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\begin{document}
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\title{Semiclassical self-consistent-field perturbation theory\\
for the hydrogen atom in a magnetic field}
% repeat the \author\address pair as needed
\author{Alexei V. Sergeev\cite{vavilov} and David Z. Goodson}
\address{ Department of Chemistry, Southern Methodist University,
Dallas, Texas  75275}
\date{\today}
\maketitle
\begin{abstract}
A recently developed perturbation theory for solving
self-consistent-field equations
is applied to the hydrogen atom in a strong magnetic field.
This system has been extensively studied using other methods
and is therefore a good test case for the new method.  The
perturbation theory yields summable large-order expansions.
The accuracy of the self-consistent-field approximation
varies according to field strength and quantum state, but it
is often higher than the accuracy from
adiabatic approximations.  A new derivation is presented for
the asymptotic adiabatic approximation, the most useful
of the adiabatic approaches.  This derivation uses
semiclassical perturbation theory without
invoking an adiabatic hypothesis.
\end{abstract}
% insert suggested PACS numbers in braces on next line
 \pacs{32.60.+i, 31.15.Md, 31.15.Ne}

% body of paper here
\narrowtext

\section{Introduction}

Large-order perturbation theory has been widely used to
solve the Schr\"odinger equation for
systems with only 1 or 2 degrees of freedom, and
occasionally for 3-coordinate problems
\cite{ijqc,morgan,popovrev,drh}, but
rarely for larger systems because it becomes increasingly
difficult to compute expansion coefficients to sufficiently
high order.  A recently developed self-consistent-field (SCF)
perturbation theory \cite{dzgscf} reduces the computational
cost by introducing a separability assumption.  Here we
apply this method to a simple 2-coordinate test case---the
hydrogen atom in an external magnetic field.


We will use this system, which has been extensively
studied using a wide variety of methods \cite{hmagbook},
to examine the accuracy of the
separability assumption.  In particular, we will compare
the accuracy of the SCF approximation to that of adiabatic
approximations.  Our method is based on a semiclassical
perturbation theory \cite{mlod1,mlod2,bender,popov1,popov2}.
At first order in this perturbation expansion the Schr\"odinger
equation becomes exactly separable in terms of a set of normal
coordinates.  These coordinates are a natural choice for a
separability approximation.  For a one-electron atom in a
magnetic field the normal coordinates are the cylindrical
coordinates $(\rho,z)$, with $z$ along the field direction.
It is well known that in the limit of infinite field strength
the exact wavefunction is a separable product in these
coordinates and that the $z$ motion is much slower than the $\rho$
motion \cite{garstang}.  This is the rationale behind adiabatic
approximations
\cite{garstang,schiff,simola,wunner,friedrich,baldereschi,starace,roesner},
in which a solution for the $\rho$ motion is obtained with $z$
treated as a fixed parameter.  These approximations
simplify the computation of energies and provide a
qualitative understanding of the energy level patterns
\cite{hmagbook,roesner}.  Approximate methods are not
as important for this problem as they used to be, now that
increasing computer power has made exact calculations feasible for
any field strength \cite{hmagbook,others}.  However,
the asymptotic adiabatic approximation
of R\"osner {\it et al.} \cite{roesner} is still useful for
calculating quantities such as partition functions that
require the computation of very large numbers of energy
levels.

Adiabatic approximations and the SCF theory are similar in that,
in principle, they both become exact in the infinite-field limit.
They differ in that the SCF method includes no assumption
about fast and slow coordinates and uses a separable
wavefunction at all values of field strength.  They differ
also in that the SCF perturbation theory becomes exact not just
in the limit of infinite field strength but also in the limit
of infinite magnetic quantum number $m$.  Thus, the error
from the SCF separability assumption can in principle remain small
at field strengths at which the adiabatic hypothesis breaks
down.  We find in fact that the SCF results for energies are
accurate even at small values of $m$ and at relatively smaller
field strengths.

Sections \ref{sec:2} and \ref{sec:3} describe our calculation
method and Section \ref{sec:4} shows representative
results.  In Section \ref{sec:5} we compare the accuracy of
the SCF theory with that of the primitive adiabatic
approximation of Schiff and Snyder \cite{schiff}, the true
adiabatic approximation of Baldereschi and Bassani
\cite{baldereschi}, and the asymptotic adiabatic approximation
of R\"osner {\it et al.} \cite{roesner}.  The asymptotic
method was originally derived as an approximation to the
primitive adiabatic method, yet it tends to be the more
accurate of these two approaches.  We show that the asymptotic
method can be derived as a second-order semiclassical
perturbation expansion, without assuming an adiabatic
separation of variables.  In Section \ref{sec:6} we discuss
our results and describe some potential advantages of the SCF
theory.


\section{Semiclassical Perturbation Theory}
\label{sec:2}

Mlodinow's semiclassical perturbation expansion \cite{mlod1}
is an example of a class of perturbation theories in which
a discrete dimensionality parameter is treated as a continuous
variable \cite{yaffe}.  These ``dimensional'' perturbation theories
are being applied to an increasingly wide variety of problems
in condensed-matter physics, statistical mechanics,
particle physics, polymer physics, and chemical physics
\cite{reviews}.  In the case of the Schr\"odinger equation the
perturbation expansion parameter is asymptotically proportional
to $1/D$, where $D$ is the dimensionality of coordinate space.
This expansion can often be computed to very large order and
accurately summed at $D=3$
\cite{popov1,popov2,dzg,watson,kais,dunn,dzghyd}.

The nonrelativistic energy levels, in atomic units, of a
hydrogen atom in a uniform magnetic field ${\bf B}$ are
\hbox{$E+mB/2$}, where $m$ is the magnetic
quantum number and $E$ is an eigenvalue of the operator
$H=T+V$, with
\begin{equation}
T=-{1\over 2}\left({\partial^2\over\partial\rho^2}\!
                    + {\partial^2\over\partial z^2}\right)
+ {\big(2|m|+D-2\big)\big(2|m|+D-4\big)\over 8\rho^2}
\label{Tphys}
\end{equation}
and
\begin{equation}
V=-(\rho^2+z^2)^{-1/2}+{1\over 8}B^2\rho^2.
\label{Vphys}
\end{equation}
\noindent
In these units $B=1$ corresponds to a field strength of
$m_e^2ce^3/\hbar^3$, which is approximately $2.35\times 10^5$ T.
The coordinates $(\rho,z)$ are cylindrical
coordinates with the $z$ axis oriented in the direction of $\bf B$.
The kinetic energy operator, Eq.~(\ref{Tphys}), is expressed
in a form valid for arbitrary $D$ \cite{frantz}.

Let
\begin{equation}
\delta=[ |m|+a+(D-3)/2 ]^{-1},
\label{delta}
\end{equation}
where $a$ is an arbitrary constant.  $\delta$ will be the expansion
parameter for the perturbation theory.  The main purpose of
the shift constant $a$ is to preserve $\delta$ as a finite
parameter at $m=0$.  The dimensional scalings
\begin{equation}
\rho=\delta^{-2}\tilde{\rho},\quad z=\delta^{-2}\tilde{z},
\quad E=\delta^2 \widetilde{E},\quad B=\delta^{3}\widetilde{B},
\label{dimscalings}
\end{equation}
of the units of distance, energy, and field strength,
respectively, yield the eigenvalue equation
\begin{eqnarray}
&&\bigg[ -{1\over 2}\delta^2
\left({\partial^2\over\partial\tilde{\rho}^{\, 2} }
                    + {\partial^2\over\partial \tilde{z}^2}\right)
+{1-2a\delta+(a^2-1/4)\delta^2\over 2\tilde{\rho}^{\, 2} }
\nonumber\\
&&\qquad\qquad\quad
-(\tilde{\rho}^{\, 2}+\tilde{z}^2)^{-1/2}
+{1\over 8}\widetilde{B}^2\tilde{\rho}^{\, 2}
- \widetilde{E} \bigg]\Psi = 0.
\label{scheq}
\end{eqnarray}
\noindent
Note that the only $D$ dependence in Eq.~(\ref{scheq}) is in
the parameter $\delta$.  Therefore, we can set
\begin{equation}
\delta=1/\big(|m|+a\big)
\label{deltam}
\end{equation}
and consider $\delta\to 0$ as equivalent to the limit of large
$|m|$ \cite{Ddegeneracies}.

In the limit of small $\delta$ the factor of $\delta^2$ plays
the role of $\hbar^2$ in the Schr\"odinger equation for a
particle of unit mass subject to an effective potential
\begin{equation}
V_{\rm eff}(\tilde{\rho},\tilde{z})
={1\over 2}\tilde{\rho}^{-2}
-(\tilde{\rho}^{\, 2}+\tilde{z}^2)^{-1/2}
+{1\over 8}\widetilde{B}^2\tilde{\rho}^{\, 2}.
\label{Veff}
\end{equation}
In the limit $\delta\to 0$ all of the eigenvalues collapse to
the value
$\widetilde{E}_0
=V_{\rm eff}(\tilde{\rho}_{\rm min},\tilde{z}_{\rm min})$,
where $(\tilde{\rho}_{\rm min},\tilde{z}_{\rm min})$ is the
minimum of the effective potential.  This minimum corresponds
to $\tilde{z}_{\rm min}=0$ with $\tilde{\rho}_{\rm min}$ equal
to a positive root of the equation
$\widetilde{B}^2\tilde{\rho}_{\rm min}^{\, 4}
=4(1-\tilde{\rho}_{\rm min})$.

In order to have a formulation suitable for the entire range of
field strengths, we introduce a further scaling of distance and
energy with the substitutions
\begin{equation}
\xi=\tilde{\rho}/\tilde{\rho}_{\rm min},\quad
\eta=\tilde{z}/\tilde{\rho}_{\rm min},
\label{substitutions}
\end{equation}
which yields the effective potential
\begin{equation}
W(\xi,\eta)=
{1\over 2\xi^{\, 2}}
-{1-g\over (\xi^2+\eta^2)^{1/2}}
+{1\over 2}g\xi^{\, 2},
\label{Veffscaled}
\end{equation}
where
\begin{equation}
g=1-\tilde{\rho}_{\rm min}
=\widetilde{B}^2\tilde{\rho}_{\rm min}^{\, 4}/4.
\label{gdef}
\end{equation}
Variation of $g$ from 0 to 1 corresponds to variation of
$\widetilde{B}$ from zero to infinity, according to the relation
\begin{equation}
\widetilde{B}=2g^{1/2}(1-g)^{-2}.
\label{Bfromg}
\end{equation}

Higher orders in the perturbation theory are obtained by
introducing displacement coordinates $x$ and $y$, defined by
$\xi=1+\delta^{1/2}x$, $\eta=\delta^{1/2}y$,
and then expanding $W$ in powers of
$\delta^{1/2}$, and collecting terms of given order.
At order $(\delta^{1/2})^2$ we have a Schr\"odinger equation
for a harmonic oscillator.  In general, the next step in the
analysis would be to diagonalize the harmonic Hamiltonian with
a normal-mode transformation.  However, in this case there is
no coupling between $x$ and $y$ at this order.  Therefore,
the normal coordinates are simply $q_1=x$ and $q_2=y$.
The Schr\"odinger equation is
\begin{equation}
\left[ -{\delta\over 2}
\left({\partial^2\over\partial q_1^2}
 +{\partial^2\over\partial q_1^2}\right) +W
 -\tilde{\rho}_{\rm min}^{\, 2}\widetilde{E}\right] \Psi = 0
\end{equation}
and the perturbation expansions for the effective potential,
the wavefunction, and the scaled eigenvalues, respectively,
have the forms
\begin{eqnarray}
&&W=\tilde{\rho}_{\rm min}^{\, 2}\widetilde{E}_0+
\delta\!\left[{\scriptstyle{1\over 2}}\omega_1^2q_1^2
+{\scriptstyle{1\over 2}}\omega_2^2q_2^2
-a+v(q_1,q_2)\right],\!\!\!\\
&&v(q_1,q_2)=\sum_{k=1}^\infty\delta^{k/2}v_k(q_1,q_2),
\label{wexpansion}\\
&&\Psi(q_1,q_2)=\sum_{k=0}^\infty\delta^{k/2}\Psi_k(q_1,q_2),
\label{psiexpansion}\\
&&\epsilon\equiv \tilde{\rho}_{\rm min}^{\, 2}
(\widetilde{E}-\widetilde{E}_0)/\delta
=\sum_{k=0}^\infty \delta^{k/2} \epsilon_k.
\label{epsilon}
\end{eqnarray}
The $v_k$ in Eq.~(\ref{wexpansion}) are polynomials of degree
$k+2$, and $\epsilon_k=0$ for odd $k$.

At first order in $\delta$ the expansion coefficient for the
eigenvalue is
\begin{equation}
\epsilon_0=\epsilon_0^{(1)}+\epsilon_0^{(2)}-a,
\end{equation}
in terms of the harmonic eigenvalues
\begin{equation}
\epsilon_0^{(i)}=(n_i+{\scriptstyle{1\over 2}})\omega_i
\end{equation}
with harmonic frequencies
\begin{equation}
\omega_1=(1+3g)^{1/2},\quad \omega_2=(1-g)^{1/2}.
\end{equation}


\section{Self-Consistent-Field Theory}
\label{sec:3}

Let us express the wavefunction as a separable product in
terms of the normal coordinates,
\begin{equation}
\Psi(q_1,q_2)=\psi^{(1)}(q_1)\psi^{(2)}(q_2).
\end{equation}
The SCF equations are
\begin{equation}
\left[ -{1\over 2}{d^2\over dq_i^2}+{1\over 2}\omega_i^2 q_i^2
         +\bar{v}^{(i)}(q_i)-\epsilon^{(i)}\right]
        \psi_i(q_i) = 0,
\label{SCFeqs}
\end{equation}
where $i=1$ or 2, and
\begin{equation}
\bar{v}^{(i)}(q_i)
 =\big\langle\psi^{(j)}(q_j)\big|
    v(q_1,q_2)\big|\psi^{(j)}(q_j)\big\rangle_{q_j},
\label{Vbar}
\end{equation}
with $(i,j)=(1,2)$ or (2,1).
Eq.~(\ref{SCFeqs}) yields the ``best'' solutions for the
$\psi^{(i)}$ according to the variational principle for the
energy.  The SCF approximation for the total energy can be
calculated as
\begin{equation}
\epsilon=\epsilon^{(1)}+\epsilon^{(2)}
-\big\langle\psi^{(2)}(q_2)\big|\bar{v}^{(2)}(q_2)\big|
   \psi^{(2)}(q_2)\big\rangle_{q_2}.
\label{SCFepsilon}
\end{equation}

Let
\begin{equation}
\psi^{(i)}(q_i)=\sum_{k=0}^\infty\delta^{k/2}\psi^{(i)}_k(q_i),
\quad
\epsilon^{(i)}=\sum_{k=0}^\infty\delta^k\epsilon^{(i)}_k.
\end{equation}
Substituting these expansions into the two SCF equations and
collecting terms by order in $\delta^{1/2}$ yields two sets
of perturbation equations that are coupled implicitly
through the dependence of $\bar{v}^{(i)}$ on $\psi^{(j)}$.
However, the equations can be uncoupled, order by order,
using an algorithm described in Ref.~\cite{dzgscf}.

By assuming separability in terms of the normal coordinates,
we ensure that the SCF theory will be exact to order $\delta$
in the energy expansion.  The error due to the separability
assumption will be of order $\delta^2$.


\section{Results}
\label{sec:4}

The SCF energy levels can be uniquely labeled by the magnetic
quantum number $m$ and by the harmonic quantum numbers
$(n_1,n_2)$.  The expression for the eigenvalue $E$, in
unscaled units, is
\begin{equation}
E=\big(|m|+a\big)^{-2}\widetilde{E}_0
+\big(|m|+a\big)^{-3}\tilde{\rho}_{\rm min}^{\, -2}\epsilon,
\label{eunscaled}
\end{equation}
where $a$ is the arbitrary shift parameter.  $\epsilon$ is
calculated as an asymptotic expansion.  In general, we choose
\begin{equation}
a=n_1+n_2+1,
\end{equation}
so that in the limit $B\to 0$ the zeroth-order term
in Eq.~(\ref{eunscaled}) yields the exact hydrogenic energy.

For the exact nonseparable problem, $(n_1,n_2)$ are true
quantum numbers only in the limit $\widetilde{B}\to\infty$.
As $\widetilde{B}$ is decreased, all but the $(0,0)$ and $(1,0)$
states will undergo avoided crossings, which occur at
$\widetilde{B}$ values for which the harmonic frequencies are
in integer ratios.  At very
high field strengths the avoided crossings are sharp and
well defined while at lower field strengths they can be broad
and complicated \cite{dunnprl,avs}.

One can expect that the major qualitative difference between
the SCF eigenvalues and the exact eigenvalues will be the
behavior at avoided crossings.  The separability assumption
ensures that for the SCF results $(n_1,n_2)$ will be good quantum
numbers for all values of $\widetilde{B}$.  Thus, the SCF
eigenvalues will cross diabatically.  Figure~1 compares
exact and SCF results as functions of the
parameter $g$ for a manifold of states with $|m|=30$.  ($g=0$
corresponds to $\widetilde{B}=0$ and $g=1$ corresponds to
$\widetilde{B}=\infty$, according to Eq.~(\ref{gdef}).)
This Figure gives the energy as the scaled quantity
\begin{equation}
\epsilon'=|m|^3 \big[ E-|m|^{-2}\widetilde{E}_0(a=0)\big],
\label{eprime}
\end{equation}
which is simply the value of $\epsilon$ that corresponds to
$a=0$.  The advantage of $\epsilon'$ is that it omits the
uninteresting but strongly field dependent zero-point energy
and, in contrast to $\epsilon$, it uses the same scaling
factor for all values of $n_1$ and $n_2$.  We obtain
convergent SCF results for $\epsilon$ over almost
the full range of $g$ using Pad\'e summation.
It can be seen that the SCF results
are very close to the exact results except in the vicinity of
broad avoided crossings.

Figure~2 shows the accuracy of the SCF approximation for the
binding energy (the difference between the total energy and the
energy of a free electron in the magnetic field) in unscaled units
as a function of $B$ and $m$ for the ``circular'' states,
$(n_1,n_2)=(0,0)$.  For given value of $B$ the
accuracy increases significantly with increasing $|m|$.  This
is consistent with the fact that the underlying separability
assumption is exact within first-order perturbation theory in
$\delta\sim |m|^{-1}$.

For given $|m|$, the error from the SCF approximation
at first holds steady at the field-free value as $B$ increases.
Then at a critical value of $B$, the accuracy begins to improve,
with the error diminishing approximately as $B^{-1/2}$.
The increase in accuracy begins when the field-dependent term
$\widetilde{B}^2\tilde{\rho}^{\, 2}/8$ starts to dominate the
coupling term $-(\tilde{\rho}^{\, 2}+\tilde{z}^2)^{-1/2}$ in
Eq.~(\ref{scheq}).  The
critical value is smaller at large $|m|$, since
$B=(|m|+a)^{-3}\widetilde{B}$.  At this critical field
strength the magnetic field and the Coulomb field on the
electron are of the same magnitude, since the radius of the
electron orbit increases with $|m|$ as $(|m|+a)^2$.

We have used Pad\'e approximants to sum the perturbation
expansions.  For the results from which Fig.~2 was prepared,
the approximants converge with a precision that is greater
than the accuracy of the underlying separability assumption.
The rate of convergence slows as $B$ increases.  This appears
to be due to the fact that $\tilde{\rho}_{\rm min}$ approaches
zero in the large-$B$ limit, which causes the expansion of the
Coulomb term in Eq.~(\ref{Veff}) to diverge.


\section{Comparison with Adiabatic Approximations}
\label{sec:5}

There are three different computational techniques that have been
referred to in the literature as ``adiabatic approximations''
for the one-electron atom in a magnetic field.  The earliest is
the method developed by Schiff and Snyder \cite{schiff},
which we will call the ``primitive'' adiabatic approximation.
That approach is based on the approximation
\begin{equation}
\Psi(\rho,z)=f_{n_1,n_2,m}(z)\,\Phi^{(Landau)}_{n_1,m}(\rho),
\label{primadiab}
\end{equation}
where the $\Phi^{(Landau)}_{n_1,m}$ are the well-known Landau
functions \cite{garstang}, which result from solving the
Schr\"odinger equation without the Coulomb potential.
The Landau quantum number $n_1$ is equivalent to our harmonic
quantum number for $\rho$ motion in the limit $\delta\to 0$,
while the quantum number $n_2$, which is equal to the number
of longitudinal nodes, is equivalent to our harmonic quantum
number for $z$ motion.  Solutions for the longitudinal
function $f$ are determined by solving Schr\"odinger equations
with effective potentials
\begin{equation}
V_{n_1,m}^{(Landau)}(z)
=-\Big\langle\! \Phi^{(Landau)}_{n_1,m}
  \big|(\rho^2+z^2)^{-1/2}\big|
  \Phi^{(Landau)}_{n_1,m}\Big\rangle .
\label{primV}
\end{equation}

The second approach \cite{baldereschi,starace},
which we call the ``true'' adiabatic approximation, uses the
expression
\begin{equation}
\Psi(\rho,z)=W_{n_1,m}(z)F_{n_1,m}(z;\rho).
\label{trueadiab}
\end{equation}
In the function $F$, the $z$ dependence is treated
parametrically, in a manner analogous to the Born-Oppenheimer
approximation in molecular physics.  The value of $W_{n_1,m}$
for given $z$ is obtained as the eigenvalue of a Schr\"odinger
equation in $\rho$.  This is carried out over a range of values
for $z$ and then the resulting $W(z)$ is used as the potential
in a second Schr\"odinger equation, the eigenvalues of which are
approximate solutions for $E$.

Probably the most useful of the adiabatic approximations is the
asymptotic approximation of R\"osner, {\it et al.} \cite{roesner}.
Consider the case
$n_1=0$.  (These are the most important states, since eigenvalues
for $n_1>0$ in strong fields lie above the ionization
threshold.)  The integrand in Eq.~(\ref{primV}) contains the
product
\begin{equation}
{\Phi^{(Landau)}_{0,m}}^*\Phi^{(Landau)}_{0,m}
\propto e^{-B\rho^2/2}\rho^{2|m|+1},
\end{equation}
which is a bell-shaped curve peaked at the point
$\rho_0=\sqrt{(2|m|+1)/B}$.
Replacing $(\rho^2+z^2)^{-1/2}$ in Eq.~(\ref{primV}) with its
asymptotic expansion about $\rho_0$ and then
retaining only the leading term after evaluating the integral
leads to the expression
\begin{equation}
V_p^{(Landau)}(z)\sim V_p^{(asymp)}
=-2\,(1/p+z^2)^{-1/2},
\label{asympV}
\end{equation}
\begin{equation}
p=B\big/\big(2|m|+1\big).
\label{pdef}
\end{equation}
The striking feature of Eq.~(\ref{asympV}) is the fact that
it no longer depends on $B$ and $m$ separately, but only on
the quantity $p$.  This reduction in the dimensionality of
the manifold of eigenvalues greatly facilitates the
calculation of quantities such as partition functions
that require the computation of very large numbers of
energy levels.

Figures~3 and 4 compare the accuracy of the SCF theory with
that of the various adiabatic approximations for states
with $(n_1,n_2)=(0,0)$ and (0,2), respectively.
For the (0,0) manifold, the SCF results are in general
significantly more accurate than the primitive and asymptotic
adiabatic results, especially at lower field strengths.
For the $(0,2)$ manifold the trends are less clear.  The
primitive adiabatic, asymptotic adiabatic, and SCF results
are comparable in accuracy.  The accuracy increases with
$|m|$ for all three methods but the increase in general is
greatest for the SCF method, especially at lower $B$.

True adiabatic results are available only for states with
$|m|=0$ or 1 \cite{starace}.
Although the true adiabatic theory is in
principle the most accurate (and laborious) of the adiabatic
methods, it yields energies that are furthest from the exact
results for $m=0$.

It is interesting, also, that for states with $n_2>0$ the
primitive adiabatic theory is generally {\it less} accurate
than the asymptotic theory, despite
the fact that the latter was derived as an approximation to
the former.  In fact, it is possible to derive the asymptotic
theory from a semiclassical perturbation analysis, without
invoking the adiabatic hypothesis.  Let us make the
substitutions
\begin{equation}
E= \delta^{-2}\widetilde{E},\quad
B=\delta^{-1} \widetilde{B}.
\label{otherscaling}
\end{equation}
Note that these differ from the dimensional scalings in
Eq.~(\ref{dimscalings}).  The Schr\"odinger equation is now
\begin{eqnarray}
&&\bigg[ -{1\over 2}\delta^2
\left({\partial^2\over\partial\rho^2}
                    + {\partial^2\over\partial z^2}\right)
+{1-2a\delta+(a^2-1/4)\delta^2\over 2\rho^2}
\nonumber\\
&&\ \qquad\qquad
-\delta^2(\rho^2+z^2)^{-1/2}+{1\over 8}\widetilde{B}^2\rho^2 
- \widetilde{E} \bigg]\Psi = 0,
\label{otherscheq}
\end{eqnarray}
\noindent
instead of Eq.~(\ref{scheq}).  In the $\delta\to 0$ limit the
Coulombic potential drops out and $\Psi$ concentrates along
the cylinder defined by $\rho=\rho_{\rm min}$, with
$\rho_{\rm min}=\big(2/\widetilde{B}\big)^{1/2}$.  The scaled
energy approaches the value $\widetilde{E}_0=\widetilde{B}/2$.
If we introduce the displacement coordinate
$x=(\rho-\rho_{\rm min})/\delta^{1/2}$ and expand
Eq.~(\ref{otherscheq}) in powers of $\delta^{1/2}$, keeping
terms up to order $\delta^2$, we obtain the uncoupled equations
\begin{eqnarray}
&&
\bigg[ -{\delta\over 2}{d^2\over dx^2}
 + {1-2a\delta+(a^2-1/4)\delta^2
      \over 2(\rho_{\rm min}+\delta^{1/2}x)^2}
\nonumber\\
&&\qquad\quad
+{1\over 8}\widetilde{B}^2(\rho_{\rm min}+\delta^{1/2}x)^2
  - \widetilde{E}^{(1)}\bigg] \psi^{(1)}(x) = 0,
\label{otherxeq}
\end{eqnarray}
\begin{equation}
\bigg[ -{\delta^2\over 2}{d^2\over dz^2}
-\delta^2(\rho_{\rm min}+z^2)^{-1/2}-\widetilde{E}^{(2)}\bigg]
\psi^{(2)}(z) = 0.
\label{otherzeq}
\end{equation}
To solve Eq.~(\ref{otherxeq}) we remove all of the dimensional
scalings.  The result is simply the Landau equation, and
\begin{eqnarray}
E^{(1)}&=&\delta^{-2}
  \left[{1\over 2}\widetilde{B}
   +{1\over 2}(2n_1+1-a)\widetilde{B}\delta\right]
\nonumber\\
&=&(2n_1+|m|+1)(B/2)
\end{eqnarray}
is the Landau energy.
If the shift parameter $a$ is chosen to be $1/2$,
then $\widetilde{B}/2=p$ and the potential in
Eq.~(\ref{otherzeq}) is identical to $V_p^{(asymp)}$ in
Eq.~(\ref{asympV}).  In that case, $\psi^{(2)}$ is equivalent
to the function $f_{0,n_2,m}$ as given by the asymptotic
adiabatic approximation.
Of course, the value of $a$ is arbitrary in the
perturbation theory.  One can justify the choice $a=1/2$
by requiring that the perturbation theory agree with the
primitive adiabatic approximation in the limit of large $B$
and $|m|$.


\section{Discussion}
\label{sec:6}

In general the asymptotic adiabatic approximation and the SCF
theory for this system appear to be complementary, in that the
former is best for large values of the longitudinal quantum
number $n_2$ and very large $B$ while the SCF method is better
for smaller $n_2$ and relatively smaller $B$.  We have shown
that the asymptotic method is in fact a type of second-order
semiclassical perturbation theory, and does not depend on
an adiabatic hypothesis for its validity.  The advantage of
the SCF approach is that it allows for very efficient
computation of the perturbation expansion coefficients.
This eliminates the error that arises from the need to
truncate the asymptotic expansion at low order.  However, it
requires a separability approximation that is based on the
assumption that
the dynamics of the system is approximately harmonic.
High excitation in the longitudinal or radial quantum numbers
increases the importance of anharmonic effects.  In contrast,
the asymptotic approximation does not assume that the dynamics
in the longitudinal coordinate is harmonic.  For
longitudinally excited states this advantage can outweigh
the error from the second-order asymptotic truncation.

Our derivation of the asymptotic adiabatic approximation
uses dimensional perturbation
theory with an unconventional dimensional scaling.  This
illustrates the fact that the dimensional continuation of the
Schr\"odinger equation is arbitrary as long as the Hamiltonian
is correct for $D=3$.  The standard dimensional continuation
\cite{popovrev,drh,mlodjmp}, which we described in
Section~\ref{sec:2}, is certainly the most widely used approach,
but various other nonstandard definitions for the
$D$-dimensional Hamiltonian have been suggested
\cite{watson,march,avery,nobo}.  Our use of dimensional
perturbation theory to explain the success of an adiabatic
approximation is somewhat analogous to Goscinski and
Mujica's analysis of the hyperspherical adiabatic
approximation for the two-electron atom \cite{goscinski}.
It is possible that the particular approach we have used
here, in which part of the problem (in this case the
$z$ dependence of the potential) is not expanded as a
polynomial, will be useful in other contexts.

Both SCF and adiabatic approximations are widely used in
molecular physics, and the general experience in that field is
that SCF theory is easier to apply than adiabatic
approximations and the computational effort grows less quickly
as the number of degrees of freedom increases, but that an
adiabatic calculation is more accurate than the corresponding
SCF calculation \cite{farrelly}.  Thus, our finding that the
SCF theory is often the more accurate approach is notable.

For the one-electron atom in an external field,
approximate methods are not as important as they were
previously, since various methods can now yield essentially
exact numerical results on today's computers
\cite{hmagbook,others}.  However, such accurate
calculations, with no approximations, are not currently
feasible for many-electron systems in strong fields.  It
should be reasonably straightforward to apply SCF theory,
with dimensional perturbation expansions, to systems with
more than one electron.

SCF calculations for an atom in a magnetic field can also be
carried out using an iterative Hartree-Fock procedure
\cite{proeschel,ceperley}.  Dimensional perturbation theory
provides an efficient noniterative algorithm for solving SCF
equations.  Furthermore, the normal coordinates implied
by the large-dimension limit are an asymptotically optimal
coordinate system for the separability assumption.  For
many-electron systems these coordinates yield a separable
model that includes electron-correlation effects
\cite{dzgjcp,loeser}.  The error due to this separability
assumption can be expected to be smaller than the error from
an independent-electron separability assumption.

Some particular advantages of dimensional perturbation theory
are becoming clear.  It is in principle well suited to describing
correlated electron dynamics \cite{dzgjcp,loeser,zphys} and
provides a unified approach to bound states and resonances
\cite{popov1,kais}.
The method does not introduce approximations into the potential
energy operator, and therefore is relatively insensitive to
wide variations in physical parameters such as external field
strengths.  However, the computational cost of calculating the
expansion coefficients to sufficiently high order increases
prohibitively with the number of coupled degrees of freedom.
SCF approximations offer the possibility of bypassing these
computational difficulties without a severe cost in accuracy.


\acknowledgments{We thank Prof. P. A. Braun for helpful
discussions and suggestions.  This work was supported in part
by grants from the Robert A. Welch Foundation, the National
Science Foundation, and the Soros Foundation.}


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\begin{figure}
\caption{Energies for even parity states ($n_2$ even),
in scaled units, as function of field
strength for $|m|=30$.  $g$ is the field-strength parameter
given by Eq.~(\protect{\ref{Bfromg}}).  The scaled energy,
$\epsilon'$, is given by Eq.~(\protect{\ref{eprime}}).  Each
curve is labeled by the quantum numbers $n_1$ and $n_2$.}
\end{figure}

\begin{figure}
\caption{Accuracy of the SCF approximation for binding
energies, as function of field strength for the manifold
of circular states, $(n_1,n_2)=(0,0)$.  The ordinate is
\hbox{$-\log_{10}|(E_{\rm approx}-E_{\rm exact})/E_{\rm exact}|$}.
The abscissa is the magnetic field strength in atomic units
such that $B=1$ corresponds to $2.35\times 10^5$~T.  The
curves are labeled by the value of the magnetic quantum
number.}
\end{figure}

\begin{figure}
\caption{Accuracy of binding energies calculated by various
methods, as function of field strength for the manifold of
circular states, $(n_1,n_2)=(0,0)$.  The axes are labeled as
in Fig.~2.  The calculation methods are: SCF theory (solid
curves); primitive adiabatic approximation
\protect{\cite{roesner}} (dashed curves); asymptotic adiabatic
approximation \protect{\cite{roesner}} (dash-dot curves);
true adiabatic approximation \protect{\cite{starace}}
($\bullet$).  The panels are labeled by the value of the
magnetic quantum number.}
\end{figure}

\begin{figure}
\caption{Accuracy of binding energies calculated by various
methods, as function of field strength for the manifold
of states with $(n_1,n_2)=(0,2)$.  The axes are labeled as in
Fig.~2.  The calculation methods are: SCF theory (solid
curves); primitive adiabatic approximation
\protect{\cite{roesner}} (dashed curves); asymptotic adiabatic
approximation \protect{\cite{roesner}} (dash-dot curves);
true adiabatic approximation \protect{\cite{starace}}
($\bullet$).  The panels are labeled by the value of the
magnetic quantum number.}
\end{figure}


\end{document}