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The Journal of Chemical Physics

On the existence of bound helium trimers by the equivalent two-body method

T. K. Lim, Michael A. Zuniga

Published Sep 1, 1975• 16 citations
Publisher

Abstract

The binding energies of the 3,4He trimers are computed for a range of realistic interatomic potentials using the equivalent two-body method of Bodmer and Ali. We find a 4He trimer bound with a binding energy of about 0.25°K and a 3He trimer which is but loosely bound. We also confirm that the Hall–Post–Stenschke method gives poor lower bounds to the energy and that (4He)3 is large, having a root mean square interparticle spacing of 23 Å.

DOI
10.1063/1.431609

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Annalen der Physik•Jan 1, 1991•0 citations
The binding energies of the 3,4He trimers are computed for a range of realistic interatomic potentials using the equivalent two-body method of Bodmer and Ali. We find a 4He trimer bound with a binding energy of about 0.25°K and a 3He trimer which is but loosely bound. We also confirm that the Hall–Post–Stenschke method gives poor lower bounds to the energy and that (4He)3 is large, having a root mean square interparticle spacing of 23 Å.
Variational calculations of the ground state energy of 4He trimers are reported and are compared with the Hall-Post-Stenschke lower bound on the ground state energy. The variational result for the ground state energy for a range of model pair potentials is in the range −0.05 to −0.2 K; the effect of the triplet nonadditive term is estimated to be less than 1% of the contribution of the pair potentials. A study of the variation of the ground state energy of 3 bosons with pair potential coupling constant is also reported for a Lennard-Jones and a Morse potential model. There is no bound spin 3/2 3He trimer; a lower bound is given for the ground state energy of the spin 1/2 trimer and analogy with the boson results is used to argue that the spin 1/2 trimer is probably not bound. This paper contains computational applications of formal results derived elsewhere.
The three-body Schrdinger equation is approximately solved in the hyperspherical coordinates and the binding energies of the three-body weakly bound systems are calculated with the purpose to find if He-He-Ba trimers could exist. Using the special feature of the B-spline function like the flexible and highly localized properties, hypersphercial potentials are obtained by modifying the knots distribution of the B-spline basis of different weakly bound three-atom systems. Employing the best empirical interaction potentials between each pair of particles, we obtain that in the ground state binding energies of the weakly bound typical three-atom systems, the bindings of the molecules, 4He-4He-138Ba, 4He-3He-138Ba and 3He-3He-138Ba are possible. The binding energies of these systems are shown in the order of 1 Kelvin, each system could support only one bound state. These weakly bound molecules can exist only in a very cold environment. To get insight into the geometry of the molecules, the features of the channel functions associated with the hyperspherical potential curves of each system are investigated.
The best available intermolecular potential for helium by Aziz, Nain, Carley, Taylor, and McConville is shown here for the first time to have a 4He2 bound state. Two numerical analyses, namely, eigenvalue solution and effective range theory, are used to support this conclusion. Unlike usual chemically bound species, the binding energy of this very weakly bound level is found to be only 8.3×10−4 K, which is four orders of magnitude smaller than the potential well depth ε = 10.8 K. The scattering length for He+He collisions, determined from effective range theory, is used to calculate the elastic cross section in the very low energy limit. The results (1.878×105 Å2 for 4He+4He and 6.035×102 Å for 3He+3He) are consistent with measurements at the lowest velocities yet attained. In terms of the estimated uncertainties of the parameters of the potential of Aziz and co-workers, it is shown that it is very likely that a bound state of the 4He2 molecule does in fact exist.
Abstract We extend Blatt's and Lim's calculations for the energy of the helium trimer (4He)3, for a Herzfeld potential. We use the unitary pole approximation. We constrain the effective range and scattering lenght to values of 7.4 A and infinity, respectively. As the core radius increases from 0 to 2.8 A, the trimer energy increases monotonically from −97 to −59 millikelvin. The Yamaguchi potential gives a trimer energy of −123 mK. Two additional two‐body parameters suffice to fit these, and earlier, calculations of the trimer energy: i.e., the expectation value of the kinetic energy for the dimer, and Sprung's wound integral Io.