††thanks: These two authors have contributed equally.
<EMAIL_ADDRESS><EMAIL_ADDRESS>These two authors
have contributed equally.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
# High Resolution Photoexcitation Measurements Exacerbate
the Long-standing Fe xvii Oscillator Strength Problem

Steffen Kühn  Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117
Heidelberg, Germany  Heidelberg Graduate School of Fundamental Physics,
Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 226, 69120
Heidelberg, Germany  Chintan Shah  NASA/Goddard Space Flight Center, 8800
Greenbelt Rd, Greenbelt, Maryland 20771, USA  Max-Planck-Institut für
Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany  José R. Crespo
López-Urrutia  Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117
Heidelberg, Germany  Keisuke Fujii  Department of Mechanical Engineering and
Science, Graduate School of Engineering, Kyoto University, Kyoto 615-8540,
Japan  René Steinbrügge  Deutsches Elektronen-Sychrotron DESY, Notkestraße 85,
22607 Hamburg, Germany  Jakob Stierhof  Dr. Karl Remeis-Sternwarte,
Sternwartstraße 7, 96049 Bamberg, Germany  Moto Togawa  Max-Planck-Institut
für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany  Zoltán Harman
Natalia S. Oreshkina  Max-Planck-Institut für Kernphysik, Saupfercheckweg 1,
69117 Heidelberg, Germany  Charles Cheung  Department of Physics and
Astronomy, University of Delaware, Newark, Delaware 19716, USA  Mikhail G.
Kozlov  Petersburg Nuclear Physics Institute of NRC ”Kurchatov Institute”,
Gatchina 188300, Russia  St. Petersburg Electrotechnical University ”LETI”,
Prof. Popov Str. 5, St. Petersburg, 197376, Russia  Sergey G. Porsev
Petersburg Nuclear Physics Institute of NRC ”Kurchatov Institute”, Gatchina
188300, Russia  Department of Physics and Astronomy, University of Delaware,
Newark, Delaware 19716, USA  Marianna S. Safronova  Department of Physics and
Astronomy, University of Delaware, Newark, Delaware 19716, USA  Joint Quantum
Institute, National Institute of Standards and Technology and the University
of Maryland, Gaithersburg, Maryland 20742, USA  Julian C. Berengut  School of
Physics, University of New South Wales, Sydney, New South Wales 2052,
Australia  Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117
Heidelberg, Germany  Michael Rosner  Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, 69117 Heidelberg, Germany  Matthias Bissinger  Erlangen
Centre for Astroparticle Physics (ECAP), Erwin-Rommel-Straße 1, 91058
Erlangen, Germany  Dr. Karl Remeis-Sternwarte, Sternwartstraße 7, 96049
Bamberg, Germany  Ralf Ballhausen  Dr. Karl Remeis-Sternwarte, Sternwartstraße
7, 96049 Bamberg, Germany  Natalie Hell  Lawrence Livermore National
Laboratory, 7000 East Avenue, Livermore, California 94550, USA  SungNam Park
Moses Chung  Ulsan National Institute of Science and Technology, 50 UNIST-gil,
44919 Ulsan, South Korea  Moritz Hoesch  Jörn Seltmann  Deutsches Elektronen-
Sychrotron DESY, Notkestraße 85, 22607 Hamburg, Germany  Andrey S. Surzhykov
Physikalisch-Technische Bundesanstalt, Bundesalle 100, 38116 Braunschweig,
Germany  Braunschweig University of Technology, Universitätsplatz 2, 38106
Braunschweig, Germany  Vladimir A. Yerokhin  Peter the Great St.Petersburg
Polytechnic University, 195251 St. Petersburg, Russia  Jörn Wilms  Dr. Karl
Remeis-Sternwarte, Sternwartstraße 7, 96049 Bamberg, Germany  F. Scott Porter
NASA/Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA
Thomas Stöhlker  Institut für Optik und Quantenelektronik, Friedrich-Schiller-
Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany  Helmholtz-Institut
Jena, Fröbelstieg 3, 07743 Jena, Germany  GSI Helmholtzzentrum für
Schwerionenforschung, Planckstraße 1, 64291 Darmstadt, Germany  Christoph H.
Keitel  Thomas Pfeifer  Max-Planck-Institut für Kernphysik, Saupfercheckweg 1,
69117 Heidelberg, Germany  Gregory V. Brown  Lawrence Livermore National
Laboratory, 7000 East Avenue, Livermore, CA 94550, USA  Maurice A. Leutenegger
NASA/Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA
Sven Bernitt  Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117
Heidelberg, Germany  Institut für Optik und Quantenelektronik, Friedrich-
Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany  Helmholtz-
Institut Jena, Fröbelstieg 3, 07743 Jena, Germany  GSI Helmholtzzentrum für
Schwerionenforschung, Planckstraße 1, 64291 Darmstadt, Germany

###### Abstract

For more than 40 years, most astrophysical observations and laboratory studies
of two key soft X-ray diagnostic $2p-3d$ transitions, 3C and 3D, in Fe XVII
ions found oscillator strength ratios $f(\mathrm{{3C}})/f(\mathrm{{3D}})$
disagreeing with theory, but uncertainties had precluded definitive statements
on this much studied conundrum. Here, we resonantly excite these lines using
synchrotron radiation at PETRA III, and reach, at a millionfold lower photon
intensities, a 10 times higher spectral resolution, and 3 times smaller
uncertainty than earlier work. Our final result of
$f(\mathrm{{3C}})/f(\mathrm{{3D}})=3.09(8)(6)$ supports many of the earlier
clean astrophysical and laboratory observations, while departing by five
sigmas from our own newest large-scale _ab initio_ calculations, and excluding
all proposed explanations, including those invoking nonlinear effects and
population transfers.

Figure 1: (Top) Experimental setup: an electron beam (orange) aimed at the
trap center produces Fe xvii ions, which are then resonantly excited by a
monochromatic photon beam (red). Subsequent X-ray fluorescence is registered
by a silicon drift detector. (Bottom) Fluorescence yield and photon energy vs.
time with three different methods: (1) line scans with open photon shutter;
(2,3) by closing it at each step (purple areas), we subtract the electron-
beam-induced background at (2) each line center and (3) scanning over a one-
full-width-at-half-maximum range.

Space X-ray observatories, such as _Chandra_ and _XMM-Newton_ , resolve
$L$-shell transitions of iron dominating the spectra of many hot astrophysical
objects Paerels and Kahn (2003); Canizares  _et al._ (2000); Behar  _et al._
(2001); Mewe  _et al._ (2001). Some of the brightest lines arise from Fe xvii
(Ne-like iron) around 15 Å: the resonance line 3C
($[(2p^{5})_{1/2}\,3d_{3/2}]_{J=1}\rightarrow[2p^{6}]_{J=0}$) and the
intercombination line 3D
($[(2p^{5})_{3/2}\,3d_{5/2}]_{J=1}\rightarrow[2p^{6}]_{J=0}$). Appearing over
a broad range of plasma temperatures and densities, they are crucial for
diagnostics of electron temperatures, elemental abundances, ionization
conditions, velocity turbulences, and opacities Parkinson (1973); Doron and
Behar (2002); Xu  _et al._ (2002); Brickhouse and Schmelz (2005); Sanders and
Fabian (2011); Kallman  _et al._ (2014); de Plaa  _et al._ (2012); Bailey  _et
al._ (2015); Beiersdorfer  _et al._ (2018); Nagayama  _et al._ (2019); Gu, L.
_et al._ (2019). However, for the past four decades, their observed intensity
ratios persistently disagree with advanced plasma models, diminishing the
utility of high resolution X-ray observations. Several experiments using
electron beam ion trap (EBIT) and tokamak devices have scrutinized plausible
astrophysical and plasma physics explanations as well as the underlying atomic
theory Brown  _et al._ (1998); Beiersdorfer  _et al._ (2001); Brown  _et al._
(2001); Beiersdorfer  _et al._ (2002, 2004); Brown  _et al._ (2006); Brown and
Beiersdorfer (2012); Shah  _et al._ (2019), but also revealed clear departures
from predictions while broadly agreeing with astrophysical observations
Beiersdorfer  _et al._ (2002, 2004); Gu (2009). This has fueled a long-lasting
controversy on the cause being a lack of understanding of astrophysical
plasmas, or inaccurate atomic data.

A direct probe of these lines using an EBIT at the Linac Coherent Light Source
(LCLS) X-ray free-electron laser (XFEL) found again their oscillator strength
ratio $f(\mathrm{{3C}})/f(\mathrm{{3D}})$ to be lower than predicted, but
close to astrophysical observations Bernitt  _et al._ (2012). This highlighted
difficulties with oscillator strengths calculations in many-electron systems
Safronova  _et al._ (2001); Chen and Pradhan (2002); Loch  _et al._ (2006);
Chen (2007, 2011); Gu (2009); Bernitt  _et al._ (2012); Santana  _et al._
(2015). Nonetheless, at the high peak brilliance of the LCLS XFEL, nonlinear
excitation dynamics Oreshkina  _et al._ (2014, 2016) or nonequilibrium time
evolution Loch  _et al._ (2015) might have affected the result of Bernitt  _et
al._ (2012). An effect of resonance-induced population transfer between Fe xvi
and Fe xvii ions was also postulated Wu and Gao (2019) since the Fe xvi line C
($\left[(2p^{5})_{1/2}3s3d_{3/2}\right]_{J=1/2}\to\left[2p^{6}3s\right]_{J=1/2}$)
appeared blended with the Fe xvii line 3D. A recent semiempirical calculation
Mendoza and Bautista (2017) reproduces the LCLS results Bernitt  _et al._
(2012) by fine-tuning relativistic couplings and orbital relaxation effects,
but its validity has been disproved Wang  _et al._ (2017).

In this Letter, we report on new measurements of resonantly excited Fe xvi and
Fe xvii with a synchrotron source at tenfold improved spectral resolution and
millionfold lower peak photon flux than in Bernitt  _et al._ (2012),
suppressing nonlinear dynamical effects Oreshkina  _et al._ (2014); Loch  _et
al._ (2015); Li  _et al._ (2017) and undesired ion population transfers Wu and
Gao (2019). We also carry out improved large-scale calculations using three
different advanced approaches Kozlov  _et al._ (2015); Fischer  _et al._
(2019); Kahl and Berengut (2019), all showing a five-sigma departure from our
experimental results.

Figure 2:  Fluorescence photon yield and energy vs excitation-photon energy
for the Fe xvi C, Fe xvii 3C, and 3D transitions recorded by a silicon-drift
detector. Black dots: total fluorescence within a 50-eV region of interest.
Red solid lines: Fits to 3C and 3D. Red dashed line: fit to C.

We used the compact PolarX-EBIT Micke  _et al._ (2018), in which a
monoenergetic electron beam emitted by an off-axis cathode (see Fig. 1) is
compressed by a magnetic field. At the trap center, it collides with a beam of
iron-pentacarbonyl molecules, dissociating them, and producing highly charged
Fe ions with a relative abundance of Fe XVI to Fe XVII close to unity. These
ions stay radially confined by the negative space charge of the $\sim$2-mA,
1610-eV ($\approx 3$ times the Fe xvi ionization potential) electron beam, and
axially by potentials applied to surrounding electrodes.

Monochromatic, circularly polarized photons from the P04 beamline Viefhaus
_et al._ (2013) at the PETRA III synchrotron photon source enter through the
electron gun, irradiate the trapped ions, and exit through the collector
aperture. They can resonantly excite X-ray transitions on top of the strong
electron-induced background due to ionization, recombination, and excitation
processes. A side-on-mounted energy-resolving windowless silicon drift photon
detector (SDD) equipped with a 500-nm thin aluminum filter registers these
emissions.

By scanning the P04 monochromator between 810 and 830 eV, we excite the Fe
xvii lines 3C and 3D, as well as the Fe xvi lines B
($\left[(2p^{5})_{1/2}(3s3d_{3/2})_{1}\right]_{J=1/2}\to\left[2p^{6}3s\right]_{J=1/2}$)
and C. They are also nonresonantly excited by electron-impact as the electron
beam energy is well above threshold Brown  _et al._ (2006); Shah  _et al._
(2019). This leads to a strong, nearly constant X-ray background at the same
energies as the photoexcited transitions but independent of the exciting
photon-beam energy. In our earlier work Bernitt  _et al._ (2012), we rejected
this background by detecting the fluorescence in time coincidence with the
sub-picosecond-long LCLS pulses of $\approx 10^{11}$ photons each at 120/s.
Due to much longer and weaker 50-ps-long pulses ($\approx 10^{3}$ photons,
$6\times 10^{6}$/s repetition rate) at P04 and the limited time resolution of
SDD, we could not use the coincidence method, and reached a signal-to-
background ratio of only $\sim$5%. To improve this, we used a shutter to
cyclically turn on and off the P04 photon beam.

Using a 50-$\mu$m slit width, we reached a resolving power of ${E}/{\Delta
E}\approx 10000$, ten to fifteen times higher than that of  _Chandra_ and
_XMM-Newton_ grating spectrometers Den Herder  _et al._ (2001); Canizares  _et
al._ (2005), and tenfold that of our previous experiment Bernitt  _et al._
(2012) (See Supplemental Material (SM)). We find a separation of 3C from 3D of
$\Delta{E}_{3C-3D}={13.398(1)}~{}{\mathrm{eV}}$, and resolve for the first
time the Fe xvii 3D line from the Fe xvi C one at
$\Delta{E}_{3D-C}={154.3(1.3)}~{}{\mathrm{meV}}$. This gives us the 3C/3D
intensity ratio without having to infer a contribution of Fe xvi line C (in
Bernitt  _et al._ (2012) still unresolved) from the intensity of the well-
resolved Fe xvi A line. Thereby, we largely reduce systematic uncertainties
and exclude the resonance-induced population transfer mechanism Wu and Gao
(2019) that may have affected the LCLS result Bernitt  _et al._ (2012).

We apply three different methods (Fig. 1) to systematically measure the 3C/3D
oscillator strength ratio. In method 1, we did not operate the photon shutter;
instead, we repeatedly scanned the lines C and 3D (812.0 –
$812.5\text{\,}\mathrm{eV}$), as well as 3C (825.5 –
$826.0\text{\,}\mathrm{eV}$), in both cases using scans of 100 steps with 20-s
exposure each (see Fig. 2). The SDD fluorescence signal was integrated over a
50-eV wide photon-energy region of interest (ROI) comprising 3C, 3D, and C,
and recorded while scanning the incident photon energy. By fitting Gaussians
to the scan result, we obtain line positions, widths, and yields, modeling the
electron-impact background as a smooth linear function Shah  _et al._ (2019).
The ratio of 3C and 3D areas is then proportional to the oscillator strength
ratio Oreshkina  _et al._ (2016). However, given the low 5% signal-to-
background ratio and long measurement times, changes in the background cause
systematic uncertainties. In method 2, we fixed the monochromator energy to
the respective centroids of C, 3D, and 3C found with method 1, and cyclically
opened and closed the shutter for equal periods of $20\text{\,}\mathrm{s}$ to
determine the background. The background-corrected fluorescence yields at the
line peaks were multiplied with the respective linewidths from method 1, to
obtain the 3C/3D ratio. Still, slow monochromator shifts from the selected
positions could affect the results. To address this, in method 3, we scanned
across the FWHM of C, 3D, and 3C in 33 steps with on-off exposures of
$20\text{\,}\mathrm{s}$, which reduced the effect of possible monochromator
shifts. After background subtraction, we fit Gaussians to the lines of
interest fixing their widths to values from method 1.

All three methods share systematic uncertainties caused by energy-dependent
filter transmission and detector efficiency ($\sim$1%) and by the incident
photon beam flux variation ($\sim$2%). Additionally, for method 1, we estimate
systematic uncertainties from background ($\sim$1.2%) and ROI selection
($\sim$2.7%). In method 2, possible monochromator shifts from (set) line
centroids and widths taken from method 1 cause a systematic uncertainty of
$\sim$3.5%. Analogously, for method 3, we estimate a $\sim$3% uncertainty due
to the use of linewidth constraints from method 1. The weighted average of all
three methods is
$f(\mathrm{3C})/f(\mathrm{3D})=3.09(8)_{\mathrm{sys}}(6)_{\mathrm{stat}}$, see
Fig. 3 (see SM for individual ratio and uncertainties). Note that the circular
polarization of the photon beam does not affect these results, since 3C and 3D
(both ${\Delta}{J}=1$) share the same angular emission characteristics
Balashov  _et al._ (2000); Rudolph  _et al._ (2013); Shah  _et al._ (2015,
2018).

Calculations using a density-matrix approach by Oreshkina  _et al._ (2016,
2014), and the time-dependent collisional-radiative model of Loch  _et al._
(2015), pointed to a possible nonlinear response of the excited upper state
populations in Bernitt  _et al._ (2012), reducing the observed oscillator
strength ratio Li  _et al._ (2017), which would depend on photon pulse
parameters, like intensity, duration, and spectral distribution. It has been
estimated that peak intensities above 1012 W/cm2 would give rise to nonlinear
effects Oreshkina  _et al._ (2014); Li  _et al._ (2017). Fluctuations of the
self-amplified spontaneous emission process at LCLS can conceivably generate
some pulses above that threshold. At P04, we estimate a peak intensity of
$\approx$105 W/cm2, more than 6 orders of magnitude below that threshold (see
SM). This could explain why the 3C/3D ratio in Bernitt  _et al._ (2012) is in
slight disagreement with the present result. Nonetheless, our experiment
validates the main conclusion of that work with reduced uncertainty. Our work
implies that future experiments at ultra-brilliant light sources should take
possible nonlinear effects into account.

Figure 3:  Present experimental 3C/3D ratios compared with previous
predictions and experiments. Red band: combined results of the three different
methods. Blue open circles: values from databases Kramida  _et al._ (2018);
Kaastra  _et al._ (1996); Foster  _et al._ (2012). Blue crosses: predictions
Bhatia and Doschek (1992); Chen  _et al._ (2003); Dong  _et al._ (2003); Chen
(2007); Oreshkina  _et al._ (2014); Jönsson  _et al._ (2014); Santana  _et
al._ (2015); Mendoza and Bautista (2017); Wu and Gao (2019). Note that the
validity of theory Mendoza and Bautista (2017) has been disputed Wang  _et
al._ (2017). Blue solid circles: present FAC Gu (2008), large-scale CI Kozlov
_et al._ (2015), MR-MCDHF Fischer  _et al._ (2019), and AMBiT Kahl and
Berengut (2019) calculations. Blue band: observed line ratios in astrophysical
sources Blake  _et al._ (1965); McKenzie  _et al._ (1980); Mewe  _et al._
(2001); Behar  _et al._ (2001); Xu  _et al._ (2002); Ness, J.-U.  _et al._
(2003), with color shades coding the distribution of values weighted by their
reported accuracies. Purple band: spread of tokamak results Beiersdorfer  _et
al._ (2004). Open green diamonds: previous EBIT results Brown  _et al._ (2001,
2006); Bernitt  _et al._ (2012). Note that the spread seen in various
astrophysical sources and in tokamak in part arises from insufficient removal
of Fe XVI C line contamination of Fe XVII 3D line, at varying Fe XVI/Fe XVII
abundance ratios Doron and Behar (2002); Beiersdorfer  _et al._ (2004);
nonlinear dynamical effects Oreshkina  _et al._ (2014); Loch  _et al._ (2015);
Li  _et al._ (2017) (see main text) may have reduced the LCLS ratio Bernitt
_et al._ (2012) .

In the present work, we also carried out relativistic calculations using a
very-large-scale configuration interaction (CI) method, correlating all ten
electrons, including Breit and quantum electrodynamical Tupitsyn  _et al._
(2016) corrections. We implemented a message passing interface (MPI) version
of the CI code from Kozlov  _et al._ (2015) to increase the number of
configurations to over 230,000, saturating the computation in all possible
numerical parameters. Basis sets of increasing size are used to check for
convergence, with all orbitals up to $12sp17dfg$ included in the largest
version (the contributions of $n>12\,\,sp$ orbitals are negligible). We start
with all possible single and double excitations from the $2s^{2}2p^{6}$,
$2s^{2}2p^{5}3p$ even and $2s^{2}2p^{5}3s$, $2s^{2}2p^{5}3d$, $2s2p^{6}3p$,
$2s^{2}2p^{5}4d$, $2s^{2}2p^{5}5d$ odd configurations, correlating eight
electrons. We separately calculate triple excitations and fully correlate the
$1s^{2}$ shell, and also included dominant quadruple ones, finding them
negligible. The line strengths $S$ and 3C/3D oscillator strength ratio after
several computation stages are summarized in the SM to illustrate the small
effect of all corrections. Theoretical uncertainties are estimated based on
the variance of results from the smallest to largest runs, size of the various
effects, and small variances in the basis set construction. We verified that
the energies of all 18 states considered, counted from the ground state, agree
with the National Institute for Standards and Technology Kramida  _et al._
(2018) database well within the experimental uncertainty of 0.05%. The
theoretical 3C-3D energy difference of 13.44 eV is in agreement with the
experiment to 0.3%.

We also carried out entirely independent large-scale calculations using the
multireference multiconfiguration Dirac-Hartree-Fock (MR-MCDHF) approach
Fischer  _et al._ (2019) with up to 1.2 million configurations. First, the
$2s^{2}2p^{5}3s$, $2s^{2}2p^{5}3d$ and $2s2p^{6}3p$ $J=1$ levels were used as
reference states to generate the list of configuration state functions with
single and double exchanges from all occupied orbitals up to $12spdfghi$.
Virtual orbitals were added in a layer-by-layer manner. Subsequently, the role
of triple excitations was studied by the CI method. In a second step, the
multireference list was extended to include all $J=1$ odd parity states,
generated from the Ne-like ground state by single and double electron
exchanges. Monitoring the convergence of the results for the addition of
layers of virtual orbits, we arrive at an oscillator strength ratio of
3.55(5), and to a 3C-3D energy splitting of 13.44(5) eV. Another full-scale CI
calculation with more than a million configurations was carried out in the
particle-hole formalism using AMBiT Kahl and Berengut (2019), agreeing well
with the other theoretical results. Full details of all calculations can be
found in SM.

We emphasize that there are no other known quantum mechanical effects or
numerical uncertainties to consider within the CI and MCDHF approaches. With
modern computational facilities and MPI codes, we have shown that all other
contributions are negligible at the level of the quoted theoretical
uncertainties. The significant improvements in experimental and theoretical
precision reported here have only further deepened this long-standing problem.
This work on the possibly so far most intensively studied many-electron ion in
experiment and theory, finally demonstrates convergence of the dedicated
atomic calculations on all possible parameters, excluding an incomplete
inclusion of the correlation effects as potential explanation of this puzzle.

Our result is the presently most accurate on the 3C/3D oscillator strength
ratio. Its excellent resolution suggests promising direct determinations of
the natural linewidth. They depend on the Einstein $A$ coefficients, hence, on
the oscillator strengths Weisskopf and Wigner (1997). Thus, future accurate
measurements of individual natural linewidths of 3C and 3D not only would test
theory more stringently than their oscillator strength ratio does, but also
deliver accurate oscillator strengths.

Moreover, 3C and 3D with their, among many transitions, strong absorption and
emission rates can also dominate the Planck and Rosseland mean opacity of hot
plasmas Rogers and Iglesias (1994); Seaton  _et al._ (1994); Beiersdorfer  _et
al._ (1996). Therefore, an accurate determination of their oscillator
strengths may help elucidating the iron opacity issue Bailey  _et al._ (2007,
2015); Nagayama  _et al._ (2019), if, e. g., Rosseland mean opacity models
Fontes  _et al._ (2015); Pain  _et al._ (2017) were found to use predicted
oscillator strengths also in departure from experiments. Our result exposes in
simplest dipole-allowed transitions of Fe XVII a far greater issue, namely,
the persistent problems in the best approximations in use, and calls for
renewed efforts in further developing the theory of many-electron systems.

Shortcomings of low-precision atomic theory for $L$-shell ions had already
emerged in the analysis of high resolution _Chandra_ and _XMM-Newton_ data
Beiersdorfer  _et al._ (2002, 2004); Netzer (2004); Gu (2009); Beiersdorfer
_et al._ (2018). Similar inconsistencies were recently found in the high
resolution $K$-shell X-ray spectra of the Perseus cluster recorded with the
Hitomi microcalorimeter Hitomi Collaboration  _et al._ (2016, 2018). Moreover,
recent opacity measurements Bailey  _et al._ (2015); Nagayama  _et al._ (2019)
have highlighted serious inconsistencies in the opacity models used to
describe the interiors of stars, which have to rely on calculated oscillator
strengths.

All this shows that the actual accuracy and reliability of the opacity and
turbulence velocity diagnostics are still uncertain, and with them, the
modeling of hot astrophysical and high-energy density plasmas. The upcoming
X-ray observatory missions XRISM Tashiro  _et al._ (2018) and Athena Barret
_et al._ (2016) will require improved and quantitatively validated modeling
tools for maximizing their scientific harvest. Thus, benchmarking atomic
theory in the laboratory is vital. As for the long-standing Fe xvii oscillator
strength problem, our results may be immediately used to semiempirically
correct spectral models of astrophysical observations.

###### Acknowledgements.

Financial support was provided by the Max-Planck-Gesellschaft (MPG) and
Bundesministerium für Bildung und Forschung (BMBF) through Project No.
05K13SJ2. Work by C.S. was supported by the Deutsche Forschungsgemeinschaft
(DFG) Project No. 266229290 and by an appointment to the NASA Postdoctoral
Program at the NASA Goddard Space Flight Center, administered by Universities
Space Research Association under contract with NASA. Work by LLNL was
performed under the auspices of the U.S. Department of Energy under Contract
No. DE-AC52-07NA27344 and supported by NASA grants. M.A.L. and F.S.P.
acknowledge support from NASA’s Astrophysics Program. The work of M.G.K. and
S.G.P. was supported by the Russian Science Foundation under Grant No.
19-12-00157. The theoretical research was supported in part through the use of
Information Technologies resources at the University of Delaware, specifically
the high-performance Caviness computing cluster. The work of C.C. and M.S.S.
was supported by U.S. NSF Grant No. PHY-1620687. Work by UNIST was supported
by the National Research Foundation of Korea (No. NRF-2016R1A5A1013277).
J.C.B. acknowledges support from the Alexander von Humboldt Foundation. We
acknowledge DESY (Hamburg, Germany), a member of the Helmholtz Association
HGF, for the provision of experimental facilities. Parts of this research were
carried out at PETRA III.

## References

  * Paerels and Kahn (2003) F. B. S. Paerels and S. M. Kahn, Annu. Rev. Astron. Astrophys. 41, 291 (2003).
  * Canizares  _et al._ (2000) C. R. Canizares, D. P. Huenemoerder, D. S. Davis, D. Dewey, K. A. Flanagan, J. Houck, T. H. Markert, H. L. Marshall, M. L. Schattenburg, N. S. Schulz, _et al._ , Astrophys. J. Lett. 539, L41 (2000).
  * Behar  _et al._ (2001) E. Behar, J. Cottam, and S. M. Kahn, Astrophys. J. 548, 966 (2001).
  * Mewe  _et al._ (2001) R. Mewe, A. J. J. Raassen, J. J. Drake, J. S. Kaastra, R. L. J. van der Meer, and D. Porquet, Astron. Astrophys 368, 888 (2001).
  * Parkinson (1973) J. H. Parkinson, Astron. Astrophys 24, 215 (1973).
  * Doron and Behar (2002) R. Doron and E. Behar, Astrophys. J. 574, 518 (2002).
  * Xu  _et al._ (2002) H. Xu, S. M. Kahn, J. R. Peterson, E. Behar, F. B. S. Paerels, R. F. Mushotzky, J. G. Jernigan, A. C. Brinkman, and K. Makishima, Astrophys. J. 579, 600 (2002).
  * Brickhouse and Schmelz (2005) N. S. Brickhouse and J. Schmelz, Astrophys. J. Lett. 636, L53 (2005).
  * Sanders and Fabian (2011) J. S. Sanders and A. C. Fabian, Mon. Notices Royal Astron. Soc 412, L35 (2011).
  * Kallman  _et al._ (2014) T. Kallman, A. E. Daniel, H. Marshall, C. Canizares, A. Longinotti, M. Nowak, and N. Schulz, Astrophys. J. 780, 121 (2014).
  * de Plaa  _et al._ (2012) J. de Plaa, I. Zhuravleva, N. Werner, J. S. Kaastra, E. Churazov, R. K. Smith, A. J. J. Raassen, and Y. G. Grange, Astron. Astrophys 539 (2012).
  * Bailey  _et al._ (2015) J. E. Bailey, T. Nagayama, G. P. Loisel, G. A. Rochau, C. Blancard, J. Colgan, P. Cosse, G. Faussurier, C. Fontes, F. Gilleron, _et al._ , Nature 517, 56 (2015).
  * Beiersdorfer  _et al._ (2018) P. Beiersdorfer, N. Hell, and J. Lepson, Astrophys. J. 864, 24 (2018).
  * Nagayama  _et al._ (2019) T. Nagayama, J. E. Bailey, G. P. Loisel, G. S. Dunham, G. A. Rochau, C. Blancard, J. Colgan, P. Cossé, G. Faussurier, C. J. Fontes, _et al._ , Phys. Rev. Lett. 122, 235001 (2019).
  * Gu, L.  _et al._ (2019) Gu, L., Raassen, A. J. J., Mao, J., de Plaa, J., Shah, C., Pinto, C., Werner, N., Simionescu, A., Mernier, F., and Kaastra, J. S., A&A 627, A51 (2019).
  * Brown  _et al._ (1998) G. V. Brown, P. Beiersdorfer, D. A. Liedahl, K. Widmann, and S. M. Kahn, Astrophys. J. 502, 1015 (1998).
  * Beiersdorfer  _et al._ (2001) P. Beiersdorfer, S. von Goeler, M. Bitter, and D. B. Thorn, Phys. Rev. A 64, 032705 (2001).
  * Brown  _et al._ (2001) G. V. Brown, P. Beiersdorfer, H. Chen, M. H. Chen, and K. J. Reed, Astrophys. J. Lett. 557, L75 (2001).
  * Beiersdorfer  _et al._ (2002) P. Beiersdorfer, E. Behar, K. Boyce, G. Brown, H. Chen, K. Gendreau, M.-F. Gu, J. Gygax, S. Kahn, R. Kelley, _et al._ , Astrophys. J. Lett. 576, L169 (2002).
  * Beiersdorfer  _et al._ (2004) P. Beiersdorfer, M. Bitter, S. Von Goeler, and K. Hill, Astrophys. J. 610, 616 (2004).
  * Brown  _et al._ (2006) G. V. Brown, P. Beiersdorfer, H. Chen, J. H. Scofield, K. R. Boyce, R. L. Kelley, C. A. Kilbourne, F. S. Porter, M. F. Gu, S. M. Kahn, and A. E. Szymkowiak, Phys. Rev. Lett. 96, 253201 (2006).
  * Brown and Beiersdorfer (2012) G. V. Brown and P. Beiersdorfer, Phys. Rev. Lett. 108, 139302 (2012).
  * Shah  _et al._ (2019) C. Shah, J. R. C. López-Urrutia, M. F. Gu, T. Pfeifer, J. Marques, F. Grilo, J. P. Santos, and P. Amaro, Astrophys. J. 881, 100 (2019).
  * Gu (2009) M. F. Gu, arXiv e-prints , arXiv:0905.0519 (2009), arXiv:0905.0519 [astro-ph.SR] .
  * Bernitt  _et al._ (2012) S. Bernitt, G. V. Brown, J. K. Rudolph, R. Steinbrugge, A. Graf, M. Leutenegger, S. W. Epp, S. Eberle, K. Kubicek, V. Mackel, _et al._ , Nature 492, 225 (2012).
  * Safronova  _et al._ (2001) U. I. Safronova, C. Namba, I. Murakami, W. R. Johnson, and M. S. Safronova, Phys. Rev. A 64, 012507 (2001).
  * Chen and Pradhan (2002) G. X. Chen and A. K. Pradhan, Phys. Rev. Lett. 89, 013202 (2002).
  * Loch  _et al._ (2006) S. D. Loch, M. S. Pindzola, C. P. Ballance, and D. C. Griffin, J. Phys. B: At., Mol. Opt. Phys. 39, 85 (2006).
  * Chen (2007) G.-X. Chen, Phys. Rev. A 76, 062708 (2007).
  * Chen (2011) G. Chen, Phys. Rev. A 84, 012705 (2011).
  * Santana  _et al._ (2015) J. A. Santana, J. K. Lepson, E. Träbert, and P. Beiersdorfer, Phys. Rev. A 91, 012502 (2015).
  * Oreshkina  _et al._ (2014) N. S. Oreshkina, S. M. Cavaletto, C. H. Keitel, and Z. Harman, Phys. Rev. Lett. 113, 143001 (2014).
  * Oreshkina  _et al._ (2016) N. S. Oreshkina, S. M. Cavaletto, C. H. Keitel, and Z. Harman, J. Phys. B: At., Mol. Opt. Phys. 49, 094003 (2016).
  * Loch  _et al._ (2015) S. D. Loch, C. P. Ballance, Y. Li, M. Fogle, and C. J. Fontes, Astrophys. J. Lett. 801, L13 (2015).
  * Wu and Gao (2019) C. Wu and X. Gao, Sci. Rep 9, 7463 (2019).
  * Mendoza and Bautista (2017) C. Mendoza and M. A. Bautista, Phys. Rev. Lett. 118, 163002 (2017).
  * Wang  _et al._ (2017) K. Wang, P. Jönsson, J. Ekman, T. Brage, C. Y. Chen, C. F. Fischer, G. Gaigalas, and M. Godefroid, Phys. Rev. Lett. 119, 189301 (2017).
  * Li  _et al._ (2017) Y. Li, M. Fogle, S. Loch, C. Ballance, and C. Fontes, Can. J. Phys. 95, 869 (2017).
  * Kozlov  _et al._ (2015) M. Kozlov, S. Porsev, M. Safronova, and I. Tupitsyn, Comput. Phys. Commun 195, 199 (2015).
  * Fischer  _et al._ (2019) C. F. Fischer, G. Gaigalas, P. Jönsson, and J. Bieroń, Comput. Phys. Commun 237, 184 (2019).
  * Kahl and Berengut (2019) E. V. Kahl and J. C. Berengut, Comput. Phys. Commun 238, 232 (2019).
  * Micke  _et al._ (2018) P. Micke, S. Kühn, L. Buchauer, J. R. Harries, T. M. Bücking, K. Blaum, A. Cieluch, A. Egl, D. Hollain, S. Kraemer, _et al._ , Rev. Sci. Instrum. 89, 063109 (2018).
  * Viefhaus  _et al._ (2013) J. Viefhaus, F. Scholz, S. Deinert, L. Glaser, M. Ilchen, J. Seltmann, P. Walter, and F. Siewert, Nucl. Instrum. Methods Phys. Res., Sect. A 710, 151 (2013).
  * Den Herder  _et al._ (2001) J. Den Herder, A. Brinkman, S. Kahn, G. Branduardi-Raymont, K. Thomsen, H. Aarts, M. Audard, J. Bixler, A. den Boggende, J. Cottam, _et al._ , Astron. Astrophys 365, L7 (2001).
  * Canizares  _et al._ (2005) C. R. Canizares, J. E. Davis, D. Dewey, K. A. Flanagan, E. B. Galton, D. P. Huenemoerder, K. Ishibashi, T. H. Markert, H. L. Marshall, M. McGuirk, _et al._ , Publ. Astron. Soc. Pac 117, 1144 (2005).
  * Balashov  _et al._ (2000) V. V. Balashov, A. N. Grum-Grzhimailo, and N. M. Kabachnik,  _Polarization and Correlation Phenomena in Atomic Collision_ (Kluwer Academics/ Plenum publishers, 2000).
  * Rudolph  _et al._ (2013) J. K. Rudolph, S. Bernitt, S. Epp, R. Steinbrügge, C. Beilmann, G. Brown, S. Eberle, A. Graf, Z. Harman, N. Hell, _et al._ , Phys. Rev. Lett. 111, 103002 (2013).
  * Shah  _et al._ (2015) C. Shah, H. Jörg, S. Bernitt, S. Dobrodey, R. Steinbrügge, C. Beilmann, P. Amaro, Z. Hu, S. Weber, S. Fritzsche, _et al._ , Phys. Rev. A 92, 042702 (2015).
  * Shah  _et al._ (2018) C. Shah, P. Amaro, R. e. Steinbrügge, S. Bernitt, J. R. C. López-Urrutia, and S. Tashenov, Astrophys. J. Suppl. 234, 27 (2018).
  * Kramida  _et al._ (2018) A. Kramida, Yu. Ralchenko, J. Reader, and and NIST ASD Team, NIST Atomic Spectra Database (ver. 5.6.1), [Online]. Available: https://physics.nist.gov/asd [2019, September 6]. National Institute of Standards and Technology, Gaithersburg, MD. (2018).
  * Kaastra  _et al._ (1996) J. S. Kaastra, R. Mewe, and H. Nieuwenhuijzen, in  _11th Colloq. on UV and X-ray Spectroscopy of Astrophysical and Laboratory Plasmas_, edited by K. Yamashita and T. Watanabe (Tokyo: Universal Academy Press, 1996) pp. 411–414.
  * Foster  _et al._ (2012) A. R. Foster, L. Ji, R. K. Smith, and N. S. Brickhouse, Astrophys. J. 756, 128 (2012).
  * Bhatia and Doschek (1992) A. Bhatia and G. Doschek, At. Data Nucl. Data Tables 52, 1 (1992).
  * Chen  _et al._ (2003) G.-X. Chen, A. K. Pradhan, and W. Eissner, J. Phys. B: At., Mol. Opt. Phys. 36, 453 (2003).
  * Dong  _et al._ (2003) C. Dong, L. Xie, S. Fritzsche, and T. Kato, Nucl. Instrum. Methods Phys. Res., Sect. B 205, 87 (2003), 11th International Conference on the Physics of Highly Charged Ions.
  * Jönsson  _et al._ (2014) P. Jönsson, P. Bengtsson, J. Ekman, S. Gustafsson, L. Karlsson, G. Gaigalas, C. F. Fischer, D. Kato, I. Murakami, H. Sakaue, _et al._ , At. Data Nucl. Data Tables 100, 1 (2014).
  * Gu (2008) M. F. Gu, Can. J. Phys. 86, 675 (2008).
  * Blake  _et al._ (1965) R. Blake, T. Chubb, H. Friedman, and A. Unzicker, Astrophys. J. 142, 1 (1965).
  * McKenzie  _et al._ (1980) D. L. McKenzie, P. B. Landecker, R. M. Broussard, H. R. Rugge, R. M. Young, U. Feldman, and G. A. Doschek, Astrophys. J. 241, 409 (1980).
  * Ness, J.-U.  _et al._ (2003) Ness, J.-U., Schmitt, J. H. M. M., Audard, M., Güdel, M., and Mewe, R., A&A 407, 347 (2003).
  * Tupitsyn  _et al._ (2016) I. I. Tupitsyn, M. G. Kozlov, M. S. Safronova, V. M. Shabaev, and V. A. Dzuba, Phys. Rev. Lett. 117, 253001 (2016).
  * Weisskopf and Wigner (1997) V. Weisskopf and E. P. Wigner, “Berechnung der natürlichen Linienbreite auf Grund der Diracschen Lichttheorie,” in  _Part I: Particles and Fields. Part II: Foundations of Quantum Mechanics_, edited by A. S. Wightman (Springer Berlin Heidelberg, Berlin, Heidelberg, 1997) pp. 30–49.
  * Rogers and Iglesias (1994) F. J. Rogers and C. A. Iglesias, Science 263, 50 (1994).
  * Seaton  _et al._ (1994) M. J. Seaton, Y. Yan, D. Mihalas, and A. K. Pradhan, Mon. Not. R. Astron. Soc 266, 805 (1994).
  * Beiersdorfer  _et al._ (1996) P. Beiersdorfer, A. L. Osterheld, V. Decaux, and K. Widmann, Phys. Rev. Lett. 77, 5353 (1996).
  * Bailey  _et al._ (2007) J. E. Bailey, G. A. Rochau, C. A. Iglesias, J. Abdallah, J. J. MacFarlane, I. Golovkin, P. Wang, R. C. Mancini, P. W. Lake, T. C. Moore, _et al._ , Phys. Rev. Lett. 99, 265002 (2007).
  * Fontes  _et al._ (2015) C. Fontes, C. Fryer, A. Hungerford, P. Hakel, J. Colgan, D. Kilcrease, and M. Sherrill, High Energy Density Physics 16, 53 (2015).
  * Pain  _et al._ (2017) J.-C. Pain, F. Gilleron, and M. Comet, Atoms 5 (2017).
  * Netzer (2004) H. Netzer, Astrophys. J. 604, 551 (2004).
  * Hitomi Collaboration  _et al._ (2016) Hitomi Collaboration _et al._ , Nature 535, 117 (2016).
  * Hitomi Collaboration  _et al._ (2018) Hitomi Collaboration _et al._ , Publ. Astron. Soc. Jpn. 70, 12 (2018).
  * Tashiro  _et al._ (2018) M. Tashiro, H. Maejima, K. Toda, R. Kelley, L. Reichenthal, J. Lobell, R. Petre, M. Guainazzi, E. Costantini, M. Edison, _et al._ , Proc. SPIE 10699, 1069922 (2018).
  * Barret  _et al._ (2016) D. Barret, T. L. Trong, J.-W. Den Herder, L. Piro, X. Barcons, J. Huovelin, R. Kelley, J. M. Mas-Hesse, K. Mitsuda, S. Paltani, _et al._ , in  _Space Telescopes and Instrumentation 2016: Ultraviolet to Gamma Ray_, Vol. 9905 (International Society for Optics and Photonics, 2016) p. 99052F.

High resolution Photoexcitation Measurements Exacerbate
the Long-standing Fe xvii Oscillator Strength Problem: Supplemental Material

##  I Experiment and Data Analysis

Table S1: Comparison between experimental values and theoretical predictions of the $3C/3D$ oscillator strength ratio and relative line energy positions achieved within this work. | Experiment | CI | MCDHF | AMBiT
---|---|---|---|---
3C/3D oscillator strength ratio | $3.09(8)_{\mathrm{sys}}(6)_{\mathrm{stat}}$ | 3.55(5) | 3.55(5) | 3.59(5)
Energy 3C (eV) |  | 825.67 | 825.88(5) | 825.923
Energy 3D (eV) |  | 812.22 | 812.44(5) | 812.397
$\Delta$Energy 3C-3D (eV) | 13.398(1) | 13.44(5) | 13.44(5) | 13.526
$\Delta$Energy 3D-C (eV) | 0.1543(13) |  |  |
Natural linewidth 3C (meV) |  | 14.74(3) | 14.75(3) | 14.90
Natural linewidth 3D (meV) |  | 4.02(5) | 4.01(6) | 4.04

###  I.1 Individual methods data and their uncertainities

Figure S1: Experimental 3C/3D ratio for the three different methods. The individual measurements are shown as solid green circles. The weighted mean experimental values and associated 1-$\sigma$ statistical and systematic uncertainties are indicated as red band. Table S2: $3C/3D$ oscillator strength ratios obtained from three diffferent measurement methods and their statistical and systematic uncertainties. | Method 1 | Method 2 | Method 3
---|---|---|---
$3C/3D$ oscillator strength ratio | 2.960 | 3.080 | 3.210
Uncertainty Budget
Statistical | 0.106 | 0.140 | 0.095
Systematics due to: |  |  |
(1) ROI width selection on 2D histogram (Fig. 2 of the main paper) | 0.030 |  |
(2) ROI centroid selection on 2D histogram | 0.044 |  |
(3) Filter transmission and efficiency of the detector | 0.030 | 0.031 | 0.032
(4) Time-dependent background variation due to the electron-impact excitation | 0.036 |  |
(5) Monochromator shifts in the (set) energy position |  | 0.092 |
(6) Linewidth constraints in Gaussian fits |  |  | 0.048
Total systematic uncertainty | 0.071 | 0.097 | 0.058
Total (statistical + systematic) uncertainties | 0.127 | 0.170 | 0.111
Common systematics for all three methods: |  |  |
Flux variation of the incident photon beam at P04/PETRA III | 0.0618
Final $3C/3D$ oscillator strength ratio | 3.09 $\pm$ 0.08stat. $\pm$ 0.06sys.

###  I.2 Resolving Fe XVII 3D and Fe XVI C lines

Figure S2: Spectral resolution comparison between the present work at PETRA
III/P04, our previous work at LCLS/XFEL Bernitt  _et al._ (2012), and the high
resolution Chandra High Energy Transmission Grating (HETG) Canizares  _et al._
(2000) spectrum of Capella [ObsId: 1103]. In this work, for the first time, we
have resolved previously-blended Fe xvi C from Fe xvii 3D line, which are
154.3(1.3) meV apart from each other. This enables us to obtain the line
intensity ratio of Fe xvii 3C and 3D without having to subtract the
contribution of the Fe xvi C line, in contrast to all other previous works.
Moreover, this has largely reduced the systematic uncertainties and eliminated
the need for taking resonance-induced population transfer Wu and Gao (2019)
into account, which may have affected the accuracy of our LCLS work Bernitt
_et al._ (2012).

###  I.3 Comparison between experimental data, theoretical predictions and
astrophysical observations

Figure S3: Present experimental 3C/3D ratios – combined results of three
different methods (red band), compared with previous predictions and
experiments. Blue open circles: Values from spectral databases. Blue crosses:
theoretical predictions. Blue solid circles: our present FAC Shah  _et al._
(2019); Gu (2008), large-scale CI (Sec. II.1), MR-MCDHF (Sec. II.2)), and
AMBiT (Sec. II.3) calculations. Open green diamonds: previous EBIT results.
Purple band: spread of Tokamak results Beiersdorfer  _et al._ (2004). Light
blue band: range of ratios observed in the Sun Blake  _et al._ (1965);
McKenzie  _et al._ (1980), Capella Mewe  _et al._ (2001); Behar  _et al._
(2001), and NGC4636 Xu  _et al._ (2002). Orange solid circles: 46
astrophysical observations of 26 stellar coronae Ness, J.-U.  _et al._ (2003)
sampled by the RGS onboard XMM-Newton and the LETGS/HETGS onboard Chandra
observatories. Note that the numbers in the square bracket corresponds to the
references cited in the main paper.

###  I.4 Estimation of photon peak intensity on sample at P04 PETRA III

For an estimation of the photon beam intensity on the plasma sample, we refer
to the technical parameters of beamline P04 Viefhaus  _et al._ (2013) and
generously assume a total time-averaged photon flux after monochromatization
of the beam of
$\Phi_{\mathrm{Beam}}=$4\text{\times}{10}^{12}\text{\,}\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{/}\mathrm{s}$,$
a focal spot size of $1\text{\times}{10}^{-6}\text{\,}{\mathrm{cm}}^{2}$, and
a photon energy of $825\text{\,}\mathrm{eV}$. The $16\text{\,}\mathrm{ns}$
pulse separation ensures that the upper state population has sufficient time
to completely relax into the ground state in between the pulses. We obtain a
total energy per bunch of

$E_{\mathrm{bunch}}\approx$8.5\text{\times}{10}^{-6}\text{\,}\mathrm{J}\text{\,}{\mathrm{cm}}^{-2}$.$
(S1)

Combined with the typical pulse duration of $44\text{\,}\mathrm{ps}$ given by
the official PETRA III datasheet (see 111https://photon-
science.desy.de/facilities/petra_iii/beamlines/p04_xuv_beamline/unified_data_sheet_p04/index_eng.html
and 222https://photon-
science.desy.de/facilities/petra_iii/beamlines/p04_xuv_beamline/beamline_parameters/index_eng.html),
we deduce an average intensity per pulse of

$I\approx$1.9\text{\times}{10}^{5}\text{\,}\mathrm{W}\text{\,}{\mathrm{cm}}^{-2}$.$
(S2)

For typical pulse shapes, the peak intensity can be expected to be well below
$1\text{\times}{10}^{6}\text{\,}\mathrm{W}\text{\,}{\mathrm{cm}}^{-2}$.

##  II Calculations of 3C and 3D Oscillator Strengths

###  II.1  Very large-scale CI calculations

We start from the solution of the Dirac-Hartree-Fock equations in the central
field approximation to construct the one-particle orbitals. The calculations
are carried out using a configuration interaction (CI) method, correlating all
10 electrons. The Breit interaction is included in all calculations. The QED
effects are included following Ref. Tupitsyn  _et al._ (2016). The basis sets
of increasing sizes are used to check for convergence of the values. The basis
set is designated by the highest principal quantum number for each partial
wave included. For example, $[5spdf6g]$ means that all orbitals up to $n=5$
are included for the $spdf$ partial waves and $n=5,6$ orbitals are included
for the $g$ partial waves. We find that the inclusion of the $6,7h$ orbitals
does not modify the results of the calculations and omit higher partial waves.
The CI many-electron wave function is obtained as a linear combination of all
distinct states of a given angular momentum $J$ and parity Dzuba  _et al._
(1996):

$\Psi_{J}=\sum_{i}c_{i}\Phi_{i}\,.$ (S3)

The energies and wave functions are determined from the time-independent
multiparticle Schrödinger equation $H\Phi_{n}=E_{n}\Phi_{n}.$

Table S3:  Contributions to the energies of Fe16+ calculated with increased size basis sets and a number of configurations. The results are compared with experiment. All energies are given in cm-1 with exception of the last line that shows the difference of the 3C and 3D energies in eV. The basis set is designated by the highest quantum number for each partial wave included. For example, $12spdfg$ means that all orbitals up to $n=12$ are included for $spdfg$ partial waves. Contributions from triple excitations, excitations from the $1s^{2}$ shells, and QED contributions are given separately. Configuration | Expt. Kramida  _et al._ (2018) | Expt. S_A  | $[5spdf6g]$ | Triples | $1s^{2}$ | $+[12spdfg]$ | $+[17dfg]$ | QED | Final | Diff. Kramida  _et al._ (2018) | Diff. S_A  | Diff. S_A
---|---|---|---|---|---|---|---|---|---|---|---|---
$2p^{6}$ | ${}^{1}S_{0}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
$2p^{5}3p$ | ${}^{3}S_{1}$ | 6093450 | 6093295 | 6087185 | 6 | 254 | 3876 | 772 | 67 | 6092159 | 1291 | 1136 | 0.02%
$2p^{5}3p$ | ${}^{3}D_{2}$ | 6121690 | 6121484 | 6116210 | -21 | 24 | 2886 | 701 | 43 | 6119842 | 1848 | 1642 | 0.03%
$2p^{5}3p$ | ${}^{3}D_{3}$ | 6134730 | 6134539 | 6129041 | -23 | 25 | 3015 | 711 | 94 | 6132864 | 1866 | 1675 | 0.03%
$2p^{5}3p$ | ${}^{1}P_{1}$ | 6143850 | 6143639 | 6138383 | -11 | 41 | 2825 | 704 | 82 | 6142025 | 1825 | 1614 | 0.03%
$2p^{5}3s$ | $2$ | 5849490 | 5849216 | 5842248 | -10 | 108 | 3408 | 735 | 787 | 5847276 | 2214 | 1940 | 0.03%
$2p^{5}3s$ | $1$ | 5864770 | 5864502 | 5857770 | -10 | 70 | 3303 | 708 | 784 | 5862626 | 2144 | 1876 | 0.03%
$2p^{5}3s$ | $1$ | 5960870 | 5960742 | 5953697 | -10 | 74 | 3364 | 717 | 1042 | 5958883 | 1987 | 1859 | 0.03%
$2p^{5}3d$ | ${}^{3}P_{1}$ | 6471800 | 6471640 | 6466575 | -11 | 16 | 2384 | 665 | 87 | 6469717 | 2083 | 1923 | 0.03%
$2p^{5}3d$ | ${}^{3}P_{2}$ | 6486400 | 6486183 | 6481385 | -13 | 16 | 2250 | 658 | 86 | 6484383 | 2017 | 1800 | 0.03%
$2p^{5}3d$ | ${}^{3}F_{4}$ | 6486830 | 6486720 | 6482549 | -12 | 27 | 1745 | 622 | 97 | 6485028 | 1802 | 1692 | 0.03%
$2p^{5}3d$ | ${}^{3}F_{3}$ | 6493030 | 6492651 | 6488573 | -14 | 26 | 1740 | 607 | 84 | 6491016 | 2014 | 1635 | 0.03%
$2p^{5}3d$ | ${}^{1}D_{2}$ | 6506700 | 6506537 | 6502481 | -17 | 21 | 1696 | 627 | 88 | 6504895 | 1805 | 1642 | 0.03%
$2p^{5}3d$ | ${}^{3}D_{3}$ | 6515350 | 6515203 | 6511163 | -18 | 18 | 1762 | 604 | 87 | 6513617 | 1733 | 1586 | 0.02%
$2p^{5}3d$ | ${}^{3}D_{1}$ | 6552200 | 6552503 | 6548550 | -16 | -3 | 1747 | 616 | 134 | 6551029 | 1171 | 1474 | 0.02%
$2p^{5}3d$ | ${}^{3}F_{2}$ | 6594360 | 6594309 | 6589977 | -16 | 22 | 1729 | 629 | 335 | 6592676 | 1684 | 1633 | 0.02%
$2p^{5}3d$ | ${}^{3}D_{2}$ | 6600950 | 6600998 | 6596316 | -17 | 14 | 1947 | 641 | 334 | 6599235 | 1715 | 1763 | 0.03%
$2p^{5}3d$ | ${}^{1}F_{3}$ | 6605150 | 6605185 | 6600744 | -17 | 19 | 1803 | 610 | 343 | 6603501 | 1649 | 1684 | 0.03%
$2p^{5}3d$ | ${}^{1}P_{1}$ | 6660000 | 6660770 | 6656872 | -8 | -52 | 1743 | 619 | 288 | 6659462 | 538 | 1308 | 0.02%
3C-3D |  | 13.3655 | 13.4234 | 13.4302 | 0.0009 | -0.0061 | -0.0005 | 0.0004 | 0.0191 | 13.4440 | -0.0785 | -0.0206 | 0.15%

We start with all possible single and double excitations to any orbital up to
$5spdf6g$ from the $2s^{2}2p^{6}$, $2s^{2}2p^{5}3p$ even and $2s^{2}2p^{5}3s$,
$2s^{2}2p^{5}3d$, $2s2p^{6}3p$, $2s^{2}2p^{5}4d$, $2s^{2}2p^{5}5d$ odd
configurations, correlating 8 electrons. We verified that inclusion of the
$2s2p^{6}3s$, $2s^{2}2p^{5}4f$, $2s^{2}2p^{5}5f$ even and $2s2p^{6}4p$,
$2s^{2}2p^{5}4s$, and $2s^{2}2p^{5}5s$ odd configurations as basic
configurations have negligible effect on either energies of relevant matrix
elements.

The only unusually significant change in the ratio, by 0.07, is due to the
inclusion of the $2s^{2}2p^{3}3d^{3}$ and $2p^{5}3d^{3}$ configurations. These
are obtained as double excitations from the $2s^{2}2p^{5}3d$ odd
configuration, prompting the inclusion of the $2s^{2}2p^{5}4d$,
$2s^{2}2p^{5}5d$ to the list of the basic configurations.

Contributions to the energies of Fe16+ calculated with different size basis
sets and a number of configurations are listed in Table S3. The results are
compared with experimental data from the NIST database Kramida  _et al._
(2018) and from a revised analysis of the experimental data S_A . We use LS
coupling and NIST data term designations for comparison purposes, but note
that $jj$ coupling would be more appropriate for this ion. Contributions to
the E1 reduced matrix elements $D(3D)=D(2p^{6}$ ${}^{1}S_{0}-2p^{5}3d$
${}^{3}D_{1})$ and $D(3C)=D(2p^{6}$ ${}^{1}S_{0}-2p^{5}3d$ ${}^{1}P_{1})$ and
the ratio of the respective oscillator strengths

$R=\left(\frac{D(3C)}{D(3D)}\right)^{2}\times\frac{\Delta E(3C)}{\Delta
E(3D)}$

are listed in Table S4. The energy ratio is 1.01655.

Table S4:  Contributions to the $E1$ reduced matrix elements $D(3D)=D(2p^{6}$ ${}^{1}S_{0}-2p^{5}3d$ ${}^{3}D_{1})$ and $D(3C)=D(2p^{6}$ ${}^{1}S_{0}-2p^{5}3d$ ${}^{1}P_{1})$ (in a.u.) and the ratio of the respective oscillator strengths $R$. See caption of Table S3 for designations. $L$ and $V$ rows compared results obtained in length and velocity gauges for the $[12spdfg]$ basis. All other results are calculated using the length gauge. Transition rates are listed in the last row in s-1.  | $D(3C)$ | $D(3D)$ | Ratio
---|---|---|---
$[5spdf6g]$ |  | 0.33492 | 0.17842 | 3.582
$[5spdf6g]$ |  $+$Triples | 0.33493 | 0.17841 | 3.583
| Triples | 0.00001 | -0.00001 |
$[5spdf6g]$ | +$1s^{2}$ | 0.33480 | 0.17849 | 3.577
| $1s^{2}$ | -0.00012 | 0.00007 |
$[12spdfg]$ | $L$ | 0.33527 | 0.17884 | 3.573
| $V$ | 0.33551 | 0.17894 | 3.574
$+[12spdfg]$ |  | 0.00036 | 0.00042 |
$+[17dfg]$ |  | -0.00001 | 0.00001 |
QED |  | -0.00017 | 0.00030 |
Final |  | 0.33498 | 0.17921 | 3.552
Recomm. |  |  |  | 3.55(5)
Transition rate |  | 2.238$\times 10^{13}$ | 6.098$\times 10^{12}$ |
Table S5:  Contributions to the 3C and 3D line strengths $S$ and the 3C/3D oscillator strength ratios (energy ratio 1.01655 is used). Energies in eV, transition rates $A$ in s-1 and natural linewidths $\Gamma$ in meV are listed in the last three rows of the tables.  | $S(3C)$ | $S(3D)$ | Ratio
---|---|---|---
Small basis | 0.11217 | 0.03183 | 3.582
Medium basis | 0.11241 | 0.03198 | 3.573
Large basis | 0.11240 | 0.03199 | 3.572
\+ triple excitations | 0.11241 | 0.03198 | 3.573
+$1s^{2}$ shell excitations | 0.11233 | 0.03201 | 3.567
+QED | 0.11221 | 0.03212 | 3.552
Final | 0.1122(2) | 0.0321(4) | 3.55(5)
Energies (eV) | 825.67 | 812.22 |
$A$ (s-1) | $2.238(4)\times 10^{13}$ | $6.10(7)\times 10^{12}$ |
$\Gamma$ (meV) | 14.74(3) | 4.02(5) |

We include a very wide range of configurations obtained by triple excitations
from the basic configurations as well as excitations from the $1s^{2}$ shell
and find negligible corrections to both energies and matrix elements as
illustrated by Tables S3 and S4. These contributions are listed as “Triples”
and “$1s^{2}$” in both tables. A significant increase of the basis set from
$[5spdf6g]$ to $[12spdfg]$ improves the agreement of energies with experiment
but gives a very small, -0.009, contribution to the ratio. We find that the
weights of the configurations containing $12fg$ orbitals are several times
higher than those containing $12spd$ orbitals, so we expand the basis to
include more $dfg$ orbitals. We also include $2s^{2}2p^{3}nd^{3}$ and
$2p^{5}nd^{3}$ configurations up to $n=14$. The contributions to the energies
of the orbitals with $n=13-17$ are $3-5$ times smaller than those with
$n=6-12$, clearly showing the convergence of the values with the increase of
the basis set. The effect on the ratio is negligible. The uncertainty of the
NIST database energies, 3000 cm-1 is larger than our differences with the
experiment. The energies from the revised analysis of Fe16+ spectra S_A  are
estimated to be accurate to about 90 cm-1 and the scatter of the differences
of different levels with the experiment is reduced. The last line of Table S3
shows the difference of the $3C$ and $3D$ energies in eV, with the final value
13.44(5)eV. We explored several different ways to construct the basis set
orbitals. While the final results with an infinitely large basis set and
complete configurations set should be identical, the convergence properties of
the different basis sets vary, giving about 0.04 difference in the ratio and
0.04 eV in the $3C-3D$ energy difference at the $12spdfg$ level. Therefore, we
set an uncertainty of the final value of the ratio to be $0.05$. As an
independent test of the quality and completeness of the current basis set, we
compare the results for $D(3C)$ and $D(3D)$ obtained in length and velocity
gauges for the $[12spdfg]$ basis, see rows $L$ and $V$ in Table S4. The
difference in the results is only 0.001. The final results for the line
strengths $S$ and the 3C/3D oscillator strength ratio after several stages of
computations are summarised in Table S5, which clearly illustrates a very
small effect of all corrections.

This work was supported in part by U.S. NSF Grant No. PHY-1620687 and RFBR
grants No. 17-02-00216 and No. 18-03-01220.

###  II.2  Multiconfiguration Dirac-Hartree-Fock calculations

In the multiconfiguration Dirac-Hartree-Fock (MCDHF) method, similarly to the
CI approach outlined in the previous section, the many-electron state is given
as an expansion in terms of a large set of $jj$-coupled configuration state
functions [see Eq. (S3)]. In contrast to the CI calculations, in the case of
MCDHF, the single-electron wave functions (orbitals) are self-consistently
optimized. We use the method in one of its most recent implementations,
namely, applying the GRASP2018 code package Fischer  _et al._ (2019). For the
virtual orbitals, the optimization of the orbitals was done in a layer-by-
layer approach, i.e. when adding a new layer of orbitals (in our case,
orbitals in the same shell) in the configuration expansions, the lower-lying
single-electron functions are kept frozen.

Figure S4:  Convergence of the MCDHF calculations: The X-ray transition
energies of the 3C and 3D lines, their difference, the weighted oscillator
strengths $gf$ and their ratio vs. the maximal principal quantum number
$n_{\rm max}$ used. Oscillator strengths are given both in the relativistic
length (Babushkin) and velocity (Coulomb) gauges to numerically control gauge
invariance. The different columns display results from different sets of
calculations, as described in the text.

In a first set of calculations, we use the $2s^{2}2p^{6}$ configuration for
the ground state and the $2s^{2}2p^{5}3s$, $2s^{2}2p^{5}3d$, and
$2s^{1}2p^{6}3p$ $J=1$ odd configurations for the excited states to generate
the configuration lists. Single and double electron exchanges from the $n=2,3$
spectroscopic (occupied) orbitals were taken into account up to virtual
orbitals $n_{\rm max}spdfg$, where the maximal principal quantum number
$n_{\rm max}$ is varied in the computations to study the convergence of the
results. Such an approach is helpful in estimating the final theoretical
uncertainty. Test calculations also using virtual orbitals with $h$ and $i$
symmetry have shown that these high angular momenta do not play a noticeable
role. The ground and excited states were treated separately, i.e. two
independently optimized sets of orbitals were used. After these multireference
MCDHF calculations, the possible effects of further higher-order electron
exchanges were included in a subsequent step, when a CI calculation was
performed with the extended configuration lists (triple excitations from the
multireference states up to $n=4$ orbitals, yielding approx. 800 thousand
configurations for $n_{\rm max}=13$), employing the radial wave functions
obtained from the previous MCDHF calculations. Furthermore, the effects of the
frequency-independent Breit relativistic electron interaction operator, the
normal and specific mass shift, and approximate radiative corrections are
accounted for (see Fischer  _et al._ (2019) and references therein). The QED
effects have been included in the calculation of transition energies (which
also enter the oscillator strengths), however, not in the electric dipole
matrix elements, as such corrections are anticipated to be on the 1% level and
thus can be neglected. The oscillator strengths were evaluated with the
biorthogonal basis sets, each optimized separately for the ground- and excited
states, to include orbital relaxation effects. The results of these
calculations are presented in the first column of Fig. S4. The bottom panel
shows that the oscillator strength ratio is converged from $n_{\rm max}=9$ on.

In a subsequent set of calculations, the multireference set describing the
ground and excited levels were expanded to include all $J=0$ even and $J=1$
odd states with 1 or 2 electrons in the M shell. The maximal principal quantum
number of the virtual orbitals was set to 9 to limit the computational expense
of calculations. Results are shown in the 2nd column of Fig. S4. In a third
setting, calculations were performed with the smaller multireference list as
described in the previous paragraph, however, with all spectroscopic orbitals
(those with $n=1,2,3$) included in the active set of orbitals when generating
the configuration list. With the triple electron exchanges also included, this
procedure yielded approx. 1.2 million configurations in the description of the
excited states. The energies and strengths are shown in the last column of the
figure. The converged 3C/3D oscillator strength ratios agree well for all 3
calculations. Comparing the different results, the final value for the ratio
is 3.55(5), which agrees well with earlier large-scale MCDHF results Bernitt
_et al._ (2012); Jönsson  _et al._ (2014), and also with the results of the
other theoretical methods described in this Supplement. For the difference of
the energies of the 3C and 3D lines – which can be more accurately determined
in the experiment than the absolute X-ray transition energies – we obtain
13.44(5) eV.

###  II.3 AMBiT: particle-hole CI method calculations

A separate CI calculation of the 3C and 3D lines in Fe16+ has been performed
with the AMBiT code Kahl and Berengut (2019). Our calculation begins with a
Dirac-Hartree-Fock calculation of Ne-like Fe to construct the core $1s$, $2s$,
and $2p$ orbitals in the $V^{N}$ potential. The Breit interaction is included
throughout the calculation. We diagonalize a set of $B$-splines in the Dirac-
Fock potential to obtain valence orbitals. Configuration interaction is
performed using the particle-hole CI method Berengut (2016), however, this can
be mapped exactly to the electron-only approach described in Section II.1.

Our basic calculation is presented on the first line of Table S6. The CI space
consists of all possible single and double excitations up to $10spdf$ from the
same set of leading configurations presented previously: $2s^{2}\ 2p^{6}$,
$2p^{-1}\ 3p$, $2p^{-1}\ 3s$, $2p^{-1}\ 3d$, $2p^{-1}\ 4d$, $2p^{-1}\ 5d$, and
$2s^{-1}3p$. At this stage, we do not include excitations from the frozen
$1s^{2}$ core. Even for this calculation, the matrix size for the odd-parity
$J=1$ levels is $N=479\,075$. To reduce the number of stored matrix elements
we use emu CI Geddes  _et al._ (2018), where interactions between high-lying
configuration state functions are ignored. We limit the smaller side of the
matrix to only including double excitations up to $5spdf$ and limit the number
of $2s$ and $2p$ holes to single removals from an expanded set of leading
configurations which include, in addition to those listed above, $2p^{-2}\
3d^{2}$, $2p^{-1}\ 2s^{-1}\ 3d^{2}$, and $2s^{-2}\ 3d^{2}$. This results in a
reduced small side $N_{small}=80497$. We have checked that expanding the
configuration state functions included in $N_{small}$ makes no difference to
our results at the displayed accuracy.

All of our calculations include the Breit interaction at all stages, and the
dipole matrix elements are calculated in the relativistic formulation with
$\omega=30$ a.u. In the second and third lines of Table S6, we show the
effects of removing the Breit interaction and using the static dipole matrix
element ($\omega=0$), respectively.

We then expand our calculation to include $g$-wave excitations, up to basis
$10spdfg$. The difference from $10spdf$ is shown in the fourth line of Table
S6. We see in the AMBiT calculation very little effect from the inclusion of
these waves. In the fifth line, we show the effect of allowing excitations
from $1s^{2}$, and in the sixth line we see the effect of including the
Uehling potential Ginges and Berengut (2016a) and self-energy Ginges and
Berengut (2016b) using the radiative-potential method Flambaum and Ginges
(2005). This broadly agrees with the model-operator QED presented in Table S4.

The final row of Table S6 gives the results including excitations to
$g$-waves, excitations from $1s^{2}$, and QED effects. The total number of
configuration state functions accounted for is over 1.25 million for the odd-
parity $J=1$ symmetry. Nevertheless, the CI is not quite converged with
respect to including orbitals with $n>10$. We estimate based on calculations
for $8spdf$ and $12spdf$ that the uncertainty in level energies is
conservatively of order $3000~{}\textrm{cm}^{-1}$ and for the ratio
$gf_{3C}/gf_{3D}$ is of order 0.05. These results are consistent with the CI
calculation presented in Sec. II.1 and the MCDHF results presented in Sec.
II.2.

Table S6:  Particle-hole CI calculations using AMBiT of the level energies $E_{3C}$ and $E_{3D}$ (cm-1), reduced matrix elements $D(3C)$ and $D(3D)$ (a.u.), and ratio of oscillator strengths of the $3C$ and $3D$ transitions in Fe16+.  | $E_{3C}$ | $E_{3D}$ | $D(3C)$ | $D(3D)$ | $gf_{3C}/gf_{3D}$
---|---|---|---|---|---
[$10spdf$] | 6661400 | 6552424 | 0.33893 | 0.17992 | 3.607
without Breit | 6668847 | 6557636 | 0.33758 | 0.18164 | 3.511
$\omega=0$ |  |  | 0.33937 | 0.18009 | 3.610
+[$10g$] | -15 | -13 | -0.000001 | 0.000003 |
+$1s^{2}$ | -103 | -85 | -0.000040 | 0.000016 |
+QED | 235 | 94 | -0.00012 | 0.00031 |
Final | 6661517 | 6552420 | 0.33877 | 0.18025 | 3.591

## References

  * Bernitt  _et al._ (2012) S. Bernitt, G. V. Brown, J. K. Rudolph, R. Steinbrugge, A. Graf, M. Leutenegger, S. W. Epp, S. Eberle, K. Kubicek, V. Mackel, _et al._ , Nature 492, 225 (2012).
  * Canizares  _et al._ (2000) C. R. Canizares, D. P. Huenemoerder, D. S. Davis, D. Dewey, K. A. Flanagan, J. Houck, T. H. Markert, H. L. Marshall, M. L. Schattenburg, N. S. Schulz, _et al._ , Astrophys. J. Lett. 539, L41 (2000).
  * Wu and Gao (2019) C. Wu and X. Gao, Sci. Rep 9, 7463 (2019).
  * Shah  _et al._ (2019) C. Shah, J. R. C. López-Urrutia, M. F. Gu, T. Pfeifer, J. Marques, F. Grilo, J. P. Santos, and P. Amaro, Astrophys. J. 881, 100 (2019).
  * Gu (2008) M. F. Gu, Can. J. Phys. 86, 675 (2008).
  * Beiersdorfer  _et al._ (2004) P. Beiersdorfer, M. Bitter, S. Von Goeler, and K. Hill, Astrophys. J. 610, 616 (2004).
  * Blake  _et al._ (1965) R. Blake, T. Chubb, H. Friedman, and A. Unzicker, Astrophys. J. 142, 1 (1965).
  * McKenzie  _et al._ (1980) D. L. McKenzie, P. B. Landecker, R. M. Broussard, H. R. Rugge, R. M. Young, U. Feldman, and G. A. Doschek, Astrophys. J. 241, 409 (1980).
  * Mewe  _et al._ (2001) R. Mewe, A. J. J. Raassen, J. J. Drake, J. S. Kaastra, R. L. J. van der Meer, and D. Porquet, Astron. Astrophys 368, 888 (2001).
  * Behar  _et al._ (2001) E. Behar, J. Cottam, and S. M. Kahn, Astrophys. J. 548, 966 (2001).
  * Xu  _et al._ (2002) H. Xu, S. M. Kahn, J. R. Peterson, E. Behar, F. B. S. Paerels, R. F. Mushotzky, J. G. Jernigan, A. C. Brinkman, and K. Makishima, Astrophys. J. 579, 600 (2002).
  * Ness, J.-U.  _et al._ (2003) Ness, J.-U., Schmitt, J. H. M. M., Audard, M., Güdel, M., and Mewe, R., A&A 407, 347 (2003).
  * Viefhaus  _et al._ (2013) J. Viefhaus, F. Scholz, S. Deinert, L. Glaser, M. Ilchen, J. Seltmann, P. Walter, and F. Siewert, Nucl. Instrum. Methods Phys. Res., Sect. A 710, 151 (2013).
  * Note (1) https://photon-science.desy.de/facilities/petra_iii/beamlines/p04_xuv_beamline/unified_data_sheet_p04/index_eng.html.
  * Note (2) https://photon-science.desy.de/facilities/petra_iii/beamlines/p04_xuv_beamline/beamline_parameters/index_eng.html.
  * Tupitsyn  _et al._ (2016) I. I. Tupitsyn, M. G. Kozlov, M. S. Safronova, V. M. Shabaev, and V. A. Dzuba, Phys. Rev. Lett. 117, 253001 (2016).
  * Dzuba  _et al._ (1996) V. A. Dzuba, V. V. Flambaum, and M. G. Kozlov, Phys. Rev. A 54, 3948 (1996).
  * Kramida  _et al._ (2018) A. Kramida, Yu. Ralchenko, J. Reader, and and NIST ASD Team, NIST Atomic Spectra Database (ver. 5.6.1), [Online]. Available: https://physics.nist.gov/asd [2019, September 6]. National Institute of Standards and Technology, Gaithersburg, MD. (2018).
  * (19) A. Kramida, Preliminary critical analysis of Fe XVII spectral data, Private Communication (2019).
  * Fischer  _et al._ (2019) C. F. Fischer, G. Gaigalas, P. Jönsson, and J. Bieroń, Comput. Phys. Commun. 237, 184 (2019).
  * Jönsson  _et al._ (2014) P. Jönsson, P. Bengtsson, J. Ekman, S. Gustafsson, L. Karlsson, G. Gaigalas, C. F. Fischer, D. Kato, I. Murakami, H. Sakaue, _et al._ , At. Data Nucl. Data Tables 100, 1 (2014).
  * Kahl and Berengut (2019) E. V. Kahl and J. C. Berengut, Comput. Phys. Commun 238, 232 (2019).
  * Berengut (2016) J. C. Berengut, Phys. Rev. A 94, 012502 (2016).
  * Geddes  _et al._ (2018) A. J. Geddes, D. A. Czapski, E. V. Kahl, and J. C. Berengut, Phys. Rev. A 98, 042508 (2018).
  * Ginges and Berengut (2016a) J. S. M. Ginges and J. C. Berengut, J. Phys. B: At., Mol. Opt. Phys. 49, 095001 (2016a).
  * Ginges and Berengut (2016b) J. S. M. Ginges and J. C. Berengut, Phys. Rev. A 93, 052509 (2016b).
  * Flambaum and Ginges (2005) V. V. Flambaum and J. S. M. Ginges, Phys. Rev. A 72, 052115 (2005).