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\documentstyle[twocolumn,aps]{revtex}
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\begin{document}
\draft
\title{Resonance States of Atomic Anions}
\author{Alexei V. Sergeev and Sabre Kais}
\address{Department of Chemistry, Purdue University, 1393 Brown Building,
West Lafayette, IN 47907}
\date{\today}
\maketitle
\begin{abstract}

Two methods are proposed to treat resonance states of an atom with
a nuclear charge less than a ''critical'' value, which is the
minimum charge necessary to bind $N$ electrons. The first method
represents a reformulated variational approach in order to
consider resonance and bound states on an equal footing. The
second method represents an extrapolating scheme which is based on
a one-particle model. The energy of a two-electron atom was found
in the entire range $0\leq Z<\infty $. In the region
$0.877<Z<0.901$ near the critical charge $Z_{\rm c}\approx 0.911$
our results agree with the numerical calculations of the complex
energy by Dubau and Ivanov. Using the one-particle model, we
estimated resonance energies of doubly charged negative ions of
atoms with $Z\leq 18$. Results for ${\rm O}^{--}$ and ${\rm
S}^{--}$ are in good agreement with earlier estimates.

\end{abstract}
\pacs{31.15.-p, 31.15.Pf, 32.10.Hq, 02.60.Gf}
%% http://www.th.physik.uni-frankfurt.de/~cbest/pacs.numbers.00, http://www.physica.org/pacs9830.html
%%    31.15.-p       Calculations and mathematical techniques in atomic and molecular physics (excluding electron correlation calculations) (for computational techniques, see 02.70)
%%    31.15.Pf       Variational techniques
%%    32.10.Hq       Ionization potentials, electron affinities
%%    02.30.Mv Approximations and expansions
%%    02.60.-x Numerical approximation and analysis
%%    02.60.Gf Algorithms for functional approximation
%%    03.65.-w Quantum theory; quantum mechanics
%%    03.65.Db Functional analytical methods
%%    03.65.Ge Solutions of wave equations: bound states

\narrowtext

\section{Introduction}

There has long been an interest in the existence of long lived
doubly charged negative atomic ions\cite{buc,bates,scheller}. The
possibility of doubly charged negative ion resonances has been
raised in the case of oxygen by experiments of Peart et al.
\cite{peart} who observed resonance like structures in
electron-impact detachment cross section at energies of 19.5 and
26.5 eV. A Hartree Fock calculation of the closed-shell electronic
configuration shows that the resonance energy of ${\rm O}^{--}$ is
about 8 eV above ${\rm O}^-$, which later was confirmed and
modified by configuration interaction and other
methods\cite{bates}. More recently, Sommerfeld et al.\cite{som}
performed a large scale multireference configuration interaction
calculation using the complex rotation technique to investigate
resonance states of ${\rm H}^{--}$. Their results predict the
existence of a $(2p^3)^4S$ resonance state of ${\rm H}^{--}$ with
a resonance position of about $1.4$ eV above the $(2p^2)^3P$ state
of ${\rm H}^{-}$ and a lifetime of $3.8\cdot
10^{-16}$~sec\cite{sommerfeld}. Until now, to the best of our
knowledge, all observation of long lived resonances of doubly
negative ions are associated exclusively with excited electronic
configurations.

Considering resonance energy as an analytic continuation of bound
state energy into an instability region, with the change of some
parameters of the system, allows one to use essentially the same
computational methods for both bound and the resonance states. For
example, Rayleigh-Schr\"odinger perturbation theory originally
designed to approximate bound states has been successfully used
for resonance states together with methods of analytic
continuation such as conformal mapping \cite{vai} and quadratic
Pad\'e approximants \cite{sershe}. Applicability of variational
methods was effectively extended to unstable quasistationary
states by the method of complex coordinate rotation
\cite{nimrod,ho}.

In this paper, we consider an atom in its ground electronic
configuration state with the nuclear charge $Z$ as a variable.
Integer values of $Z$ with $Z>N$ correspond to  positive ions,
with $Z<N$ to negative ions and with $Z=N$ to neutral atoms, where
$N$ is the number of electrons.

In Sec.\ \ref{sec:two}, we consider the ground state of the
simplest two-electron system as a function of the nuclear charge.
Here, we use precise variational calculations to determine the
energy in the entire range of positive $Z$ and to analyze its
singularities. Exact results for this system are compared with
approximation that is used later in Sec.\ \ref{sec:man} for
multi-electron atoms.

In Sec.\ \ref{sec:man}, we find an approximate dependence of the energy on
the nuclear charge by extrapolation from the known
energies of a neutral atom and the corresponding negative ion.
Energies and widths of doubly negative ions for isoelectronic series
exhibiting a bound singly charged anion are estimated
systematically for $N\le 18$. Results for ${\rm O}^{--}$ and ${\rm
S}^{--}$ are in good agreement with a number of earlier theoretical
predictions. Estimated positions and widths can be used to search for
these resonances experimentally.


\section{Two-electron atoms}
\protect \label{sec:two}

The variational method gives accurate results for bound states of He and ${\rm H}%
^{-}$, but it fails when the nuclear charge is less than the critical value $%
Z_{{\rm c}}\approx 0.911\,028\,225$ when the state turns to a
resonance. In
order to understand why the variational method becomes inadequate when $Z<Z_{%
{\rm c}}$, let us perform the following  numerical test. We use
the Hylleraas trial wave function of the form
\begin{eqnarray}
\psi_N=\sum_{i+j+k\leq N}C_{i,j,k}[ r_1^ir_2^j\exp
(-ar_1-br_2)\nonumber \\ \label{hylfun}+r_2^ir_1^j\exp
(-ar_2-br_1)] r_{12}^k\exp (-cr_{12})
\end{eqnarray}
Minimization of the energy functional with  respect to the
coefficients $C_{i,j,k}$ reduces the calculation to an eigenvalue
problem for a finite matrix whose size increases as $\sim N^3/6$.
We found several lowest eigenvalues for $N=20$ with $a=0.1$,
$b=0.8$, and $c=0$.

The ionization energy $E_{{\rm I}}=-E-Z^2/2$ as a function of the
nuclear charge is shown in Fig.~\ref{helen} (solid lines). If
$Z>Z{\rm _c}$ then the lowest level corresponding to maximum
ionization energy, the upper curve in Fig.~\ref{helen}, gives the
bound state energy. Fig.~\ref{helen} shows that the upper curve
rapidly bends to zero after going below $Z_c$. This means that the
variational method gives a trivial result $E_{{\rm I}}=0$ when the
bound state ceases to exist. However, all the curves corresponding
both to the minimum and to higher eigenvalues  exhibit a typical
avoided-crossing ladder pattern of proliferation of the bound
state into continuum as a resonance. This situation which is
similar to the two-electron problem in finite space\cite{sti} is
the result of using of variational functions satisfying the
boundary conditions of a bound state but not the boundary
conditions of a resonance.

In order to calculate the resonance by variational method without
encountering avoided-crossings, we make the boundary conditions
more flexible by introducing a complex trial function. Until now,
we considered the exponential parameters $a$, $b$, and $c$ as real
numbers independent of $Z$, and minimized the energy functional
with respect to the linear coefficients $C_{i,j,k}$.
Alternatively, we can minimize the energy with respect to both the
linear and the non-linear parameters. In this way, we found that
the optimized parameters $a$, $b$, and $c$ are real for
sufficiently large charges. If the charge is lower than some value
(see Table \ref{cricha}), then the minimum of the energy
functional no longer exists. This  situation is different from
minimizing over the linear parameters only when the minimum of the
energy functional {\it always} exists because a real symmetric
matrix always has a minimum real eigenvalue. An analytic
continuation of a minimum of some function, when this minimum
ceases to exist, represents a complex stationary point. We found
numerically the parameters $a$, $b$, and $c$ as complex stationary
points in the range $0\leq Z\leq 1$ with up to $N=5$. The result
for $N=5$ is shown in Fig.~\ref{helen}. The real part is a dashed
line, and the imaginary part of the ionization energy is a
dot-dashed line. By allowing the parameters of the trial function
to be complex-valued, we eliminated the avoided-crossings and made
the results to converge with increase of $N$. It is interesting
that the traditional variational method, with real parameters $a$,
$b$, and $c$, gives very accurate results at the inflection
points, between adjacent avoided-crossings (see Fig.~\ref{helen}),
but it never reproduces the imaginary part  of the resonance.


Let us consider the variational results from the point of view of
analytic structure of the energy as a function of the nuclear
charge. If the exponential parameters $a$, $b$, and $c$ are real,
then the energy (shown by solid lines in Fig.~\ref{helen}) is real
and does not have singularities at the real axis. However, for
sufficiently small charge there is a pair of complex conjugate
square root branch points close to the real axis joining each
branch of the energy function (shown as a continuous solid curve
 in Fig.~\ref{helen}) with the neighbor branch (the nearest curve
that lies above or below). In contrast, if the exponential
parameters $a$, $b$, and $c$ are allowed to have complex values
then the variational energy (shown by dashed and dot-dashed
curves) has a single singularity at the real axis at the point
where the minimum of the energy functional disappears and turns to
a complex stationary point. This singularity models a singularity
of the exact energy at the ''critical'' charge where the system
goes from a bound state to a quasistationary state. Positions of
this singularity $Z_{*}^{(N)}$ for different $N$
are listed in Table \ref{cricha}.

The numerical evidence is that most of the variational
singularities $Z_{*}^{(N)}$ give lower bounds for the critical
charge $Z_{{\rm c}}\approx 0.911\,028$ and converge with the
increase of $N$ although the convergence is not monotonous. Table
\ref{cricha} lists also variational ''critical'' charges $Z_{{\rm
c}}^{(N)}$ defined as the zeroes of the ionization energy
$-E^{(N)}(Z)-Z^2/2$. The ''critical'' charges $Z_{{\rm c}}^{(N)}$
could be calculated by solving a generalized eigenvalue problem by
a variational method \cite{ser}, they
always give upper bounds for $Z{\rm _c}$. We found that convergence of $Z_{%
{\rm c}}^{(N)}$ to the critical charge is much faster than that of $%
Z_{*}^{(N)}$. By extending variational calculations of $Z_{{\rm
c}}^{(N)}$ to higher $N$, the most accurate estimation of the
critical charge was found earlier \cite{ser}. Calculations of
$Z_{*}^{(N)}$ are generally more difficult than that of $Z_{{\rm
c}}^{(N)}$ because they represent a
singularity. They converge to the singularity $Z_{*}$ of the function $E(Z)$%
, which is believed to limit the radius of convergence of the
$1/Z$ expansion to $1/Z_{*}$. According to an
earlier hypothesis based on the analysis of the $1/Z$ perturbation
series \cite{sti}, $Z_{*}$ is slightly smaller than $Z_c$ (see
Table \ref{cricha}) which means that $E(Z_{*})$ lies above the
continuum, but still corresponds to a localized wave function. More
elaborate computations of the $1/Z$ series and its analysis by
Baker et al. \cite{bak} show that $Z_{*}$ and $Z_{{\rm c}}$ are
equal.

We used the complex parameters $a$, $b$, and $c$ calculated for the particular case of $%
N=5$ in order to extend calculations to higher $N$ by optimizing
only the linear coefficients $C_{i,j,k}$. We found that ''almost
exact'' variational energy calculated at $N=25$ differs from the
variational energy at $N=5$ shown in Fig.~\ref{helen} in the
amount of less than $0.5\cdot 10^{-4}$. Calculations show that the
behavior of the parameters $a$, $b$, and $c$ which are a
stationary point of the energy functional is more unpredictable
than that of the energy. Dependence of $a$, $b$, and $c$ on $N$ at
$Z=Z_{{\rm c}}$ is shown in Fig.~\ref{abcn}. It seems that the
parameters  oscillate as $N$ increases. In our previous paper
\cite{ser}, we used near-average parameters $a/Z=0.35$,
$b/Z=1.03$, and $c/Z=0.03$ shown by dashed lines on
Fig.~\ref{abcn} to perform large-$N$ calculations of $Z_{{\rm
c}}$.

Dependence of $a$, $b$, and $c$ on $Z$ for $N=5$ is shown in
Fig.~\ref{abcz}. The parameters are continuous functions of $Z$
with a square root singularity at $Z=Z_{*}$, below which they
become complex-valued. Numerical results show many erratic swerves
on the curves, this fact probably indicates the existence of many
singularities close to the real axis.

Most of the above features are typical for any system passing from
a bound to a quasistationary state that is treated variationally,
for example for Ne isoelectronic series with a nuclear charge
below $Z_{*}=8.74$ \cite{her}.

The above method is a more general version of the complex rotation
or the complex stabilization method \cite{ho}. Instead of one
non-linear complex variational parameter, the rotation angle, we
are using three non-linear variational parameters $a$, $b$, and
$c$.

Dubau and Ivanov \cite{dub} calculated the two-electron atom
resonance in the vicinity of the critical charge using $1/Z$
expansion and the complex rotation method. Their results agree
with our calculation, see Table \ref{reshel}.

We extended calculations of the resonance to the range of $0\leq
Z\leq 1$. Results are shown in Fig.~\ref{helres}. The real part of
the ionization energy is always negative at $0<Z<Z_{{\rm c}}$. It
reaches its minimum at $Z\approx 0.45$, and the width reaches its
maximum approximately at the same point. For small $Z$,
convergence of the variational method becomes worse. It seems that
both real and imaginary parts tend to zero when $Z\rightarrow 0$.

Our results for small $Z$ apparently contradict that of Herrick
and Stillinger \cite{her}. They found that a small negative charge
corresponds after suitable scaling to a tightly bound
''di-electron'' moving in a Coulomb field of the nucleus. The
behavior of $E(Z)$ as $Z\rightarrow -0$ has the form

\begin{equation}
\label{stiexp}E(Z)=-1/4-4Z^2+256Z^4+O(Z^5)
\end{equation}

Also by considering the simplest
trial function

\begin{equation}
\label{stitri}\phi _{{\rm var}}({\bf r}_1,{\bf r}_2)=\exp \left[ -\alpha
(r_1+r_2)-\beta r_{12}\right]
\end{equation}
 they found that $E_{{\rm var}}(Z)=-1/4-4Z^2+448Z^4+...$ at $%
Z\rightarrow 0$ in a good agreement with (\ref{stiexp}).

We re-examined the variational calculations with the trial function
(\ref{stitri}) and found that Herrick and Stillinger actually dealt with
two different branches of the variational energy which are
displayed in Fig.~\ref{helsti}. The first branch, the lower curve in
Fig.~\ref{helsti}, corresponds to a physical wave function with a
positive exponential parameter $\alpha $ at sufficiently large
$Z$. It goes almost to zero as the charge decreases and becomes
complex below $Z_{*}=0.084\,6$. The second branch corresponds to
a divergent wave function with a negative $\alpha $. It goes to
$-1/4$ as the
charge decreases and then turns to a physical branch with $\alpha >1$ at negative $%
Z $. The second branch is also present in our calculations with
the trial function (\ref{hylfun}), but we always disregard it.
According to our numerical results, the energy goes to zero at
$Z\rightarrow 0$ (see Fig.~\ref{helres}).

% It seems that our results don't support the hypothesis of
% existence of one more singular point at $Z\sim 0.1$ below which
% the energy becomes real. This hypothesis was derived by Stillinger
% \cite{sti} and later confirmed
% by Dubau and Ivanov \cite{dub}.


\section{Many-electron atoms}
\protect \label{sec:man}

Applying the complex rotation method to systems of more than three
charged particles faces slow convergence because of the difficulty to
simulate the oscillatory character of the wave functions
\cite{ho}.

The present study deals with the ground state ionization energy of
a multi-electron atom considered as a function of a nuclear
charge. Since  the size of the variational basis set grows
exponentially with the increase of  the number of electrons, we
choose here to follow a simpler path. We use the reliable data for
the ionization energy of a negative ion and a neutral atom, which
were calculated or experimentally measured. We are going to use
here an extrapolating technique in order to find a complex energy
of a doubly charged negative ion. In contrast to simple
extrapolating such as polynomial fits or analytic formulas with a
few fitting parameters \cite{eld}, we are solving here a
one-particle Schr\"odinger equation with a potential that models
the  movement of a loosely bound valence electron that is going to
dissociate when the charge approaches its critical value. This
model is realistic in the vicinity of the critical charge and
effectively reproduces the non-trivial singularity \cite{iva} of
the ionization energy at the critical charge. Herric and
Stillinger \cite{her} used for Ne isoelectronic series a
polynomial fitting formula plus a singular term $\sim
(Z-Z_{*})^{3/2}$. Their
method correctly reproduces a similar singularity of a variational energy $%
\sim (Z-Z_{*}^{(N)})^{3/2}$, but it fails for the exact energy,
which has a less trivial singularity as it was established by
Dubau and Ivanov \cite{dub}.

\subsection{Description of the one-particle model}
\protect \label{sec:des}

For a given atom with $N$ electrons and a nuclear charge $Z$, let
us consider a spherically-symmetric potential (also known as
Hellmann potential \cite{hel}) of the form

\begin{equation}
\label{helpot}V(r)=-\frac 1r+\frac \gamma r\left( 1-e^{-\delta r}\right)
\end{equation}
with $\gamma =(N-1)/Z$.

Since in the neighborhood of the critical charge, particularly for
the negative hydrogen ion \cite{rau}, one of the electrons is held
much farther from the nucleus than the others, we suggest a
one-particle model of this electron in an effective potential of
the atomic core comprising of the nucleus and $N-1$ electrons. In
scaled coordinates $r\rightarrow Zr$, this potential is
approximated by our model potential, Eq. (\ref{helpot}). Our
approximation is asymptotically correct both at small and at large
distances from the nucleus where the scaled atomic core potential
tends to $-1/r$ and to $-(Z-N+1)/(Zr)$ respectively. The
transition between the two different asymptotic regimes occurs at
distances roughly equal to $1/\delta $ that is about the atomic
core radius.

The second parameter of the model potential, Eq. (\ref{helpot}),
$\delta $, is chosen so that the ionization energy in the
potential (\ref{helpot}) is equal to the scaled ionization energy
$Z^{-2}E{\rm _I}(Z)$ of the atom. Note that for atoms with more
than two electrons, we consider here an excited state in the
potential (\ref{helpot}) with the same spherical quantum numbers
$(n,l)$ as quantum numbers of the loosely bound electron on an
external atomic shell (in this aspect our approach differs from
the method of pseudo-potentials \cite{cal} that deals with the
ground state in a potential with an additional repulsive term
necessary to take into account orthogonality conditions). In this
way, we map an arbitrary atom, which is
characterized by a pair of numbers $(N,Z)$ to the model one-particle system (%
\ref{helpot}), which is characterized by a pair of parameters $(\gamma
,\delta )$. Results of fitting the parameter $\delta $ for elements with $%
N\leq 10$ in our previous study \cite{unp} give evidence that
$\delta $ depends on $1/Z$ almost linearly.

In summary, our model (\ref{helpot}) effectively eliminates the
singularity in the energy function $E_{{\rm I}}(Z) $ to be
extrapolated by replacing it with a weakly varying function
$\delta (Z)$ that can be accurately extrapolated by a linear
dependence on $1/Z$ without taking into account a complex
singularity at $Z=Z_{*}$. In our previous study \cite{unp}, we
fitted the parameter $\delta $ to meet the known binding energy of
the neutral atom and its isoelectronic negative ion and then found
$\delta $ as a function of $1/Z$ by a linear extrapolation.
After that, we solved Schr\"odinger equation with the potential (\ref{helpot}%
) and found some kind of extrapolation of the ionization energy of an atom
to the range of $Z<N-1$. Finally, by locating a zero of the ionization
energy, we found critical charges for most of atoms with $N\leq 86$. In the
present paper, we use the same technique to find resonances of doubly
charged negative ions by calculating the ionization energy at $Z=N-2$.

\subsection{Results}
\protect \label{sec:res}

For two-electron isoelectronic series, the parameter $\delta $ is 1.066 and
0.881 for He and ${\rm H}^{-}$ respectively. Decreasing of $\delta $ for   $%
{\rm H}^{-}$ means increasing of the core radius that is induced
by stronger repulsion between electrons. Results of extrapolating
the ionization energy to the range $Z<1$ with $\delta $ assumed as
a linear function of $1/Z$ are shown in Fig.~\ref{helres} together
with more accurate variational results. As it was expected, the
one-particle model gives fairly accurate results in the vicinity
of the critical charge where it models the threshold singularity.
Here, the real part is approximated more accurately than the
imaginary part because the singularity manifests mostly in the
imaginary part. Numerical results for the imaginary part are given
in Table \ref{reshel}. Since in the latter case the charge is
close to the critical charge, we slightly corrected
location of the singularity in our model by fitting $\delta $ at points $%
Z=Z_{{\rm c}}\approx 0.911$ and $Z=2$ instead of fitting it at
points $Z=1$ and $Z=2$.

For three-electron isoelectronic series, our model cannot be used
in the same way because ${\rm He}^{-}$ does not exist and we don't
know the behavior near the singular point $Z_{{\rm c}}=2$. It is
reasonable to expect that the resonance ${\rm H}^{--}$ is highly
unstable because the isoelectronic ion ${\rm He}^{-}$
is
already unstable. We tried to extrapolate the ionization energy by fitting $%
\delta $ at points $Z=4$ (${\rm Be}^{+}$) and $Z=3$ (Li) with
subsequent extrapolation of $\delta $ to $Z=1$ by a linear
function in $1/Z$ and found a broad resonance with $E{\rm
_I}=-0.109-0.089i$ for  ${\rm H}^{--}$.

For four-electron isoelectronic series, results of using our model
for extrapolating the ionization energy are shown in
Fig.~\ref{berres}. We found that the
curve hits the border of continuum spectrum $E_{{\rm I}}=0$ at $Z_{{\rm c}%
}=2.864$ \cite{unp} and again at $Z_{{\rm c}}^{\prime }=2.17$.
Very small imaginary part exists in the interval $Z_{{\rm
c}}^{\prime }<Z<Z_c$. The
extrapolation gives {\it positive} ionization energy for ${\rm He}^{--}$, $%
E_{{\rm I}}=0.064$. Since a bound state of ${\rm He}^{--}$ does
not exist, it means that we deal with a virtual state with an
exponentially growing wave function. It may be also possible, that
because of its limitations our method fails for ${\rm He}^{--}$
ion. We found positive ionization energies
for another doubly charged negative ions of noble gases, for example ${\rm Ne%
}^{--}$ (see Table \ref{resomm}).

The doubly charged negative ion of oxygen in the same electronic
configuration as a closed-shell 10-electron configuration of a
neutral atom of neon is the system that was studied theoretically
using various methods, see Table \ref{resomm}. However, this
relatively stable ion was never observed experimentally. The
extrapolated ionization energy as a function of $Z$ is shown in
Fig.~\ref{neores}.

Comparison of the ionization energy of  ${\rm O}^{--}$ found by
different methods is given in Table \ref{resomm}. For critical
discussion of some of these results, see \cite{her}. Our result
for resonance position is in agreement with most of the above
results. However, our prediction of the resonance width is
significantly lower than that of Herrick. It may be attributed to
the fact that Herrick's 3/2-power term, which responsible for the width,
is far from the actual singular behavior or to the fact that our
model systematically underestimates the width as it happens for
two-electron atoms.

We extended calculations of resonances of doubly charged negative
ions to the first two rows of the Periodic Table. Here, we
considered only that ions for which an isoelectronic singly
charged negative ion is stable, for example we omitted  ${\rm
C}^{--}$ because  ${\rm N}^{-}$ does not exist. Results are
listed in Table \ref{resion}. For the closed shell ion ${\rm
S}^{--}$, our ionization energy is slightly smaller than the
previous estimates. The width for ${\rm S}^{--}$ is considerably
smaller than for the ion ${\rm O}^{--}$ in agreement with an
argument of Herrick and Stillinger \cite{her}.


\section{Discussion}

We have investigated the existence of resonance states for doubly
charged atomic negative ions using two methods. The first method
is a reformulated variational approach to treat resonance states.
In Sec.\ \ref{sec:two}, we traced  the position of the ground
state energy level of two-electron atoms as the nuclear charge
decreases. When the charge drops below the critical charge, the
level goes into a complex plane and becomes a resonance. With only
minor change of computer program, the Hylleraas-basis variational
technique is used for accurate determination of position and width
of the resonance inside instability region of small charges. These
results are used to check the accuracy of the less accurate
approximation developed in Sec.\ \ref{sec:man} necessary to study
destabilization of the ground state of a multi-electron atom with
decrease of its nuclear charge. The results of our second
approach, the approximate one particle model, agree with  earlier
estimates of closed-shell resonances of ${\rm O}^{--}$ and ${\rm
S}^{--}$. In addition, the model predicts similar resonances of
another di-anions which are isoelectronic to some bound
singly-charged anion.

Our method is in fact some kind of refinements of Herrick and
Stillinger method of analytic continuation \cite{her}. We believe
that our model reproduces the threshold singularity of the energy
more realistically than 3/2-power singularity of Herrick and
Stillinger. Generally, the method of analytic continuation from
bound to resonance states appears very accurate when the threshold
behavior of the energy is incorporated into continuation scheme,
as it was demonstrated earlier for two-particle resonances in
nuclear physics, see Chapter 5 of the book \cite{kuk}.

Although no multiple negative ions exist in a bound
state\cite{kal,hog,unp}, some long-lived resonances of doubly
negative ions are observed experimentally\cite{buc}. Our results
give evidence for some resonances that remain to be observed.

\begin{acknowledgements}

We would like to acknowledge
the financial support of
the Office of Naval Research (N00014-97-1-0192). S.K. acknowledges
support of an NSF Early-Career Award.

\end{acknowledgements}

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\begin{figure}
\caption{Variational results for the energy of the helium isoelectronic
series. Non-linear parameters of the Hylleraas trial function
(\protect\ref{hylfun}) are $a=0.1$, $b=0.8$, and $c=0$. The size
of the basis set corresponds to $N=20$. The restriction on the
summation indexes $i+j^2+k^2\leq N$ is used instead of the more
common restriction $i+j+k\leq N$ in order to decrease the number
of terms in the sum (\protect\ref{hylfun}) from $\sim
\frac{1}{6}N^3$ to $\sim\frac{\pi}{8}N^2$ as it was suggested in
\protect\cite{ser}. These results are shown by solid lines.
Variational results with parameters $a$, $b$, and $c$ determined
as a complex stationary point of the energy functional for $N=5$
are shown by dashed and dot-dashed lines, real and imaginary parts
respectively. \label{helen}}
\end{figure}

\begin{figure}
\caption{Dependence of the optimal variational parameters $a$, $b$,
and $c$ calculated at $Z=Z_{\rm c}$ on $N$.
\label{abcn}}
\end{figure}

\begin{figure}
\caption{Optimal variational parameters $a$, $b$, and $c$ for
$N=5$ as a function of the nuclear charge $Z$.\label{abcz}}
\end{figure}

\begin{figure}
\caption{Ionization energy for the helium isoelectronic series as
a function of the nuclear charge including an instability region
$0\le Z<Z_{{\rm c}}$. Bold lines are accurate results found by the
variational method with a large basis set. Thin lines are results
of the simplified one-particle model described in Sec.\
\ref{sec:des}. Solid and dashed lines are real and imaginary parts
respectively.\label{helres}}
\end{figure}

\begin{figure}
\caption{Ionization energy for the helium isoelectronic series
found using Stillinger's trial function (\protect\ref{stitri}).
The lower curve approximates real ionization energy when $Z\ge
Z_{{\rm c}}$ and resonance energy when $Z< Z_{{\rm c}}$ (for very
small $Z< 0.085$ it has an imaginary part shown by a dashed line).
The upper curve is an unphysical branch of the function which
tends to $1/4$ at $Z=0$ and after continuation to negative $Z$
turns to ionization energy of a hypothetical system with Coulomb
attraction between the particles.\label{helsti}}
\end{figure}

\begin{figure}
\caption{Ionization energy for the beryllium isoelectronic series
extrapolated from bound systems Be and ${\rm Li}^{-}$ to a
resonance state of ${\rm He}^{--}$. A dashed line is an imaginary
part of the energy times 100. The system ${\rm He}^{--}$ with a
zero imaginary part is presumably in a virtual
state.\label{berres}}
\end{figure}

\begin{figure}
\caption{Ionization energy for neon isoelectronic series
extrapolated from bound systems Ne and ${\rm F}^{-}$ to a resonance
state of ${\rm O}^{--}$. A dashed line is an imaginary part of the
energy which is half-width of the level.\label{neores}}
\end{figure}

\narrowtext

\begin{table}
\caption{Singularities and critical charges for the helium
isoelectronic series.\label{cricha}}

\begin{tabular}{ccc}

$N$&$Z_{*}^{(N)}$&$Z_{{\rm c}}^{(N)}$\\ \tableline

0&0.883\,998&0.925\,879\\

1&0.868\,302&0.915\,729\\

2&0.889\,957&0.913\,198\\

3&0.891\,584&0.911\,369\\

4&0.919\,327&0.911\,265\\

5&0.899\,586&0.911\,081\\

6&&0.911\,070\\

&&\\

Stillinger \cite{sti}&0.894\,1&0.911\,2\\

Baker et al. \cite{bak}&0.911\,028&

\end{tabular}
\end{table}

\onecolumn

\mediumtext

\begin{table}

\caption{\label{reshel}Resonances in the helium isoelectronic series
for charges below the critical charge}

\begin{tabular}{ccccc}

$1/Z$&$-Z^{-2}{\rm Re}E$&$-Z^{-2}{\rm Im}E$&$-Z^{-2}{\rm Im}E$
(CRM \cite{dub})&$-Z^{-2}{\rm Im}E$ (model)\\ \tableline

1.11&0.497\,131&0.000\,050&0.000\,06&$\sim$0.000\,03\\

1.12&0.494\,953&0.000\,286&0.000\,28&0.000\,24\\

1.13&0.492\,792&0.000\,686&0.000\,70&0.000\,57\\

1.14&0.490\,616&0.001\,207&0.001\,21&0.000\,98

\end{tabular}
\end{table}

\widetext

\begin{table}
\caption{\label{resomm}Energy of ${\rm O}^{--}$ resonance
calculated by various methods}

\begin{tabular}{ccc}
{\bf Reference}&{\bf Method}&$-E_{\rm I}$, eV\\ \tableline

Clementi and McLean \cite{cle}&Hartree-Fock&8.30\\

Huzinaga and Hart-Davis
\cite{huz}&Hartree-Fock-Roothaan\tablenotemark[1]&7.68\\

Cantor \cite{can}&Combining thermochemical data&7.94\\

Herrick and Stillinger \cite{her}&Extrapolating formula including
3/2 power term&$5.38-0.65i$\\

&Eld\'en's \cite{eld} three-parameter formula&5.31\\

&Kaufman's \cite{kau} two-parameter formula&7.17\\

&Kaufman's \cite{kau} three-parameter formula&6.53\\

&Quadratic fit&7.09\\

Gadzuk and Clark \cite{gad}&Extrapolating of isoelectronic
cases&8.8\\

Robicheaux et al. \cite{rob}&Polynomial fit&7.2\tablenotemark[2]\\

Present paper&One-particle model&$7.65-0.22i$

\end{tabular}
\tablenotetext[1]{Huzinaga and Hart-Davis listed the total energy.
We found ionization energy by subtracting Hartree-Fock total
energy of the corresponding singly charged negative ion that is
listed in papers of Clementi et al. \cite{cle,cle2}}

\tablenotetext[2]{Estimated width is greater than 1 eV.}
\end{table}

\narrowtext

\begin{table}
\caption{\label{resion} Energies\protect \tablenotemark[1] of
doubly charged negative ions\protect \tablenotemark[2]}

\begin{tabular}{ccc}
$Z$&$-E_{\rm I}$, eV&Lifetime, $10^{-14}$ sec\\ \tableline

2 ${\rm He}^{--}$&-1.73&\\

4 ${\rm Be}^{--}$&3.43&1.1\\

5 ${\rm B}^{--}$&4.88&0.3\\

6 ${\rm C}^{--}$&6.11\tablenotemark[3]&\\

7 ${\rm N}^{--}$&7.48&0.12\\

&7.97\tablenotemark[4]&\\

&7.07\tablenotemark[3]&\\

10 ${\rm Ne}^{--}$&-0.92&\\

12 ${\rm Mg}^{--}$&1.79&\\

13 ${\rm Al}^{--}$&3.44&3.0\\

14 ${\rm Si}^{--}$&4.76&0.5\\

&4.12\tablenotemark[3]&\\

15 ${\rm P}^{--}$&4.95&0.5\\

&4.47\tablenotemark[3]&\\

16 ${\rm S}^{--}$&4.91&0.7\\

&4.62\tablenotemark[3]&\\

&4.7\tablenotemark[5]&

\end{tabular}
\tablenotetext[1]{Tabulated energies are results of the present
paper unless marked otherwise.}

\tablenotetext[2]{For energy of ${\rm O}^{--}$ ion, see Table
\ref{resomm}.}

\tablenotetext[3]{Hartree-Fock-Roothaan calculations of Huzinaga
and Hart-Davis\cite{huz}.}

\tablenotetext[4]{Hartree-Fock calculations of Clementi and McLean
\cite{cle}.}

\tablenotetext[5]{Polynomial fit of Robicheaux et al. \cite{rob}.}

\end{table}

\end{document}