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International Journal of Quantum Chemistry

Models for low‐temperature helium dimers and trimers

LAWRENCE L. JUN. LOHR, S. M. Blinder

Published Oct 2, 2001• 5 citations
Publisher

Abstract

Abstract We have shown that a simple model based on pairwise deltafunction potentials can, with an appropriate choice of parameters, represent the general behavior of ultra‐low temperature helium dimers and trimers, including all possible isotopomers. The species 4He2, 4He3, and 4He23He are stable, with binding energies of approximately 1.3, 100, and 10 mK, respectively. An Efimov state for 4He23He is also predicted, bound by 2.4 mK. The remaining dimers and trimers, 4He 3He, 3He2, 4He 3He2, and 3He3, are evidently unstable. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001

DOI
10.1002/qua.1807

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Abstract We have shown that a simple model based on pairwise deltafunction potentials can, with an appropriate choice of parameters, represent the general behavior of ultra‐low temperature helium dimers and trimers, including all possible isotopomers. The species 4He2, 4He3, and 4He23He are stable, with binding energies of approximately 1.3, 100, and 10 mK, respectively. An Efimov state for 4He23He is also predicted, bound by 2.4 mK. The remaining dimers and trimers, 4He 3He, 3He2, 4He 3He2, and 3He3, are evidently unstable. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001
Abstract The helium dimer 4He2 has recently been detected, confirming earlier ab initio predictions of stability for a single bound state with binding energy of 1.310 mK. The predicted potential minimum is at 2.96 Å, with a radial distribution function peaking at 6.96 Å. We model this system using a Dirac bubble potential, which also admits just one bound j = 0 state. With the bubble located at 6.96 Å, an overlap of 0.9994 with the ab initio wave function is obtained. An average internuclear distance of 52.6 Å is calculated, in good agreement with the ab initio result. The root mean square deviation from the mean, 48.0 Å, indicates an enormous spread of the radial wave function. Also consistent with our model is the absence of bound states for the isotopic variants 3He4He and 3He2. Cross sections for helium‐helium scattering are also computed, using both a partial‐wave expansion and the Born approximation. General trends in the energy dependence of the total cross section are accounted for, in qualitative agreement with experimental results. © 1995 John Wiley & Sons, Inc.
The Dirac bubble potential, previously used to model the helium dimer, is applied to the trimer. It is shown that the quantum mechanical kinetic energy operator for a three-body system contains terms over and above the analogues of classical pairwise contributions. The additional terms, the ‘Borromean couplings’, are responsible for a dramatic increase in the binding energy of the trimer. For example, 4He3 is bounded by two orders of magnitude more strongly than 4He2, 279 mK versus 1.31 mK, respectively, according to our calculations. Moreover, the trimer is considerably more compact than the dimer, with (r12) decreasing to 9.01 Å from 51.9 Å.
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.