ELECTRON-IMPACT EXCITATION OF Ni ii: EFFECTIVE COLLISION STRENGTHS FOR OPTICALLY ALLOWED FINE-STRUCTURE TRANSITIONS

The study of astrophysics is governed by observation rather than experiment. In the present era of scientific discovery, state-of-the-art research facilities are enabling astronomers to acquire astrophysical images and spectra of exceptional quality and quantity over a broad spectral range. Absorption and emission lines of Ni ii, the first ion of the heaviest and second most abundant iron-peak element, are commonly observed in nebular spectroscopy. A wealth of observations of this species have been identified in the spectra of a myriad of astronomical sources at all wavelengths ranging from the infrared to the ultraviolet. For example, spectra of the peculiar metal-rich star HD 135485 between 3793 Å and 6913 Å revealed absorption lines attributed to a number of metallic species including Ni ii (Trundle et al. 2001). Local thermodynamic equilibrium absolute and differential abundances were presented for each ion identified in the spectrum of HD 135485 and used to deduce the chemical composition and evolutionary process of this mid-B-type star. Recent Space Telescope Imaging Spectrograph (STIS) observations of the strontium filament, a largely neutral emission nebulosity lying close to the luminous blue variable symbiotic star η Carinae, have revealed an uncommon spectrum of over 600 emission lines, including allowed Ni ii contributions (Hartman et al. 2004). Almost 350 weak emission features, mostly associated with allowed transitions from high-excitation states of first ions, were identified in the optical and near-infrared spectral region of the chemically peculiar He-weak star 3 Cen A (HD 120709; Wahlgren & Hubrig 2004). Prominent among these features was the Ni ii spectra. The more recent work of Vreeswijk et al. (2007) presents high-resolution spectroscopic observations of the γ-ray burst GRB 060418 between 3300 Å and 6700 Å, obtained with Very Large Telescope/UVES. These spectra show clear evidence for time variability of allowed transitions involving metastable levels of Ni ii. This is the first report of absorption lines arising from metastable levels of Ni ii along any GRB sightline. In addition, the observations of Véron-Cetty et al. (2006) have shown that the spectrum of the narrow-line Seyfert galaxy IRAS 07598+6508 is dominated by lines of the iron-peak elements including Ni ii. High-resolution spectroscopic observations, both UV Hubble Space Telescope/STIS and optical, were used to characterize the physical state and velocity structure of the multiphase interstellar medium seen toward the nearby star HD 102065 (Nehmé et al. 2008). Four groups of species were identified including lowly ionized states of atoms such as Ni ii. Fossati et al. (2009) analyzed abundances of Ni, including lines of the first ion, using a vast amount of spectral lines observed in three slowly rotating early-type stars, namely HD 145788, 21 Peg, and π Cet. Most recently, Fynbo et al. (2010) measured metallicities from Ni ii after detecting absorption lines from this ion in the spectrum of the quasi-stellar object Q2222-0946. This synopsis of the vibrant observational enterprises of the past decade actualizes the prominence of Ni ii in the astronomical environment and emphasizes its fundamental importance to our comprehension of star formation, stellar structure, and the early universe. In order to facilitate the meaningful interpretation of these observations, an ability to properly understand and meticulously model these spectra is paramount. The provision of accurate and extensive atomic data, both collisional and radiative, is therefore indispensable to the aggregate observational and theoretical effort of determining a reliable spectral synthesis. These data constitute the infrastructure of our quantitative knowledge of the universe. While some of these data can be obtained experimentally, they are frequently of insufficient accuracy or limited to a small number of transitions. Computational approaches therefore represent the only means by which atomic data of the required quality and quantity can be provided.

Within the past three decades, a number of theoretical works investigating the electron-impact excitation of Ni ii have emerged, highlighting the unwavering demand for accurate collisional data for this ion. An early calculation by Nussbaumer & Storey (1982) reported the first computation of electron excitation rates for Ni ii. This was later superseded by the works of Bautista & Pradhan (1996), Watts et al. (1996), and Bautista (2004), each calculation gradually increasing in sophistication simultaneously with the development of more powerful computing resources. To date, the most reliable and extensive set of collisional data for Ni ii has been calculated by Cassidy et al. (2010). This computation transcends antecedent theoretical efforts by retaining all 113 LS terms identified with the five Ni ii basis configurations, specifically 3d⁹, 3d⁸4s, 3d⁷4s², 3d⁸4p, and 3d⁷4s 4p, including all doublet, quartet, and sextet terms in the ionic target representation. This corresponds to a substantial 295 jj-level, 1930 coupled channel scattering complex, involving a total of 43,365 individual transitions. In the work of Cassidy et al. (2010), the authors concentrated specifically on the 153 low-lying forbidden fine-structure transitions between the energetically lowest 18 levels of Ni ii associated with the 3d⁹, 3d⁸4s, and 3d⁷4s² even parity basis configurations. It has been well established that convergence of these transitions, analogous to other forbidden transitions, is much more accelerated than for allowed transitions. Cassidy et al. (2010) concluded that the electron-impact excitation of Ni ii for low-lying forbidden transitions is sufficiently described by the inclusion of partial wave contributions with total angular momentum L ⩽ 15 (J ⩽ 12), with contributions to the collision strengths of these transitions from each of the Jπ partial waves having fully converged before the aforementioned total angular momentum value is reached. In contrast to the forbidden transitions, convergence of the allowed lines is significantly slower and accordingly they necessitate further endeavor in order to achieve suitably converged collision strengths. This slow convergence is directly attributable to the additional contributions to the total collision strength for optically allowed dipole transitions which come from large values of incident electron angular momenta, larger values than would be considered within a full exchange evaluation. In order to adequately account for these supplementary contributions, the initial exchange calculation incorporating all partial waves with total angular momenta L ⩽ 15 (J ⩽ 12) must be augmented with a non-exchange calculation incorporating contributions from higher partial waves, followed by a "top-up" procedure to complete the summation of partial collision strengths over yet higher values of L.

There is currently a paucity of reliable atomic data for dipole allowed transitions among fine-structure levels of Ni ii. This dearth is directly attributable to the complexity of the open 3d shell of the ionic target—a universal problem when considering iron-peak elements. In the present theoretical effort, we hope to enhance the framework of spectroscopic astrophysics by concentrating expressly on the computation of converged total collision strengths and corresponding effective collision strengths for the optically allowed transitions between the 3d⁹, 3d⁸4s and 3d⁷4s² even parity levels and the 3d⁸4p and 3d⁷4s 4p odd parity levels.

This paper is structured as follows. The following section describes the specifics of the atomic calculation. Section 3 is devoted to conveying a graphical and quantitative synopsis of the results and comparisons are made with previous theoretical works where possible. The convergence of the collision strengths is exhaustively investigated. Finally, conclusions are inferred in Section 4.

The theoretical target model adopted in the present calculation has been comprehensively discussed by Cassidy et al. (2010) and thus we present only a brief summary for current purposes. All 113 LS terms identified with the five basis configurations 3d⁹, 3d⁸4s, 3d⁷4s², 3d⁸4p, and 3d⁷4s 4p were included in the target wavefunction expansions, retaining all doublet, quartet, and sextet terms. These target states were optimally represented using conventional configuration interaction (CI) type wavefunctions. The orbital set comprised nine orthogonal one-electron functions, specifically eight spectroscopic (1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p), in addition to one non-physical ${\overline{4{d}}}$ pseudo-orbital included to represent additional electron correlation effects. The theoretical target state energies have been discussed in detail in the works of Cassidy et al. (2010). In the present collision calculation, the energies of the spectroscopic target states were shifted during diagonalization of the Hamiltonian in order to bring them into line with the experimental thresholds indicated by the NIST databank.

The parallel RMATRXII R-matrix package (Burke et al. 1994) was utilized in conjunction with the parallel PSTGF program (Ballance & Griffin 2004) to compute the total electron-impact excitation collision strengths for all fine-structure transitions considered in this study. The former suite of codes was used to complete the R-matrix inner region calculation in LS coupling. The resulting Hamiltonian matrices were then transformed to Jπ coupling using the program FINE, and the external region calculation carried out using PSTGF (see Cassidy et al. 2010 for a detailed discussion). The collision strengths computed in the present work have been evaluated over a very fine mesh of incident electron-impact energies, in order to ensure the explicit delineation of the intricate resonance structures which are found to dominate the low-energy scattering region and converge to the target state thresholds. Approximately 12,000 individual energy points were considered in this resonance region. We note that convergence of the collision strengths with decreasing mesh spacing was thoroughly tested. The (N + 1) symmetries were established by considering all total angular momenta L ⩽ 15 (J ⩽ 12). The coupling of the incident electron to the doublet, quartet, and sextet target spin symmetries yields singlet, triplet, quintet, and septet (N + 1) multiplicities. Considering both even and odd parities, a total of 128 independent (N + 1)-electron symmetry full exchange contributions are included. This representation proved sufficient in ensuring convergence of the collision strengths for the low-lying optically forbidden transitions considered by Cassidy et al. (2010). This full exchange analysis for the 113 LS state, 295 jj-level approximation was augmented with a supplementary non-exchange calculation comprising contributions from higher partial waves up to L = 50. This non-exchange calculation was performed using the parallel intermediate-coupling frame transformation PSTGICF external region package of Griffin et al. (1998). A very coarse mesh of incident impact energies was employed across the entire external region as such high values of L would be devoid of any resonance activity. As a direct consequence of the long-range nature of the Coulomb potential, a further contribution to the dipole allowed transitions arises from even higher partial waves. This top-up contribution is accounted for by employing physically viable extrapolation techniques which have been incorporated into the PSTGICF program (Griffin et al. 1998). The top-up contribution to the dipole allowed transitions is computed using the Burgess sum rule, while a geometric series is included to calculate the additional contribution to the long-range non-dipole transitions. In this way, converged total collision strengths were generated for all 43,365 transitions, both allowed and forbidden, for the electron-impact energy range of interest from 0 to 10 Ryd. The corresponding Maxwellian averaged effective collision strengths have been acquired by averaging the finely resolved collision strengths over a Maxwellian distribution of electron velocities. These thermally averaged effective collision strengths have been computed at 30 individual electron temperatures ranging from 30 to 1,000,000 K. We note that the temperature of maximum abundance for Ni ii in ionization equilibrium is log  T_e(K) = 4.1 (Mazzotta et al. 1998), and hence the considered range is more than sufficient to encompass all temperatures significant to diverse astrophysical and plasma applications. It is important to consider a large range of high temperatures in order to properly emphasize the effects in the high-temperature region attributed to the explicit inclusion of high-order partial waves.

Aspiring to investigate the effects of high partial wave contributions at high energies and necessarily, high temperatures, and simultaneously to assess the convergence of the collision strengths for the dipole allowed transitions in the high-energy region, we present a synoptic evaluation of the results obtained following the non-exchange and top-up calculations. Attention is concentrated on the allowed transitions for which additional contributions from high partial waves have the most sizeable effect at high energies and subsequently high temperatures. The effective collision strengths are compared with the rates of Bautista (2004). It is pertinent to remark that these data were not published however and have been obtained directly from the author to facilitate a comparative analysis. We note that the transitions are labeled according to the index values assigned to each level in Table 1.

In Figure 1, we present the collision strength as a function of incident electron energy in Rydbergs for the low-lying allowed transition from the ground state, 3d^{9 2}D_5/2, to the odd parity excited state, 3d⁸(³F)4p²D^o_5/2 (index 1–37). Comparisons are made between the collision strengths as determined from the preliminary exchange calculation incorporating all partial waves up to and including a total angular momentum of L = 15, the exchange calculation augmented with the non-exchange calculation which explicitly includes all partial wave contributions up to and including L = 50, and additionally the accretion of the top-up calculation which encompasses contributions from even higher partial waves. Evidently the collision strength pertaining to the exchange-only evaluation lies considerably lower than those obtained from the other calculations across the entire energy range, thereby emphasizing the significance of including the additional contributions from higher partial waves for these allowed lines. The non-exchange contributions to the total collision strength begin to have an effect at approximately 0.82 Ryd. The resulting exchange plus non-exchange approximation appears to accurately represent the collision process up until approximately 3.6 Ryd. Beyond this energy, the collision strength exhibits deviations attributed to the effect of further contributions emerging from the top-up procedure. These top-up contributions equate to a 1% difference at 5 Ryd, with an increase to 5% at 7.5 Ryd and finally a sizeable 12% difference at the maximum electron-impact energy of 10 Ryd. The combined effect of the non-exchange and top-up contributions on the corresponding Maxwellian averaged effective collision strength is demonstrated in Figure 2, where the aforesaid is plotted as a function of the logarithm of electron temperature. A comparison is made between the effective collision strengths computed using both the exchange-only and exchange plus non-exchange plus top-up approximations. The rate profiles agree faultlessly for the majority of temperatures considered. However, in the high-temperature region ranging from log  T_e(K) = 4.6–6.0 discernible disparities arise, highlighting the substantial influence of high partial wave contributions. Differences of approximately 83% are noted at the highest temperature considered, log  T_e(K) = 6.0. Although this high-energy region may not necessarily be significant to Ni ii applications, the importance of augmenting the exchange calculation with the contributions imputed to both the non-exchange and top-up approximations is affirmed for other systems for which high temperatures hold particular astrophysical interest.

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The topped-up rates of Bautista (2004) consistently overestimate the present predictions across all comparable temperatures. Differences are at best a factor of two and at worst a factor of four. In the absence of the associated collision strength data, it is difficult to understand the nature of these discrepancies. The effective collision strengths of Bautista (2004) were derived at only 11 electron temperatures, the highest temperature considered being 30,000 K (log  T_e(K) = 4.48). Consequently, a comparison of behavior in the high-temperature region is not possible.

In Figure 3, we present the collision strength as a function of incident electron energy in Ryd for the transition from the ground state to the same 3d⁸4p²D^o multiplet discussed previously, namely, 3d^{9 2}D_5/2–3d⁸(³F)4p²D^o_3/2 (index 1–39). Contributions from the non-exchange calculation manifest at approximately 0.85 Ryd. The combinative inclusion of exchange contributions up to L = 15 and the supplementary non-exchange contributions as far as L = 50 satisfactorily describes the collision process up to roughly 3.8 Ryd. Beyond this energy the top-up contributions materialize, increasing the total collision strength by 4% at 7.5 Ryd and 10% at 10 Ryd. The differences observed in the collision strength due to the inclusion of the non-exchange and top-up contributions translate into disparities in the affiliated thermally averaged effective collision strength as displayed in Figure 4. The high partial wave contributions have an appreciable effect at temperatures above log  T_e(K) = 4.7, with differences of 62% recorded at the highest temperature considered.

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Analogous to the previous transition, the predicted rates of Bautista (2004) lie consistently higher than the present topped-up data set, with an average difference of a factor of two noted at temperatures where a comparison can be made.

In Figures 5 and 6, we plot the collision strengths for the allowed transitions from the ground state to the next 3d⁸4p²D^o multiplet; 3d^{9 2}D_5/2–3d⁸(¹D)4p²D^o_3/2 (index 1–46) and 3d^{9 2}D_5/2–3d⁸(¹D)4p²D^o_5/2 (index 1–50), respectively. Examination of these transitions uncovers consonant characteristics to those observed in the previous considerations. Each exchange calculation predicts a lower collision strength to those obtained using the augmented data sets.

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In Figure 5, non-exchange effects become valid at approximately 1.1 Ryd. The exchange plus non-exchange calculation exhibits excellent accord with the concluding calculation which incorporates the additional top-up contributions as far as an electron-impact energy of roughly 4.6 Ryd. For the remainder of the energy range of interest, the top-up contributions effectuate an increased collision strength which deviates from the combined exchange plus non-exchange data set by just 3% at 7.5 Ryd and 7% at 10 Ryd. Comparable fluctuations occur for the allowed transition presented in Figure 6. The completion of the summation of partial collision strengths over high values of L as performed using the top-up procedure has an apparent influence on the collision strength in the non-resonance region above all thresholds, most notably as we progress to higher incident impact energies. Differences of 3% are noted at 7.5 Ryd and 8% at the maximum electron-impact energy of interest of 10 Ryd. The fundamental importance of including the additional non-exchange and top-up contributions from higher partial waves in the calculation is evident in Figures 7 and 8, where we plot the associated Maxwellian averaged effective collision strengths. The present rate profiles as computed using the exchange-only and augmented approximations coincide directly for the majority of temperatures. Deviations begin to emerge in the high-temperature region, with differences of 50% and 68% observed, respectively, at the highest temperature considered. In Figure 7, the effective collision strengths of Bautista (2004) exhibit equivalent trends to those formerly noted and continue to overestimate the current predictions. The rates presented in Figure 8 demonstrate the best agreement with the present data set, with the worst discrepancies recorded at the highest temperatures considered in the Bautista (2004) analysis. The predictions of Bautista (2004) lie approximately 34% lower than the present at the highest comparable temperature and appear to be decreasing which is an uncharacteristic feature of allowed lines. This perhaps suggests that by including all partial waves up to and including a total angular momentum value of only L = 25 before allowing for top-up, Bautista (2004) has failed to comprehensively account for contributions from higher partial waves. We can thus surmise that imperative in achieving convergence of the electron-impact excitation collision strengths for the allowed transitions in Ni ii is the explicit inclusion of contributions from partial waves up to a total angular momentum value of approximately L = 50 in the calculation, before employing a top-up procedure.

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The overall quality of the collisional data presented in Figures 1, 3, 5, and 6 is necessarily of relevance, especially so in the absence of priorly published theoretical determinations. It is possible to critically assess the accuracy of the current data in the medium- to high-energy region by ascertaining that our results conform with the expected infinite-energy limits. In order to perform such checks, we have employed the method introduced by Burgess & Tully (1992). We utilize this "C-plot" method to condense the entire variation of a collision strength onto a single plot. A reduced collision strength (Ω_r) is plotted against reduced energy (E_r), thereby mapping the complete range of incident electron energies onto the interval (0, 1). The reduced energy E_r is calculated as

$$\begin{eqnarray} &&E_{r}=1-\frac{\ln C}{\ln \Big (\frac{E_{j}}{E_{ij}}+C\Big)}, \end{eqnarray} \tag{ 1 }$$

where C is an adjustable parameter permitting flexibility, E_j is the incident electron energy following excitation, and E_ij is the transition energy for excitation from state i to state j. The collision strength as a function of the reduced energy is represented by

$$\begin{eqnarray} &&\Omega (E_{r})=\frac{\Omega (E_{j})}{\ln \Big (\frac{E_{j}}{E_{ij}}+e\Big)}. \end{eqnarray} \tag{ 2 }$$

We note that the reduced collision strength behaves asymptotically as

$$\begin{eqnarray} &&\Omega (E_{r}=1)=\frac{4\omega _{i}f_{ij}}{E_{ij}}, \end{eqnarray} \tag{ 3 }$$

where f_ij denotes the absorption oscillator strength. Equation (3) is deemed the infinite-energy limit and will naturally vary in accordance with the chosen oscillator strength. The true behavior is assumed to be consistent with the experimentally determined oscillator strength. A selection of the currently available theoretical and experimental LSJ oscillator strengths for the allowed transitions discussed are presented in Table 2. We note that the present values constitute those calculated using the wavefunctions of the current scattering model. It is evident that discrepancies exist across the board for all transitions. Even the f-values determined by experimental and observational means exhibit disaccord. This makes it difficult to interpret the correct infinite-energy limit point. The complete set of LSJ oscillator strengths associated with the present target model is available from the authors upon request.

In pursuance of examining the behavior of the collision strengths in the high-energy region and thereby verifying the accuracy of the collisional data associated with this work, Figure 9 presents the reduced collision strengths generated by the present approximation including exchange, non-exchange, and top-up contributions, for the two allowed transitions from the ground state to the levels of the 3d⁸(³F)4p²D^o multiplet. We note that the reduced collision strength has been extended to the infinite-energy limit point at E_r = 1. Attention is concentrated on the aforesaid value derived using the present oscillator strengths noted in Table 2. For completeness, we also include the infinite-energy limit points computed using other currently available LSJ oscillator strength data, both theoretical and experimental. For the allowed transition 3d^{9 2}D_5/2–3d⁸(³F)4p²D^o_3/2 (index 1–39) synonymous with the lower plot, the present predicted infinite-energy limit point lies midway between the medley of other theoretically and experimentally determined values. In the upper plot, epitomizing the transition 3d^{9 2}D_5/2–3d⁸(³F)4p²D^o_5/2 (index 1–37), the present result lies between the theoretical prediction of Kurucz & Bell (1995) and the various other experimental and theoretical considerations, with slightly better concurrence with the latter results exhibited.

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In Figure 10, we plot the reduced collision strengths as a function of reduced energy for the two allowed transitions from the ground state to the levels of the 3d⁸(¹D)4p²D^o multiplet. In the lower plot, representing the 3d^{9 2}D_5/2–3d⁸(¹D)4p²D^o_3/2 transition (index 1–46), the present infinite-energy limit point demonstrates excellent agreement with those derived from the experimental determinations of Fedchak et al. (2000) and the observed values of Zsargó & Federman (1998). Considering the transition 3d^{9 2}D_5/2–3d⁸(¹D)4p²D^o_5/2 (index 1–50) as illustrated in the upper plot, the present results lie somewhat higher than those based on experimental f-values.

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The apparent deviations in the infinite-energy limit points emphasizes the need for more accurate experimental and theoretical oscillator strength data to facilitate the precise identification of the correct high-energy behavior. In the current analysis however, the reasonable to excellent agreement demonstrated between the present predictions and the infinite-energy limit points derived using experimental f-values assigns credence to the accuracy and quality of the collisional data computed in this work.

In Table 3, we tabulate the effective collision strengths computed in the present 295 jj-level approximation for all of the Ni ii allowed lines with which this study is associated, with full account being taken of the exchange calculation incorporating partial waves up to L = 15, the non-exchange calculation incorporating contributions from higher partial waves up to L = 50, plus the addition of contributions from yet higher partial waves included via a top-up procedure. A total of 2436 individual transitions between the 3d⁹, 3d⁸4s, and 3d⁷4s² even parity levels and the 3d⁸4p and 3d⁷4s 4p odd parity levels are tabulated at 30 electron temperatures ranging from 30 to 1,000,000 K. The transitions are labeled according to the index values assigned to each level in Table 1. We note that Table 3 is available in its entirety online only.

In this paper, we present the computation of converged total collision strengths and effective collision strengths for the optically allowed transitions between fine-structure levels of Ni ii. The theoretical target model included the 3d⁹, 3d⁸4s, 3d⁷4s², 3d⁸4p, and 3d⁷4s 4p basis configurations, giving rise to a sophisticated 295 jj-level, 1930 coupled channel scattering complex. We believe this to be the most extensive and complete collisional evaluation performed on this ion to date. Collision strengths and effective collision strengths have been tabulated for a total of 2436 allowed transitions between the 3d⁹, 3d⁸4s, and 3d⁷4s² even parity levels and the 3d⁸4p and 3d⁷4s 4p odd parity levels. The astrophysically significant effective collision strengths have been calculated over a large range of electron temperatures ranging from 30 to 1,000,000 K. The convergence of the allowed transitions, which are at the focus of this study, has been thoroughly investigated. We found that intrinsic in achieving convergence of the allowed lines was the combined inclusion of contributions from the (N + 1) partial waves extending to a total angular momentum value of L = 50 and further contributions from even higher partial waves accomplished by employing a top-up procedure. Comparisons with the rates of Bautista (2004) affords a miscellany of results. The rates of Bautista (2004) consistently overestimate the present topped-up data set at all temperatures where a comparison is possible for the majority of transitions considered. The paucity of prior theoretical works on dipole allowed transitions among fine-structure levels of Ni ii hinders a more comprehensive comparative evaluation. However, the caliber and precision of the present collisional data at medium to high energies has been verified by checking that our results are consistent with the expected infinite-energy limits. We note that the infinite-high energy limit points predicted using the present theoretical model exhibit satisfactory to excellent accord with those derived from other analyses.

A conclusive assessment of the accuracy of the presented effective collision strengths proves difficult. The definitive test will necessarily emanate from any subsequent astrophysical or diagnostic applications. We, however, maintain the reliability of the present data set and would accordingly recommend it to astrophysicists for use in their ongoing application work. The reliability of the data follows directly from the level of sophistication of the current calculation, where great diligence has been exercised in the inclusion of adequate correlation and CI, in the accurate delineation of complex resonance effects and in the proper consideration of contributions from the high partial waves to the collision strengths of the allowed lines.

The authors acknowledge the years of work of V. M. Burke and C. J. Noble at Daresbury Laboratory (UK) who have been responsible for the development of the internal region codes. Special acknowledgment is extended to M. A. Bautista for supplying unpublished data for comparison purposes. The authors thank N. R. Badnell for his guidance on the PSTGICF package. Thanks is extended to the referee for their constructive comments and suggestions. The work presented in this paper is supported by STFC and C. M. Cassidy is supported by a DEL Studentship. The computations were carried out on the IBM HPCx facility at the CLRC Daresbury Laboratory.