present a sample of the f -values of the free-bound transitions for some bound symmetries as obtained by integrating photoionization cross section over photon energy where these data are obtained from stage STGBF of the R-matrix code. The columns in these tables stand for the bound states identified by their indices as given in Table (1) while the rows stand for the resonances represented by their indices as given in Table (2). An entry of '0' in the f -value tables indicates that no peak was observed in the photoionization cross section data.
Finally, Table (6) contains a sample of the effective recombination coefficients for transitions extracted from our list in a wavelength range where several lines have been observed in the spectra of planetary nebulae [18].

Conclusions
In this study, a list of line effective recombination coefficients is generated for the atomic ion C + using the R-matrix, Autostructure and Emissivity codes. These lines are produced by dielectronic capture and subsequent radiative decays of the low-lying autoionizing states above the threshold of C 2+ 1 S e with a principal quantum number n < 5 for the captured electron. The line list contains 6187 optically-allowed transitions which include many C ii lines observed in astronomical spectra, notably the lines of C ii in the observational list of Zhang et al [18] for the spectrum of planetary nebula NGC 7027. Beside the effective recombination coefficients, the list also include detailed data of level energies for bound and resonance states, and oscillator strengths.
The theoretical results for energy and fine structure splitting agree very well with the available experimental data for both resonances and bound states. The complete data set of our line list can be obtained from the Centre de Données astronomiques de Strasbourg (CDS) database.
In the course of this investigation, a method for finding and analyzing resonances was developed as an alternative to the QB [44,45] and Time-Delay [42] methods. In this one-channel case, the method offers a superior alternative in terms of numerical stability, computational viability and comprehensiveness in the search for the low-lying autoionizing states.    The seven orbitals used to construct the C 2+ target and their STO parameters. The bar marks the pseudo orbitals.
Orbital       f Theoretical Energy in Rydberg from R-matrix calculations relative to the C 2+ 1 S e 0 limit. g Fine Structure Splitting from R-matrix in cm −1 . The minus sign indicates that the theoretical levels are in reverse order compared to the experimental.

Table 2
The available experimental data from NIST for the resonance states of C + above the C 2+ 1 S e 0 threshold alongside the theoretical results as obtained by K-matrix and QB methods.   Table 3 Free-bound f -values for bound symmetry J π = 1/2 o obtained by integrating photoionization cross sections from Rmatrix calculations. The superscript denotes the power of 10 by which the number is to be multiplied.  Table 4 Free-bound f -values for bound symmetry J π = 3/2 o obtained by integrating photoionization cross sections from R-matrix calculations. The superscript denotes the power of 10 by which the number is to be multiplied.
In .  2  13  14  18  21  23  28  33  35  50  65  78  89  100  111  122 Table 6 Sample of our line list where several lines have been observed astronomically. The first column is for experimental/theoretical energy identification for the upper and lower states respectively where 1 refers to experimental while 0 refers to theoretical. The other columns are for the atomic designation of the upper and lower states respectively, followed by the air wavelength in angstrom and effective dielectronic recombination rate coefficients in cm 3 .s −1 for the given logarithmic temperatures. The superscript denotes the power of 10 by which the number is to be multiplied.