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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
$0.877N$ correspond to positive ions,
with $ZZ{\rm _c}$ 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 $01$ 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