Variational study of fermionic helium dimer
and trimer in two dimensions
It is dedicated to Academician Krunoslav Ljolje
in honor of his 70th birthday
L.Vranješ and S. Kilić
Faculty of Natural Science, University of Split,
21000 Split, Croatia
March 15, 2000
Abstact
In variational calculation we obtain binding energy of helium 3 dimer in two
dimensions. The existence of one bound state, with binding energy -0.014 mK,
has been definitively found. Also, the existence of a binding state of helium 3
trimer having spin-1/2 with the energy below -0.0057 mK is indicated. This
reopens the question of the existence of the gas phase of many helium 3 atoms on
a surface of superfluid helium 4.
PACS: 36.90. +f, 31.20. Di
1
1 Introduction
About thirty years ago it was demonstrated [1] that in dilute bulk 3He - 4He
solution atoms of 3He prefer to float on the surface of the 4He rather than to be
dissolved in the bulk. All atoms in the solution are pulled down by gravity. A 3He
atom is less massive than a 4He atom and therefore its zero point motion energy is
greater than that of 4He (for a factor 1.3 approximately). Due to this motion it
tends to have no 4He nearby. This tendency leads it to sit on the surface of the
4
He, where it has empty space above. Thus a 3He atom at low temperatures
(below 0.1 K), on the surface of bulk liquid 4He behaves as a spin-1/2 Fermi
particle in two dimensions.
In our recent papers [2, 3, 4] we have considered binding of helium diatomic
molecules in confined and unconfined geometries. It has been shown that in
infinite space helium fermionic dimer exists only in two dimensions. In confined
geometry two helium atoms were studied in 2 and 3 dimensions. Motion of atoms
has been confined by spherically external holding potentials [2]. Using similar
procedure diatomic helium molecules have been studied in external holding
potential that depends on one coordinate as well [4]. All considered systems
might be thought as models for the interactions between helium atoms in specific
real physical environment. For example, in solid matrices, where helium dimers
form the condensation seed for helium clusters, in nanotubes, with a diameter
between 10 and 100 Å , and in "condensation" on a solid or liquid substrate.
We are not convinced that the atoms of 3He form a gas on a surface. This doubt is
based on the fact that there is one bound state of two 3He atoms in 2 D space with
binding energy of about -0.02mK [2]. This result was achieved after numerical
solving Schrödinger equation. Of course a variational calculation is desired as
well. A successful variational calculation showing binding of helium 3 dimer in 2
D, has not been done so far.
The first goal of this paper is to derive a trial radial wave function and perform
variational calculation in finding binding energy of fermionic helium 3 dimer in 2
D and mean value of the internuclear distance (Sec. 2 and 3.). The second goal is
to examine the possibility of the existence of helium 3 trimer with spin-1/2 (Sec.
4). In Sec. 5 a discussion of our results is presented.
2 A derivation of trial wave function
Very good trial wave functions describing ground state of helium 4 dimer and
molecule consisting of one atom of helium 4 and one atom of helium 3, were
obtained and used in ref. [2]. They describe short-range correlation between two
atoms, like in Jastrow wave function for liquid helium state. Long-range
correlations are described by decreasing exponential function. Comparing our
results with the numerical solution of Schröedinger eq. we found that the best
2
form was a product of the functions, which describe short and long range
correlations divided by the square root of the distance:
  a 

1
(1)

exp      sr  ,
r
  r 

where a,  and s are variational parameters.
Our experience showed that this function, although very good for helium 4 dimer
and 4He-3He molecule, was not enough good to give bound state of the fermionic
helium dimer. This dimer is very large (the largest molecule we know) and
behaviour of the wave function in between short and long range is very
important. Using Gnuplot graphics and data from numerical solution of
Schrödinger equation we were able to construct the following trial wave function.
1
1 6
(2)
( r ) 
f (r) 
fi ( r ) ,
r
r i 1
where
  a 2  a3 
f1 ( r )  a1 exp     ,
r r1 ,r2 
  r  

r  r2 ,r3 
f 2 ( r )  b1 (ln( r  b2 )) b3 ,
  r  r  c3 
  ,
f 3 ( r )  c1 exp   4
  c2  
  r  r  d3 
4
  ,
f 4 ( r )  d1 exp  
  d 2  
  r  r  e3 
4
  ,
f 5 ( r )  e1 exp  
  e2  
f 6  g1 exp(  sr ) ,
r  r3 ,r4 
r  r4 ,r5 
r  r5 ,r6 
r  r6 , 
and r1=1 Å , r2=2.97 Å, r3=34.57 Å, r4=165.1 Å, r5=228.5 Å, r6=2000 Å. It has
17 parameters a1, a2, a3, b1, b2, b3, c1, c2, c3, d1, d2, d3, e1, e2, e3, g1, s and
8 of them are independent. Namely, using the continuity of the wave function and
first derivative in points r2, to r6, one finds the following nine equations among
the parameters; there would be ten, but our wave function has its maximum at
point r  r4 and the equation which demands continuity of the first derivative
disappears (with the constraint that d 2  1 ):
  a  a3 
b
a1 exp   2    b1 (ln( r2  b2 )) 3 ,
  r2  
(3)
b3
 a2  a2
b3
1
 

,
 r2  r2 r2  b2 ln( r2  b2 )
3
(4)
b1 (ln( r3  b2 ))
b3
  r  r  c3 
 c1 exp   4 3   ,
  c2  
b3
c r r 
1
 3  4 3 
r2  b2 ln( r3  b2 ) c2  c2 
c1  d1 ,
c3 1
,
d 3 1
e r r 
 3  5 4 
e2  e2 
(6)
(7)
  r  r  d3 
  r r  e3 
5
4
   e1 exp   5 4   ,
d1 exp  
  d 2  
  e2  
d 3  r5  r4 


d 2  d 2 
(5)
(8)
e3 1
,
  r r  e3 
e1 exp   6 4    g1 exp(  sr6 ) ,
  e2  
(9)
(10)
e3 1
e3  r6  r4 


 s.
(11)
e2  e2 
We choose the coefficients a2, a3, b2, c1, c3, d2, d3 and s as variational
parameters, and for the others, using relations (3-11), we obtained the following
expressions:
(12)
d1  c1 ,
a3
 a  a3
b3   2 
( r2  b2 ) ln( r2  b2 ) ,
 r2  r2
(13)


c
c2  ( r3  b3 ) ln( r3  b2 ) 3 ( r4  r3 ) c3 1 
b3


  r  r  c3 
exp   4 3  
  c2  
,
b1  c1
(ln( r3  b2 )) b3
1 / c3
,
(14)
(15)
(ln( r2  b2 )) b3
,
a1  b1
  a  a3 
exp   2  
  r2  
(16)
 d2
ln s ( r6  r4 )
 r5  r4
e3 
 r  r4 

ln  6
 r5  r4 
4



d3
1
d3
,
(17)
1 / e3
e 
e2   3 
s
( r6  r4 )11 / e3 ,
(18)
 r  r  e3  r  r  d3 
e1  d1 exp  5 4    5 4   ,
 e2 
 d 2  
  r  r e3 
sr6
g1  e1e exp   6 4   .
  e2  
The coefficients are given in order in which they are calculated.
(19)
(20)
3 Variational calculation of the dimer
Having derived the trial wave function we performed a variational calculation
ˆ r dr
*H
E 
,
(21)
 * r dr


where
 
2
Hˆ  
  Vˆ (| r1  r2 |) ,
2
(22)
 
  m / 2 is the reduced mass of 3He, m = 5.00649231 10-27 kg and r | r1  r2 | .
Than, the expression for the energy can be written in the form

E

0
rdr (V ( r ) 2 ( r ) 


0
2
2
rdr 2 ( r )
(( r )) 2 )
.
(23)
For the interatomic potential we used ab initio SAPT potential by Korona et al.
[6]. After adding the retardation effects (SAPT1 and SAPT2 versions) Janzen and
Aziz [13] showed that SAPT potential recovers the known bulk and scattering
data for helium more accurately than all other existing potentials. To calculate the
integrals we used the Romberg extrapolation method [5] and by a minimization
procedure obtained the binding energy of -0.014 mK. Values of variational
parameters for this energy are: a2=2.873 Å, a3=3.698, b2=1.55 Å, c3=5.9, d2=573
Å, d3=2.0, s=0.0009318 Å-1 . The value of the parameter c1 doesn't affect the
binding energy, but only the normalization integral, in our calculation it has the
value c1 =0.03588. We also used the boundary points r3, r4, r5 and r6 as
variational parameters. Their final values are r3=19.5 Å, r4=199 Å, r5=282 Å and
r6=1200 Å. Other parameters, when calculated from the expressions (12-20) are:
a1=0.01988, b1=0.01456, b3=0.547, c2=217.6 Å, d1=0.03588, e1=0.03634,
e2=1262.9 Å, e3=3.634 and g1=0.05257.
5
In the limit r   the wave function has the asymptotic form e  s0r , where s0 is
determined by relation s0 
2
. The value of s0 coincides with the value of
2 (  )

s, what confirms the correct asymptotic behaviour of the wave function .
The function f(r) and its first derivative are shown in Fig.1.
We also calculated the mean value of internuclear distance < r > and the rootmean-square (rms) deviation r for the (3He)2.
 r ( r )rdr ,
 r 
  ( r )rdr
 r  (r )rdr ,
 r 
  ( r )rdr
2
2
2
(24)
2
2
2
(25)
and
(26)
( r )   r 2    r  2 .
3
The obtained values of < r > = 651 Å and r = 562 Å show that ( He)2 is a
really huge molecule. Our results for the energy and average radius < r > confirm
the results of the numerical calculations from the paper [2]. Since the value of our
binding energy is a bit higher than the one in ref. [2], which is to be expected
from a variational calculation, we also obtained a bigger value of average radius.
This small energy requires a great numerical precision. To verify our numerical
procedure we repeated the whole calculation, with a slightly redefined wave
function and using an equivalent but different expression for the energy. Namely,
the function f3(r) now reads,
  | r  r | c3 
  ,
f 3 ( r )  c1 exp   4
r  r3 ,r4   ,
(27)
  c2  
and the function f4(r) is defined for r  r4   ,r5  where  =1.1 Å. From the
condition that the function and the first derivative are continuous in r4   two
relations for parameters d1 and d2 are obtained,
1/ d3
d 
(28)
d 2   3  c2c3 / d3 11/ e3 ,
 c3 
   d 3    c3 
d1  c1       .
(29)
 d 2 
 c2  
The relations for other parameters (13-20) are left unchanged. With the wave
function defined in this way no singularities in the second derivative of the
function  are expected, and therefore the variational calculation can be
performed using the relation (23) as well as the following relation for the energy
6

E

0
rdr (V ( r ) 2 ( r ) 


0
2
2
( r )( r ))
rdr ( r )
,
(30)
2
where the kinetic energy is expressed through Laplace operator, . The
minimization of energy in both cases gave the same value of - 0.014 mK, which
is the same as the one obtained using the function where there is no displacement
 from the maximum in r4. Thus, we can be certain in applied numerical
procedures.
4 Calculation of trimer with spin-1/2
In 1979 Cabral and Bruch [7] considered the binding of 3He2 and 3He3. They
performed a variational calculation, with the interatomic potentials available at
the time, and concluded that both molecules are probably not bound in 2 D. Our
results for the dimer led us to extend variational calculation to trimer binding.
Since 3He atoms are fermions they form spin-1/2 trimers and spin-3/2 trimers.
The results from [7] indicated that spin-1/2 trimer has a lower energy and
therefore we studied only that case. The chosen form of the variational wave
function, following [7, 8] is
  a X s   s X a ,
(31)
where Xs and Xa are spin doublets symmetric and antisymmetric, respectively,
under exchange of particles 1 and 2 while a and s are space wave functions
which are respectively, antisymmetric and symmetric under exchange of
particles 1 and 2. Spin +1/2 projections of the doublets are
1
(32)
X a ( s z  1 / 2) 
(1  2 3  1 2 3 ) ,
2
1
(33)
X s ( s z  1 / 2) 
(21 2  3  1  2 3  1 2 3 ) ,
6
where  i (  i ) are the usual spin up (down) eigenstates of a spin 1/2 particle and
the subscript i is particle label. In the calculation for the space wave functions we
combine the following forms:
a  ay 0  ( y1  y2 )0
and
a  ax 0  ( x1  x2 )0 ,
(34)
then
 s   sy 0 
1
( y3  y1 )  ( y3  y2 )0
3
 s   sx 0 
1
( x3  x1 )  ( x3  x2 )0 ,
3
and
7
(35)
0 
f ( r12 ) f ( r13 ) f ( r23 )
,
r12
r13
r23
(36)
where f(rij) is the new dimer wave function (2), with the  modification. The
constructed wave function is antisymmetric under exchange of particles 1 and 2
and symmetric under cyclic exchange of particles 1,2 and 3. Therefore [8] it is
also antisymmetric under the exchange of particles 2 and 3 as well as 1 and 3.
The Hamiltonian of the system is
2
(37)
H 
( 1   2   3 )  V ( r12 )  V ( r13 )  V ( r23 ) .
2m
Again a variational ansatz was used to calculate the binding energy (21). Using
the fact that the Hamiltonian is spin independent and symmetric under the
exchange of x and y coordinates we managed to express energy by the following
relations:
I p  I ka  I kb  I kc
,
(38)
E
In
where

2
 


2 
I n  C 1d1  2 d 2
d ( 12  1  2   22 ) 02 ( 1 ,  2 , | 1   2 |) , (39)
0
0
3 0


2
 
 
I p  2C 1d1V ( 1 )  2 d 2
d ( 12  1  2   22 )02 ( 1 ,  2 , | 1   2 |) ,(40)


0
I ka  2C
0
0
 f ( 1 )
2 
1  
  d 2 
1d1 

2  0 2
m 0
f
(

)
4

1
1 



2
0
I kb  2C

 


d ( 12  1  2   22 ) 02 ( 1 ,  2 , | 1   2 |) ,
 f ( 1 )
 f (  2 )
2 
1  
1 
  2 d 2 
 
1d1 


m 0
 f ( 1 ) 2 1  0
 f ( 2 ) 22 
2
 


d cos ( 12  1  2   22 ) 02 ( 1 ,  2 , | 1   2 |) ,

(41)

0
(42)
2
 f ( 1 )


1  
  2 d 2 d 02 ( 1 ,  2 , | 1   2 |) (43)

0
0
 f ( 1 ) 2 1  0
   


 
C is a constant, 1  r1  r2 ,  2  r1  r3 and  is the angle between 1 and  2 .
The fact that expressions for the energy were reduced to three-dimensional
integrals enabled us to perform the calculations using the same numerical
methods as in the dimer case. After time consuming numerical calculations we
found that the upper bound of binding energy of 3He3 trimer was -0.0057mK.
Using the same wave function we derived the average distance among atoms of
 1  = 4503 Å and 1 = 3633 Å.
I kc  3C
2
m


12 d1 

8

5 Discussion
To the best our knowledge the function (2) is the first trial form which in
variational calculation led to the binding of helium 3 dimer. In this case onedimensional Romberg integration with high accuracy has been performed. The
results are in good agreement with those obtained by numerical solving of
Schrödinger equation.
Having appropriate two-body function we were able to construct a special form
of Cabral-Bruch trimer wave function describing state with spin 1/2. We
performed a very accurate Romberg integration of three-dimensional integrals
and found an upper bound to the binding energy of -0.0057 mK. This shows that
helium 3 trimer with spin-1/2 is bound in two dimensions. (As it was showed in
the paper [4] binding of diatomic helium molecules is significantly increased if
they are close (about 3 Å) to the surface of liquid helium. It means that binding of
trimers could be experimentally observed in future.) This result is quite a new
one. It opens the question of the phase of many helium 3 atoms on a surface of
liquid helium 4. So far it is believed that they form two-dimensional gas.
A qualitative estimation of our result for trimer may be done as well. The
obtained values of  1  = 4503 Å and 1 = 3633 Å show that 3He3 is a large
molecule. Let us assume that there is a homogenous monolayer gas (or liquid)
with the average distance between particles as in helium 3 trimer, then its
concentration is 4.9 1012 m-2 . This concentration is several orders lower then one
of 3%, what is the upper limit for the attractive interaction between two 3He
atoms in helium 3 - helium 4 film [14]. Consequently, it may be concluded that in
our case a necessary condition for binding of three helium atoms is satisfied.
Recently, new interatomic helium potentials appeared. Van Mourik and Dunning
computed a new ab initio potential energy curve [9] that lies between the HFDB3-FCI1 [12] and SAPT2 [13] potentials, being closer to SAPT2 potential. Other
authors [10, 11] conclude that, according to their calculations, SAPT potential is
insufficiently repulsive at short distances.
In the papers [2, 3] the binding energy of helium molecules was calculated using
two different potentials, HFD-B3-FCI1 and SAPT. The obtained results didn't
differ much for these two cases. Therefore, we don't expect that the calculations
with new, more precise potentials would change our results appreciably.
6 Acknowledgements
We are indebted Professor E. Krotscheck for many stimulating discussions and R.
Zillich for providing us with data concerning numerical solution of Schrödinger
equation for helium 3 dimer in 2 D.
9
References
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10
f(r)
f'(r)
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
1
10
100
1000
r (Å)
10000
Figure 1: The figure shows the radial wave function f(r) (solid line) and the first
derivative f'(r) (dashed line) of helium 3 in 2 D for the parameters determined by
minimization of the energy.
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SAŽETAK
Varijacijska analiza dimera i trimera helija 3 u dvije dimenzije
L. Vranješ i S. Kilić
Varijacijskim proračunom dobijena je energija vezanja dimera helija 3 u dvije
dimenzije. Postojanje jednog vezanog stanja, s energijom vezanja od -0.014 mK
je definitivno utvrđeno. Također je određena energija vezanja odgovarajućeg
trimera spina-1/2 od -0.0057 mK. Ovaj rezultat otvara dvojbe o tome da atomi
helija 3 formiraju plinsku fazu na ravnoj površini suprafluidnog helija 4.
12