|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
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
About thirty years ago it was demonstrated  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 4He, 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 . Using similar procedure diatomic helium molecules have been studied in external holding potential that depends on one coordinate as well . 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 . 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. . 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 form was a product of the functions, which describe short and long range correlations divided by the square root of the distance:
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.
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 and the equation which demands continuity of the first derivative disappears (with the constraint that):
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:
The coefficients are given in order in which they are calculated.
3 Variational calculation of the dimer
Having derived the trial wave function we performed a variational calculation
is the reduced mass of 3He, m = 5.00649231 10-27 kg and . Than, the expression for the energy can be written in the form
For the interatomic potential we used ab initio SAPT potential by Korona et al. . After adding the retardation effects (SAPT1 and SAPT2 versions) Janzen and Aziz  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  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.
In the limit the wave function has the asymptotic form, where s0 is determined by relation . The value of s0 coincides with the value of 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 root-mean-square (rms) deviation r for the (3He)2.
The obtained values of < r > = 651 Å and r = 562 Å show that (3He)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 . Since the value of our binding energy is a bit higher than the one in ref. , 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,
, , (27)
and the function f4(r) is defined for where =1.1 Å. From the condition that the function and the first derivative are continuous in two relations for parameters d1 and d2 are obtained,
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
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  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  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
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
where 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:
and , (34)
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  it is also antisymmetric under the exchange of particles 2 and 3 as well as 1 and 3.
The Hamiltonian of the system is
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:
C is a constant, , and is the angle between and . 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 = 4503 Å and = 3633 Å.
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 one-dimensional 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  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 = 4503 Å and = 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 . 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  that lies between the HFD-B3-FCI1  and SAPT2  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.
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.
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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.
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.