# R-Matrix calculations for opacities: II. Photoionization and oscillator
strengths of iron ions Fe xvii , Fe xviii and Fe xix

S N Nahar1, L Zhao2, W Eissner3, A K Pradhan1 1 Department of Astronomy, The
Ohio State University, Columbus, Ohio 43210, USA  2 Department of Physics, The
Ohio State University, Columbus, Ohio 43210, USA  3 Institut für Theoretische
Physik, Teilinstitut 1, 70550 Stuttgart, Germany111Deceased

###### Abstract

Iron is the dominant heavy element that plays an important role in radiation
transport in stellar interiors. Owing to its abundance and large number of
bound levels and transitions, iron ions determine the opacity more than any
other astrophysically abundant element. A few iron ions constitute the
abundance and opacity of iron at the base of the convection zone (BCZ) at the
boundary between the solar convection and radiative zones, and are the focus
of the present study. Together, Fe xvii , Fe xviii and Fe xix contribute 85%
of iron ion fractions 20%, 39% and 26% respectively, at the BCZ physical
conditions of temperature T $\sim$ $2.11\times 10^{6}$K and electron density
Ne = $3.1\times 10^{22}$cc. We report heretofore the most extensive R-matrix
atomic calculations for these ions for bound-bound and bound-free transitions,
the two main processes of radiation absorption. We consider wavefunction
expansions with 218 target or core ion fine structure levels of Fe xviii for
Fe xvii , 276 levels of Fe xix for Fe xviii , in the Breit-Pauli R-matrix
(BPRM) approximation, and 180 LS terms (equivalent to 415 fine structure
levels) of Fe xx for Fe xix calculations. These large target expansions which
includes core ion excitations to n=2,3,4 complexes enable accuracy and
convergence of photoionization cross sections, as well as inclusion of high
lying resonances. The resulting R-matrix datasets include 454 bound levels for
Fe xvii , 1,174 levels for Fe xviii , and 1,626 for Fe xix up to $n\leq$ 10
and $l$=0 - 9. Corresponding datasets of oscillator strengths for
photoabsorption are: 20,951 transitions for Fe xvii , 141,869 for Fe xviii ,
and 289,291 for Fe xix . Photoionization cross sections have obtained for all
bound fine structure levels of Fe xvii and Fe xviii , and for 900 bound LS
states of Fe xix . Selected results demonstrating prominent characteristic
features of photoionization are presented, particularly the strong Seaton PEC
(photoexcitation-of-core) resonances formed via high-lying core excitations
with $\Delta n=1$ that significantly impact bound-free opacity.

††: J. Phys. B: At. Mol. Phys.

Keywords: Solar opacity, Photoionization, Oscillator Strengths, R-matrix
method

##  1 Introduction

As described in the first paper in this series R-Matrix calculations for
opacities RMOP1 [1], solar elemental abundances are uncertain which, in turn,
are related to the accuracy of atomic opacities in stellar interiors. Opacity,
which is measure of radiation absorption during its transport, is determined
by two main processes, absorption by photo-excitations and photoionization via
all bound states for all ionization stages of all elements that exist in the
plasma, and hence requires extensive amount of atomic data. The present work
focuses on high precision atomic data for these two radiative processes.
Opacity also depends by photon scattering and free-free transitions, but their
contributions are generally small in most of the frequency range at high
temperatures and densities.

This work is an extension of the Opacity Project (hereafter OP [2]) which
reported findings under the series of Atomic data for opacities (hereafter
ADOC) papers. We first describe the approximations employed in the OP and
other prior works and their limitations, and extensions and improvements in
the present series.

###  1.1 Atomic data calculations for opacities

Other methods besides the R-matrix method used for large-scale calculations of
atomic data for opacities are based on atomic structure calculations for the
bound-bound transitions and the distorted wave (DW) approximation for
photoionization. Under such approximations, oscillator strengths and
photoionization cross sections are computed for all possible bound-bound and
bound-free transitions among levels specified by electronic configurations
included in atomic calculations.

The DW approximation based on an atomic structure calculation couples to the
continuum and yields complete and readily computable datasets for opacities.
However, since the DW approximation includes only the coupling between initial
and final states, it is unable to produce autoionizing (AI) resonances embeded
in the continua formed from the complex interference between the bound and
continuum wavefunction expansions involving other levels. DW models employ the
independent resonance approximation that treats the bound-bound transition
probability independently from coupling to the continuum. In this paper, we
report developments in the R-matrix calculations with new features that impact
the opacity in contrast to the original OP R-matrix works.

###  1.2 Opacity Project R-matrix calculations

In contrast to atomic structure and DW calculations, the R-matrix method
accounts inherently for coupling effects due to electron-electron correlation
and introduce autoionizing resonance profiles in an ab initio manner. However,
R-matrix calculations are computationally laborious and entail multiple steps.
The OP work by M.J. Seaton and collaborators [3, 4] was devoted to the
development of the R-matrix method using the close coupling (CC) approximation
based on implementation framework by P.G. Burke and collaborators (e.g. [5, 6,
7]). The R-matrix method was employed extensively for accurate calculations of
radiative data for energies, oscillator strengths and photoionization cross
sections systematically for most astrophysically abundant atoms and ions from
hydrogen to iron [2]. The objective of the OP was to determine the stellar
plasma opacity using high accuracy radiative atomic data. The atomic data
under the OP are available through OP database, TOPbase [8] and NORAD-Atomic-
Data [9].

Despite unprecedented effort and advances the OP data are not of sufficient
precision or extent, as revealted with the advent of sophisticated high
resolution observational and experimental set-ups. The original atomic data
from the OP for oscillator strengths ($f$-values) and photoionization cross
sections ($\sigma_{PI}$) were found to be inadequate and of insufficient
accuracy, primarily because those data were computed in LS coupling without
relativistic fine structure effects and with very limited configuration
interaction. While OP data included highly excited states with $n\leq$10, the
typical angular momentum limit was $l\leq$3\. Many of these calculations were
later repeated for more complete data for $l\leq$9 using larger wavefunction
expansions, and using the BPRM method (these data are available from the
NORAD-Atomic-Data database [9]). The main problem for discrepancies was found
to be the missing physics manifest at high energies. The OP R-matrix work
reported in the ADOC series used small CC wavefunction expansion which
included a few LS terms of the ground configuration, or a few configurations
of the same n-complex of the target or the core ion. Such considerations
missed important physical effects of Seaton PEC resonances first introduced in
[10], and Rydberg series resonances corresponding to highly excited core
states (demonstrated extensively for oxygen ions [11] and subsequent works).
In addition, with high resolution observations fine structure data were needed
in contrast to the OP LS coupling data.

###  1.3 Breit-Pauli R-matrix method

The dynamic package of R-matrix codes has been revised and expanded several
times. In the follow-up to OP, the Iron Project [12], the R-matrix package was
extended to include fine structure with relativistic effects in the Breit-
Pauli approximation (the BPRM method [13]). Other physical effects such as
radiation damping of AI resonances were also incorporated [14, 15]. Of
particular relevance to this work is one of the latest calculations on the
convergence of resonant core ion excitations with increasing $n$ for Fe xvii
[16].

##  2 Theoretical framework

The BPRM framework is described with particular emphasis on atomic absorption
of radiation in plasma sources.

###  2.1 Radiative processes for opacities

The two main processes are photo-excitation for a bound-bound transition and
photo-ionization for a bound-free transition. Photo-excitation of an ion
$X^{+}$,

$h\nu+X^{+}\rightarrow X^{+*},$ (1)

is related to oscillator strength (or $f$-value) which gives the probability
of transition. Photoionization can occur directly as

$h\nu+X^{+}\rightarrow e+X^{++},$ (2)

which is described by a smooth background cross section or through an
intermediate AI state as

$h\nu+X^{+}\leftrightarrow(X^{+})^{**}\leftrightarrow e+X^{++}$ (3)

which introduces a resonance in the cross section. A doubly excited AI state
is formed when the photon energy matches that of a Rydberg level,
$E_{R}=E_{c}^{*}-z^{2}/\nu^{2}$ where $E_{c}^{*}$ is an excited energy of the
core ion, $z$ is the ion charge and $\nu$ is the effective quantum number with
respect to $E_{c}^{*}$. The state may autoionize into the continuum, or
undergo dielectronic recombination if a free electron is captured by emission
of a photon via radiative decay of the core ion. The AI resonance can be
produced theoretically by including the core excitations in the wave function
expansion in the close coupling (CC) approximation.

As mentioned above, opacity can also be caused by scattering of photons by
atoms at all frequencies of radiation prevalent in a given environment.
However, they are much less significant compared to bound-bound and bound-free
transitions and can be taken care more easily as described in the first paper
of the series RMOP.I.

###  2.2 Close coupling approximation and the R-matrix method

The CC approximation describes the atomic system of (N+1) electrons by a
’target’ or the ’core’ or the residual ion of N-electrons interacting with the
(N+1)th electron. The total wave function, $\Psi_{E}$, of the (N+1) electrons
system in a symmetry $SL\pi$ is represented by an expansion as (e.g. [3])

$\Psi_{E}(e+ion)=A\sum_{i}\chi_{i}(ion)\theta_{i}+\sum_{j}c_{j}\Phi_{j},$ (4)

where the target ion eigenfunction $\chi_{i}$ is coupled with the (N+1)th
electron function $\theta_{i}$ in a bound or continuum state. The summation is
over the ground and as many excited ion states as practical in the CC
calculation. $A$ is the anti-symmetrization operator. The (N+1)th electron
with kinetic energy $k_{i}^{2}$ corresponds to a channel labeled as
$S_{i}L_{i}\pi_{i}k_{i}^{2}\ell_{i}(SL\pi)$, where $S_{i}L_{i}\pi_{i}$ is the
symmetry of the target state $i$. For $k_{i}^{2}<0$ the channel is closed and
the $\Psi_{E}$ represents a bound state (all channels closed), and for
$k_{i}^{2}>0$ the channel is open and $\Psi_{E}$ represents a continuum state.
In the second sum, the $\Phi_{j}$s are bound channel functions of the
(N+1)-electron system that account for short-range electron correlation, and
subject to an orthogonality condition between the continuum and the bound
electron spin-orbital functions. Autoionizing resonances arise from
interference effects among th closed and open channels including core ion
excitations in the CC wave function expansion.

In the BPRM method [12, 17] relativistic effects are included in the Breit-
Pauli approximation where the Hamiltonian of the (N+1)-electron system is

$H_{N+1}^{\rm
BP}=\sum_{i=1}^{N+1}\left\\{-\nabla_{i}^{2}-\frac{2Z}{r_{i}}+\sum_{j>i}^{N+1}\frac{2}{r_{ij}}\right\\}+H_{N+1}^{\rm
mass}+H_{N+1}^{\rm Dar}+H_{N+1}^{\rm so}$ (5)

The curly bracketed term is the non-relativistic Hamiltonian and the
additional terms are the relativistic one-body correction terms, the mass
correction, $H^{\rm mass}=-{\alpha^{2}\over 4}\sum_{i}{p_{i}^{4}}$, Darwin,
$H^{\rm Dar}={Z\alpha^{2}\over 4}\sum_{i}{\nabla^{2}({1\over r_{i}})}$, and
the spin-orbit interaction, $H^{\rm so}=Z\alpha^{2}\sum_{i}{1\over
r_{i}^{3}}{\bf l_{i}.s_{i}}$ where $p_{i}$ is the momentum of an electron,
$\alpha$ is the fine structure constant, and ${\bf l_{i},s_{i}}$ are the
orbital and spin angular

Substitution of the CC wavefunction $\Psi_{E}(e+ion)$ in the Schrodinger
equation

$H_{N+1}\mit\Psi_{E}=E\mit\Psi_{E}$ (6)

results in a set of coupled equations that are solved using the R-matrix
method [7, 3, 13, 13, 17]. The BPRM method implements an intermediate coupling
scheme. The set of ${SL\pi}$ terms is recoupled for $SLJ\pi$ fine structure
levels of the (e + ion) system, followed by diagonalization of the
(N+1)-electron BP Hamiltonian. The solutions are either a continuum
wavefunction $\Psi_{F}$ for an electron with positive energies (E $\geq$ 0),
or a bound state wavefunction $\Psi_{B}$ for negative total energies (E $<0$).

###  2.3 R-matrix method for radiative data

With wavefunction expansions in the R-matrix formulation as above, transition
matrix element for a radiative transition to an excited bound state or for
photoionization is given by (e.g. [17])

$<\Psi_{j}||{\bf D}||\Psi_{k}>,$ (7)

where the photon-ion interaction is represented by the dipole operator, ${\bf
D}=\sum_{i}{{\bf r}_{i}}$, and the sum is over the number of active electrons.

The generalized line strength ${\bf S}$ is expressed as

${\bf S}=|<\Psi_{j}||{\bf
D}_{L}||\Psi_{k}>|^{2}=\left|\left\langle{\mit\psi}_{f}|\sum_{j=1}^{N+1}r_{j}|{\mit\psi}_{i}\right\rangle\right|^{2},$
(8)

where $\mit\Psi_{j}$ and $\mit\Psi_{k}$ are initial and final state
wavefunctions. The line strengths are energy independent quantities. The
oscillator strengths ($f_{ij}$) and radiative decay rates or Einstein
A-coefficients for E1 dipole transitions are given by

$f_{ij}={E_{ji}\over{3g_{i}}}S(ij),~{}~{}A_{ji}(a.u.)={1\over
2}\alpha^{3}{g_{i}\over g_{j}}E_{ji}^{2}f_{ij}.$ (9)

$E_{ji}$ is the energy difference between initial and final states, and
$g_{i}$, $g_{j}$ are statistical weight factors, respectively.

The photoionization cross section ($\sigma_{PI}$) is obtained as,

$\sigma_{PI}={4\pi^{2}\over 3c}{1\over g_{i}}\omega{\bf S},$ (10)

where $g_{i}$ is the statistical weight factor of the initial bound state and
$\omega$ is the incident photon energy. The complex resonant structures in
photoionization result from channel couplings between open continuum channels
($k_{i}^{2}~{}\geq$ 0) and closed channels ($k_{i}^{2}~{}<$ 0) at electron
energies $k_{i}^{2}$ corresponding to autoionizing states along Rydberg series
$S_{i}L_{i}~{}J_{i}\pi_{i}\nu_{i}\ell_{i}$, where $\nu_{i}$ is the effective
quantum number relative to the target threshold $S_{i}L_{i}~{}J_{i}\pi_{i}$.
We note that present work also includes radiation damping of the resonances
using the approach of [14], but is found to be insignificant for the Fe ions
considered herein.

##  3 R-matrix Computations for atomic processes

The relativistic R-matrix calculations are carried out through a package of
BPRM codes [13, 23, 14] in several stages of computations as shown in Figure 1
(left branch).

Figure 1:  Schematic diagram for various stages of R-matrix computations. The
data obtained are: 1) energy Levels, 2) oscillator strengths, 3)
photoionization cross sections, 4) recombination rate coefficients, and 5)
electron impact collision strengths, astrophysical models and opacities
calculations.

BPRM computations begin with the STG1 program for which the program reads the
orbital wavefunctions and potentials of the core ion as the first input and
computes one- and two-electron radial integrals for the output. In the present
cases for Fe xvii , Fe xviii and Fe xix , these wavefunctions are obtained
from configuration interaction atomic structure calculations using the code
SUPERSTRUCTURE (SS) [18, 19]. The second program STG2 computes spin-angular
algebraic coefficients in LS coupling for the ion and (e + ion) Hamiltonian
matrices. Program RECUPD recouples the $LS$ coefficients in intermediate
coupling to introduce fine structure $SLJ$. Using SS wavefunctions, RECUPD
also recomputes fine structure energies of the core ion. Typically the energy
values and order from RECUPD match closely with those from SS. However, for
complex ions they may differ in the third or fourth significant digits and
energy order for some levels, particularly those highly excited ones. The
program STGH diagonalizes the (e + ion) Hamiltonian to obtain R-matrix basis
functions that are used to compute subsequent parameters as follows. Program
STGB computes bound energy levels and wavefunctions and STGF computes
continuum wavefunctions and electron impact excitation collision strengths.
STGBB computes oscillator strengths for bound-found transitions, and STGBF
computes photoionization cross sections for bound-free transitions.

###  3.1 Core ion wavefunctions from SUPERSTRUCTURE

As mentioned above, computation using R-matrix codes starts with wavefunctions
of the core ion obtained from code SUPERSTRUCTURE (SS). Similar to
intermediate coupling in the BPRM method, SS includes relativistic
contributions in Breit-Pauli approximation [19]. SS includes several terms of
the Breit interaction in addition to one-body terms in BPRM calculations, such
as the full Breit interaction and part of other two-body interactions.
Configuration interaction calculations are carried out using the Thomas-Fermi-
Dirac-Amaldi central potential to compute one-orbitals functions, scaled
according to a variational minimization scheme [18, 19, 17].

The list of configurations and Thomas-Fermi orbital scaling parameters for
core ions of each of the three Fe ions Fe xvii , Fe xviii , Fe xix are given
and discussed in following subsections. All configurations for each ion are
treated as spectroscopic; that is, all energies are optimized in SS
iteratively. Each table quotes the total number of core ion excitation
produced by the spectroscopic configurations, all of which were included in
wavefunction expansions.

The SS run itself computes a large number of the transitions of types E1, E2,
E3, M1, and M2 among all possible levels possible within configurations
specified in atomic structure calculations. Although not presented in this
paper which focuses on R-matrix results for E1 transitions that primarily
contribute to opacities, SS results for all energy levels and all types of
transitions stated here are available through atomic database NORAD-Atomic-
Data [9].

Progran RECUPD of the R-matrix codes use orbital wavefunctions obtained from
SS to recompute the energies of the core ion. These reproduced energies match
almost exactly to those from SS for most ions. However, with large complex
atomic systems the SS and RECUPD energies can show differences in the third or
fourth decimal figures, and also the order of the higher energy levels. In the
present work, for both Fe xvii and Fe xviii , the RECUPD energies are used in
the Hamiltonian matrix diagonalization in STGH.

####  3.1.1 CC wavefunction expansion for Fe xvii :

Table 1 lists the optimized set of 17 configurations with Thomas-Fermi scaling
parameters of orbitals that produced the 218 energy evels for the core ion Fe
xviii included in the first summation term on RHS of Eq. (4) to represent the
(e + ion) wavefunction expansion for Fe xvii . Table 1 provides only a small
sample set of energies of the ground and a few excited levels of Fe xviii from
the full set of 218 levels obtained from SS. SS energies are compared with
measured values by [20] available from NIST [21], and found to be in good
agreement.

Table 1:  Optimized set of 17 spectroscopic configurations of the core ion Fe xviii belonging to complexes n=2,3,4 are: $1s^{2}2s^{2}2p^{5}$(1), $1s^{2}2s2p^{6}$(2), $1s^{2}2s^{2}2p^{4}3s$(3), $1s^{2}2s^{2}2p^{4}3p$(4), $1s^{2}2s^{2}2p^{4}3d$(5), $1s^{2}2s2p^{5}3s$(6), $1s^{2}2s2p^{5}3p$(7), $1s^{2}2s2p^{5}3d$(8), $1s^{2}2s^{2}2p^{4}4s$(9), $1s^{2}2s^{2}2p^{4}4p$(10), $1s^{2}2s^{2}2p^{4}4d$(11), $1s^{2}2p^{6}3s$(12), $1s^{2}2p^{6}3p$(13), $1s^{2}2p^{6}3d$(14), $1s^{2}2s2p^{5}4s$(15), $1s^{2}2s2p^{5}4p$(16), $1s^{2}2s2p^{5}4d$(17). Thomas-Fermi orbital scaling parameters are: 1.39944(1s), 1.20409(2s), 1.14074(2p), 1.13092(3s), 1.08125(3p), 1.11030(3d), 1.13092(4s), 1.08125(4p), 1.1103(4d). The table present a small sample set of energies from the 218-level set, being compared with those of [20] available at NIST webpage [21]. The number within parentheses next to the LS term corresponds to configuration number for the term.  i | LS term | 2J+1 |  ET(Ry) | E(Ry) | i | LS term | 2J+1 |  ET(Ry) | E(Ry)
---|---|---|---|---|---|---|---|---|---
Total number of levels = 218
1 |  ${}^{2}P^{o}$(1) | 4 | 0.0 | 0.0 | 2 |  ${}^{2}P^{o}$(1) | 2 | 0.94116 | 0.9348
3 |  ${}^{2}S^{e}$(2) | 2 | 9.8248 | 9.70 | 4 |  ${}^{4}P^{e}$(3) | 6 | 56.798 | 57.70
5 |  ${}^{2}P^{e}$(3) | 4 | 57.054 | 56.94 | 6 |  ${}^{4}P^{e}$(3) | 2 | 57.502 | 57.50
7 |  ${}^{4}P^{e}$(3) | 4 | 57.668 | 57.57 | 8 |  ${}^{2}P^{e}$(3) | 2 | 57.906 | 57.80
9 |  ${}^{2}D^{e}$(3) | 6 | 58.436 | 58.32 | 10 |  ${}^{2}D^{e}$(3) | 4 | 58.471 | 58.36
… |  |

The Hamiltonian matrix for Fe xvii is set up and diagonalized in STGH using
these energies and energy order of the core ion reproduced in RECUPD. Partial
waves 0$\leq\ell\leq$ 9 for the interacting free electron form singlet,
triplet, and quintet spins for (e + ion) $LS\pi$ up to $L$=0-4 of even and odd
parities, recoupled in RECUPD to yield total $SLJ\pi$ symmetries with $J\leq
12$. The R-matrix boundary was chosen to be $a_{o}=6$ a.u., large enough to
ensure all bound orbitals to have decayed to at least $P_{n\ell}<10^{-3}$.

The second term in RHS of Eq. (4), which represents (e + ion) bound state
correlation functions as bound channels in the Hamiltonian, included 42
($N$+1)-particle configurations with minimum to a maximum number electron
occupancies in orbitals as given within parentheses: 1s(2-2), 2s(0-2),
2p(3-6), 3s(0-2), 3p(0-2), 3d(0-2), 4s(0-1), 4p(0-1), 4d(0-1). The total
angular momenta of Fe xvii selected are J=0-9 of even and odd parities,
sufficient for radiative excitations via dipole E1 transitions.

####  3.1.2 CC wavefunction expansion for Fe xviii :

The ground and 275 excited fine structure levels of the core ion Fe xix
included with configuration complexes $n$=2,3,4, were optimized using a set of
12 configurations given in Table 2 along with the Thomas-Fermi scaling
parameters of orbitals, and a small subset of the 276-level expansion for Fe
xviii . The calculated energies from SS are compared with measured values from
NIST [21].

Table 2:  The set of 12 spectroscopic configurations of the core ion Fe xix (as in table 1), optimized for energies and orbitals with complexes n=2,3,4 in the CC wavefunction expansion of Fe xviii are: $1s^{2}2s^{2}2p^{4}$(1), $1s^{2}2s2p^{5}$(2), $1s^{2}2p^{6}$(3), $1s^{2}2s^{2}2p^{3}3s$(4), $1s^{2}2s^{2}2p^{3}3p$(5), $1s^{2}2s^{2}2p^{3}3d$(6), $1s^{2}2s^{2}2p^{5}4s$(7), $1s^{2}2s^{2}2p^{3}4p$(8), $1s^{2}2s^{2}2p^{3}4d$(9), $1s^{2}2s2p^{4}3s$(10), $1s^{2}2s2p^{4}3p$(11), $1s^{2}2s2p^{4}3d$(12). The set of Thomas-Fermi scaling parameters for the orbitals are 1.35(1s), 0.9009(2s), 1.12(2p), 1.07(3s), 1.05(3p), 1.10(3d), 1.0(4s), 1.0(4p), 1.0(4d).  i | LS term | 2J+1 |  ET(Ry) | E(Ry) | i | LS term | 2J+1 |  ET(Ry) | E(Ry)
---|---|---|---|---|---|---|---|---|---
Total number of levels = 276
1 |  ${}^{3}P^{e}$(1) | 4 | 0.0 | 0.0 | 2 |  ${}^{3}P^{e}$(1) | 0 | 0.6681 | 0.6857
3 |  ${}^{3}P^{e}$(1) | 2 | 0.8003 | 0.8150 | 4 |  ${}^{1}D^{e}$(1) | 4 | 1.5675 | 1.5387
5 |  ${}^{1}S^{e}$(1) | 0 | 2.9543 | 2.9629 | 6 |  ${}^{3}P^{o}$(2) | 4 | 8.4592 | 8.4100
7 |  ${}^{3}P^{o}$(2) | 2 | 9.0267 | 8.9736 | 8 |  ${}^{3}P^{o}$(2) | 0 | 9.4316 | 9.3862
9 |  ${}^{1}P^{o}$(2) | 2 | 11.714 | 11.5512 | 10 |  ${}^{1}S^{e}$(3) | 0 | 19.706 | 19.4481
… |  |

The CC expansion for Fe xviii included 0$\leq\ell\leq$ 9 partial waves with
doublet, quartet, sextet $LS$ symmetries and $L$=0-4 of even and odd parities
of the core ion Fe xix . The R-matrix basis sets for orbitals contained 20
continuum functions inside the R-matrix boundary of radius 4 $a_{o}$. The
second bound-channel correlation function term in Eq. (4) included 96
($N$+1)-electron configurations with minimum to a maximum occupancies in
orbital as given within parentheses: 1s(2-2), 2s(0-2), 2p(2-6), 3s(0-2),
3p(0-2), 3d(0-2), 4s(0-2), 4p(0-2), 4d(0-1). The total (e + ion) angular
momenta $SLJ\pi$ for Fe xviii was chosen to be $J$=1/2-17/2 of even and odd
parities.

####  3.1.3 CC wavefunction expansion for Fe xix :

The atomic data for transition probabilities with fine structure of Fe xix was
computed using BPRM codes. Line and oscillator strengths for bound-bound
transitions were computed an 18CC wavefunction expansion for the core ion Fe
xx (details in [28]) since no bound states of Fe xix are formed with higher
core excitations. The computation of photoionization cross sections was
initially set with a very large 415CC wavefunction expansion which included
n=2,3,4 core ion excitations and resonances up to the high energy region.
However, owing to computational limits BPRM computations proved to be
impractical. Hence $\sigma_{PI}$ were obtained in the LS coupling
approximation. Table 3 lists the optimized set of configurations with Thomas-
Fermi scaling parameters of orbitals of Fe xx , and provides a sample set of
energies for the ground and a number of excited states, compared with energies
from NIST [21]. A large number of odd parity states exist in the high energy
region, but they are not allowed for dipole transitions from the ground state
$2s^{2}2p^{3}(^{4}S^{o})$ and therefore corresponding series of strong
resonances would not manifest themselves. Hence, a concise set of 56 LS states
was chosen which includes all dipole allowed and forbidden transitions from
the ground and low-lying states, where the basic physics of transitions with
the full set of 415 fine structure levels is retained.

Table 3:  The set of 16 spectroscopic configurations of the core ion Fe xx that was optimized for Fe xix CC calculations with complexes n=2,3,4 are: $2s^{2}2p^{3}$(1), $2s2p^{4}$(2), $2p^{5}$(3), $2s^{2}2p^{2}3s$(4), $2s^{2}2p^{2}3p$(5), $2s^{2}2p^{2}3d$(6), $2s^{2}2p^{2}4s$(7), $2s^{2}2p^{2}4p$(8), $2s^{2}2p^{2}4d$(9), $2s^{2}2p^{2}4f$(10), $2s2p^{3}3s$(11), $2s2p^{3}3p$(12), $2s2p^{3}3d$(13), $2s2p^{3}4s$(14), $2s2p^{3}4p$(15), $2s2p^{3}4d$(16) with filled $1s$ orbital. Thomas-Fermi scaling parameters are: 1.35(1s), 1.25(2s), 1.12(2p), 1.07(3s), 1.05(3p), 1.0(3d), 1.0(4s), 1.0(4p), 1.0(4d), 1.0(4f). A sample set of energies $E_{T}$ is compared with $E$ [20] available from NIST [21].  i | LS term |  ET(Ry) | E(Ry) | i | LS term |  ET(Ry) | E(Ry)
---|---|---|---|---|---|---|---
Total number of levels = 415 and states =180
1 | $2s^{2}2p^{3}(^{4}S^{o})$ | 0.0 | 0.0 | 2 | $2s^{2}2p^{3}(^{2}D^{o})$ | 1.5091 | 1.4683
3 | $2s^{2}2p^{3}(^{2}P^{o})$ | 2.7698 | 2.7549 | 4 | $2s2p^{4}(^{4}P)$ | 7.2235 | 7.2029
5 | $2s2p^{4}(^{2}D)$ | 9.6995 | 9.5869 | 6 | $2s2p^{4}(^{2}S)$ | 10.9879 | 10.8920
7 | $2s2p^{4}(^{2}P)$ | 11.7840 | 11.6184 | 8 | $2p^{5}(^{2}P^{o})$ | 18.3096 | 18.1380
9 | $2s^{2}2p^{2}3s(^{4}P)$ | 66.4611 | 66.2742 | 10 | $2s^{2}2p^{2}3s(^{2}P)$ | 66.6114 |

The LS term energies computed in STG2 are in similar order as that os SS but
with slight differences in values in the 3rd or 4th figure and a few energies
with different order. Partial waves for the Fe xix calculations included
0$\leq\ell\leq$ 9 and formed 89 singlets, triplets, quintets and septets, with
the target ion total $L$=0-4 of even and odd parities. The R-matrix basis sets
contained 14 continuum wavefunctions, R-matrix boundary was $a_{o}$ = 6 a.u.
The bound channel correlation functions included 104 ($N$+1)-particle
configurations with a minimum to a maximum number electron occupancies in the
orbitals as given within parentheses: 1s(2-2), 2s(0-2), 2p(2-6), 3s(0-2),
3p(0-2), 3d(0-2), 4s(0-1), 4p(0-1), 4d(0-1), 4f(0-1).

###  3.2 Bound states and oscillator strengths

The bound energies were obtained by numerically scanning through eigenvalues
of the (e + ion) Hamiltonian with a sufficiently fine energy mesh in effective
quantum number, typically 0.001-0.005, and corresponding wavefunctions were
computed using program STGB of the R-matrix codes (Figure 1). Comparisons show
good agreement between the observed NIST compilation and calculated energies
[19, 27, 28]. It may be noted that the R-matrix calculations encompass a large
number of configurations for the (N+1)-electrons atomic system, and generally
more accurate and yield more extensive data sets than atomic structure codes
such as SS.

The transition parameters, such as, oscillator strengths, line strengths, and
radiative decay rates for the iron ions Fe xvii -Fe xix [19, 27, 28] were
obtained using program STGBB of the R-matrix codes which uses the Hamiltonian
matrix and dipole transition matrices computed by STGH and bound wavefunctions
computed by STGB.

###  3.3 Bound-free photoionization cross sections

Basic physical features and illustrative results from large-scale computations
of photoionization cross sections ($\sigma_{PI}$) of the three Fe ions Fe xvii
, Fe xviii and Fe xix were obtained using the STGBF program of the R-matrix
package of codes. The $\sigma_{PI}$ of Fe xvii and Fe xviii reported herein
have been obtained from new BPRM calculations. Whereas the $\sigma_{PI}$ of Fe
xix are taken from [22] but features relevant to opacities calculations are
highlighted and discussed for comparison, consistency and completeness.

Owing to large sizes of arrays and matrices, the BPRM codes went through an
extensive revisions for opacities calculations. Often the computations could
be carried out only for few energy levels and photon energies at a time, and
required several years to complete.

Photoionization cross sections are obtained with consideration of radiation
damping implemented in BPRM codes [13, 23, 14], although not eventually found
to be significant for opacities for these Fe ions. Autoionizing resonances in
photoionization span wide energy ranges, and were resolved with variable and
appropriately fine energy meshes.

##  4 Results and discussion

Opacity calculations require complete datasets for any and all atomic species.
We report more extensive studies of the three Fe ions, Fe xvii , Fe xviii and
Fe xix , than previous works and which also reveal reveal new features in
photoionization not observed before. With the objective of obtaining high
accuracy and complete sets of atomic data we calcauted transition probablities
and photoionizatio cross sections of levels with $n\leq 10$ and $l\leq 9$, and
all associated $SLJ\pi$ spin-orbital symmetries, with large wavefunction
expansions that show important features in high energy regions. Selected
results and prominent characteristic features are discussed below.

###  4.1 Energy levels and Oscillator strengths

We obtained large sets of fine structure energy levels for the three Fe ions
Fe xvii , Fe xviii and Fe xix [19, 27, 28]. The number of energy levels and
oscillator strengths obtained from BPRM method for each ion are given in Table
4. For calculating oscillator strengths between bound-bound transitions, size
of the CC wavefunction expansions was smaller compared to those for
photoionizatio cross sections, since very high excited core ion levels do not
lead to additional bound levels of the (e + ion) system. However, the larger
number of core ion thresholds included herein for photoionization give rise to
many more high lying Rydberg series of autoionizing states that can be much
stronger than those from the low lying excited states.

Table 4: The table lists the number of energy levels and oscillator strengths for bound-bound transitions obtained for three iron ions. The numbers are nearly the same as those obtained in earlier works with much smaller wavefunction expansions.  Ion | No. of energies | No. of oscillator strengths | Reference
---|---|---|---
Fe XVII | 454 | 20,951 |  Nahar et al (2003)[19]
Fe XVIII | 1174 | 141,869 |  Nahar (2006)[27]
Fe XIX | 1626 | 289,291 |  Nahar (2011)[28]

###  4.2 Photoionization cross sections

The BPRM method reveals several characteristic features in photoionization
that are of importance in opacity calculations. The features can be
characterized based on type of states or levels of the particular ion, and can
impact opacities differently depending on energy, temperature and density of
plasma in a given region. The following subsections focus on these features.

Resonances in photoionization may play a dominant role as they can increase
radiation absorption by orders of magnitude. In particular, the present work
shows that one of the main reasons for discrepancy in photoabsorption from
past studies is the lack of consideration of resonances due to high lying core
excitations. The only way to obtain these resonances inherently is through the
close coupling approximation from a large wavefunction expansion that includes
sufficiently high excitations. Hence, we include many excited levels belonging
to n=2, 3 and 4 complexes for the 3 Fe ions to satisfy this convergence
criterion (see also RMOP.III).

####  4.2.1 Photoionization of ground states

The accuracy of the ground level and associated transitions are obviously
important in all applications. However, in a plasma at different temperatures
and densities the photoionization cross section $\sigma_{PI}$ of the ground
state is not necessarily the most significant one, and in fact may not
dominate bound-free opacity [32]. Typically, $\sigma_{PI}$ has a slowly
varying background cross section up to the highest threshold energy in the CC
expansion, and then decreases with energy. The Rydberg series of resonances
are superimposed up on the background. However, it is the $n$ -complex of the
core ion ground and low-lying configurations that produce more prominent
resonances compared to higher ones, as their magnitudes weaken. Figure 2
presents photoionization cross sections of the ground states of the three ions
Fe XVII-XIX from the present work (blue) and from TOPbase [8] (magenta) for Fe
XVII and Fe XVIII. No other detailed $\sigma_{PI}$ for Fe XIX are found in
literature. Our study finds that ground state core excitations beyond $n$ =2
complex are not important as they do not produce strong resonances, and
inclusion of $n$ = 3 and 4 levels does not impact cross sections
significantly.

Figure 2:  $\sigma_{PI}$ of the ground level of a) Fe XVII (218CC FS), b) Fe
XVIII (276CC FS), c) Fe XIX (180CC LS). Present computations with large
wavefunction expansion for $\sigma_{PI}$ in blue basically agree with those in
magenta from earlier TOPbase calculations with only the ground $n$ =2
configuration CC expansion [8]. Hence, photoionization of the ground level is
largely unaffected by inclusion of high-level excitations.

####  4.2.2 Relativistic fine structure effect:

In LS coupling resonances in $\sigma_{PI}$ are approximately averaged over
their fine structure components. Accuracy increases with inclusion of
relativistic fine structure channels as they provide more resolved resonances,
more accurate positions of resonances spread over somewhat more extended
energy region, as well as additional resonances not allowed in LS coupling.
With splitting of LS terms of the core ion states into fine structure levels,
a much larger number of excited core ion thresholds is produced, resulting in
correspondingly larger number of Rydberg series of resonances. However, the
resulting accuracy in $\sigma_{PI}$ may not be significant when the resonances
are statistically averaged to obtain integrated quantities such as
recombination rates or photoionization rates for plasma opacity at high
temperature-density. But exceptions are noticeable at low energies and in low
temperature plasmas when fine structure coupling creates resonant features,
which are actually observed in experiments (e.g. [24, 25]), but missing from
LS coupling calculations.

Figure 3 demonstrates the effect of coupling of relativistic fine structure
channels on photoionization in the near ionization energy for the ground
$2s^{2}2p^{6}(^{1}S_{0})$ state of Fe xvii . The upper panel is the present
BPRM $\sigma_{PI}$, and the lower panel from a non-relativistic R-matrix
calculation in LS coupling [16]. Features in both cross sections are very
similar. However, the upper panel shows resonances created by fine structure
channels $2s^{2}2p^{5}(^{2}P^{o}_{1/2})\epsilon s,d$ in the energy region
between the two ground state core ion levels $2s^{2}2p^{5}(^{2}P^{o}_{3/2})$
and $2s^{2}2p^{5}(^{2}P^{o}_{1/2}$, and an enhancement at the
$2s^{2}2p^{5}(^{2}P^{o}_{1/2})$ threshold, unlike in coupling without fine
structure splitting of $2s^{2}2p^{5}(^{2}P^{o})$. Also, the ionization
threshold is lowered to the $2s^{2}2p^{5}(^{2}P^{o}_{3/2})$ (pointed arrow),
whereas in LS coupling the threshold is at the statistical average of the two
levels. These resonances would affect applications in low energy-temperature
plasma sources. In addition, it may be noted that fine structure has split the
resonances in LS coupling in lower panel into its component resonances in the
upper paner. It has been found for the ion that relativistic fine structure
effect near the ionoization threshold exists in $\sigma_{PI}$ of most of the
excited levels of the ion.

Figure 3: Photoionization cross sections $\sigma_{PI}$ in BPRM method of the
ground level $2s^{2}2p^{6}(^{1}S_{0})$ of Fe XVII in the near ionization
threshold region (upper panel), and non-relativistic LS coupling R-matrix
cross sections (lower panel, [16]), showing the coupling effect of fine
structure channels in the region between $2s^{2}2p^{5}(^{2}P^{o}_{3/2})$ and
$2s^{2}2p^{5}(^{2}P^{o}_{1/2})$ in the upper panel with resonances and
background jump at $2s^{2}2p^{5}(^{2}P^{o}_{1/2})$, not formed in LS coupling.

####  4.2.3 Photoionization of equivalent electron states:

Equivalent electron levels/states, with more than a single electron in the
outer orbit, typically have photoionization cross sections with smooth
background with some enhancement at core ion thresholds, and then decrease
slowly with energy. These levels, particularly the ones formed from excited
configurations, produce high-peaked closely-spaced Rydberg series of
resonances at lower energies. These resonances typically belong to the core
ion excitations of the same $n$ -complex as the ground state. Hence these
states can have significant impact on applications in relatively low
temperature plasmas.

For the present three ions, there is no equivalent electron state for Fe xvii
, one for Fe xviii and three for Fe xix . Photoionization cross sections
($\sigma_{PI}$) of these levels for Fe xviii and Fe xix are shown in Figures 4
and 5. Figure 4 presents $\sigma_{PI}$ of the equivalent electron state
$2s2p^{6}(^{2}S_{1/2})$ of Fe xviii . The arrows point to positions of various
core ion thresholds where Rydberg series of resonances converge and enhance
the background. It may be noted that closely-spaced Rydberg resonances are
highly-peaked and strong only for n=2 core ion thresholds. Higher ones are
very weak resonances merging with the background.

Figure 4: $\sigma_{PI}$ of the equivalent electron state
$2s2p^{6}(^{2}S_{1/2})$ of Fe xviii . Arrows at the bottom indicate excitation
thresholds to which Rydberg series of resonances converge. Arrows at n=2 and
n=3 indicate the highest core ion excitation energy for the respective shell.
Features show formation of very strong series of $n$ = 2 resonances; higher
ones are much weaker.

Figure 5 presents $\sigma_{PI}$ of the three equivalent electron states of Fe
XIX: a) $2s2p^{5}(^{3}P^{o})$, b) $2s2p^{5}(^{1}P^{o})$, and c)
$2p^{6}(^{1}S)$. Features are similar to those of Figure 4, with almost no
resonance structures beyond $n$ =2 threshold excitations. However, note that
$\sigma_{PI}($$2s2p^{5}(^{3}P^{o}))$ in panel a) shows closely-spaced strong
resonances, those are sparse for the other two levels in panels b) and c). The
reason for the latter is lack of the channel couplings with the singlet states
$2s2p^{5}(^{1}P^{o})$ and $2p^{6}(^{1}S)$.

Figure 5: $\sigma_{PI}$ of the equivalent electron states a)
$2s2p^{5}(^{3}P^{o})$, b) $2s2p^{5}(^{1}P^{o})$, and c) $2p^{6}(^{1}S)$ of Fe
XIX. Arrows at n=2 indicate the highest core ion excitation energy threshold
for the shell and appearance of resonances below it. Features show formation
of very strong dense series of resonances for $2s2p^{5}(^{3}P^{o})$ for
stronger coupling while it they are less for the singlets due to lack of
coupling couplings to the states.

####  4.2.4 Photoionization of low-lying excited levels:

Photoionization features change dramatically for single-electron excited
states in comparison to those of the ground and equivalent electron states.
Core excitations to high lying levels beyond the ground $n$ -complex exhibits
enriched variations. For the three Fe ions, considerable impact can be seen in
$\sigma_{I}$ of excited states in forming strong resonances and enhancing the
background beyond the ground n-complex (n=2), even for the first excited level
illustrated in Figure 6.

Figure 6 presents $\sigma_{PI}$ of the first single valence electron excited
level $2s^{2}2p^{5}3s(^{3}p^{o}_{2})$ of Fe xvii ; blue represents this work
and black the OP data obtained by M. P. Scott (unpublished) available in
TOPbase [8]. Regions of resonant features belonging to core excitation to $n$
=2, $n$ =3 and $n$ =4 complexes are marked by arrows which point to energies
of the highest excited core of the respective $n$ -complex; those associated
with $n$ =2 are very weak and the background cross section is decreasing.
However, above $\sim$ 57 Ry strong resonance structures appear and dominate
until $n$ =3 thresholds around $\sim$97 Ry, where the background rises by more
than on order of magnitude. Beyond $n$ =3, resonances become weak and merge
with the smooth background which decreases smoothly. The strength of the $n$
=3 resonance complex, relative to $n$ =2, indicates high photoabsorption and
enhanced background missing in OP data [8].

Figure 6:  $\sigma_{PI}$ of the first single valence electron excited level of
$2s^{2}2p^{5}3s(^{3}p^{o}_{2})$ of Fe xvii , demonstrating impact of the $n$
=3 complex of resonances forming strong resonance features that are several
orders of magnitude higher than those due to the $n$ =2 complex and background
enhancement near the $n$ =3 threshold; however, beyond the $n$ =3 resonances
are very weak.

Figure 7 presents $\sigma_{PI}$ of the first single valence electron excited
level $2s^{2}2p^{4}3s(^{4}Po_{5/2})$ of Fe xviii where the arrows point to the
energy positions of the highest excited core of the respective shells $n$
=2,3,4. Although it has one less electron, F-like instead of Ne-like Fe xvii ,
the features are similar indicating characteristics of the level: large
enhancement due to $n$ =3 complex of resonances relative to $n$ =2. Couplings
to the $n$ =4 complex are very weak and the background decreases with energy.
Overall, between $\sim$60-102 Ry resonant excitations raise the background by
more than an order of magnitude, and merge into the background above the $n$
=3 thresholds.

Figure 7:  $\sigma_{PI}$ of the first single electron excited level
$2s^{2}2p4^{4}3s(^{4}P_{5/2}$ level of Fe xviii demonstrating core excitations
of the $n$ =3 complex similar to that of Fe xvii in Figure 6.

Figure 8 presents $\sigma_{PI}$ of the first single electron excited state,
$2s^{2}2p^{3}3s^{3}S^{o}$ of Fe xix demonstrating relative magnitudes of
resonant complexes due to $n$ =2 and $n$ =3 thresholds, However, similar to Fe
xvii and Fe xviii the impact is negligible for $n$ =4 complex. Therefore, it
may be concluded that resonance structures due to core excitations beyond $n$
=4 for all three Fe ions have converged.

Figure 8:  $\sigma_{PI}$ of the first excited level of
$2s^{2}2p^{3}3s^{3}S^{o}$ of Fe xix , as for Figures 6 and 7.

####  4.2.5 Seaton PEC resonances:

In addition to dense and strong Rydberg series of resonances, the high lying
excited states typically show dominance by huge resonances due to photo-
excitation-of core via strong dipole transitions over extended energy regions
[10, 17, 26]. Figure 9 presents $\sigma_{PI}$ of the high-lying single valence
electron excited level $2s^{2}p^{3}(^{2}D^{o})4d(^{3}S^{o})$, of Fe xix .
Generally, Seaton PEC resonances are formed as the core ion is excited from
the ground state through a dipole allowed transition, while the outer electron
remains a spectator as it photoionizes [33]. The interference of Seaton and
Rydberg resonances usually form a combined resonance feature over a wide
energy range, as seen around $\sim$70 Ry and $\sim$95 Ry (red arrows) in
Figure 9. Since the transition energies remain the same irrespective of
ionization thresholds, Seaton resonances appear at the same photon energies in
$\sigma_{PI}$ of all excited levels, but not in $\sigma_{PI}$ of equivalent
electron states shown in Figure 5.

Figure 9:  Characteristic features of a high-lying excited state,
$2s^{2}p^{3}(^{2}D^{o})4d(^{3}S^{o})$, of Fe xix . Photoionization of such
levels is typically dominated by wide Seaton PEC resonances, as seen at energy
positions pointed by red arrows. Interference of Seaton and Rydberg resonances
can impact the background over a wide energy range up to and above 100 eV, and
enhancement by more than an order of magnitude above the background.

Figure 10 illustrates other characteristics of Seaton resonances, the
progressive behaviour and commonality in positions at the same PEC energies.
Figure 10 shows $\sigma_{PI}$ of three excited levels with different
ionization energies: a) $2s^{2}p^{4}3p(^{4}D^{o}_{3/2})$ at $\sim$41 Ry, b)
$2s^{2}2p^{4}4f(^{2}P^{o}_{1/2})$ at $\sim$19.07 Ry and c)
$2s^{2}2p^{4}7f(^{2}P^{o}_{1/2})$ at $\sim$5.06 Ry of Fe XVIII. The first two
resonances appear around 10 Ry, but seen only for the third excited state
(panel c) since the ionization threshold for the level is lower than for these
PEC excitations. Other Seaton resonances are at higher energies and appear in
$\sigma_{PI}$ of all levels exactly at the same transition energies. The
general feature is that Seaton resonances become more prominent with exciation
level, from a) to c) in Figure 10. Finally, we also find that Seaton
resonances with $\Delta n$=1 are stronger than than those with $\Delta n$=0,
but weaker for higher core transition energies $\Delta n$.

Figure 10: Progression of chararcterstic features with excitation level: a)
$2s^{2}p^{4}3p(^{4}D^{o}_{3/2})$, b) $2s^{2}2p^{4}4f(^{2}P^{o}_{1/2})$ and c)
$2s^{2}2p^{4}7f(^{2}P^{o}_{1/2})$ of Fe XVIII. Seaton resonances appear at the
exact energies of core transitions via dipole allowed transitions, and become
more prominent in $\sigma_{PI}$ with higher level of $n$ -value of the
spectator electron.

Finally, these PEC phenomena should be detectable experimentally owing to
their extended energy ranges. By virtue of their immense magnitude and extent,
Seaton PEC resonances are the largest contributor to bound-free opacity.

####  4.2.6 Convergence of resonance series:

One crucial outcome of having a vary large wavefunction expansion which
includes many core ion excitations is the convergence of highly-peaked
prominent resonant features with increasing principle quantum number n.
Exceptions are the $\sigma_{PI}$ of the ground level and equivalent electron
levels for which resonances become weaker and start to converge to the
background beyond $n$ =2 complex. However, altogether the ground and
equivalent electron states are relatively few in number compared to hundreds
to over a thousand other excited bound states of each ion where AI resonances
dominate. With increasing $n$ , the excitation probability of the core ion
first increases and then starts to decrease and weaker channel couplings
merging on to the background. In $\sigma_{PI}$ presented in Figures 3-10 for
the three Fe ions, we see various progressions with $n$ =2, 3 and 4. The n=4
AI resonance structures are reduced considerably and show the trend towards
convergence. Past computations of $\sigma_{PI}$ either did not consider
resonances beyond the $n$ -complex of the ground configuration, or prominent
Rydberg and Seaton resonances and their convergence.

##  5 Conclusion

We have presented detailed albeit limited sets of photoionization cross
sections of three Fe ions Fe xvii , Fe xviii and Fe xix that are the dominant
iron opacity source in the solar interior at the boundary of the radiative and
convection zones (paper RMOP.I). We explore features in the high energy region
that include core ion excitations in hundreds of levels of these three ions.
These features were not heretofore studied primarily owing to presumption of
of weak couplings of interacting channels and AI resonance interference
effects, as well as practical computational limits. The present study reveals
characteristic features of photoionization based on the type of initial bound
states and convergence criteria of AI resonant phenomena, and relativistic
fine structure effects not produced in LS coupling. Application of these data
are expected to provide high-precision plasma opacity in stellar interiors for
these ions.

Data availability

All atomic data for energies, radiative transitions, and collisional
excitations will be available online at the NORAD-Atomic-Data database at the
Ohio State University at: https://norad.astronomy.osu.edu/

###  5.1 Acknowledgments

We acknowledge the support of the Ohio Supercomputer Center for extensive
computational resources for the present work. Computations were carried out
over several years on high performance computer clusters at the Ohio
Supercomputer Center (OSC) and on the UNITY cluster at the Ohio State
University.

References

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