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The Journal of Chemical Physics

Faddeev–UPE theory and the three 4He-atom system: Bound states and zero-energy atom–diatom scattering

T. K. Lim, K E Duffy

Published Jan 15, 1978• 14 citations
Publisher

Abstract

Faddeev–UPE theory is applied to the trimer bound states and atom–diatom scattering of 4He. (4He)3 has two bound states while the scattering length of 4He– (4He)2 is 207 Å.
DOI
10.1063/1.435735

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The ground-state energies of systems of three and four identical heavy rare-gas atoms, treated as spinless bosons and assumed to interact via pairwise Morse potentials, are calculated using five different methods. Self-bound trimers and tetramers are found in every case with upper and lower bound formulas yielding energies which are extremely close. For these near-classical and massive systems, the Faddeev–UPE method is decidedly poor. On the other hand, the simple generalization of Bruch–Sawada’s upper bound is particularly effective. The plight of Faddeev–UPE here implies that great caution must be exercised in using the unitary pole expansion in chemical physics. ATMS is excellent and appears the method of choice for bound-state problems.

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arXiv:cond-mat/0203421•Physical Review A•Apr 11, 2003•70 citations
Using the HFDHE2 He-He potential of Aziz et al. [J. Chem. Phys. 90, 4330 (1979)] with the Faddeev-Yakubovsky theory, we have calculated the 4He^{4}\mathrm{He}4He atom-triatom scattering length and found it to be - 116 \AA{}. We reduced the sss-wave two-body ttt matrix to a separable form through the unitary-pole approximation and did the same for the sss-wave [3+1] and [2+2] subamplitudes with the energy-dependent-pole approximation in our work. Also, the binding energies of all the states in the trimer and tetramer, and the atom-diatom scattering length, already determined in our previous work with other potentials, were recalculated for the new interaction. We found an excited trimer bound state, apparently not an Efimov state, near the atom-diatom threshold and an atom-diatom scattering length of 195 \AA{}. A feature of our results was the absence of an excited state in the tetramer, but in its place, was a resonancelike state which led to the large atom-triatom scattering length.
The system of three 4He atoms interacting through the intermolecular potential HFDHE2 is solved exactly within the Faddeev scheme. We find that the trimer has two bound states at E=−0.11 K and E=−0.16(−2) K, where the excited state shows the property of an Efimov state. It is overrun by the two-body fragmentation threshold for an artificially enhanced overall interaction strength. Uncertainties in the potential parameters, possibly combined with solvent effects, may allow a slightly weaker force, which supports a second Efimov state. In any case dynamical effects of the Efimov states will be the same whether it lies close to E=0 on the physical or unphysical sheet.
The ground-state energies of systems of three and four identical heavy rare-gas atoms, treated as spinless bosons and assumed to interact via pairwise Morse potentials, are calculated using five different methods. Self-bound trimers and tetramers are found in every case with upper and lower bound formulas yielding energies which are extremely close. For these near-classical and massive systems, the Faddeev–UPE method is decidedly poor. On the other hand, the simple generalization of Bruch–Sawada’s upper bound is particularly effective. The plight of Faddeev–UPE here implies that great caution must be exercised in using the unitary pole expansion in chemical physics. ATMS is excellent and appears the method of choice for bound-state problems.
The two-body scattering length a for 4He{}^{4}\mathrm{He}4He atoms is much larger than their effective range rs.{r}_{s}.rs​. As a consequence, low-energy few-body observables have universal characteristics that are independent of the interaction potential. Universality implies that, up to corrections suppressed by rs/a,{r}_{s}/a,rs​/a, all low-energy three-body observables are determined by a and a three-body parameter ${\ensuremath{\Lambda}}_{*}.$ We give simple expressions in terms of a and ${\ensuremath{\Lambda}}_{*}$ for the trimer binding-energy equation, the atom-dimer scattering phase shifts, and the rate for three-body recombination at threshold. We determine ${\ensuremath{\Lambda}}_{*}$ for several 4He{}^{4}\mathrm{He}4He potentials from the calculated binding energy of the excited state of the trimer and use it to obtain the universality predictions for the other low-energy observables. We also use the calculated values for one potential to estimate the effective range corrections for the other potentials.