\documentstyle[preprint,prb,aps,epsf,eqsecnum]{revtex}
\tolerance=10000
\def\comment#1{\bigskip\hrule\smallskip#1\smallskip\hrule\bigskip}
\begin{document}
\title{Helium Dimers in Confined Geometry}
\author{S. ~Kili$\acute{\rm c}^{1}$, E.~Krotscheck$^{2}$, and
L. Vranje\v{s}$^{1}$}
\address{$^1$ Faculty of Natural Sciences, University of  Split,
21000 Split, Croatia}
\address{$^2$Institut f\"ur Theoretische Physik,
Johannes Kepler Universit\"at, A 4040 Linz, Austria}
\maketitle
\begin{abstract}
We study the binding of dimers of helium isotopes in a geometry that
corresponds to the adsorption on a flat substrate. By varying the
width of the holding potential, we go continuously from a rigorously
three--dimensional to a rigorously two--dimensional geometry. It is
shown that the binding energy is significantly enhanced when the width
of the holding potential is approximately equal to the range of the
pair interaction.
\end{abstract}

\section{Introduction}
\label{introduction}

We have in a recent paper\cite{dimers1} studied the binding of dimers
of helium isotopes in two and three dimensions when their motion is
confined by a spherically symmetric external holding potential.  The
calculation was designed to provide a model for the interaction
between helium atoms in solid matrices or on a solid, corrugated
substrate.  Among others, we devised a formalism that allows to map
the problem of two particles with confinement onto the problem of two
particles in free space; the effect of confinement appeares as an
auxiliary ``confinement potential'' that supplements the pair
interaction.

In this paper, we apply the same technique to examine the problem
where two helium atoms are located in an external holding potential
that depends only on one coordinate, say $z$. This system is, for
example, the ``seed'' for the condensation of liquid $^4$He on a
substrate. It is commonly believed that this situation can be modeled
by a two--dimensional liquid. Indeed, the two--dimensional equation of
state is a reasonable approximation\cite{filmstruc} for an atomic
monolayer of $^4$He, however, many--body calculations have also
revealed that the atomic monolayer is more strongly bound than the
two--dimensional liquid. It will turn out that the situation is
somewhat more complicated in the sense that a finite width of an
external holding potential effectively {\it enhances\/} the
interaction between pairs of particles such that the dimers are more
strongly bound than both in 3D and in 2D. This may also have
significant consequences on the physical nature of $^3$He-$^4$He
mixture films.

\section{Variational Wave Function and Effective Potential}
It is the purpose of this paper to examine the binding of
$^3$He-$^4$He in a situation with one-dimensional symmetry
breaking. The Hamiltonian of our system is
\begin{equation}
H(i,j) = -{\hbar^2\over 2m_i}\nabla_1^2  -{\hbar^2\over 2m_j}\nabla_2^2 
+ V(z_1) + V(z_2) + V(|{\bf r}_1-{\bf r}_2|)
\label{eq:Hamiltonian}
\end{equation}
where $(i,j) \in (3,4)$ specifies the helium isotope under
consideration. The external potential
$V(z)$ is kept
unspecified for the time being except that we assume that it
is the same for
both particles and has at least one bound state, and that

We make a variational {\it ansatz\/} for the ground state
wave function of the dimer of the form
\begin{equation}
\psi_{ij}({\bf r}_1,{\bf r}_2)
= \phi_i(z_1)\phi_j(z_2) f(|{\bf r}_1-{\bf r}_2|)
\label{eq:wavefunction}
\end{equation}
where the $\phi_i(z)$ is the ground state wave function of species $i$
in the holding potential $V(z)$,
\begin{equation}
\left[-{\hbar^2\over 2m_i}{d^2\over dz^2}
+ V(z)\right]\phi_i(z) = \epsilon_i\phi_i(z)
\label{eq:1dequation}
\end{equation}

For zero external potential we recover, of course, the unconfined
three--dimensional case. In the case of an infinitely strong and
narrow potential, the (squares of the) single--particle functions
$\phi_i^2(z)$ collapse to $\delta$--functions and the problem reduces
to that of two particles in two dimensions.  Hence, the variational
wave function (\ref{eq:wavefunction}) will give the exact answer in
two and in three dimensions, and otherwise a rigorous upper bound of
the energy.

The energy expectation value for the wave function
(\ref{eq:wavefunction}) can, after a few manipulations\cite{dimers1},
be written in the form
\begin{equation}
E = \epsilon_i + \epsilon_j +
\frac{\int d^3r\,  W_{ij}(r)\left[{\hbar^2\over 2\mu_{ij}}
\left|\nabla f(r)\right|^2 + f^2(r) v(r)\right]}
{\int d^3r\,  W_{ij}(r) f^2(r)}
\label{eq:energy}
\end{equation}
where the $\epsilon_i$ are single particle ground state energies of
the species $i$ in the external potential, $\mu_{ij} = m_i m_j/M_{ij}$
is the reduced mass of the dimer, $M_{ij}=(m_i+m_j)$ and normalization
is assumed. The weight function $W_{ij}(r)$ is an average of the
single--particle functions,
\begin{equation}
W_{ij}(r) = {1\over2}\int_{-1}^1 d\xi \int_{-\infty}^{\infty} dZ
\phi_i^2(Z + {m_j\over M_{ij}}r\xi)\phi_j^2(Z - {m_i\over M_{ij}}r\xi)\,.
\label{eq:weight}
\end{equation}

The interesting quantity in Eq. (\ref{eq:energy}) is the last term
which is due to interactions. Minimizing this term with respect to the
two--body function $f(r)$ yields an effective Schr\"odinger equation
\begin{equation}
-{\hbar^2\over 2\mu_{ij}W_{ij}(r)}
\nabla\cdot \left[W_{ij}(r)\nabla f(r)\right] + v(r) f(r) =
\epsilon_{ij} f(r)\,.
\label{eq:schreff}
\end{equation}

Defining, finally, $u(r) = \sqrt{W(r)} f(r)$ lets us write
Eq. (\ref{eq:schreff}) in the hermitian form
\begin{equation}
-{\hbar^2\over 2\mu_{ij}}\nabla^2 u(r)
 + \left[v(r) +  V_{\rm conf}(r)\right]u(r)
= \epsilon_{ij} u(r)\,,
\label{eq:schrherm}
\end{equation}
where
\begin{equation}
V_{\rm conf}(r) = {\hbar^2\over 2\mu_{ij}}
{\nabla^2 \sqrt{W_{ij}(r)}\over \sqrt{W_{ij}(r)}}.
\label{eq:Vconf}
\end{equation}
With these manipulations, we have mapped the problem onto an ordinary
Sch\"odinger equation.

Our approach to estimate the binding energy in confined geometry is
similar to one proposed some time ago by Bashkin\cite{BashkinTreiner}
who wrote a variational ground state wave function in the form
\begin{equation}
\psi_{ij}'({\bf r}_1,{\bf r}_2)
= \phi_i(z_1)\phi_j(z_2) f(|{\bf r}_1^{\|}-{\bf r}_2^{\|}|)
\label{eq:BTwavefunction}
\end{equation}
where $|{\bf r}_1^{\|}-{\bf r}_2^{\|}|$ is the {\it two-dimensional\/}
distance in the $x-y$ plane. Variation with respect to $f(r^\|)$ leads
to a two-dimensional Schr\"odinger equation with in effective
potential which is, just as ours, exact in the 2D limit, but {\it
not\/} in 3D. While the wave function (\ref{eq:BTwavefunction}) also
yields rigorous upper bounds for the binding energy, we found that
this wave function significantly underestimates the binding energy of
the dimers by at least an order of magnitude.

\section{Harmonic Oscillator Model}
\label{sec:model}

The interaction between helium atoms and solid surfaces is reasonably
well known\cite{ZarembaKohn76,ZarembaKohn77}, it can be characterized
by its long range part going as $z^{-3}$, and the binding energy of
the ad-atom at the surface. Close to the surface, however, one has
significant effects of the atomic structure of the underlying
solid\cite{Cole}. Since we are interested mostly in the behavior close
to the substrate and in particular in the dependence of the dimer
binding energy on the width of the holding potential, we found it more
transparent and flexible to assume a family of oscillator potentials
of variable width
\begin{equation}
V(z) = {m\omega_0^2\over 2} z^2\,.
\label{eq:Vofz}
\end{equation}
The single--particle functions are then Gaussian
\begin{equation} 
\phi_i(z)  = {1\over a_i^{1/2}\pi^{1/4}}\,\exp(-z^2/2a_i^2)
\end{equation}
with $a_i = \left(\hbar/m_i\omega_0\right)^{1\over2}$.
The weight function (\ref{eq:weight}) is then
\begin{equation}
a_{ij} W(x) = {1\over2}\,{\rm erf(x)\over x}
\end{equation}
where $a_{ij} = \sqrt{a_i^2+a_j^2} =
\left(\hbar/\mu_{ij}\omega_0\right)^{1\over2}$, $x = r/a_{ij}$, and
the confinement induced effective interaction (\ref{eq:Vconf}) is
\begin{equation}
V_{\rm conf}(x) = {\hbar^2\over 2\mu_{ij} a_{ij}^2}
\left[-\frac{1}{4\,x^2}
-\frac{1}{\pi}\,\frac {e^{-2 x^2}} {\left ({\rm erf} (x)\right)^2}
+\frac{1}{\sqrt{\pi}}\,\frac {e^{-x^2}} {x\,{\rm erf}(x)}
-\frac{2}{\sqrt{\pi}}\,\frac {x e^{-x^2}}{{\rm erf}(x)}\right]\,.
\label{eq:Vgauss}
\end{equation}

Before discussing numerical results, it is worth verifying that Eq.
(\ref{eq:schrherm}) reduces to the proper two-- and three--
dimensional Schr\"odinger equation in the case $a_{ij}\rightarrow 0$
and $a_{ij}\rightarrow \infty$. For $a_{ij}\rightarrow\infty$ we find
that $V_{\rm conf}(x)\sim -1/a_{ij}^2$, {\it i.e.\/} the interaction
correction vanishes as it should. For $a_{ij}\rightarrow 0$, the first
term in Eq. (\ref{eq:Vgauss}) dominates and we find $V_{\rm conf}(x)
\sim -1/4 r^2$.  The Schr\"odinger equation (\ref{eq:schrherm}) is
then simply
\begin{equation}
-{\hbar^2\over 2\mu_{ij}} \left[\nabla^2 + {1\over 4r^2}\right]u(r) 
 + v(r)u(r) = \epsilon_{ij} u(r)\,,
\label{eq:sch2d}
\end{equation}
which is readily seen to be the two-dimensional radial Sch\"odinger
equation. Eq. (\ref{eq:sch2d}) also shows that the dimer in infinite
space is, in 2D, always more strongly bound than in 3D. The same
result can, of course, also be obtained by inserting $\phi_i^2(z) =
\delta(z)$ in Eq. (\ref{eq:weight}).

Let us now turn to our results.  We have used the SAPT potential by
Korona {\it et al.}  \cite{Korona97} as two--body interaction which
appears to be the most accurate potential (without retardation) today;
the authors assert an accuracy of about 0.1\%\ in the region of the
minimum.

The binding energy of the dimer is, as a function of the effective
width $a_{ij}$, shown in Fig. \ref{fig:DimerEnergy}.  Note that all
three dimers are bound in 2D, but only the $(^4{\rm He})_2$ dimer
exists in 3D. Hence, the binding energy of $(^4{\rm He})_2$ goes to a
finite value as $a_{44}\rightarrow\infty$, however, the asymptotic
value of -1.878~mK of the binding energy of $(^4{\rm He})_2$ is not
reached until a width of about 1000~\AA.  The enhancement of the
binding of the dimer due to the finite well--width is quite
remarkable, it ranges from a factor of about 2 for $(^4{\rm He})_2$ to
two orders of magnitude for $(^3{\rm He})_2$, when compared with the
two--dimensional limit. The additional enhancement of the binding
energy of $(^4{\rm He})_2$ due to the finite width of the holding
potential is comparable to the result found in
Ref. \onlinecite{filmstruc} where it was shown that, at equilibrium
density, an atomic monolayer has an energy that is about 0.1~K lower
than that of a rigorously two--dimensional system.

The effect is explained by looking at the confinement interaction
$V_{\rm conf}(r)$. In Fig. \ref{fig:veff}, we show $V_{\rm conf}(r)$
for three different parameters $a_{44}$ in comparison with the bare
interaction. It is seen that, for large $a_{44}$, the correction to
the interaction is very small. For {\it small\/} $a_{44}$, $V_{\rm
conf}(r)$ can be large for $r\rightarrow 0+$, but this area is
dominated by the repulsive core of the bare interaction and the
confinement has no effect. In the intermediate regime, $V_{\rm
conf}(r)$ is still sizeable where the potential is most attractive,
hence the enhancement effect is the strongest there. We also notice
that the enhancement of the binding is most pronounced in the $(^3{\rm
He})_2$ dimer because, in this case, the cancellation between kinetic
and potential energy is the largest, and any small increase of the
attraction of the potential can cause relatively large changes.

Along with the increase in binding energy comes a dramatic reduction
of the size of the dimer. Fig. \ref{fig:DimerSize} shows the
average radius
\begin{equation}
\left\langle r \right\rangle
= \frac{\int d^2r \,r W(r) f^2(r)}{\int d^2r\, W(r) f^2(r)}.
\label{eq:DimerSize}
\end{equation}
All of the dimers are, in the area of strongest binding where the
width of the single-particle wave function in $z$-direction is about
3~\AA, a bnding energy that is significantly smaller than their
rigorousy two--dimensional counterparts. In particular the size of the
$(^4{\rm He})_2$ dimer reaches, with about 8~\AA, dimensions that are
not much larger than those of a typical molecule. Note that
$\left\langle r \right\rangle$ is the size of the dimer {\it in the
symmetry plane.\/}

\section{Summary}

We have in this paper studied the binding of helium atoms on solid
substrates. By giving the substrate potential a finite width, we have
constructed a model that is more realistic than a purely
two--dimensional sytem. It was shown that the effect of a finite width
holding potential can be quite dramatic: The binding energy of the
dimer is increased and, at the same time, the size of the dimer
significantly decreased.

\section{ACKNOWLEDGMENTS}

This work was supported by the Austrian Science Fund under
grant No. P12832-TPH.

%\bibliography {papers}
%\bibliographystyle{prsty}

\begin{thebibliography}{1}

\bibitem{dimers1}
S. Kilic, E. Krotscheck, and R. Zillich, J. Low Temp. Phys. {\bf 116},  245
  (1999).

\bibitem{filmstruc}
B.~E. Clements, J.~L. Epstein, E. Krotscheck, and M. Saarela, Phys. Rev. B {\bf
  48},  7450  (1993).

\bibitem{BashkinTreiner}
E. Bashkin, N. Pavloff, and J. Treiner, J. Low Temp. Phys. {\bf 99},  659
  (1995).

\bibitem{ZarembaKohn76}
E. Zaremba and W. Kohn, Phys. Rev. B {\bf 13},  2270  (1976).

\bibitem{ZarembaKohn77}
E. Zaremba and W. Kohn, Phys. Rev. B {\bf 15},  1769  (1977).

\bibitem{Cole}
M.~W. Cole, D.~R. Frankl, and D.~L. Goodstein, Rev. Mod. Phys. {\bf 53},  199
  (1981).

\bibitem{Korona97}
T. Korona {\it et~al.}, J. Chem. Phys. {\bf 106},  5109  (1997).

\end{thebibliography}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}
\epsfxsize=6truein
\centerline{\epsffile{DimerEnergy.eps}}
\vspace{1truein}
\caption{The figure shows the binding energy of the $(^3{\rm He})_2$
(short-dashed line), the $^3{\rm He}-^4{\rm He}$ (long--dashed line) ,
and the $(^4{\rm He})_2$ (solid line) dimers as a function of the
effective width $a_{ij} = \sqrt{a_i^2 + a_j^2}$.  The binding
energy of $(^4{\rm He})_2$ in 3D is shown as a dash-dotted horizontal
line in the upper right hand corner of the figure.
\label{fig:DimerEnergy}}
\end{figure}
\newpage
\begin{figure}
\epsfxsize=6truein
\centerline{\epsffile{veff.eps}}
\vspace{1truein}
\caption{The figure shows interaction correction
$V_{\rm eff}(r)$ for three representative widths $a_{44}$ of the holding
potential as indicated in the plot (solid lines, right scale).
Also shown is the bare potential SAPT of Ref. \protect\onlinecite{Korona97}
(dashed line, left scale).
\label{fig:veff}}
\end{figure}
\newpage
\begin{figure}
\epsfxsize=6truein
\centerline{\epsffile{DimerSizeLog.eps}}
\vspace{1truein}
\caption{The figure shows the average radius $\left\langle
r\right\rangle$ of the $(^4{\rm He})_2$ (solid line), the $^3{\rm
He}-^4{\rm He}$ (long--dashed line) , and the $(^3{\rm He})_2$ (short--
dashed line) dimers as a function of the effective width $a_{ij} =
\sqrt{a_i^2 + a_j^2}$. We could here also use a logarithmic scale
to show how much they grow, and then cover a larger area.
\label{fig:DimerSize}}
\end{figure}
\newpage

\end{document}