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International Journal of Quantum Chemistry

Deltafunction model for the helium dimer

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

Published Feb 15, 1995• 9 citations

Abstract

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.

DOI: 10.1002/qua.560530407

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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.

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

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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.

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Van der Waals attraction between helium atoms can lead to the formation of weakly-bound He 2 diatomic molecules stable at temperatures in the millikelvin range. Helium dimers were first detected experimentally in 1993 and 1994. Sophisticated quantum-mechanical computations confirm that the He–He bond is the weakest known, with a binding energy corresponding to 1.176 mK, about 10 -4 eV. The average internuclear distance is computed to be in excess of 50 Å, which makes He 2 the largest known diatomic molecule as well. The authors have introduced an idealized deltafunction model for the helium dimer that succinctly represents its remarkable properties. Despite the weakness of each individual He–He bond, an intriguing mechanism exists for helium to form trimers and larger clusters with much greater binding energies.