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\begin{document}
\title{Some relations for the ground state energy 
          and helium diatomic molecules}

\author{S. Kili† and L. Vranjeç \\
Faculty of Natural Science, University of Split,\\ 21000 Split, Croatia}
\date{Received: October 20, 1999\\
Revised: January 26, 2000}
\maketitle
\begin{abstract}
\noindent
It is shown quite generally that the ground state energy of two atoms in infinite 
space, interacting {\it via} a
spherical potential which depends only on the distance between particles,
is the lowest in two dimensions. Using a variational procedure,
binding energies of helium diatomic molecules, in infinite and restricted 
space, are obtained as well. The results derived for helium atoms are in 
accordance with the lemma.
\end{abstract}

PACS: 36.90.+f, 31.20.Di 

\newpage
\section{Introduction}
Many physical phenomena in nature are related to the behaviour of 
 small systems of particles. Among them, in low temperature physics
are superconductivity, superfluidity and Bose-Einstein condensation.
Special interesting and important cases are systems in which particles
are helium atoms: helium liquids, helium films, liquid drops, atoms in 
cavities in solid matrices and in nanotubes.

The consideration of small systems begins with study of two atoms. They
can be located in both restricted and unrestricted space: in 3 dimensions 
(3 D), 2 dimensions (2 D) and 1 dimension (1 D). Of course, real physical
world has been occuring in finite 3 dimensional space. In making models 
of different physical situations we are led to consider 2 D and 1 D space.
In such circumstances many physical effects are dominant in corresponding 
dimension.

In this paper, in Sec. I, we prove a general lemma. It relates ground state 
energies of two particles in 1, 2 and 3 dimensions in infinite space. It is 
assumed that particles interact via spherical potential depending only on 
the distance between them. 
In Sec. II, using variational procedure and employing the newest 
potential of the interaction between helium atoms [1], the ground state 
energies of helium molecules are obtained. The consistency  with the lemma
is demonstrated. 

\newpage
\section{Relations between ground state energies in different dimensions}

We consider the ground state of	
 two particles which interact {\it via} a spherically 
symmetrical potential 
$\hat{V}(\vec{r}_1,\vec{r}_2)$,
in one, two and three dimensional space. 
The Hamiltonian of the system
in the relative coordinates reads
\begin{eqnarray}
   \hat{H}~ = ~-\frac{\hbar^2}{2\mu}\Delta~+~ \hat{V}(|\vec{r_1}-\vec{r_2}|)~,
\end{eqnarray}
where $\mu=\frac{m_1m_2}{m_1+m_2}$ is reduced mass of the particles,
 $m_1$ and $m_2$ are the masses of the particles. In the ground state only the 
"radial" part of the Hamiltonian is important and the operator 
$\Delta$, in this case, has the form
\begin{eqnarray}
  \Delta_1~&=&~\frac{\partial^2}{\partial r^2} ~,~~~~~~~~~~~~~~~~~~~~~in~~ 1 D\\
  \Delta_2~&=&~\frac{\partial^2}{\partial r^2} + \frac{1}{r}\frac{\partial}{\partial r}
             ~,~~~~~~~~~~~~in~~ 2 D\\
  \Delta_3~&=&~\frac{\partial^2}{\partial r^2} + \frac{2}{r}\frac{\partial}{\partial r}
             ~,~~~~~~~~~~~~in~~ 3 D.
\end{eqnarray}
Inequalities between energies in different dimensions may be obtained
by variational {\it ansatz}
\begin{eqnarray}
       E_n\leq~ \frac{\int \!\Psi^{\star}_{n}~\hat{H}_n~\!\Psi_n~r^{n-1}
 dr\,d\Omega^n}{\int \!\Psi^{\star}_{n}~\Psi_n~ r^{n-1} dr\,d\Omega^n},
\end{eqnarray}
where $n=1,2,3$ denotes the dimension of physical space and $d\Omega^1=1$,
 $d\Omega^2=2\pi$ and $d\Omega^3=4\pi$.

As we study the ground state and having in mind the symmetry of the system,
it is useful to write the trial 
wave functions in the form
\begin{eqnarray} 
 \Psi_1(r) &=& \Psi_{10}(r) \nonumber	\\
 \Psi_2(r) &=& \frac{1}{\sqrt{r}}\Psi_{20}(r) \nonumber \\
 \Psi_3(r) &=& \frac{1}{r}\Psi_{30}(r) .
\end{eqnarray}
Introducing trial wave functions in the variational {\it ansatz} (5) one finds
\begin{eqnarray}
E_1 &\leq~&  \frac{1}{I_1}\left[-\frac{\hbar^2}{2\mu}\int_0^{\infty}dr
\Psi_{1} \frac{d^2}{dr^2}\Psi_{1}\,+\, \int_0^{\infty}dr\Psi_{1}^2\,V(r) \right] \\
E_2 &\leq~&  \frac{1}{I_2}\left[-\frac{\hbar^2}{2\mu}\int_0^{\infty}dr
\Psi_{20} \{\frac{d^2}{dr^2}\Psi_{20}\,+\,\frac{1}{4r^2}\Psi_{20}\} \,
+\, \int_0^{\infty}dr\Psi_{20}^2\,V(r)\right] \\
E_3 &\leq~&  \frac{1}{I_3}\left[-\frac{\hbar^2}{2\mu}\int_0^{\infty}dr
\Psi_{30} \frac{d^2}{dr^2}\Psi_{30} \,+\, \int_0^{\infty}dr\Psi_{30}^2\,V(r)\right] ,
\end{eqnarray}
where the normalization integrals read
$I_n=\int_0^{\infty}dr\Psi_{n0}^2$, n=1,2,3.

The relations (6), (7) and (8) are general. Assuming that $\Psi_1$ is
the eigenfunction in  one dimensional case and taking $\Psi_{20}$=$\Psi_1$, 
from (6) and (7) it follows
\begin{equation}
E_2 \,<\,  E_1 ~-~\frac{1}{I_1}\frac{\hbar^2}{2\mu}\int_0^{\infty}dr
\,\frac{1}{4r^2}\Psi_{20}^2;
\end{equation}
it means ~ $E_2\,<\, E_1$. 

If it is supposed $\Psi_{30}$=$\Psi_{20}$, where $\Psi_2$ is the
eigenfunction in 2 D, then from (7) and (8) one finds
\begin{equation}
E_3 \,<\,  E_2 ~+~\frac{1}{I_2}\frac{\hbar^2}{2\mu}\int_0^{\infty}dr
\frac{1}{4r^2}\Psi_{20}^2.
\end{equation}
  
On the other hand, if $\Psi_{3}$ is the eigenfunction in 3 D and 
$\Psi_{20}$ = $\Psi_{30}$, then
\begin{equation}
E_3 \,>\,  E_2 ~+~\frac{1}{I_3}\frac{\hbar^2}{2\mu}\int_0^{\infty}dr
\frac{1}{4r^2}\Psi_{30}^2.
\end{equation}
The last two inequalities may be joined
\begin{equation}
E_2 ~+~\frac{1}{I_3}\frac{\hbar^2}{2\mu}\int_0^{\infty}dr
\frac{1}{4r^2}\Psi_{30}^2\,<\,
E_3 \,<\,  E_2 ~+~\frac{1}{I_2}\frac{\hbar^2}{2\mu}\int_0^{\infty}dr
\frac{1}{4r^2}\Psi_{20}^2.
\end{equation}
From the above relation it follows $E_2 < E_3$.	In a similar consideration
from (6) and (8) follows $E_3 = E_1$. In this way it is proved that
binding energy of two interacting particles is the lowest in 2 D. The result 
is independent on the statistics of the particles.

\section{Ground state energy of diatomic helium molecules}

In order to describe physical systems that contain helium, many potentials 
between atoms have been obtained. One of the best is ${\it ab~ initio}$
SAPT potential by Korona et al.	[1]; its enlarged forms by Janzen and Aziz
[2] are SAPT1 and SAPT2 which comprise retardation effects. 
Since the SAPT potential is so precise, it is expected that the effect of 
retardation forces could be examined experimentally.   
It reads
\begin{eqnarray}
						V(r)~&=&~\epsilon \,V^{*}(r)	\\
					 V^{*}(r)~&=&~A e^{-\alpha\,r~+~\beta\,r^{2}}~-~B\,\sum_{n=3}^{8}
               f_{2n}(b,r)\, \frac{c_{2n}}{r^{2n}}~, 
\end{eqnarray}

where  
\begin{equation}   
f_{2n}(b,r)~=~1\,-\,e^{-b\,r}~\sum_{k=0}^{2n}\frac{(b\,r)^{k}}{k!}
\end{equation}
and
\[ 
\begin{array}{ll}
            \epsilon\,=\,10^{8}mK,~~~~~&c_{6}\,=\, 0.03207856 \,\mbox{\AA}^{6}\\
            A\,=\,20.7436426     ~~~~~ &c_{8}\,=\,0.08680214 \,\mbox{\AA}^{8}\\
	    B\,=\,3.157765       ~~~~~~&c_{10}\,=\,0.31625734 \,\mbox{\AA}^{10}\\											
	   \alpha\,=\,3.56498393\,\mbox{\AA}^{-1}~~~~~	&c_{12}\,=\,1.57407624 \,\mbox{\AA}^{12} \\
            \beta\,=\,-0.22141687\,\mbox{\AA}^{-2} ~~~~~  & c_{14}\,=\,10.31938196 \,\mbox{\AA}^{14} \\  
	   b\,=\,3.68239497 \,\mbox{\AA}^{-1}  ~~~~~~	& c_{16}\,=\,86.00126516\,\mbox{\AA}^{16}.
\end{array}
\]

As first let us calculate binding energy of two helium atoms in 
infinite space. For the ground state we found that a good 
analytic form of the functions (6) in all dimensions is
\begin{equation}
\Psi_{0i}(r) = \exp\left[-\left({a\over r}\right)^\gamma - s r\right]\,,
%\label{eq:psi03d}
\end{equation}
where i=1,2,3; $a$, $\gamma$ and $s$ are variational parameters and of course 
have different minimization values in 3\,D (1\,D) and 2\,D.
The same form
of pair correlations in 3\,D has recently been used by Bruch [3]
 to examine the properties of boson trimers.  In
2\,D, we use the form employed in the paper [4] and
which provides a slight improvement over a variational wave function
introduced in Ref. [5]. Binding energy and parameters are 
obtained by a minimization procedure. The results are shown in Table I. In 
order to estimate our variational calculation, and compare the results, 
corresponding numerical solutions of Schroedinger eq. are presented for
HFD-B3-FCI1 [6] and SAPT potentials as well. 

Now, as second, we concentrate on  two helium atoms confined by a hard-walled
spherical potential in 3\,D and circular in 2\,D. As it was demonsrated in
the paper [4], good variational wave functions of the ground state are
\begin{equation}
\Psi_{03}(r;d) = \Psi_{03}(r) j_0(\pi r/d)
\end{equation}
in 3\,D, and
\begin{equation}
\Psi_{02}(r;d) = \Psi_{02}(r) J_0(2.404826 r/d)
\end{equation}
in 2\,D.  d is the diameter of the sphere and of the circle. $j_0$ is the 
spherical Bessel function and $J_0$ is the zeroth-order Bessel function.
As in infinite space, the ground state energy of the non-interacting system 
must be subtracted. The energy of two free
particles is ~$ \frac{C_i}{d^2}$,~i=2,3, where 
$C_2= \hbar^2\,(2.404826)^2/2\mu$ in 2\,D and
~ $C_3= \hbar^2\pi^2/2\mu$ in 3\,D. The results for   d=50\, \AA~ and
d=100\, \AA~ are
presented in Table II. 


\section{Discussion}

Let us mention that only the helium 4 dimer in 3 D has been observed 
experimentally [7-9] up to now.\\ 
As it is seen in Table 1, the binding
energies for all helium molecules are consistent with the lemma.
Moreover both lighter molecules are not bound at all in 3~D.   
Two particles may be kept in 2~D  space by an external potential. It can be 
realized, for example, in a space between two close, parallel big plates.  
Similarly interior of a long and thin cylinder may represent 1~D space.
Of course these "confining" external potentials are not included
in binding energies cited in Table 1.

Since in restricted geometry (in our case sphere and cylinder) the external 
potentials are included partly, the lemma can not be valid. Of 
course it is correct in this case as well, if parameters of the geometry (for 
instance in our case 
diameter of sphere or cylinder) are much biger than the effective range of the 
interaction potential. Such behaviour can be recognized in Table 2. \\
From the "exact" numerical solution of the Schroedinger eq. [4] we know that all 
combinations of two helium atoms
are bound in finite space (in the above sense); the same is true in infinite space 
except for
two atoms of $^{3}$He and one atom $^{3}$He and one atom $^{4}$He which are 
not bound in 3~D. 
Let us notice that our trial function in the case of ($^{3}$He)$_2$ is not 
good enough to reproduce binding in 2~D in both infinite and finite space.
 As comparision with numerical solution shows, it is quite 
good for other cases.

The zero point energy of the relative motion may be obtained by calculating the 
root-mean-square
deviation $\Delta$r $({\it rms})$ . For the dimer helium 4 in 3~D it is found 
numerically [4]  that the expectation 
value of the coordinate <r>=47.8 \AA~ and $\Delta$r = 44.3 \AA	, which 
leads to 
the uncertainty in energy 2.05 mK. So we may conclude that the zero-point 
energy of relative motion is of the order of binding energy of the dimer 
helium 4. The corresponding
de Broglie wavelenght and the mean speed are then 556.7 \AA~ and $4.75m/s$
respectively.\\
It is shown that in confined space the root-mean-square deviation $\Delta$r
depends on the diameter of the "box" almost linearly.
 This dependence disappears for helium 4 dimer for the diameters greater than
 about 28 \AA~ in 2~D and 100 \AA~ in 3~D.
Thus the de Broglie wavelenght of the zero-point relative motion dependens on the 
diameter of 
the "box" and increases with it up to the value which it has in infinite 
space. Let us mention that in above consideration the center of mass motion
is not included.

It seems that an interior of a cylinder is a form which could be
the easiest to be realized in an experiment. Although we haven't solved this problem 
theoretically, the main energetic characteristics are given by our 
spherical-models in 3~D and 2~D.

Finally we mention that our calculation in finite space is an
approximative one. Namely we assumed that the center of mass of two particles was
located in the center of space symmetry. It was shown in Ref. [4] that this 
approximation gives the general features of the systems considered.  

A further approximation has been performed in using the same form of the potential
in all dimensions. The potential (14) has been obtained for a pure physical 
situation in 3~D. We expect that our approximation is valid for  physical 
situations in which two- and one- dimensional motion are dominant, like in films 
and nanotubes.


\newpage
\begin{thebibliography}{99}

\bibitem[1]{}
T. Korona {\it et~al.}, J. Chem. Phys. {\bf 106},  5109  (1997).

\bibitem[2]{}
A.~R. Janzen and R.~A. Aziz, J. Chem. Phys. {\bf 107},  (1997).


\bibitem[3]{}
L. Bruch, J. Chem. Phys. {\bf 110},  2410  (1999).
											

\bibitem[4]{}
S. Kili†, E. Krotscheck and R. Zillich, accepted in J. Low Temp. Phys. (1999).

\bibitem[5]{}
S. Kili$\acute{\rm c}$ and S. Sunari$\acute{\rm c}$, Fizika {\bf 11},  225
  (1979).

\bibitem[6]{}
R.~A. Aziz, A.~R. Janzen, and M.~R. Moldover, Phys. Rev. Lett. {\bf 74},  1586
  (1995).


\bibitem[7]{}
F. Luo, G.C. McBane, G. Kim, C.F. Giese and W.R. Gentry, J. Chem. Phys.
{\bf 98}, 3564 (1993).
\bibitem[8]{}
F. Luo, C.F. Giese and W.R. Gentry, J. Chem. Phys. {\bf104}, 1151 (1996).
\bibitem[9]{}
W. Schollkopf and JP. Toennies, J. Chem. Phys. {\bf104}, 1155 (1996).

\end{thebibliography}
\section{ACKNOWLEDGEMENTS}
The authors would like to thank Professor E. Krotscheck
and R. Zillich for providing results prior to publication.
\newpage
\noindent
\begin{table}
\caption{Binding energies in infinite space (in mK) of helium 
molecules in 2\,D  and for dimer $(^4He)_2$ in 3\,D (second line), 
derived by numerical solving Schroedinger eq. 
and in variational procedure; variational values are in round brackets; 
parameters: a (in \AA), $\gamma$ (dimensionless) and s (in \AA $^{-1}$),
are shown for the SAPT potential only. Note that our variational wave 
function is not flexible enough to
predict a bound state of the $(^3$He$)_2$ dimer in 2\,D and that molecules 
$(^3He)_2$ and $^3He - ^4He$ are not bound in 3\,D.}
\vspace{1cm}
\begin{tabular}{|c|ccccc|}
\hline
Molecule & HFD-B3-FCI1$^a$ & SAPT$^b$ & a &$\gamma$ & s  \\ \hline 
$(^4He)_2$\,  & -39.4 (-37.7)& -40.7 (-39.93)  &2.758 &4.408 &0.047   \\ \cline{2-6}
              & -1.559 (-1.480)& -1.871 (-1.762)  &2.737 &4.49 &0.012 \\ \hline                 
$(^3He)_2$   & -0.016 & -0.02 &  &  & \\  												
$^3He - ^4He$ &-4.0 (-3.21) &-4.3 (-3.51) &2.761 &4.173  & 0.011 \\ \hline
\end{tabular}
\end{table}
\\
\noindent
$^a$ Ref. [6] \\ 
$^b$ Ref. [1] \\ 
%%%%%
\newpage
\noindent
\begin{table}
\caption{Binding energies (in mK) of helium 
molecules in a sphere (3\,D, first line) and in a circle (2\,D, second line)
  derived 
in variational procedure for the SAPT potential; the diameter of both 
confinements are d=50 \AA ~and
d=100 \AA; 
parameters: a (in \AA), $\gamma$ (dimensionless) and s (in \AA $^{-1}$),
are shown for d=50 \AA.}
\vspace{1cm}
\begin{tabular}{|c|ccccc|}
\hline
Molecule & 50 & 100 & a &$\gamma$ & s  \\ \hline 
$(^4He)_2$\,  & -138.713 &-40.650 & 2.753 &4.41 & -0.013 \\ \cline{2-6}
             & -61.660 & -52.133 & 2.767  & 4.36 & 0.02 \\ \hline                  
$(^3He)_2$ & -67.086 & -10.191 & 2.782  &3.91 &-0.058 \\ \cline{2-6}
            & 73.159 & 14.827 &2.798  &3.87  &-0.029   \\ \hline
$^3He - ^4He$  &-94.354  &-19.936  &2.774 & 4.10 &-0.042  \\  \cline{2-6}
    &16.718  &-7.264  &2.794   &4.04  &-0.011 \\  
\hline
\end{tabular}
\end{table}
\\
\end{document}