2 The R-matrix calculation

2.1 C²⁺ target

We calculate the the bound-bound and bound-free radiative data for C ⁺ states from a C ²⁺ $\; + \; {\rm e}^{-}$ scattering problem. The starting point for the calculations is the C ²⁺ target. In our photoionization calculations, as in the OP calculations (Yu Yan & Seaton [1987]), the C ²⁺ target given by Berrington et al. ([1977]) was used. It includes the 2s^2 1S^e, 2s2p ³P^o, ¹P^o and 2p^{2 3}P^e, ¹D^e, ¹S^e target eigenstates represented in terms of seven orthogonal basis orbitals and pseudo-orbitals: 1s, 2s, 2p, ${\bar 3}$s, ${\bar 3}$p, ${\bar 3}$d, ${\bar 4}$f. The radial parts of the basis orbitals were obtained using the CIV3 code of Hibbert ([1975]).

2.2 C⁺ bound state energies

 

 

Table 1: Energies of even parity C ⁺ bound states relative to the ionization threshold

	Energy (Ry)
State	Calculated	Experimental
	2Se
2s2p2	-0.899095	-0.912867
2s23s	-0.725740	-0.730205
2s24s	-0.358483	-0.359335
2s25s	-0.212136	-0.212489
2s26s	-0.140152	-0.140346
2s27s	-0.099483	-0.099622
2s28s	-0.074279
2s29s	-0.057592
2s210s	-0.045989
2s211s	-0.037622
2s212s	-0.031477
2s2p3p	-0.027270
2s213s	-0.024921
2s214s	-0.022077
2s215s	-0.019253
	2Pe
2s2p2	-0.767228	-0.783844
2s2p3p	-0.131191	-0.133305
	2De
2s2p2	-1.10066	-1.10937
2s23d	-0.463932	-0.465812
2s24d	-0.259290	-0.260092
2s25d	-0.165203	-0.165578
2s26d	-0.114471	-0.114731
2s27d	-0.084449
2s2p3p	-0.070199	-0.073480
2s28d	-0.062318
2s29d	-0.049872
2s210d	-0.040445
2s211d	-0.033420
2s212d	-0.028068
2s213d	-0.023903
2s214d	-0.020599
2s215d	-0.017934
	2Ge
2s25g	-0.160169	-0.160314
2s26g	-0.111251	-0.111320
2s27g	-0.081734	-0.081775
2s28g	-0.062574	-0.062612
2s29g	-0.049437
2s210g	-0.040041
2s211g	-0.033089
2s212g	-0.027802
2s213g	-0.023688
2s214g	-0.020424
2s215g	-0.017791

 

 

Table 2: Energies of odd parity C ⁺ bound states relative to the ionization threshold

	Energy (Ry)
State	Calculated	Experimental
	2Po
2s2p2	-1.78778	-1.79184
2s23p	-0.589943	-0.591739
2s24p	-0.310095	-0.311142
2p3	-0.245582	-0.254462
2s25p	-0.190823	-0.194769
2s2p3s	-0.166010	-0.172032
2s26p	-0.124169	-0.124589
2s27p	-0.090202	-0.090391
2s28p	-0.068281
2s29p	-0.053445
2s210p	-0.042958
2s211p	-0.035277
2s212p	-0.029484
2s213p	-0.025008
2s214p	-0.021479
2s215p	-0.018647
	2Do
2p3	-0.413724	-0.421035
	2Fo
2s24f	-0.252011	-0.252308
2s25f	-0.161223	-0.161382
2s26f	-0.111882	-0.112070
2s27f	-0.082144	-0.082228
2s28f	-0.062855
2s29f	-0.049638
2s210f	-0.040189
2s211f	-0.033202
2s212f	-0.027890
2s213f	-0.023758
2s214f	-0.020480
2s215f	-0.017836

We have calculated ionization energies for all doublet states with total orbital angular momentum quantum number $L \leq 4$ and with principal quantum number $n \leq 15$. These energies are compared with existing experimental energies in Tables 1 and 2. Note that the main 2s²nl series states are perturbed by states from the 2s2p3s and 2s2p3p electron configurations. The upper principal quantum number limit ($n \leq 15$) of the present calculation was chosen to include the 2s2p3p ²S term which lies energetically between the 12s ²S$\rm ^e$ and 13s ²S$\rm ^e$ terms. This perturbation is not included in the OP calculation for C⁺, which only extends to n = 10. Such perturbations significantly modify the photoionization cross-sections and hence recombination coefficients for nearby states in the main series.

2.3 C⁺ bound-bound radiative transitions

We have described most of the techniques that we use to calculate bound-bound radiative data elsewhere (Kisielius et al. [1998]). Briefly, for $n \leq 15$, $L \leq 4$, all radiative transition probabilities are obtained from the R-matrix calculation referred to above. For most other transitions we use a combination of the Coulomb approximation and hydrogenic data, the methods and codes used having been described by Storey ([1994]). For transitions $n \leftrightarrow n^{\prime}, n < n^{\prime}$ with $n \leq 5$ and $n^{\prime} \geq 16$ we use the following further technique. Using the R-matrix approach, photoionization cross-section data from a given lower term (S_i L_i) to a specific final symmetry (S_f L_f) can be computed for negative free electron energies, enabling us to obtain absorption oscillator strengths from the given lower term to high members of the 2s ² nl (S_f L_f) series. This technique has the advantage over other approximate methods, such as the Coulomb approximation, that configuration interaction is fully accounted for in the wave function of the lower term. Cascading from high Rydberg states to the low-lying terms of the ion is therefore more accurately treated.

2.4 C⁺ photoionization cross-sections

In many cases treated by the OP, the free electron energy mesh on which the photoionization cross-sections are calculated, was too coarse to accurately map narrow resonance features. This usually leads to significant contributions to the recombination coefficient being missing, since the resonance contribution is proportional to the area underneath the resonance. The photoionization cross-sections for C ⁺ generated for the OP (Yu Yan & Seaton [1987]) have poor resolution in some of the resonance structure because they were based on a mesh in which points are equally spaced in effective quantum number relative to the next threshold (hereafter referred to as a quantum defect mesh). One hundred points were provided per integer change in effective quantum number and subsequently, this number was further reduced by removing any points that could be linearly interpolated from two adjacent points within some prescribed accuracy.

In contrast to the OP data, we use a variable step energy mesh for photoionization cross-sections that delineates all resonances to a prescribed accuracy. First, we employ quantum defect theory to determine resonance positions and widths using a coarse energy mesh. We then use this information to generate an electron energy mesh which maps the resonances to a chosen accuracy (Kisielius et al. [1998]). In the third step we calculate photoionization cross-sections using the new variable step mesh.

Figure 1: Comparison of the photoionization cross-section (in Mb) for 2s2p2 2D, calculated by the OP (top) and this work (bottom)

In Fig. 1, we demonstrate the difference in resolution between an OP cross-section and one calculated by our method. The OP cross section consists of 199 points based on a quantum defect mesh whilst the latter is based on an energy mesh of 3281 points. In the OP cross-section, the resonance at 1.135 Ry consists only of 3 points, and the next one is missing completely. There is a significant difference in the peak heights of the resonances too.

Figure 2: Comparison of the area (in Mb $\cdot$ Ry) under the 2s2p2 2D cross-section. Top figure - OP data, bottom - present calculation

The effects of correctly delineating the resonances are more clearly demonstrated in Fig. 2. We show the area under the cross-sections which is a more important measure of their contribution to the recombination coefficient. The total area under the 2s2p^2 2D$\rm ^e$ photoionization cross-section from the OP is $1.585\;{Mb} \, \, {Ry}$, whereas our calculation gives $4.573\;{Mb}\, \,{Ry}$for the same energy range.

This detailed treatment of the energy mesh was undertaken for the energy region from the 2s²(¹S$\rm ^e$) limit up to 4/15² Ry below the 2s2p(³P^o) threshold, since this region contains the main contribution to the recombination at the temperatures of interest. Above the 2s2p(³P^o) threshold, a quantum defect mesh was used. A very high degree of accuracy is not required here so a step in effective quantum number was chosen that provided 1000 points per integer change in quantum number.

Finally, we note that we deal with radiative damping of resonances in the same way as described by Kisielius et al. ([1998]), in that any autoionizing states for which the radiative width is greater than the autoionization width are omitted from the list of resonances used to generate the high resolution energy mesh. As a result such resonances do not appear in the photoionization cross-sections and do not contribute to the recombination. In practice this means omitting all resonances associated with 2s2p(³P^o)nh electron configurations, as well as the ²H^o terms from the 2s2p(³P^o)ng configurations.