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The Journal of Chemical Physics

The quantum parameter and critical binding of helium dimers

H. B. Ghassib

Published May 1, 1984• 1 citation
Publisher

Abstract

A recent paper by Uang and Stwalley is taken as a starting point for relating the quantum parameter to the critical binding of diatomic helium molecules in a simple and direct manner. It is conjectured that the relations examined here may also prove to be useful in investigating the general problem of dimers and n-mers in helium fluids within a unified picture.

DOI
10.1063/1.447208

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A recent paper by Uang and Stwalley is taken as a starting point for relating the quantum parameter to the critical binding of diatomic helium molecules in a simple and direct manner. It is conjectured that the relations examined here may also prove to be useful in investigating the general problem of dimers and n-mers in helium fluids within a unified picture.
The behaviour of two bodies that are just bound or nearly bound is discussed. A class of potentials is given for which Schrödinger's equation has exact solutions at critical binding (zero binding energy). This class includes the known solution for the 6–10 potential. For a general potential characterized by a coupling parameter α, it is shown that the bound state energy tends to zero as -(α - α0)2, where α0 is the critical value of the coupling parameter. Small energy scattering of atoms which are near critical binding (e.g. helium atoms) is examined. It is shown that determination of the total cross-section up to terms of order k 2 is in principle sufficient to distinguish between bound and virtual states of the diatomic molecule.
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.
Long-range states of diatomic molecules have average values of internuclear separations $\ensuremath{\langle}R\ensuremath{\rangle}$ at least one order of magnitude larger than the equilibrium value of RRR. For example, the helium dimer 4He2{{}^{4}\mathrm{He}}_{2}4He2​ has a single bound state with $\ensuremath{\langle}R\ensuremath{\rangle}$ of about 50 \AA{}. We show that the properties of these states, such as $\ensuremath{\langle}R\ensuremath{\rangle}$, the dissociation energy, or the sss-wave scattering length, can be related by simple, yet very accurate, formulas if a potential-energy curve is known. By examining a range of ab initio and empirical helium dimer potentials as well as scaling these potentials, we found that the formulas remain accurate, even if very approximate potentials were used. In addition to 4He2{{}^{4}\mathrm{He}}_{2}4He2​, we present results for 9Be2,20Ne2{{}^{9}\mathrm{Be}}_{2},\phantom{\rule{0.28em}{0ex}}{{}^{20}\mathrm{Ne}}_{2}9Be2​,20Ne2​, and KRb.