# R-Matrix calculations for opacities: I. Methodology and computations

A K Pradhan1,2,3, S N Nahar1, W Eissner4111Deceased 2022 1 Department of
Astronomy, 2 Biophysics Graduate Program, 3 Chemical Physics Program, The Ohio
State University, Columbus, Ohio 43210, USA  4 Institüt für Theoretische
Physik, Teilinstitut 1, 70550 Stuttgart, Germany

###### Abstract

An extended version of the R-matrix methodology is presented for calculation
of radiative parameters for improved plasma opacities. Contrast and
comparisons with existing methods primarily relying on the Distorted Wave (DW)
approximation are discussed to verify accuracy and resolve outstanding issues,
particularly with reference to the Opacity Project (OP). Among the
improvements incorporated are: (i) large-scale Breit-Pauli R-matrix (BPRM)
calculations for complex atomic systems including fine structure, (ii)
convergent close coupling wave function expansions for the (e + ion) system to
compute oscillator strengths and photoionization cross sections, (iii) open
and closed shell iron ions of interest in astrophysics and experiments, (iv) a
treatment for plasma broadening of autoionizing resonances as function of
energy-temperature-density dependent cross sections, (v) a ”top-up” procedure
to compare convergence with R-matrix calculations for highly excited levels,
and (vi) spectroscopic identification of resonances and bound (e + ion)
levels. The present R-matrix monochromatic opacity spectra are fundamentally
different from OP and lead to enhanced Rosseland and Planck mean opacities. An
outline of the work reported in other papers in this series and those in
progress is presented. Based on the present re-examination of the OP work, it
is evident that opacities of heavy elements require revisions in high
temperature-density plasma sources.

##  1 Introduction

Opacity is due to interaction of radiation with matter. It is a fundamental
parameter in plasma, astrophysics, and atomic physics that determines
radiation transport, and entails absorption and scattering of photons by atoms
at all frequencies of radiation prevalent in a given environment. Methods for
calculating opacities are well-established, and essentially involve the atomic
physics of bound-bound and bound-free transition probabilities incorporated
within an equation-of-state (EOS) of the plasma. However, in practice
complexities arise owing to several physical factors that influence the
accurate determination of opacity, and are addressed in this series of papers.
As this work is an extension of the Opacity Project (hereafter OP), we first
briefly outline OP and its calculations described under the Atomic data for
opacities (hereafter ADOC) series of papers, and their limitations. Next, we
describe the extensions and improvements over OP in the present series
R-Matrix calculations for opacities (hereafter RMOP), subsequently referred to
as papers RMOP1, RMOP2, RMOP3, RMOP4.

###  1.1 The Opacity Project

The OP work by M.J. Seaton and collaborators [1, 2, 3] and reference therein]
was devoted to the development of a framework for calculation of opacities
based on the close coupling approximation implemented in the powerful R-Matrix
(RM) method by P.G. Burke and collaborators, and employed extensively for
accurate calculations of a variety of radiative and collisional atomic
processes [4, 5, 6]. The OP work entailed an EOS for stellar interior plasmas
based on the ”chemical picture” by D. Mihalas, D.G. Hummer and W. Däppen
(named MHD-EOS [7]), that connects physically with OP atomic data via an
occupation probability factor of ionization fractions, level populations, and
partition function in the modified Saha-Boltzmann equations that accounts for
plasma interactions.

Despite unprecedented effort and advances, the OP R-matrix work reported in
ADOC faced several then intractable difficulties that limited the scope of
atomic calculations. Primarily, the limitations were due to computational
constraints which, in turn, did not enable accounting for important physical
effects and a complete R-matrix calculation of atomic opacities. The main
features and deficiencies of OP are as follows: (I) The calculations were in
LS coupling neglecting relativistic fine structure, (II) The close coupling
(hereafter CC) wavefunction expansion for the target or the core ion in the (e
+ ion) system included only a few ground configuration LS terms, (III) Inner-
shell excitations could not be included owing to the restricted target ion
expansion, (IV) While autoionizing resonances in bound-free photoionization
cross sections were delineated within the few excited target terms, (V) Total
angular and spin (e + ion) symmetries with large orbital angular-spin quantum
numbers were not computed. All of these factors are crucial for a complete and
accurate opacity calculation. Therefore, the OP work incorporated a relatively
small subset of R-matrix data. Rather, most of the opacities contributions
were obtained using atomic structure codes and the Distorted Wave (hereafter
DW) approximation, similar to other opacity models [6-10].

In addition to the limitations of ADOC work mentioned above, new physical
issues emerge in extending R-matrix calculations towards a complete
calculation of opacities. There are three major problems that need to be
solved: (A) convergence of large coupled channel wavefunction expansions
necessary to include sufficient atomic structures manifest in opacity spectra,
(B) completeness of high $n\ell$ contributions up to $n\equiv\infty$, and (C)
attenuation of resonance profiles due to intrinsic autoionization broadening
(included in RM calculations in an ab initio manner) and extrinsic plasma
effects due to temperature and density, as generally considered for bound-
bound line opacity.

###  1.2 Scientific problems

The erstwhile OP work summarized above concluded that the agreement between OP
and another independent calculation OPAL [15, 3] do not differ by more than
2.5%, implying that a further revision of opacities was not needed [34].
However, there are outstanding problems related to opacities derived from the
OP and all other opacity models. The foremost among them is related to a
downward revision of solar abundances of common volatile elements such as
carbon, nitrogen, oxygen and neon, relative to earlier ones by up to $\sim$50%
[12, 13]. Thereupon, astrophysicists suggested that an upward revision of
opacities by $\sim$10% [18, 33] would countenance the lower solar abundances,
since abundances are inversely linked to opacities which affect the radiation
field in non-local thermodynamic equilibrium (NLTE) models employed to analyze
observed line profiles of elements. In particular, the iron opacity plays a
crucial role owing to relatively high abundance of iron. Also, recent
experimental measurements of iron opacity were higher than given by OP and
other models [29, 30]. Whereas opacity models have been improved by including
additional transition arrays resonances, etc., the discrepancies with
astrophysical and experimental results remain outstanding.

This series describes the work carried out since the OP opacities reported in
2003 and available via database OPServer [28].

###  1.3 BPRM and DW Methods

Current opacity models employ the DW approximation or variants thereof. In
order to compare and contrast the present BPRM results, as well as to test
complementarity and completeness of atomic data, we have also carried out
relativistic distorted wave calculations reported in paper RMOP3 of this RMOP
series. In principle, the DW approximation based on an atomic structure
calculation coupled to the continuum yields complete sets of opacities.
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 the atomic calculation. However, since
the DW approximation includes only the coupling between initial and final
states, the complexity of interference between the bound and continuum
wavefunction expansions involving other levels is neglected. That manifests
itself as quasi-bound levels and autoionizing resonances embedded in the
continua. DW models employ the independent resonance approximation that treats
the bound-bound transition probability independently from coupling to the
continuum. Apart from relative simplicity of atomic computations, the
advantages of DW models is that well-established line broadening treatments
may be employed to account for plasma interactions. Another advantage is ease
of completeness of datasets that can be augmented by including additional
configurations with multiple-electron excitations. Furthermore, high angular-
spin momenta do not pose a computational problem commonly encountered in CC
calculations. For these reasons the DW method is generally employed for
opacities calculations. In contrast, RMOP calculations are computationally
laborious and time-consuming. However, coupling effects can affect atomic
parameters significantly.

###  1.4 Prior work

Opacity in the bound-free continuum is dominated by autoionizing resonances,
as shown in recently completed works cited above and present results.
Hitherto, they have been treated generally as lines akin to bound-bound
transitions. The most important consequence, and likely source of missing
opacity, is the intrinsic autoionizing broadening and the extrinsic plasma
broadening thereof. The much wider spread of resonances in the continuum than
lines raises the opacity significantly [24, 25].

Recent work [26] extended Fe xvii R-matrix calculations by including more
configurations than NP16a. Whereas that confirmed our earlier results for
photoionization cross sections, there are several issues: (i) D21 do not
consider plasma broadening of autoionizing resonances that enhance opacities
significantly, (see papers II and III), (ii) the D21 comparison between DW and
unbroadened RM appears to agree, although fundamentally different since the DW
method treats autoionizing levels and broadening thereof as for lines, (iii)
D21 do not compare unbroadened RM cross sections for Fe xvii previously
available from database NORAD [31], (iv) inexplicably, D21 RM Fe xvii
Rosseland mean opacities are 10% below below the primarily DW results from
OP2005, whereas all other DW models yield values up to 1.5 times higher [25];
there is no reason why RM opacities, even without broadening, should be lower
than OP and other DW models, except that D21 might have an incomplete number
of initial Fe xvii levels in their RM calculations. Other issues such as
radiative data, cross sections, and shapes of autoionizing resonances due to
plasma broadening are addressed in this series.

Experimental opacity measurements at the Sandia Z facility for Fe, Ni, and Cr
have highlighted deficiencies in theoretical models [29, 30]. However,
experimental results need to be viewed in the context of the very limited
energy range where monochromatic iron opacity is actually measured. Indeed,the
experimental energy range does not include the region of maximum opacity from
Fe ions around $\sim$ 1 Kev (well-known in X-ray spectroscopy). Therefore,
experimental opacities per se contribute only about 20% to the Rosseland mean
opacities directly. However, extrapolating the differences between OP and
experimental data in that limited range, B15 estimate a solar mixture opacity
enhancement of 7$\pm$3% (the large error bars imply a factor of 2.5
discrepancy between the low and high experimental values).

Opacity is a sensitive function of temperature and density, and an incomplete
tabulation in a limited range may give inconsistent results since different
ionization states of Cr, Fe and Ni contribute. For example, N-like Cr ions
with 3 active p-electrons make the largest contribution at the Z
temperature/density, whereas F-like Fe with 5 p-electrons is the largest
contributor; for Ni it is Ne-like, a closed p-shell configuration. These
issues need to be examined individually at a much wider range of energy-
temperature-density to ascertain the source of discrepancies. Thus, although
experimental results might point to ”missing physics”, it is first important
to include physics that is known but missing, such as plasma broadening
described in this series of papers.

###  1.5 Overview of RMOP calculations

Sections of this first paper P1 cover the following topics, as well as general
features of subsequent papers in the series: (i) opacities and solar
temperature-density structure, (ii) local-thermodynamic-equilibrium (LTE)
plasma equation-of-state valid in stellar interiors, (iii) relativistic
effects using the Breit-Pauli R-Matrix (BPRM) approximations, (iv) DW and BPRM
calculations, and (v) plasma broadening of autoionizing resonances in bound-
free opacity, (vi) convergence and completeness of atomic data.

##  2 Monochromatic and mean opacities

The atomic parameters comprising the monochromatic opacity are due to bound-
bound (bb), bound-free (bf), free-free (ff), and photon scattering (sc)
contributions:

$\kappa_{ijk}(\nu)=\sum_{k}a_{k}\sum_{j}x_{j}\sum_{i,i^{\prime}}[\kappa_{bb(}(i,i^{\prime};\nu)+\kappa_{bf}(i,\epsilon
i^{\prime};\nu)+\kappa_{ff}(\epsilon
i,\epsilon^{\prime}i^{\prime};\nu)+\kappa_{sc}(\nu)]\ ,$ (1)

where $a_{k}$ is the abundance of element $k$, $x_{j}$ the $j$ ionization
fraction, $i$ and $i^{\prime}$ are the initial bound and final bound/continuum
states of the atomic species, and $\epsilon$ represents the electron energy in
the continuum. The atomic absorption coefficients are related to the local
radiation field at temperature T described by the Planck function

$B_{\nu}(T)=\frac{(2h\nu^{3}/c^{2})}{e^{h\nu/kT}-1}.$ (2)

Macroscopic quantities such as radiative forces and fluxes may be computed in
terms of mean opacities, such as the Planck Mean Opacity (PMO)

$\kappa_{P}B(T)=\int\kappa_{\nu}B_{\nu}d\nu.$ (3)

Of particular interest to opacity calculations is the Rosseland Mean Opacity
(RMO), $\kappa_{R}$ RMO defined as the harmonic mean of monochromatic opacity
$\kappa_{ijk}(\nu)$ as

$\frac{1}{\kappa_{R}}=\frac{\int_{0}^{\infty}g(u)\kappa_{\nu}^{-1}du}{\int_{0}^{\infty}g(u)du}\
\ \ \hbox{\rm; }\ \ \ g(u)=u^{4}e^{-u}(1-e^{-u})^{-2},$ (4)

where $g(u)=dB_{\nu}/dT$ is the derivative of the Planck weighting function
(corrected for stimulated emission). Eq. 4 is mathematically and physically a
complex quantity to evaluate. Whereas the opacity determines radiative
transfer through the stellar interior, the RMO is related to the total
radiation flux that eventually escapes the star and observed [22]. Although
the singularity in the denominator $1/\kappa_{n}u$ is generally avoided owing
to overlapping spectral features, the RMO depends critically on the precise
distribution of monochromatic opacity at all frequencies at a given (T,$\rho$)
at each point inside the star. The opacity spectrum is a complex quantity with
superimposed dips or windows and large peaks that vary by orders of magnitude
due to energy dependence of atomic parameters, $\kappa_{bb}(i,i^{\prime})=(\pi
e^{2}/m_{e}c)N_{i}f_{ii^{\prime}}\phi_{\nu}$, and
$\kappa_{bf}=N_{i}\sigma_{\nu}$. The $\kappa_{\nu}$ is then primarily a
function of the bb oscillator strengths $f$, bf photoionization cross sections
$\sigma_{\nu}$, level populations $N_{i}$, and the line-profile factor
$\phi_{\nu}$. The RMOP framework for large-scale computations comprises mainly
the first two components of the opacity in Eq. (1): (i) the bb transition
probabilities and (ii) the bf photoionization cross sections.

###  2.1 Solar structure and opacity

Tables 1 and 2 provide a numerical glimpse of solar interior structure and
related plasma and atomic parameters. In Table 1 we focus on the region
outside of the nuclear fusion core in the radiative zone up to the boundary at
the base of the convection zone (BCZ) [21]. Helioseismological analysis of
thousands of modes of solar oscillations yields a precise measurement of the
BCZ at solar radius R⊙ = 0.713$\pm$0.001. At and above the BCZ outward energy
transport via radiative diffusion gives way to convection which becomes more
efficient since $(dT/dr)_{diff}>(dT/dr)_{ad}$, the adiabatic temperature
gradient. There are two main reasons for convective motions to be more
efficient at the BCZ: the weight of the outer layers is less than the
radiation pressure from the interior below, and the increase in opacity from
higher to lower temperatures. Opacity increases due to the prevalence of lower
stages of ionization, as more bound electrons are active in absorption of
radiation via larger number of bound-bound and bound-free transitions than at
higher temperatures below the BCZ.

Table 2 shows the ionization states of the dominant elements that determine
opacity at the BCZ: O, Ne and Fe. Almost 90% oxygen is in H-like or fully
ionized, and 86% of neon is in H-like and He-like ionization states. But one
and two electron K-shell ionization states do not contribute as much to
opacity, as lower ones such as the partially filled L-shell Fe ions with
percentage contributions given in Table 2. Just three Fe ions constitute 85%
of iron at BCZ temperatures and densities. Those ions, Fe XVII, XVIII, XIX
have very complex atomic structure and large number of radiative transitions
that need to be accounted for. Large-scale calculations are necessary to
compute accurate opacities, and detailed calculations for these three ions are
reported in RMOP2.

In table 3 we present a sample of the lowest and highest levels for the Fe
xviii that has the highest ionization fraction of all Fe ions at BCZ
conditions (table 2). As described in paper RMOP2, RMOP calculations for Fe
xviii yield 1,174 bound levels, and a total of 1,604 levels including high-
lying levels with $n$ $>$ 4 described in paper RMOP4 calculations to test
convergence and completeness. The MHD-EOS parameters given in table 3
demonstrate the typical distribution of occupation probabilities and level
populations across the bound-level spectrum of complex Fe ions. Very high-
lying levels make insignificant contribution to opacity calculations. Previous
works have discussed the inexplicable differences of orders of magnitude in
occupation probabilities between OP and OPAL [32]. The EOS issue therefore
remains open for future study.

Table 1: Solar opacity parameters derived from [20] using elemental abundances from [12, 13] (numbers in parenthesis are powers of 10), with the exception of the central temperature at $r/R_{\odot}$ = 0 from several other sources. The boundary of the radiative zone and base of the convection zone(BCZ) is accurately determined from helioseismology to be 0.713$\pm$0.001 [14, 16]. $r/R_{\odot}$ |  $\rho$(g/cc) | T(K) | $N_{e}(cm^{-3})$
---|---|---|---
0.00 | 162.2 | 1.58(7) | 1.0(26)
0.35 | 6.89 | 5.75(6) | 3.57(24)
0.40 | 3.88 | 5.01(6) | 2.01(24)
0.45 | 2.29 | 4.47(6) | 1.19(24)
0.50 | 1.31 | 3.89(6) | 6.82(23)
0.55 | 0.82 | 3.47(6) | 4.25(23)
0.60 | 0.51 | 3.09(6) | 2.67(23)
0.65 | 0.33 | 2.69(6) | 1.71(23)
0.71 | 0.20 | 2.24(6) | 1.02(23)
Table 2: Main solar BCZ atomic opacity contributing elements and ionization states and fractions $>$0.03, at T = $2.24\times 10^{6}$K and $N_{e}=10^{23}cm^{-3}$, obtained from [19] using the Q-form of the MHD-EOS [7, 8]. The actual elemental opacity contributions depend significantly on the theoretical model employed with respect to solar abundances and the EOS [23]. Element | Ionization state (fraction)
---|---
Oxygen | O VII (0.11), O VIII (0.47), O IX (0.42)
Neon | Ne VIII (0.10), Ne IX (0.51), Ne X (0.35)
Iron | Fe XVI (0.031), Fe XVII (0.196), Fe XVIII (0.372), Fe XIX (0.284), Fe XX (0.098)

###  2.2 LTE equation-of-state

Stellar interiors are generally assumed to be characterized by a local
temperature-density (TD) parameter in LTE at any given point in the star.
However, TD tracks vary by orders of magnitude as nuclear energy produced in
the core is transported through the radiative diffusion zone and the
materially convective zones up to the atmosphere where radiation escapes. A
realistic EOS must therefore account for atomic-plasma effects all throughout.
In the first paper on OP opacities ([2], hereafter SYMP), the authors defined
”stellar envelopes to be regions where atoms are not markedly perturbed by the
plasma environment”; the stellar envelope generally comprising of radiative
and convection zones.

The MHD-EOS is a modified version of the Saha-Boltzmann equations, based on
the concept of occupation probability $w$ of an atomic level being populated,
taking into account perturbations of energy levels by the plasma environment,

$N_{ij}=\frac{N_{j}g_{ij}w_{ij}e^{-E_{ij}/kT}}{U_{j}}.$ (5)

The $w_{ij}$ are the occupation probabilities of levels $i$ in ionization
state $j$. The occupation probabilities do not have a sharp cut-off, but
approach zero for high-$n$ as they are ”dissolved” due to plasma interactions.
The partition function is re-defined as

$U_{j}=\sum_{i}g_{ij}w_{ij}e^{(-E_{ij}/kT)}.$ (6)

$E_{ij}$ is the excitation energy of level $i$, $g_{ij}$ its statistical
weight, and $T$ the temperature. The $w_{ij}$ are determined upon free-energy
minimization in the plasma at a given temperature-density. An atomic level $i$
is considered dissolved by the plasma microfield when its highest Stark sub-
level overlaps with the lowest sub-level of the $i+1$ level (discussed further
in RMOP3).

The original version of MHD-EOS estimated the range of validity to $\rho<$
)0.02 g/cc. That rather restrictive density limit is less than prevalent at
the BCZ (c.f. Table 1), and most of the solar interior. The later version
called the Q-MHD [8] has been employed in all present calculations. However,
the EOS employed by OPAL differs considerably from OP; these differences and
approximations made in the OP work, have been previously discussed,
particularly for H-like ions for which data have been available [3, 32].
Nevertheless, the agreement between OP and OPAL to $<5$% [33] seems to
indicate that the differences in EOS do not affect final results. But the EOS
redistributes level populations significantly; further work on improving the
MHD-EOS is in progress.

Table 3: MHD Equation-of-state parameters for Fe xviii at solar BCZ: $T=2\times 10^{6}K,\ N_{e}=10^{23}/cc$. Out of 1604 bound levels calculated, the lowest six levels, energies, occupation probabilities W(OP), and percentage level populations are given. The highest bound levels approaching the first ionization threshold E $\rightarrow$ 0 are also given. The rapid decrease in W and N(%pop) by orders of magnitude is evident. Notation: 4.06(-5) = 4.06 $\times 10^{-5}$. Level | Energy (Ry) | W(OP) | N(% pop)
---|---|---|---
$1s^{2}2s^{2}2p^{5}(^{2}P^{o}_{3/2})$ | -99.924 | 1.00 | 8.79
$1s^{2}2s^{2}2p^{5}(^{2}P^{o}_{1/2})$ | -99.010 | 1.00 | 4.09
$1s^{2}2s^{2}2s(^{1}S_{0})$ | -90.156 | 1.00 | 2.03
$1s^{2}2s^{2}2p^{4}3s(^{4}P_{5/2})$ | -43.203 | 0.99 | 0.15
$1s^{2}2s^{2}2p^{4}3s(^{4}P_{3/2})$ | -42.957 | 0.99 | 0.05
$1s^{2}2s^{2}2p^{4}3s(^{4}P_{1/2})$ | -42.477 | 0.99 | 0.05
Highest levels $n>4$ | -0.500 | 0.56 | 4.06(-5)
Non-hydrogenic | -0.343 | 0.01 | 1.10(-7)

##  3 R-matrix opacity calculations

In this section we describe in some detail the differences from the OP work
mentioned previously. In addition, a description of the revised RMOP codes,
related extensions, and the new set of opacity codes is described.

###  3.1 Convergent close coupling calculations

Owing to the fact that the OP R-matrix calculations included only a few LS
terms of the target or core ion, the (e + ion) wavefunction expansions were
far from convergence of computed quantities and completeness. In the present
work, particularly for iron ions reported in RMOP2, an effort is made to
ensure convergence of photoionization calculations in the close coupling (CC)
approximation using the R-matrix method as developed in the OP [3], and later
in the Iron Project (IP) [10]. In the CC approximation, the atomic system is
represented as the ’target’ or the ’core’ ion of N-electrons interacting with
the (N+1)th electron. The (N+1)th electron may be bound in the electron-ion
system, or in the electron-ion continuum depending on its energy to be
negative or positive. The total wavefunction, $\Psi_{E}$, of the
(N+1)-electron system in a symmetry $SL\pi$ or $J\pi$ is an expansion over the
eigenfunctions of the target ion, $\chi_{i}$ in specific state
$S_{i}L_{i}(J_{i})\pi_{i}$, coupled with the (N+1)th electron function,
$\theta_{i}$:

$\Psi_{E}(e+ion)=A\sum_{i}\chi_{i}(ion)\theta_{i}+\sum_{j}c_{j}\Phi_{j},$ (7)

where the sum is over the ground and excited states of the target or the core
ion. The (N+1)th electron with kinetic energy $k_{i}^{2}$ corresponds to a
channel labeled $S_{i}L_{i}(J_{i})\pi_{i}k_{i}^{2}\ell_{i}(SL(J)\pi)$. The
$\Phi_{j}$s are bound channel functions of the (N+1)-electron system that
account for short range correlation not considered in the first term and the
orthogonality between the continuum and the bound electron orbitals of the
target.

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

$H_{N+1}\mit\Psi_{E}=E\mit\Psi_{E}$ (8)

introduces a set of coupled equations that are solved using the R-matrix
method. The solution is a continuum wavefunction $\Psi_{F}$ for an electron
with positive energies (E $>$ 0), or a bound state $\Psi_{B}$ at a negative
total energy (E $\leq$ 0). The complex resonance structures in photoionization
cross sections result from channel couplings between the continuum channels
that are open ($k_{i}^{2}~{}>$ 0), and ones that are closed ($k_{i}^{2}~{}<$
0). Resonances occur at electron energies $k_{i}^{2}$ corresponding to
autoionizing states belonging to Rydberg series, $S_{i}L_{i}\pi_{i}\nu\ell$
where $\nu$ is the effective quantum number, converging on to the target
threshold $S_{i}L_{I}$.

Convergence of the (e + ion) expansion in Eq. 1 is a difficult computational
problem in CC calculations, since the numerical size of the Hamiltonian
increases as square of the total number of channels in both the first and the
second sum on the RHS. For example, the calculations reported in RMOP2 there
are hundreds of target levels for each iron ion and thousands of corresponding
channels.

###  3.2 Relativistic effects and BPRM codes

The limited OP R-matrix calculations did not consider fine structure. However,
subsequent IP work employed the BPRM framework [10, 11] including fine
structure target levels and recoupling scheme $LS\rightarrow LSJ$. The
relativistic BPRM Hamiltonian is given by

$H_{N+1}^{\rm
BP}=\sum_{i=1}^{N+1}\left\\{-\nabla_{i}^{2}-\frac{2Z}{r_{i}}+\sum_{j>i}^{N+1}\frac{2}{r_{ij}}\right\\}+H_{N+1}^{\rm
mass}+H_{N+1}^{\rm Dar}+H_{N+1}^{\rm so}.$ (9)

where the last three terms are relativistic corrections:

$\begin{array}[]{l}{\rm the~{}mass~{}correction~{}term},~{}H^{\rm
mass}=-{\alpha^{2}\over 4}\sum_{i}{p_{i}^{4}},\\\ {\rm
the~{}Darwin~{}term},~{}H^{\rm Dar}={Z\alpha^{2}\over
4}\sum_{i}{\nabla^{2}({1\over r_{i}})},\\\ {\rm the~{}spin-
orbit~{}interaction~{}term},~{}H^{\rm so}=Z\alpha^{2}\sum_{i}{1\over
r_{i}^{3}}{\bf l_{i}.s_{i}},\end{array}$ (10)

respectively.

The BPRM codes used for the present opacity calculations are shown in Fig. 1
which is modified from the LS coupling version given in ADOC II [5]. The
atomic structure codes Superstructure (SS), CIV3, STG1, STG2, STGH, STGB,
STGF, STGBB and STGBF are described in [5]. Briefly, SS and CIV3 are atomic
structure codes; either one is first employed to obtain reasonably accurate
target wavefunctions, eigenenergies, and oscillator strengths for the target
ion. The target ion orbital radial functions are then used by STG1 to
reconstruct the target and calculate R-matrix basis functions and radial
integrals for the (e + ion) system. With radial integrals from STG1 as input,
STG2 computes angular coefficients and matrices of the Hamiltonian and dipole
operators. The BP recoupling $LS\rightarrow LSJ$ is implemented in the code
RECUPD [11]. STGH diagonalizes the BP Hamilitonian and produces the H and D
files required to obtain physical parameters such as energy levels and
radiative data such as oscillator strengths and photoionization cross
sections.

Figure 1: R-matrix codes for opacity calculations.

Other new codes or extended versions (bracketed by asterisks) comprise of the
following.

###  3.3 Level identification

Energy levels from R-matrix calculations in STGB are obtained as eigenvalues
of bound states without spectral designation, as in atomic structure
calculations. The new code BPID is employed to assign spectroscopic
identification of all computed fine structure levels. Following diagnolization
of the Hamiltonian matrix, the R-matrix basis functions are obtained and used
to compute energy levels in STGB. BPID then analyzes the parameters computed
in STGB to determine spectroscopic identification. Those are the channel
percentage weights and quantum defects, complemented independently by atomic
structure calculations. Level identification is necessary not only for
spectroscopic designations required in practical applications, but also for
matching and high $n\ell(SLJ)$ ”top-up” of computed oscillator strengths from
STGBB and photoionization cross sections from STGBF to test completeness of
atomic data.

###  3.4 Radiation damping

For highly charged ions, in particular H-like and He-like ions of Fe-group
elements, radiative damping of autoionizing resonances is important (e.g.
[35]), and may considered using the extended code STGBF-RD. However, for
opacities calculations this is not needed since total photon absorption cross
section regardless of subsequent radiative decays is required.

###  3.5 Unified (e + ion) recombination

Level-specific and total (e + ion) recombination cross sections may be
computed employing the unified method subsuming both radiative and di-
electronic recombination in ab initio manner within the R-matrix CC
formulation [36], using the code STGRC ([6], and references therein).

###  3.6 Convergence and Completeness

Even when the CC calculations are converged to practically acceptable
accuracy, as discussed in RMOP2, completeness may not have been achieved with
respect to all of the bound-bound and bound-free transitions in an ion. But
R-matrix calculations become computationally intensive with increasing energy
as successive thresholds of the target ion are exceeded and more channels open
up. At the same time computations need to be done at all energies with a
sufficiently fine energy mesh to resolve autoionzation resonance structures.
However, above the highest target level all channels are open and there are no
more resonances. Although the number of open channels may be large, the cross
sections are featureless and slowly varying with energy. Moreover, for high
$n\ell(SLJ)$ levels resonance structure are weak and may be neglected. In such
cases a ”top-up” procedure using DW methods may be employed to test if
convergence has been achieved to ensure completeness of atomic data for
opacities. One such ”top-up” procedure is described in paper RMOP4. Generally,
we find that the ”top-up” contribution to opacities is small and does not
exceed $\sim$5% for any given ion.

###  3.7 Plasma effects

The OP and RMOP calculations are carried out for isolated atomic systems. As
such, external effects due to plasma environment at specific temperature,
density, abundances, etc. need to be considered in opacity calculations. Those
effects determine the EOS as discussed above. But in addition, they alter
computed atomic features such as line shapes of bound-bound atomic transitions
significantly.

In OP work quasi-bound levels that give rise to resonances in the continuum
are treated as bound levels a priori, and plasma broadening of autoionizing
resonances is neglected due to, (i) difficulty in including pressure
broadening, and (ii) because quantum interference between resonances and the
continuum is considered to be small [32]. Therefore, a perturbative approach
in the independent resonance approximation, akin to independent treatment of
radiative and di-electronic recombination, is employed. However, as we
demonstrate in paper RMOP3 in detail, plasma broadening of autoionizing
resonances fundamentally different from that of line broadening of bound
levels. Practically, plasma broadening not only has a significant but large
effect on bound-free opacity and derivative quantities such as the Rosseland
and Planck mean opacities.

###  3.8 Opacities calculations

The opacity codes employed in RMOP calculations have not heretofore been
published, and are different from those in the OP work. In the initial stages
of OP, both sets of codes have been extensively checked against each other for
opacities reported in [2]. However, most of OP data was from sources other
than R-matrix calculations and processed to compute opacities in a different
manner than described herein. Fig. 2 shows the schematic diagram of the codes
and datasets in RMOP calculations (codes bracketed by ’$\ast$’ have not been
heretofore presented).

Figure 2: Plasma opacity codes.

####  3.8.1 Atomic data

The input RMOP data for opacity calculations are the final products from codes
shown in Fig. 1. Each ion is treated as an (e + ion) system characterized by
(Z,N), the atomic number Z and the number of electrons in the target ion.

The input atomic datasets consists of four files: (i) t-file — target level
energies and statistical weights, (ii) e-file — energy levels as computed by
STGB and further processes using BPID, (iii) f-file — oscillator strengths for
E1 transitions computed in STGBB, (iv) p-file — photoionization cross sections
from STGBF. These files are input to the code INTFACE that interfaces the
atomic data, and maps out at a photon frequency mesh of 100,000 frequencies
(in contrast the OP work is at 10,000 frequencies), into bound-bound (bb) and
bound-free (bf) files for opacity calculations separately for each ion. Prior
to input into INTFACE, the p-files are pre-processed by the code PBRO for
plasma broadening of autoionizing resonances in photoionization cross sections
to produce broadened bf-files. PBRO computes plasma broadened cross sections
(described in RMOP3) for each temperature and density. This results in a large
number of pb-files for all TD pairs from a single unbroadened p-file for each
level of each ion of each element in opacities calculations. INTFACE then
processes either the unbroadened p-files or broadened pb-files and produces
corresponding bf-files mapped on to the opacity frequency mesh. Thus, a huge
amount of data is produced as result of the interface of atomic and plasma
parameters, most of it too large to be stored and therefore treated as
intermediate files that are recreated for each TD.

####  3.8.2 Equation-of-state

The MHD-EOS parameters are taken from OPCD codes using the Q-form [7, 8, 3],
so that there are no inconsistencies owing to the EOS between RMOP and OP. The
input EOS parameters consist of: a — abundances of elements, x — ionization
fractions of each ion, and w — EOS data to obtain occupation probabilities.
The opacity code OPAC computes monochromatic, Planck and Rosseland mean
opacities, $\kappa_{\nu},\kappa_{P},\kappa_{R}$ respectively, using the
INTFACE bb and bf files, and EOS parameters, independently for each T-D, or
ranges thereof. Since one of the primary motivation of the RMOP calculations
is to solve the aforementioned solar abundances problem, different sets of
abundances may be used to ascertain differences among them.

####  3.8.3 Bound and continuum opacities

The most important difference between RMOP and other opacity calculations is
the treatment of bound-bound as distinct from bound-free continuum opacity.
There is a clear division between lines as strictly the transitions among
negative energy bound levels, and autoionizing resonances in the bound-free
continua. Practically, this difference manifests itself in the code OPAC (Fig.
2). The bb-opacity consists of negative energy bound levels only, and
corresponding oscillator strengths, and the bound-free opacity consists of
photoionization cross sections with resonances that are otherwise treated as
lines in DW opacity calculations. The DW calculations may couple lines a
posteori to single-channel feature-less continuum perturbatively, but not in a
fully coupled manner as in RMOP opacities. The combined bb and bf opacity
spectra therefore are quite different in detail, which reflects in the
calculation of mean opacities.

There are additional steps necessary in order to ensure completeness, as
discussed in RMOP4, relating to the division between negative and positive
energy levels and to ensure that there is no double-counting of levels if it
is necessary to include high-$n\ell$ contributions, although they are found to
matter little since high-lying levels have insignificant populations.

The treatment of free-free contribution to plasma broadening, discussed in
RMOP3, is also implemented in OPAC. Large datasets of $f$-values for
transitions among positive energy levels, obtained from atomic structure codes
such as Superstructure or variants, are required to compute this contribution.
Although, small relative to electron impact, Stark and Doppler broadening, it
nevertheless needs to be included for completeness.

##  4 Acknowledgments

This work has been partially supported by grants from the US National Science
Foundation, NASA, and the Department of Energy. Most of the computational work
was carried out at the Ohio Supercomputer Center.

## References

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