In the earlier sections, we have provided a few examples in which, by systematic inclusion of CSFs, sometimes in conjunction with the fine-tuning process, it has been possible to reach levels of accuracy in the calculated oscillator strengths which are sufficient for the needs of those who use this data in their modelling of, for example, stellar atmospheres or the determination of elemental abundances in stars or the interstellar medium. However, these examples required just a small number of levels to be calculated simultaneously, even though in some cases they interact strongly. While it is true that the Ar II work did require a larger number of levels to be considered, many of them were spectroscopically fairly pure: they did not exhibit strong interactions. However, difficulties arise when a much larger number of strongly interacting levels have to be included in the same calculation, either because they do interact strongly, or because they are quite high-lying in energy and all levels with lower energy also need to be included in the calculation. Ions with open d-shells are particularly challenging, and we consider now some calculations which have proven and continue to prove difficult for theorists in their attempts to provide oscillator strengths of sufficient accuracy, and with an accuracy which can be substantiated.
4.1. The 2507.552, 2509.097 Å Lines in Fe II
These two lines are prominent in the spectrum of
Carinae (Verner et al. 2002) [
26]. Notionally, they correspond to the transitions [
7]
| Transition |
2507.552Å | c F–x F |
2509.097Å | c F–w G |
One would expect the former, being an intercombination line, to have a small oscillator strength, while the latter, being an allowed line, would be expected to have a comparatively large oscillator strength. On this assumption, Verner et al. (2002) [
26] found they could not properly model this part of the spectrum, but could do so if the oscillator strengths were chosen to be equal.
The oscillator strengths of the two transitions have been calculated by a number of researchers, as shown in
Table 4.
The results of Kurucz (2010) [
27] fit the expected pattern of allowed and intercombination lines. Those of Raassen and Uylings (1998) [
28] do not; indeed, their values reverse the expected size order. Our own calculations (Corrégé and Hibbert) (2005) [
29] were completed, though not published, before we became aware of the proposal of Verner et al. [
26] to abandon the data already published at that time and adopt equal oscillator strengths for the two transitions in their modelling. We sought to understand how it was that our calculated oscillator strengths were indeed of comparable size, contrary to expectations but in accord with the needs of Verner et al. [
26]. We notice that the total oscillator strength from these two transitions is quite similar in all three sets of results. This suggests that we are seeing a different distribution of the total oscillator strength in each of the three calculations. We found that the A-values were entirely dependent on the proportion of
G symmetry in the upper states. The two transitions have a common lower level, so that the upper level of 2507Å is 90,067.4 cm
, while the upper level of 2509Å is 90,042.8 cm
.
In
Table 5, we show the main percentage compositions given by Corrégé and Hibbert (2005) [
29] and those of Raassen and Uylings [
28]. During our studies of these transitions, it became apparent that while the upper levels did exhibit strong configuration mixing, a third main component, corresponding to the 3d
5p
F
configuration, also interacted strongly. It also became apparent that it was the size of the component of 4p
G
in the wave function of each of the three levels that determined the size of the oscillator strengths of the transitions. It can be seen in
Table 5 that our fine-tuning process, following on from an already extensive ab initio calculation, resulted in the two upper levels of these lines having almost equal components of the
G
configuration, resulting in our two calculated oscillator strengths being almost equal.
Our fine-tuning process allowed us also to fine-tune to the energy levels given by Raassen and Uylings. In doing so, we found fairly good quantitative agreement with the percentage compositions which Raassen and Uylings obtained, as displayed in
Table 5.
On reflection, we were perhaps rather fortunate in arriving at results which, unbeknown to us at the time, fitted the requirements of Verner et al. [
26]. We would wish to highlight that, for most of the transitions which both Kurucz and Raassen and Uylings studied, their results are of good quality and can be used with confidence. In this instance though, the upper levels were so close in energy that small modifications to the fine-tuning corrections led to very different mixings and therefore oscillator strengths. In all three sets of results, a large-scale calculation had been undertaken, with the wave functions of many levels having to be determined simultaneously. That always leaves open the possibility that for a small proportion of the wave functions, when CI mixing is particularly strong, the mixing coefficients can be wrong. Oscillator strengths determined using heavily mixed wave functions need to be treated with caution. In particular, extrapolation such as our fine-tuning process, needs to be undertaken very carefully, moving from the ab initio calculation in small steps.
4.2. Correlation in Open 3d Subshells
The capture of the effect of electron correlation in wave functions is generally a slowly-converging process. This is particularly true in the case of open 3d subshells, as found in iron group elements, and therefore of considerable importance when the wave functions are used in the calculation of oscillator strengths of transitions of astrophysical importance. The key configurations are of the form 3dml, 3d, 3dmlk. Challenges in calculations arise because both the radial dependence of the 3d orbitals and also the level of correlation in the 3d subshell are state-dependent.
The state-dependency of electron correlation implies that even for a large-scale calculation, there is some level of inaccuracy in the calculated energy levels, and therefore in the determination of oscillator strengths in an ab initio calculation. The scale of such calculations is so extensive that undertaking a systematic enlargement of the CI expansions becomes a prohibitive exercise, and so alternative extrapolation procedures, such as fine-tuning or the scaling of Slater integrals [
12], is then a better approach, even though (as we have seen) much care has to be exercised in using these extrapolation processes, as they can lead to erroneous results.
Difficulties are also encountered by the state-dependence of the radial functions. For example, in their studies of E1 Fe II transitions, Corrégé and Hibbert (2005) [
29], as well as Deb and Hibbert (2014) [
30] found it better to choose a 3d function optimised on the energy of the ground 3d
4s
D state, whereas Deb and Hibbert (2010a,b,2011) [
31,
32,
33], in studying forbidden transitions involving 3d
levels, found that much better results could be obtained by optimising the 3d function on the 3d
F state. This is possible when a calculation is limited to certain types of transition, or even to a very small number of transitions. On the other hand, if a comprehensive study of both allowed and forbidden transitions is to be undertaken, the use of orthogonal orbitals (and therefore a single choice of 3d function for all transitions) requires the introduction of many CSFs to compensate for the non-optimal choice of radial functions for some of the levels. Ideally, non-orthogonal orbitals would allow different 3d functions to be used for different occupancies of the 3d subshell arising within the same calculation, and this could profitably be pursued to reduce the overall number of CSFs required.
4.3. Open d-Subshells with Differing Seniority
Open 3d-subshells with between three and seven electrons can have more than one level with the same LSJ combination, but differing by their seniority. For example, for the
F
levels in Fe II, 3d
4s have two distinct levels, which are usually written either as [
7] 3d
(
F1)4s and 3d
(
F2)4s, or as 3d
(
F)4s and 3d
(
F)4s, respectively. In the latter notation, the subscripts denote the seniority of the 3d-subshell. Even in a large-scale calculation, it is difficult to achieve a calculated energy difference between these two
F
levels which agrees with experiment.
If we include just these two configurations, the Hamiltonian matrix is of the form
The eigenvalues and eigenvectors depend solely on the magnitudes of
and
c. We find that for these
F
levels in Fe II, in atomic units,
= 0.0915,
c = 0.0643, so that the eigenvectors are respectively
with the eigenvalue difference 0.1579, compared with the experimental value of 0.1248. With fine-tuning, the smallest eigenvalue difference achievable is 2
= 0.1286, so that this simple model cannot achieve agreement with experiment.
However, when we use the radial functions listed by Deb and Hibbert (2014) [
30] in a large-scale CI calculation, and then apply our fine-tuning process, we can at least obtain agreement with experiment. Specifically, we find that the eigenvector corresponding to the lower of the two eigenvalues and dominated by these CSFs in an ab initio calculation has the following components for the two principal configurations
with the two eigenvalues differing by 0.1335. The fine-tuning process results in the lower eigenvalue having mixing coefficients with
for the two principal CSFs, and the two eigenvalues then differ by 0.1248, in agreement with experiment. Fine-tuning results in similar agreement with experiment for some other mixings between levels differing only in the seniority of the 3d subshell.
However, it is not so in every case. Another situation where there is strong mixing between levels differing only in the seniority of the 3d subshell occurs in the case of 3d
(
P1)4s
P
and 3d
(
P2)4s
P
of Fe II. In the terminology used above, we found that, in atomic units,
= 0.0037 and
c = 0.0785, so that in an ab initio calculation with just these two CSFs, the eigenvectors are
with the eigenvalue difference 0.1616, compared with the experimental value of 0.1262. With fine-tuning, the smallest eigenvalue difference achievable is 2
= 0.1570. The large-scale CI calculation, using the same orbitals as previously, gives the eigenvector components of these two CSFs for the lower of the two eigenvalues as
with the two lowest eigenvalues differing by 0.1393. Fine-tuning did not really help, since, even with equal components of the two principal configurations, we found an eigenvalue difference of 0.1332.
Thus, even though the same orbitals and CSFs were able to provide results in good agreement with experiment for some transitions [
29], it would seem that, as yet, insufficient electron correlation has been captured in these calculations to give us confidence in the results. We have considered the inclusion of another range of orbitals for each of
l = 0,1,2,3, but this would lead to a large number of additional CSFs. Instead, we noticed the absence of a 5g orbital: a 3d→5g substitution could be a significant correlation effect, though normally small for many wave functions. Accordingly, we added CSFs of the form 3d
4s5g and 3d
4s5g
to our earlier calculations. This resulted in the components of the lower eigenvalue changing to
with the two lowest eigenvalues differing by 0.1329. This ab initio energy difference is then marginally better than the best possible fine-tuned difference without the 5g orbital. However, the introduction of 5g does not entirely solve the difficulty, because we find that the smallest fine-tuned energy difference is 0.1297 a.u., still higher than the experimental value of 0.1262. Clearly, further correlation CSFs would need to be introduced to bring these closer together.
There are of course other methods of extrapolating ab initio results. One such method is customarily employed when using the Cowan code [
12]. Specifically, several radial integrals may be scaled by appropriate factors, notably the two-electron Slater integrals which are scaled customarily by factors of 0.80 to 0.85. However, both
and
c involve only the Slater integrals F
(3d,3d) and F
(3d,3d). If a common scaling parameter is used, the mixing coefficients are unchanged by this scaling. This process could indeed result in agreement with energy separations between levels differing only in the seniority of the 3d subshell, which our fine-tuning could not completely achieve. However, we saw above that, when additional correlation CSFs are used instead, the mixing coefficients do change. Hence, although the energy separations are in agreement with experiment, the mixing coefficients may still contain inaccuracies.