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arXiv:physics/0107056 · Physical Review A

Variational description of the helium trimer using correlated hyperspherical harmonic basis functions

Paolo Barletta, Alejandro Kievsky

Published Sep 19, 2001• 74 citations
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Abstract

A variational wave function constructed with correlated hyperspherical harmonic functions is used to describe the Helium trimer. This system is known to have a deep bound state. In addition, different potential models predict the existence of a shallow excited state that has been identified as an Efimov state. Using the Rayleigh-Ritz variational principle, the energies and wave functions of both bound states have been obtained by solving a generalized eigenvalue problem. The introduction of a suitable correlation factor reduces considerably the dimension of the basis needed to accurately describe the structure of the system. The most recent helium-helium interactions have been investigated.

DOI
10.1103/physreva.64.042514

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A variational wave function constructed with correlated hyperspherical harmonic functions is used to describe the Helium trimer. This system is known to have a deep bound state. In addition, different potential models predict the existence of a shallow excited state that has been identified as an Efimov state. Using the Rayleigh-Ritz variational principle, the energies and wave functions of both bound states have been obtained by solving a generalized eigenvalue problem. The introduction of a suitable correlation factor reduces considerably the dimension of the basis needed to accurately describe the structure of the system. The most recent helium-helium interactions have been investigated.
A variational wave function constructed with correlated hyperspherical harmonic functions is used to describe the Helium trimer. This system is known to have a deep bound state. In addition, different potential models predict the existence of a shallow excited state that has been identified as an Efimov state. Using the Rayleigh-Ritz variational principle, the energies and wave functions of both bound states have been obtained by solving a generalized eigenvalue problem. The introduction of a suitable correlation factor reduces considerably the dimension of the basis needed to accurately describe the structure of the system. The most recent helium-helium interactions have been investigated.
A correlated many-body basis function is used to describe the (4)He trimer and small helium clusters ((4)He(N)) with N = 4-9. A realistic helium dimer potential is adopted. The ground state results of the (4)He dimer and trimer are in close agreement with earlier findings. But no evidence is found for the existence of Efimov state in the trimer for the actual (4)He-(4)He interaction. However, decreasing the potential strength we calculate several excited states of the trimer which exhibit Efimov character. We also solve for excited state energies of these clusters which are in good agreement with Monte Carlo hyperspherical description.
Exact solutions for the low-temperature helium dimer and trimer, 4He2^{4}\mathrm{He}_{2}4He2​ and 4He3^{4}\mathrm{He}_{3}4He3​, are derived, based on our $\ensuremath{\delta}$ function model for the interatomic potential. For the trimer, the Faddeev equations are shown to be separable in hyperspherical coordinates, with the SSS-wave alone giving an exact solution. The parameters ${\ensuremath{\lambda}}_{0}$ and r0{r}_{0}r0​ are fitted to accurate computations on the dimer and trimer. Excited states of the trimer are shown to exhibit the Efimov effect, whereby artificially reducing the strength of the two-body potential causes an infinite number of weakly-bound levels to condense out of the continuum. All the features anticipated by Efimov are quantitatively reproduced within our model. Since short-range details of the intermolecular forces are not relevant, our results can be considered to be universally applicable.
In this paper, we investigate small clusters of helium atoms using the hyperspherical harmonic basis. We consider systems with A=2,3,4,5,6A=2,3,4,5,6A=2,3,4,5,6 atoms with an interparticle 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 Aziz and Slaman potential, called LM2M2. 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 of 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 $(A\ensuremath{-}1)$-atom system. Universal relations between the energies of the excited state of the AAA-atom system and the ground-state energy of the $(A\ensuremath{-}1)$-atom system are extracted, as well as the ratio between the ground state of the AAA-atom system and the ground-state energy of the trimer.
The adiabatic hyperspherical method is used to investigate the spectrum of the helium trimer and to explore effects on the spectrum of varying the helium masses as well as the interaction strength. When a realistic helium dimer potential is adopted, only two isotopic combinations are observed to have three-body bound states: 4He3^{4}\mathrm{He}_{3}4He3​ and 4He23^{4}\mathrm{He}_{2}^{3}4He23​He. By artificially adjusting the dimer potential, we also determine the range of interaction strengths for which halo states can result. The existence of Efimov states for both physical and unphysical systems is also examined. \textcopyright{} 1996 American Institute of Physics.
Abstract The structural properties of helium triatomic systems are studied using hyperspherical coordinates. A slow variable discretization approach is adopted to solve the three-body Schrödinger equation, in which the Schrödinger equation in hyperangular coordinates is solved using basis splines at a series of fixed FEM-DVR (finite-element methods–discrete variable representation) hyperradii. We focus on studying the geometrical structure of the 4 He 3 and 3 He 4 He 2 triatomic systems. Using the bound state wave functions obtained, we calculate and analyze the one-dimensional pair distribution and angle distribution functions as well as the two-dimensional angle–angle distributions. All these bound states are found to exhibit such a floppy nature that classifying them into particular geometrical shapes does not appear to be sensible. A comparison will be made with some bound states of the neon trimer, which are expected to be more tightly bound.