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The Journal of Chemical Physics

Modern He–He potentials: Another look at binding energy, effective range theory, retardation, and Efimov states

Alec R. Janzen, R. A. Aziz

Published Dec 8, 1995• 90 citations
Publisher

Abstract

We compare a number of helium–helium potentials with respect to their predictions of dimer binding energy, scattering length, effective range and Efimov states. We also study the effect of retardation on the ‘‘best’’ potential. All realistic potentials support a weakly bound dimer, while none supports an Efimov state. We agree with other authors that retardation decreases the binding energy by about 10%. Finally, we investigated the effect on the binding energy from the application of retardation over different ranges of separation. The precise effects of retardation at short range in realistic potentials require further study.

DOI
10.1063/1.469978

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We compare a number of helium–helium potentials with respect to their predictions of dimer binding energy, scattering length, effective range and Efimov states. We also study the effect of retardation on the ‘‘best’’ potential. All realistic potentials support a weakly bound dimer, while none supports an Efimov state. We agree with other authors that retardation decreases the binding energy by about 10%. Finally, we investigated the effect on the binding energy from the application of retardation over different ranges of separation. The precise effects of retardation at short range in realistic potentials require further study.
The modifications of effective-range theory, necessitated by the long-range nature of interatomic dispersion and relativistic retardation forces, are discussed. A numerical comparison is given of parameters describing $^{3}\mathrm{\ensuremath{-}}^{3}$He, $^{3}\mathrm{\ensuremath{-}}^{4}$He, and $^{4}\mathrm{\ensuremath{-}}^{4}$He scattering at very low energies calculated with several recently published interaction potentials. The effect of relativistic retardation on these parameters and on the calculated binding energies of the $^{4}\mathrm{\ensuremath{-}}^{4}$He dimer is shown. It is demonstrated that all of these quantities are extremely sensitive to the potential. Scattering lengths and effective ranges for the lowest triplet interactions of H-H and D-D are also reported.
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.
The best available intermolecular potential for helium by Aziz, Nain, Carley, Taylor, and McConville is shown here for the first time to have a /sup 4/He/sub 2/ 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 x 10/sup -4/ K, which is four orders of magnitude smaller than the potential well depth epsilon = 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 x 10/sup 5/ A/sup 2/ for /sup 4/He+/sup 4/He and 6.035 x 10/sup 2/ A for /sup 3/He+/sup 3/He) 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 /sup 4/He/sub 2/ molecule does in fact exist.
A comparison is made between the number of Efimov States present in 4He^{4}\mathrm{He}4He trimers as calculated using the Faddeev equations in the Faddeev coordinate-momentum approach with estimates predicted by N=(1/\ensuremath{\pi})ln(\ensuremath{\Vert}a\ensuremath{\Vert}/r0{r}_{0}r0​) using realistic phenomenological He-He potentials. It is found that there is good agreement for systems supporting few states and that agreement remains when the Efimov states are forced to disappear by mathematically altering the potentials.
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.