Spectra of helium clusters with up to six atoms using soft core potentials

Abstract

In this work we investigate small clusters of helium atoms using the
hyperspherical harmonic basis. We consider systems with $A=2,3,4,5,6$ atoms
with an inter-particle potential which does not present a strong repulsion at
short distances. We use an attractive gaussian potential that reproduces the
values of the dimer binding energy, the atom-atom scattering length, and the
effective range obtained with one of the widely used He-He interactions, the
LM2M2 potential. In systems with more than two atoms we consider a repulsive
three-body force that, by construction, reproduces the trimer binding energy of
the LM2M2 potential. With this model, consisting in the sum of a two- and
three-body potential, we have calculated the spectrum of clusters formed by
four, five and six helium atoms. We have found that these systems present two
bound states, one deep and one shallow close to the threshold fixed by the
energy of the $(A-1)$-atom system. Universal relations between the energies of
the excited state of the $A$-atom system and the ground state energy of the
$(A-1)$-atom system are extracted as well as the ratio between the ground state
of the $A$-atom system and the ground state energy of the trimer.

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arXiv:1106.3853v1 [physics.atm-clus] 20 Jun 2011
A ≤ 6 helium clusters using soft core potentials
M. Gattobigio,1A. Kievsky,2and M. Viviani2
1Universit´ e de Nice-Sophia Antipolis,
Institut Non-Lin´ eaire de Nice, CNRS,
1361 route des Lucioles, 06560 Valbonne, France
2Istituto Nazionale di Fisica Nucleare,
Largo Pontecorvo 3, 56100 Pisa, Italy
Abstract
In this work we investigate small clusters of helium atoms using the hyperspherical harmonic
basis. We consider systems with A = 2,3,4,5,6 atoms with an inter-particle potential which
does not present a strong repulsion at short distances. We use an attractive gaussian potential
that reproduces the values of the dimer binding energy, the atom-atom scattering length, and the
effective range obtained with one of the widely used He-He interactions, the LM2M2 potential. In
systems with more than two atoms we consider a repulsive three-body force that, by construction,
reproduces the trimer binding energy of the LM2M2 potential. With this model, consisting in
the sum of a two- and three-body potential, we have calculated the spectrum of clusters formed
by four, five and six helium atoms. We have found that these systems present two bound states,
one deep and one shallow close to the threshold fixed by the energy of the (A − 1)-atom system.
Universal relations between the energies of the excited state of the A-atom system and the ground
state energy of the (A−1)-atom system are extracted as well as the ratio between the ground state
of the A-atom system and the ground state energy of the trimer.
PACS numbers: 31.15.xj, 31.15.xt, 36.90.+f, 34.10.+x
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I.INTRODUCTION
Systems composed by few helium atoms have been object of intense investigation from
a theoretical and experimental point of view. The existence of the He-He molecule was
experimentally established using diffraction experiments [1–4]. Its binding energy E2bhas
been estimated to be around 1 mK and its scattering length a0around 190 a.u. This makes
the He-He molecule one of the biggest diatomic molecules. On the theoretical side, several
He-He potentials have been proposed; in spite of different details and derivations, all of
them share the common feature of a sharp repulsion below an inter-particle distance of
approximately 5 a.u..
Another important characteristic of the He-He interactions is their effective range r0≈
13 a.u.. Accordingly, the ratio a0/r0is large enough (> 10) to place small helium clusters into
the frame of Efimov physics [5, 6]. As shown by Efimov, when at least two of the two-body
subsystems present an infinitely large scattering length (or zero binding energy) an infinite
sequence of bound states (called Efimov states) appear in the three-body system; their
binding energies scale in a geometrical way and they accumulate at zero energy. The scaling
factor, e−2π/s0≈ 1/515.03, is universal and depends only on the ratio between particle masses
(for three identical bosons s0≈ 1.00624), not on the details of the two-body interaction (see
Ref. [7] for a review). For a finite a0/r0ratio, the number of the Efimov states has been
estimated to be N = (s0/π)ln|a0/r0| [6]; for example, the (bosonic) three4He system presents
an excited state just below the atom-dimer threshold that has been identified as an Efimov
state.
Triggered by this interesting fact, several investigations of the helium trimer have been
produced, establishing that its excited state is indeed an Efimov-like state (see for example
Refs. [8–10]). In addition, analysis of the atom-dimer collision in the ultracold regime have
also been performed [11–13].
One of the main difficulties in solving the quantum mechanical problem in the case of
three helium atoms results to be the treatment of the strong repulsion at short distances of
the He-He potential. Specific algorithms have been developed so far to solve this problem.
The Faddeev equation has been opportunely modified [14]. Moreover, the hyperspherical
adiabatic (HA) expansion has been extensively used in this case (for a review see Ref. [15]).
However, due to the difficulties in treating the strong repulsion, few calculations exist for
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systems with more than three helium atoms. For example, in Ref. [16] the diffusion Monte
Carlo method has been used to describe the ground state of He molecules up to 10 atoms.
On the other hand, description of few-atoms systems using soft-core potentials are currently
operated (see for example Ref. [17]).
Therefore, the equivalence between hard- or soft-core-potential descriptions needs some
clarification. In a recent work [18], an attractive He-He gaussian potential has been used
to investigate the three4He system. In absence of direct experimental data, the two-body
potential has been designed to reproduce the helium dimer binding energy E2b, the He-He
scattering length a0, and the effective range r0of the LM2M2 potential [19]. This gaussian
potential can be considered as a regularized-two-body contact term in an Effective Field
Theory (EFT) approximation of the physics driven by the LM2M2 potential [20]. It should
be noticed that two potentials predicting similar values of a0and r0predict similar phase
shifts in the low energy limit and, therefore, even if their shape is completely different, they
describe in an equivalent way the physical processes in that limit [20]. The equivalence is
lost as the energy is increased, when the details of the potential become more and more
important.
Extending the study to the three-body system, differences between the attractive gaussian
and the LM2M2 potentials are immediately observed. For example the trimer ground state
energies differ by more than 15% (see Table I). A natural way to restore the equivalence
between the two potentials is by the addition of a three-body soft-term force to the gaussian
potential. On the other hand in an EFT treatment of the three-boson system with large
scattering length, a three-body-contact term is required at leading order (LO). Its strength is
usually determined by fixing a three-body observable as for example one of the trimer bound
state energies. After this choice cut-off independent results are obtained [21]. Following this
ideas, and based on Ref. [18], in the present work we have considered a gaussian-hypercentral
three-body force with the strength fixed to reproduce the LM2M2 ground state binding
energy of the three-atom system. The quality of this description has been studied for
different ranges of the three-body force.
Using the two-atom and three-atom systems to fix the model interaction, we have analyzed
heavier systems, up to A = 6 atoms. The numerical calculations are performed by means of
the hyperspherical harmonic (HH) expansion method with the technique developed recently
by the authors in Ref. [22]. In this approach, the authors use the HH basis, without a
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previous symmetrization procedure, to describe bound states in systems up to six particles.
The method is based in a particular representation of the Hamiltonian matrix, as a sum
of products of sparse matrices, well suited for a numerical implementation.Converged
results for different eigenvalues, with the corresponding eigenvectors belonging to different
symmetries, have been obtained. As a novelty, in the present work we extend the formalism
to treat a three-body force. Moreover, as we are dealing with atoms of4He, only the the
spectrum corresponding to totally symmetric eigenstates are of interest.
After fixing the strength of the three-body force to correctly describe the LM2M2 three-
body ground state E(0)
3b= 126.4 mK, we have calculated the first three levels of the spectrum
with total angular momentum L = 0 of the A = 4,5,6 systems. In the three cases we have
found that the first two levels are bosonic bound states, one deep, E(0)
E(1)
Ab, and one very shallow,
Ab, close to the threshold formed by the A − 1 system plus one atom. The third state in
all cases belongs to a mixed symmetry with an energy above the threshold and therefore
not representing a bound state. The appearance of only two bound states in this systems
is in agreement to previous calculations [23]. This fact has been observed in A = 4 and
interpreted as a consequence of the Efimov like spectrum of the A = 3 system [24]. It should
be noticed that, whereas converged results can be found in the literature for the ground
state of the many atom systems, the energy of the excited states is much more difficult to
calculate and only rough estimates are available.
To gain insight on the shallow state, we have varied the range of the three-body force
(maintaining fixed the three-body ground state energy) and we have studied the effects of
that variation in the A = 4,5,6 spectrum. In the range considered, the variation produces
small effects in the eigenvalues; however, it is crucial to determine if the shallow state is
bound or not with respect to the A − 1 threshold. Interestingly, we have observed that
when the ranges of the two- and three-body forces have a ratio of about
between the shallow- and ground-state energy is E(1)
√2, the ratio
Ab/E(0)
A−1b≈ 1.01 − 1.02, in agreement
with Refs. [25, 26]. This analysis confirms previous observations that each Efimov state in
the A = 3 system produces two bound states in the A = 4 system. Furthermore, we have
found E(0)
4b/E(0)
3b≈ 4.5, E(0)
5b/E(0)
3b≈ 10.5, and E(0)
6b/E(0)
3b≈ 18.5, which is in agreement with
Refs. [26, 27].
The paper is organized as follows. In the next Section II we describe the two- and three-
body forces we used in our calculations to reproduce LM2M2 data. In Section III the results
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for the bound states of the A = 3,4,5,6 He clusters are collected whereas the conclusions
are given in the last section. Some technical details of the method have been summarized
in the Appendix.
II. SOFT-CORE TWO- AND THREE-BODY HELIUM POTENTIAL
As mentioned in the Introduction, the4He-4He interaction presents a strong repulsion at
short distances, below 5 a.u. This characteristic makes it difficult a detailed description of the
system with more than four atoms. Accordingly, in the present work we have studied small
clusters of helium interacting through a soft-core two- and three-body potentials which can
be interpreted as regularized two- and three-body contact terms in a LO-EFT approximation
of LM2M2.
Following Refs. [10, 18] we use the gaussian two-body potential
V (r) = V0e−r2/R2, (1)
with V0= −1.227 K and R = 10.03 a.u.. In the following we use ?2/m = 43.281307 (a.u.)2K.
This parametrization of the two-body potential approximately reproduces the dimer binding
energy E2b, the atom-atom scattering length a0 and the effective range r0 given by the
LM2M2 potential. Specifically, the results for the gaussian potential are E2b= −1.296 mK,
a0= 189.95 a.u. and r0= 13.85 a.u., to be compared to the corresponding LM2M2 values
E2b= −1.302 mK, a0= 189.05 a.u. and r0= 13.84 a.u.. As shown in Ref. [18], the use
of the gaussian potential in the three-atom system produces a ground state binding energy
E(0)
3b= 150.0 mK, which is appreciable bigger than the LM2M2 helium trimer ground state
binding energy of 126.4 mK. A smaller difference, though still appreciable, is observed in
the first excited state (see Table I).
In order to have a closer description to the A = 3 system obtained with the LM2M2
potential, we introduce the following three-body interaction
W(ρijk) = W0e−ρ2
ijk/ρ2
0, (2)
where ρ2
ijk=
2
3(r2
ij+ r2
jk+ r2
ki) is the three-body hyperradius in terms of the distances of
the three interacting particles. Moreover, the strength W0is fixed to reproduce the LM2M2
helium trimer binding energy of 126.4 mK. We have studied different cases by varying the
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