# Average Atom Model with Siegert States

C. E. Starrett<EMAIL_ADDRESS>Los Alamos National Laboratory, P.O. Box
1663, Los Alamos, NM 87545, U.S.A.  N. R. Shaffer  Los Alamos National
Laboratory, P.O. Box 1663, Los Alamos, NM 87545, U.S.A.  Laboratory for Laser
Energetics, University of Rochester, 250 East River Road, Rochester, NY 14623,
U.S.A.

###### Abstract

In plasmas, electronic states can be well localized bound states or itinerant
free states, or something in between. In self-consistent treatments of plasma
electronic structure such as the average atom model, all states must be
accurately resolved in order to achieve a converged numerical solution. This
is a challenging numerical and algorithmic problem in large part due to the
continuum of free states which is relatively expensive and difficult to
resolve accurately. Siegert states are an appealing alternative. They form a
complete eigenbasis with a purely discrete spectrum while still being
equivalent to a representation in terms of the usual bound states and free
states. However, many of their properties are unintuitive, and it is not
obvious that they are suitable for self-consistent plasma electronic structure
calculations. Here it is demonstrated that Siegert states can be used to
accurately solve an average atom model and offer advantages over the
traditional finite-difference approach, including a concrete physical picture
of pressure ionization and continuum resonances.

##  I Introduction

Average atom models use finite temperature density functional theory mermin65
to calculate the average electronic structure of one atom in a material. The
aim is to have a fast but reasonably accurate physical model of the electronic
structure, and yield information like the equation of state and average
ionization. There has been a long standing effort to make the numerical
implementation of these models robust yet inexpensive liberman ;
more1985pressure ; blenski95 ; wilson06 ; piron11 ; starrett19 ; scaalp .

The essential step in evaluating an average atom model is the self-consistent
solution of the one-electron Schrödinger (Kohn-Sham) equation in which the
effective potential is a functional of the total electron density. The
electron density contains contributions from both the bound and the free
states. One needs to quickly and accurately search for all the bound states,
no matter how weakly bound vcertik2013dftatom . One also needs to solve for a
large number of continuum states, especially at conditions where sharp
resonances appear in the density of states. These resonances are difficult to
track, since they can appear and disappear over the course of a self-
consistent iteration wilson06 .

Methods of solving the Kohn-Sham equations kohn1965self  in average atoms
typically involve numerical finite-difference integration schemes like Adams-
Bashforth or Runge-Kutta. A largely unexplored alternative is to use basis
functions via spectral methods, where one solves a matrix problem for the
coefficients. The primary problem is that the boundary condition for the
average atom requires that the wavefunctions match the free electron solutions
at the cell boundaries (or beyond, depending on the model). This condition
admits a discrete bound state spectrum and a continuous free electron
spectrum. Thus, to use basis function methods one would need to discretize the
continuum, introducing a physical approximation in doing so peyrusse2006use .

In general, average atom eigenfunctions are found by matching to the physical
free electron solutions, which are combinations of incoming and outgoing
waves, i.e., these are the physical scattering solutions that are familiar to
us. An interesting alternative set of eigenstates is found if one retains the
free electron boundary condition but restricts the eigenfunction to have
outgoing character only. These states were introduced in 1939 by Siegert,
motivated by the search for a formal derivation of the Breit-Wigner formula
siegert1939derivation . These Siegert states, as they are now known, encompass
the usual discrete bound states, but in place of the real-energy continuum of
free states, a discrete set of complex-energy eigenstates is found. These
Siegert States influence physical quantities but are themselves not directly
observable. However, the observable physical states are uniquely related to
the Siegert states via simple summations.

In the decades after Siegert’s paper, it proved difficult to find an efficient
method for solving for the Siegert states, until a basis function method was
ultimately derived tolstikhin97siegert . This method assumes a finite ranged
potential ($V(r)=0$ for $r>a$), and because of this and the finite number of
basis terms, the authors referred to their solutions as Siegert Pseudo-States.
The original papers tolstikhin97siegert ; tolstikhin1998siegert  were
restricted to s-waves only. However, the general solution for all orbital
angular momenta was later found batishchev2007siegert . The method was
demonstrated to work well for obtaining the bound states and scattering
properties of a given potential, but the use of Siegert states in self-
consistent field calculations has to our knowledge not yet been demonstrated.

In this work, we show that Siegert states are indeed well suited to the
solution of average atom models using the numerical scheme laid out in
references tolstikhin97siegert ; tolstikhin1998siegert ; batishchev2007siegert
. We demonstrate that the method works over a wide range of density and
temperature, and a range of materials, and is numerically efficient. We start
by reviewing the main equations and results of the Siegert state (SS)
formalism and how they can be used in average atom models. Then interesting
examples of how pressure ionization gives rise to anti-bound states and
resonances in the continuum are presented. We compare pressures from the model
to another state-of-the-art average atom implementation and finally
demonstrate convergence with respect to the number of basis functions.

While we restrict ourselves here to an application to average atoms, Siegert
states could be much more widely used in the plasma community. Some
applications of interest include evaluation of the electronic structure in
calculating the optical or transport properties of plasmas fontes15 ;
faussurier18 ; white2022charge ; iglesias1996updated ; son14 ; bar1989super ;
souza2014predictions ; hansen2007hybrid ; starrett13 ; perrot1995equation ;
shaffer17 .

##  II Siegert States for an Average Atom Model

###  II.1 Siegert States

We start with the radial Schrödinger equation in Hartree atomic units

$\frac{d^{2}P_{l}}{dr^{2}}+2\left(\epsilon-V^{eff}(r)-\frac{l(l+1)}{2r^{2}}\right)P_{l}=0$
(1)

where the radial solution $P_{l}(r;\epsilon)$ depends on the orbital angular
momentum $l$, electron energy $\epsilon$ and radial distance from the nucleus
$r$. $V^{eff}(r)$ is the effective one-electron potential, and

$V^{eff}(r)=0\mbox{ for }r\geq R$ (2)

For the average atom model, $R$ is the ion-sphere radius.

The physical states are regular at the origin, and behave asymptotically
as111We use an overbar to distinguish the physical states from the Siegert
states.

$\begin{split}\lim_{r\to\infty}\bar{P}_{l}(r;\epsilon)=e^{-ikr}-(-\imath)^{l}S_{l}(k)e^{ikr}\end{split}$
(3)

where $k=\sqrt{(2\epsilon)}$ is the complex momentum, and $S_{l}(k)$ is the
S-matrix. Poles of the S-matrix on the positive imaginary $k$ axis correspond
the the bound states. There are other poles, however, lying in the negative
imaginary $k$ plane. These, as we shall see, correspond to unphysical ‘states’
known as anti-bound, resonant and anti-resonant belchev11flow ;
chilcott21experimental ; newton2013scattering . To find the poles we search
for solutions that behave as outgoing waves only

$\begin{split}\lim_{r\to\infty}P_{l}(r;\epsilon)=e^{ikr}\end{split}$ (4)

This is the is the Siegert boundary condition siegert1939derivation ,
alternatively written as

$\begin{split}\left.\left(\frac{d}{dr}-\imath
k\right)P_{l}(r;\epsilon)\right|_{r\to\infty}=0\end{split}$ (5)

The eigenstates which satisfy this boundary condition, and are regular at the
origin, are known as Siegert (or Gamow-Siegert rosas2008primer ;
gamow1928quantentheorie ) states siegert1939derivation .

To solve for these states in a robust and efficient way proved difficult in
the decades after Siegert’s paper. Quite recently however, an efficient and
robust method has been found tolstikhin97siegert ; tolstikhin1998siegert ;
batishchev2007siegert . The basic idea is that for a limited range effective
potential $V^{eff}(r)$, satisfying the restriction (2), the solutions can be
expanded in a finite basis, reducing the problem to an algebraic form that
includes the boundary conditions. The resulting matrix equation is non-linear
in $k$ and is difficult to solve directly. This is linearized by increasing
the dimension of the Hilbert space so that it can be solved by standard
methods. Instead of the original $N$ dimensional space (for $N$ basis
functions), the dimension increases to $2N+l$, meaning that for a basis of
size $N$, there are $2N+l$ eigenstates $P_{n,l}(r)$ with eigenvalues $k_{n}$.

The Siegert states $P_{n,l}(r)$ (we drop the explicit $\epsilon$ dependence
for notational simplicity) are normalized according to

$\begin{split}\int_{0}^{R}P_{n,l}(r)P_{m,l}(r)dr+\imath\frac{P_{n,l}(R)P_{m,l}(R)}{k_{n}+k_{m}}\\\
\times\left[1+\sum_{p=1}^{l}\frac{z_{lp}}{(\imath k_{n}R+z_{lp})(\imath
k_{m}R+z_{lp})}\right]=\delta_{nm}\end{split}$ (6)

where $z_{lp}$ are the zeros of the reverse Bessel polynomial
batishchev2007siegert . If $P_{m,l}$ and $P_{n,l}$ are bound states of an
isolated atom, then this recovers the usual bound state normalization
condition blenski95

$\begin{split}\int_{0}^{\infty}P_{n,l}(r)P_{m,l}(r)dr=\delta_{nm}\end{split}$
(7)

It is worth noting that equations (6) and (7) are not missing a complex
conjugate on one of the Siegert states; this is a hallmark of Siegert states.

Some general properties of the Siegert states can be stated. If $k_{n}$ is an
eigenvalue, so is $-k_{n}^{*}$. If $\Re{k_{n}}\neq 0$, the complex conjugate
pair correspond to a resonant ($\Re{k_{n}}>0$) and anti-resonant
($\Re{k_{n}}<0$) pair. In this case the resonant and anti-resonant
eigenfunctions are complex conjugate. If $\Re{k_{n}}=0$, then for
$\Im{k_{n}}>0$, the state is a physical bound state, and otherwise
($\Im{K_{n}}<0$) it is an anti-bound state. The corresponding eigenfunctions
are either purely real or purely imaginary.

Crucially, the physical scattering states $\bar{P}_{l}(r,\epsilon)$ can be
expressed as a sum of Siegert states

$\begin{split}\bar{P}_{l}(r)=-\imath
k\frac{G_{l}(r,R;k)}{e_{l}(kr)}\end{split}$ (8)

where $e_{l}$ is a special function closely related to the Hankel function
(see equation (A8) of reference batishchev2007siegert ), and the partial-wave
Green’s function is

$\begin{split}G_{l}(r,r^{\prime};k)=\sum_{n=1}^{2N+l}\frac{P_{n,l}(r)P_{n,l}(r^{\prime})}{k_{n}(k-k_{n})}\end{split}$
(9)

The ability to represent the entire continuum of scattering states from a
discrete set of Siegert states is surprising and very powerful. It is this
property of Siegert states that makes them appealing to use in an average atom
model.

###  II.2 Average Atom Model

Here we give a brief summary of the average atom model and show how the
Siegert states are relevant to it. We consider a nucleus of charge $Z$ at the
center of a sphere whose volume is the average volume per atom in the
material, $V$, with radius $R$. The sphere is charge neutral, and the electron
density inside the sphere is given by the Mermin-Kohn-Sham finite temperature
density functional theory mermin65 ; kohn1965self ; hohenberg

$\begin{split}n_{e}(r)=&\sum_{n,l\in
Bound}f(\epsilon_{n},\mu)\frac{2(2l+1)}{4\pi}\frac{\bar{P}_{l}(r;\epsilon_{n})^{2}}{r^{2}}\\\
&+\int_{0}^{\infty}d\epsilon
f(\epsilon,\mu)\sum_{l=0}^{\infty}\frac{2(2l+1)}{4\pi}\frac{\bar{P}_{l}(r;\epsilon)^{2}}{r^{2}}\end{split}$
(10)

where $f(\epsilon,\mu)$ is the Fermi-Dirac occupation factor and $\mu$ is the
chemical potential determined by enforcing
$Z=4\pi\int_{0}^{R}r^{2}n_{e}(r)dr$. This expression is arrived at by using
the spherical symmetry of the average atom. By solving the Poisson equation
and taking into account the exchange and correlation contribution, the
effective potential is

$\begin{split}V^{eff}(r)=&-\frac{Z}{r}+\frac{4\pi}{r}\int_{0}^{r}\,dr^{\prime}\,{r^{\prime}}^{2}n_{e}(r^{\prime})\\\
&+4\pi\int_{r}^{R}\,dr^{\prime}\,{r^{\prime}}n_{e}(r^{\prime})\\\
&+V^{xc}[n_{e}(r)]-V^{xc}[n_{e}(R)]\end{split}$ (11)

Equations (1), (10), and (11) are solved self-consistently until converged.

However, the electron density can also be found from the Green’s function

$\begin{split}n_{e}(r)=-\frac{1}{\pi}\Im{\int_{-\infty}^{\infty}d\epsilon
f(\epsilon,\mu)\sum_{l=0}^{\infty}\frac{2(2l+1)}{4\pi}\frac{G_{l}(r,r;k)}{r^{2}}}\end{split}$
(12)

Using equation (9), it is clear that the electron density can be evaluated
directly from the Siegert states without ever calculating the physical
scattering states, equation (8). As pointed out in reference starrett15 , is
it numerically convenient to use Cauchy’s integral theorem to rewrite this is

$\begin{split}n_{e}(r)=&\frac{1}{\pi}\Im{\int_{C}d\epsilon
f(\epsilon,\mu)\sum_{l=0}^{\infty}\frac{2(2l+1)}{4\pi}\frac{G_{l}(r,r;k)}{r^{2}}}\\\
&+2T\Re{\sum_{j=1}^{N_{mat}}\sum_{l}\frac{2(2l+1)}{4\pi}\frac{G_{l}(r,r;\tilde{k}_{j})}{r^{2}}}\end{split}$
(13)

where $C$ is a convenient contour, and $N_{mat}$ is the number of Matsubara
poles inside the contour. The Matsubara energies are

$\begin{split}\tilde{\epsilon}_{j}=\mu+\imath\pi(2j-1)T\end{split}$ (14)

where $T$ is the temperature.

It is interesting to note that if the contour is chosen to be an infinite
semi-circle in the upper half energy plane, then the density is simply

$\begin{split}n_{e}(r)=2T\Re{\sum_{i=1}^{\infty}\sum_{l}\frac{2(2l+1)}{4\pi}\frac{G_{l}(r,r;k_{i})}{r^{2}}}\end{split}$
(15)

and no numerical integration is required. This is usually not a practical
formula due to the difficulty in evaluating the Green’s function for the large
imaginary arguments that are required for convergence watrous99green .
However, the simple dependence of the Green’s function on energy (equation
(9)) solves this problem. Nevertheless, for our calculations we have used the
formula (13) with a contour similar to that described in reference starrett15
.

For later use, we note that the density of states (DOS) is given in terms of
the Green’s function by

$\begin{split}\chi(k)=-\frac{1}{\pi}\Im{\int_{0}^{R}dr\sum_{l=0}^{\infty}2(2l+1)G_{l}(r,r;k)}\end{split}$
(16)

and the partial DOS we define as

$\begin{split}\chi_{l}(k)=-\frac{1}{\pi}\Im{\int_{0}^{R}dr2(2l+1)G_{l}(r,r;k)}\end{split}$
(17)

evaluated using Eq. (9).

##  III Results

Figure 1: Siegert eigenvalues and density of states (DOS) for aluminum at
solid density and temperatures for 10, 20 and 30 eV ($l=0$ only). Note that
the resonant and anti-resonant eigenvalues are nearly on top of each other for
the three temperatures. Figure 2: Siegert states and density of states (DOS)
for iron at 1 eV temperature and 8 g/cm3. We show only the eigenvalues for
$l=2$ states, where a resonant state lies close to the real energy axis,
leading to a resonance in the DOS.

Figure 1 shows an example of Siegert eigenvalues for an aluminum plasma at 2.7
g/cm3 and the corresponding density of states (DOS) for temperatures of 10, 20
and 30 eV. We show the eigenvalues in a restricted range of momentum space to
highlight certain features, and show the $l=0$ eigenvalues only. For all three
temperatures the $2s$ bound state is apparent on the positive imaginary axis.
Further, in the lower half-plane we see resonant and anti-resonant pairs that
are only weakly affected by temperature.

As temperature is increased, the average energy of the electrons increases and
the electrons therefore screen the nucleus more weakly, leading to a deepening
of the bound eigenvalues. This is most clearly seen in the 2$s$ bound state
increasing in $\Im k$ with increasing temperature. Likewise, we see that at 30
eV, the 3$s$ state is bound, but that at lower temperatures, it is anti-bound.
In this case, the nuclear potential is more strongly screened at lower
temperature, and the shorter-ranged potential no longer supports a bound 3$s$
state.

An anti-bound state is distinct from the more commonly recognized resonance
state, which occurs for $l\geq 0$ (see later). Figure 1 shows the effect the
anti-bound state has on the DOS. Going from 10 to 20 eV we see that the
appearance of the anti-bound state near the real $k$ axis causes a large peak
in the DOS, which quickly reduces as the 3$s$ anti-bound moves further into
the negative $k$ plane.

Figure 3: Effective potential $V^{eff}(r)$, centripetal term $-l(l+1)/2r^{2}$
and total potential for $l=2$, for iron at 8 g/cm3 and 1 eV temperature. The
energy of the $3d$ resonance state ($\epsilon=\Re{k_{n,l}}^{2}/2$) is marked
on the plot showing how, through quantum tunneling, the resonance state can
decay. Figure 4: Real and imaginary parts of the Siegert eigenvalue $k_{n,l}$
for carbon at a temperature of 10 eV as a function of mass density. The top
panel shows the evolution of the $1s$ eigenvalue, the middle panel shows the
$2s$ and the bottom the $3s$. Figure 5: Real and imaginary parts of the
Siegert eigenvalue $k_{n,l}$ for carbon at a temperature of 10 eV as a
function of mass density. The top panel shows the evolution of the $2p$
eigenvalue, the middle panel shows the $3p$ and the bottom the $4p$. Figure 6:
Excess pressure of carbon plasmas at 10, 100 and 1000 eV temperature. We
compare calculations with the Siegert method to results from the Tartarus
average atom code. The results of the two methods are on top of each other in
the plot, indicating agreement of the methods.

We can understand the behavior of the DOS due to an anti-bound state by
defining the DOS due to one Siegert state as

$\begin{split}\chi_{n,l}(k)=-\frac{2(2l+1)}{\pi}\Im{\frac{1}{k_{n}(k-k_{n})}\int_{0}^{R}drP_{n,l}(r)^{2}}\end{split}$
(18)

For bound (B) or anti-bound (AB) states the the integral term is real, and the
eigenvalue is purely imaginary. Let $k_{n}=\imath y_{n}$, where $y_{n}$ is
real, then

$\begin{split}\chi_{n,l}^{B/AB}(k)=\frac{2(2l+1)}{\pi}\int_{0}^{R}drP_{n,l}(r)^{2}\frac{k}{y_{n}(y_{n}^{2}+k^{2})}\end{split}$
(19)

We can see that the maximum of the DOS due to an anti-bound state occurs at
$k=y_{n}$, with the maximum value being proportional to $1/(2y_{n}^{2})$.
Therefore, for small $y_{n}$, the DOS becomes strongly peaked at $k=y_{n}$.

In figure 2 the eigenvalues for iron at 8 g/cm3 and 1 eV temperature are shown
for $l=2$. For the range of $k$ shown there are no bound states. There is one
anti-bound, and the rest are resonant or anti-resonant states. Of particular
interest is the resonant state that lies near the positive real axis; this
state is the 3$d$ resonance state. We can see this by looking at the DOS in
the bottom panel of the figure. There we show the total DOS and that due to
only the $l=2$ states. The large peak observed is known as a shape resonance,
and physically is due to a potential well created by the effective potential
$V^{eff}(r)$ and the centrifugal term $l(l+1)/2r^{2}$.

Figure 3 shows the resonance state with real part of energy
$\epsilon=\Re{k_{n,l}}^{2}/2$ is trapped in a potential well created by the
sum of $V^{eff}(r)$, which is purely attractive, and the centrifugal term
$l(l+1)/2r^{2}$ which is repulsive. Because the barrier of the potential well
is finite ranged, the lifetime of the resonance state is finite (in contrast
to bound states), and an electron in that state can tunnel out. This quasi-
bound character of the resonance state is what gives rise to the
characteristic shape resonance feature in the DOS. In contrast, anti-bound
states are not trapped in a potential well, so only those that are close in
energy to being trapped lead to such features in the DOS.

One conceptual advantage of Siegert states is that one can directly study the
evolution of bound states as a function of density or temperature, even after
they ionize. This is because upon ionization, bound states simply transform
either into anti-bound or resonant states. Figures 4 and 5 show the evolution
of selected bound states for a 10 eV carbon plasma over six orders of
magnitude in density. For low densities, the eigenvalue $k_{n,l}$ lies on the
imaginary axis. The more deeply bound the state is at the lowest density, the
greater increase in density is needed to pressure ionize it. Pressure
ionization occurs when the imaginary part of $k_{n,l}$ crosses the real axis
and becomes negative. For $p$ states (figure 5) this crossing is immediately
accompanied by the real part of $k_{n,l}$ becoming non-zero and positive (we
show only the resonant state eigenvalue, the anti-resonant state eigenvalue is
just its complex conjugate). For $s$ states, figure 4, the crossing of the
real axis is not accompanied by an immediate increase in the real part. There
is a delay in which the real part remains zero - the bound state becomes and
anti-bound state before splitting into a resonant/anti-resonant pair.

The physical reason for this difference in behaviors between $s$-states and
all other orbital angular momenta states is the centrifugal barrier. The
mathematical reason is explored in reference batishchev2007siegert . The
authors show that for $l\geq 1$ there is a forbidden region of $k$-space (a
‘dead-zone’ in their language) where eigenvalues cannot occur. This dead zone
touches the origin and lies in the lower half-plane. As a result, when a bound
state with $l\geq 1$ ionizes, it cannot continue down the imaginary $k$ axis
and must instead split into a resonant/anti-resonant pair.

As a demonstration of the usefulness of the method for equation of state
calculations, in figure 6 we show the excess pressure of carbon plasma over a
wide range of temperatures and densities. The excess pressure is defined as
the total pressure minus an ideal ion contribution $k_{B}T/V$. In the figure
we compare to results from the Tartarus average atom code, which implements
the same model using a more conventional finite difference scheme starrett19 .
No appreciable difference is observed between the two methods, and the
computational cost is similar for each, typically 1-2 minutes per density and
temperature. It is difficult to fairly make a more precise comparison, as
wall-times depend on convergence parameters such as radial grid size, number
of quadrature points for the energy integral, etc., that we have not
systematically optimized in either case. Nevertheless, an advantage that the
finite-basis Siegert approach has over finite-difference integration schemes
is that a sparser radial grid can be used to achieve comparable accuracy,
which lowers the memory footprint of the code. For reference, we use 300
radial grid points for the spectral Siegert method versus 3000 for the finite-
difference scheme in Tartarus.

A further advantage of the Siegert method is that only one type of solver is
need for all states. For the traditional approach, one might use a inward and
outward integration scheme combined with a search technique for finding the
bound state solutions, an outward integration method and resonance tracking
scheme for the continuum electron, or, as in Tartarus, a different method
again to find the Green’s function. This range of techniques leads to
algorithmic complexity. In the Siegert method, one matrix diagonalization, for
which standard libraries exist, gives the entire eigenspectrum.

Figure 7: Convergence of the excess pressure with respect to the size of the
basis $N$ for a carbon plasma at 10 eV temperature. Errors in the excess
pressure are calculated relative to $N=150$.

In figures 7 and 8 we show pressures from carbon and copper plasmas as a
function of the size of the basis $N$. In each case we show the ratio of the
pressure to the converged result. Increasing the size of the basis has two
distinct effects. On the one hand it allows more complicated radial functions
to be accurately represented. On the other hand, it increases the maximum
$|k_{n,l}|$ that is accurately represented222Note that we have used the usual
average atom acceleration scheme in which it is the difference of the
continuum electron density and that of truly free electrons that is important.
See section 3.3 of reference starrett19 .. It is remarkable that for carbon,
as few as 50 basis functions give errors less than 1% for a wide range of mass
densities and temperatures, and errors very much less than 1% are found for
$N=125$. For reference we used $N=150$ for all the results presented up to
this point.

For copper, figure 8, a larger basis is needed to maintain a similar accuracy.
This is expected, as more highly charged nuclei require accurate
representation of higher energy states. This is also why the figure shows that
a larger basis is needed at lower densities. Lower densities for the same
temperature correspond to a less degenerate system, meaning that the tail of
the Fermi-Dirac distribution will be longer. Presumably this does not show up
for carbon due to the error being dominated by inadequate representation of
the radial functions for $N=25$.

Figure 8: As for figure 7 but for copper plasmas at 10 eV temperature, and
errors have been calculated relative to $N=200$.

##  IV Conclusions

Siegert states have been shown to be a viable and advantageous method for use
in average atom models. It has been demonstrated that the Siegert states can
be used to accurately construct the Green’s function and in turn the electron
density needed to perform self-consistent field calculations. We investigated
two pressure ionization scenarios; that in which an $s$-wave state is ionized
to become an anti-bound and subsequently a resonant and anti-resonant pair,
and that for non-zero orbital angular momenta, in which pressure ionized bound
states directly turn into resonant and anti-resonant states. We show how both
scenarios lead to spikes in the density of states. It was also demonstrated
that the method leads to accurate pressures when compared to other state-of-
the-art average atom methods.

The Siegert state approach results in a completely discrete set of eigenstates
that can be used to represent the Green’s function and the scattering states
without approximation. The advantage of a discrete set of states could be
further exploited through their use in perturbation theory more1985pressure
or expanding time-dependent wave packets santra2005siegert , among other
applications requiring a complete set of states where it would be impractical
to use the usual continuum wave functions.

## Acknowledgments

We thank D. Saumon for useful discussions on the early part of this work. This
work was support by LANL’s ASC PEM Atomic Physics Project. LANL is operated by
Triad National Security, LLC, for the National Nuclear Security Administration
of the U.S. Department of Energy under Contract No. 89233218NCA000001. N.R.S.
acknowledges support by the Department of Energy National Nuclear Security
Administration under Award Number DE-NA0003856, and the New York State Energy
Research and Development Authority.

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