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\begin{document}
\title{Variational principle for critical parameters of quantum
systems}[Variational principle for critical parameters]
\author{A V Sergeev and S Kais}
%\ftnote{3}{To
%whom correspondence should be addressed.}}
\address{Purdue University,
Department of Chemistry, 1393 Brown Building, West Lafayette, IN
47907}
\begin{abstract}
Variational principle for eigenvalue problems with a non-identity
weight operator is used to establish upper or lower bounds on
critical parameters of quantum systems. Three problems from atomic
physics are considered as examples. Critical screening parameters
for the exponentially screened Coulomb potential are found using a
trial function with one non-linear variational parameter. The
critical charge for the helium isoelectronic series is found using
a Hylleraas-type trial function. Finally, critical charges for the
same system subject to a magnetic field are found using a product
of two hydrogen-like basis sets.
\end{abstract}
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\pacs{03.65.Ge, 03.65.Db}
%% http://www.th.physik.uni-frankfurt.de/~cbest/pacs.numbers.00
%% 03.65.-w Quantum theory; quantum mechanics
%% 03.65.Db Functional analytical methods
%% 03.65.Ge Solutions of wave equations: bound states
%\maketitle
\section{Introduction}
The variational principle for eigenvalue problems is a well-known
method and widely used in quantum calculations since the foundation of
quantum mechanics. For the eigenvalue equation of the form
\begin{equation}
\label{eig}L(\psi)=\lambda M(\psi),
\end{equation}
where $L$ and $M$ are self-adjoint operators, the variational
principle reads (Morse and Feshbach 1953):
\begin{equation}
\label{var}\delta[\lambda]=\delta\left[\int \psi M(\psi)dV/\int
\psi L(\psi)dV\right]=0.
\end{equation}
For a time-independent Schr\"{o}dinger equation, $M$ and $\lambda$
are usually considered as the identity operator and the energy, in
this case \Eref{var} represents the variational principle for the
energy. Here, we point out the usefulness of the variational
principle with a non-identity "weight" operator $M$ for
calculations of critical parameters of quantum-mechanical systems.
We consider a Hamiltonian that depends on some continuous
parameter $\gamma$. We call the parameter $\gamma=\gamma_c$ {\it
critical} if the energy of the system reaches the ionization
border $E_I$,
\begin{equation}
\label{cri}
H(\gamma_c)\psi=E_I(\gamma_c)\psi.
\end{equation}
If the Hamiltonian and the border of ionization depend on the
parameter $\gamma$ linearly, i.e. $H(\gamma)=H_0 +H_1 \gamma$,
$E_I(\gamma)=E_0 +E_1 \gamma$ then the Schr\"{o}dinger equation
for the critical parameter, \Eref{cri} takes the form of
\Eref{eig} where $L=H_0-E_0$, $M=E_1-H_1$, and $\lambda=\gamma_c$,
in which case the variational principle can be used immediately.
Often, a linear dependence on $\gamma$ should be achieved by an
appropriate scaling transformation before using the variational
principle.
A similar approach was used earlier for critical screening
parameters of the exponentially screened (Hulth\'{e}n and
Laurikainen, 1951) and of the cut-off (Dutt, Singh, and Varshni
1985) Coulomb potentials. Here we present a further demonstration
of the power of the general variational principle, \Eref{var} to
obtain very accurate critical parameters for both simple one
degree of freedom problems such as the Yukawa potential and for
problems of several degrees of freedom such as the two-electron
atoms in a magnetic field. To the best of our knowledge, this
general variational principle is used here for the first time to
calculate critical parameters for several degrees of freedom. Let
us consider several examples to illustrate this approach.
\section{Yukawa potential}
The radial Schr\"{o}dinger equation
for one particle in a Yukawa
potential, $v(r) = -\exp(-\delta r)/r$, after the scaling transformation
$r\rightarrow r/\delta$, takes the form
\begin{equation}
\label{yuk} \left[-\frac{1}{2}\frac{d^2}{dr^2} +
\frac{l(l+1)}{2r^2} - \delta^{-1}\frac{\exp(-r)}{r} - \delta^{-2}
E\right]P(r)=0,
\end{equation}
where $l$ is the azimuthal quantum number, $\delta$ is the
screening parameter, $E$ is the energy, and $P(r)$ is the radial
wave function multiplied by $r$ (we use atomic units
$\hbar=e=m=1$). For a sufficiently large critical screening
parameter $\delta=\delta_c$ the energy reaches the ionization
border $E_I =0$. The equation for the $\delta_c$ has a form of a
general eigenvalue equation \ref{eig} in which $L =
-\frac{1}{2}\frac{d^2}{dr^2} + \frac{l(l+1)}{2r^2}$, $M =
\exp(-r)/r$, $\psi=P$, and $\lambda = \delta_c^{-1}$. We are
looking for an extremum of the functional
\begin{equation}
\label{fun} W=\int \psi M(\psi)dV/\int \psi L(\psi)dV
\end{equation}
using a trial function with only one variational parameter $a$:
\begin{equation}
\label{tri1}
\widetilde{P}(r)=r^{-l}\left(1-e^{(-ar)}\sum_{n=0}^{2l}\frac{a^n}{n!}r^n\right).
\end{equation}
The function, \Eref{tri1}, behaves like $\widetilde{P}(r) \sim
r^{l+1}$ at $r\rightarrow 0$ and $\widetilde{P}(r)\sim r^{-l}$ at
$r\rightarrow \infty$ which is consistent with the behavior of the
general solution of the radial Schr\"{o}dinger equation at zero
energy.
The functional, \Eref{fun}, appears to have a minimum at a
certain $a=a_{\min}$, these values are listed in Table \ref{res1}
for different values of $l$.
\begin{table}
\noindent \caption{Results of minimization of the functional,
\Eref{fun}, for the Yukawa potential with one-parameter trial
function, \Eref{tri1}. The exact critical parameters are given in
the last column for comparison.}
\begin{indented}
\label{res1}
\item[]
\begin{tabular}{@{}cccc}
\br $l$&$a_{\min}$&$\widetilde{\delta}_c$&$\delta_c$\\ \mr
0&$1.535$&$1.190\,213$&$1.190\,612$\\
1&$2.534$&$0.219\,800$&$0.220\,217$\\
2&$3.525$&$0.091\,085$&$0.091\,345$\\
3&$4.517$&$0.049\,670$&$0.049\,831$\\
4&$5.513$&$0.031\,240$&$0.031\,344$\\
5&$6.510$&$0.021\,455$&$0.021\,525$\\
6&$7.507$&$0.015\,642$&$0.015\,691$\\
7&$8.506$&$0.011\,909$&$0.011\,945$\\ \br
\end{tabular}
\end{indented}
\end{table}
Its minimum gives an approximation for the eigenvalue
$\widetilde{\lambda}$. The corresponding approximations for the
critical screening parameter
$\widetilde{\delta}_c=\widetilde{\lambda}^{-1}$ together with the
exact critical screening parameters, which have been found
numerically by integration of the Schr\"{o}dinger equation, are
listed in the last two columns of Table \ref{res1}. The
variational method yields excellent lower bounds for the critical
screening parameters of the lowest states in each $l$ subspace.
Hulth\'{e}n and Laurikainen (1951) used, for the ground state
($l=0$), a different trial function in the form of the expansion
$\left(1-\e^{-x}\right)\sum_{\nu=0}^n h_{\nu} \e^{-\nu x}$. Since
it has several variational variables, their results are more
accurate.
\section{Two-electron atoms}
Calculation of the critical nuclear charge $Z_c$ for two-electron
atoms has long history (Br\"{a}ndas and Goscinski 1972, Stillinger
1966, Stillinger and Stillinger 1974, Stillinger and Weber 1974,
Reinhardt 1977) with controversial results of whether or not the
value of $1/Z_c$ is the same as the radius of convergence of the
perturbation series in $1/Z$, $1/Z_*$. Baker \etal (1990) have
performed a 400-order perturbation calculation to resolve this
controversy and found that $1/Z_*=1/Z_c$ where numerically
$Z_*^{-1}\approx 1.097\,66$. Using Euler transformation of the
series that accelerates its convergence, Ivanov (1995) estimated
the value of the radius of convergence as $Z_*^{-1}\approx
1.097\,660\,79$.
The Schr\"{o}dinger equation for a two-electron atom, after the
scaling transformation $r\rightarrow r/Z$, takes the form
\begin{equation}
\label{two} \left[-\frac{1}{2}\nabla_1^2 -\frac{1}{2}\nabla_2^2
-\frac{1}{r_1} -\frac{1}{r_2} +Z^{-1}\frac{1}{r_{12}}- Z^{-2}
E\right]\psi=0,
\end{equation}
where $Z$ is the charge of the nucleus and $E$ is the energy (in
atomic units). For a sufficiently small nuclear charge, at $Z=Z_c$
the energy reaches the ionization border $E_I =-Z^2/2$, which is
the energy of one-electron atom. The equation for the $Z_c$ has a
form of a general eigenvalue equation \ref{eig} in which $L =
-\frac{1}{2}\nabla_1^2 -\frac{1}{2}\nabla_2^2 -1/r_1 -1/r_2 +1/2$,
$M = 1/r_{12}$, and $\lambda = -Z_c^{-1}$. We are looking for an
extremum of the functional, \Eref{fun}, using a Hylleraas-type
trial function of the form
\begin{equation}
\label{tri2}
\begin{array}{c}
\psi_N= \sum_{i+j^2+k^2\leq N}^{} C_{i,j,k} \left[r_1^i r_2^j
\exp(-ar_1-br_2)+r_2^i r_1^j \exp(-ar_2-br_1)\right] \\
\exp(-cr_{12}) r_{12}^k.
\end{array}
\end{equation}
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 from $\sim \frac{1}{6}N^3$
to $\sim \frac{\pi}{8}N^2$ . Here, we suppose that correlation
terms with higher degrees of $r_{12}$ are relatively unimportant
and we suppress them by rising $k$ to $k^2$. We also assume that
expanding over $r_2$ is less important that expanding over $r_1$
because we are ordering the parameters $a$ and $b$ so that $a