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The Journal of Chemical Physics

Heavy rare-gas trimers and tetramers: Testing ground for few-body techniques in chemical physics

T. K. Lim, K E Duffy, S. Nakaichi, Y. Akaishi, Hajime Tanaka

Published May 15, 1979• 8 citations
Publisher

Abstract

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.

DOI
10.1063/1.437268

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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.
We calculate the universal spectrum of trimer and tetramer states in heteronuclear mixtures of ultracold atoms with different masses in the vicinity of the heavy-light dimer threshold. To extract the energies, we solve the three- and four-body problem for simple two- and three-body potentials tuned to the universal region using the Gaussian expansion method. We focus on the case of one light particle of mass mmm and two or three heavy bosons of mass MMM with resonant heavy-light interactions. We find that trimer and tetramer cross into the heavy-light dimer threshold at almost the same point and that as the mass ratio M/mM/mM/m decreases, the distance between the thresholds for trimer and tetramer states becomes smaller. We also comment on the possibility of observing exotic three-body states consisting of a dimer and two atoms in this region and compare with previous work.
Four-body systems are studied using an effective-field theory with two- and three-body contact interactions. A method to systematically address deep trimers (three-body bound states that are more tightly bound than four-body bound or resonant states) in four-body calculations is developed using a diagrammatic approach. Previous calculations were limited by the existence of deep trimers, which this work overcomes. For cold 4He^{4}\mathrm{He}4He atoms, binding energies of 526.1(5) and 128.517(1) mK are obtained at leading order for the tetramer ground and excited states, respectively, where errors come from the truncation of three-body partial waves. Tetramer binding energies and decay widths are also computed approaching the unitary limit. In the unitary limit, there are two tetramers associated with each trimer of binding energy B3(0){B}_{3}^{(0)}B3(0)​. The binding energy and decay width for the associated tetramer ground state are E4(0)=4.60(1)B3(0){E}_{4}^{(0)}=4.60(1){B}_{3}^{(0)}E4(0)​=4.60(1)B3(0)​ and ${\mathrm{\ensuremath{\Gamma}}}_{4}^{(0)}/2=0.0160(1){B}_{3}^{(0)}$, respectively, and for the associated tetramer excited state, E4(1)=1.0022(3)B3(0){E}_{4}^{(1)}=1.0022(3){B}_{3}^{(0)}E4(1)​=1.0022(3)B3(0)​ and ${\mathrm{\ensuremath{\Gamma}}}_{4}^{(1)}/2=2.57(2)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}{B}_{3}^{(0)}$, respectively. This calculation is a gateway to higher-order and/or more-body calculations in nuclear and atomic systems.
We calculated the 4{}^{4}4He trimer and tetramer ground and excited states with the LM2M2 potential using our Gaussian expansion method for ab initio variational calculations of few-body systems. The method has been extensively used for a variety of three-, four-, and five-body systems in nuclear physics and exotic atomic and molecular physics. The trimer (tetramer) wave function is expanded in terms of symmetric three- (four-) body Gaussian basis functions, ranging from very compact to very diffuse, without assumption of any pair correlation function. The calculated results for the trimer ground and excited states are in excellent agreement with values reported in the literature. The binding energies of the tetramer ground and excited states are obtained as 558.98 and 127.33 mK (0.93 mK below the trimer ground state), respectively. We found that precisely the same shape of the short-range correlation (${r}_{ij}\ensuremath{\lesssim}4$ \AA{}) in the dimer appears in the ground and excited states of the trimer and tetramer. The overlap function between the trimer excited state and the dimer ground state and that between the tetramer excited state and the trimer ground state are almost proportional to the dimer wave function in the asymptotic region (up to $\ensuremath{\sim}$1000 \AA{}). Also, the pair correlation functions of trimer and tetramer excited states are almost proportional to the squared dimer wave function. We then propose a model which predicts the binding energy of the first excited state of 4{}^{4}4HeN{}_{N}N​ ($N\ensuremath{\ge}3$) measured from the 4{}^{4}4He${}_{N\ensuremath{-}1}$ ground state to be nearly $\frac{N}{2(N\ensuremath{-}1)}{B}_{2}$ where B2{B}_{2}B2​ is the dimer binding energy.
In a previous work [Phys. Rev. A 85, 022502 (2012)] we calculated, with the use of the Gaussian expansion method for few-body systems, the energy levels and spatial structures of the 4{}^{4}4He trimer and tetramer ground and excited states using the LM2M2 potential, which has a very strong short-range repulsion. In this work, we calculate the same quantities using the currently most accurate 4{}^{4}4He-4{}^{4}4He potential [Przybytek et al., Phys. Rev. Lett. 104, 183003 (2010)] that includes the adiabatic, relativistic, QED, and residual retardation corrections. Contributions of the corrections to the tetramer ground-(excited-) state energy $\ensuremath{-}573.90\phantom{\rule{0.16em}{0ex}}(\ensuremath{-}132.70)$ mK are respectively $\ensuremath{-}4.13\phantom{\rule{0.16em}{0ex}}(\ensuremath{-}1.52)$ mK, +9.37(+3.48)+9.37\phantom{\rule{0.16em}{0ex}}(+3.48)+9.37(+3.48) mK, $\ensuremath{-}1.20\phantom{\rule{0.16em}{0ex}}(\ensuremath{-}0.46)$ mK, and +0.16(+0.07)+0.16\phantom{\rule{0.16em}{0ex}}(+0.07)+0.16(+0.07) mK. Further including other realistic 4{}^{4}4He potentials, we calculated the binding energies of the trimer and tetramer ground and excited states, B3(0),B3(1),B4(0){B}_{3}^{(0)},\phantom{\rule{0.16em}{0ex}}{B}_{3}^{(1)},\phantom{\rule{0.16em}{0ex}}{B}_{4}^{(0)}B3(0)​,B3(1)​,B4(0)​, and B4(1){B}_{4}^{(1)}B4(1)​, respectively. We found that the four kinds of the binding energies for the different potentials exhibit perfect linear correlations between any two of them over the range of binding energies relevant for 4{}^{4}4He atoms (namely, six types of the generalized Tjon lines are observed). The dimerlike-pair model for 4{}^{4}4He clusters, proposed in the previous work, predicts a simple interaction-independent relation B4(1)B2=B3(0)B2+23\frac{{B}_{4}^{(1)}}{{B}_{2}}=\frac{{B}_{3}^{(0)}}{{B}_{2}}+\frac{2}{3}B2​B4(1)​​=B2​B3(0)​​+32​, which precisely explains the correlation between the tetramer excited-state energy and the trimer ground-state energy, with B2{B}_{2}B2​ being the dimer binding energy.
The momentum-space structure of the Faddeev-Yakubovsky (FY) components of weakly bound tetramers is investigated at the unitary limit using a renormalized zero-range two-body interaction. The results, obtained by considering a given trimer level with binding energy B3{B}_{3}B3​, provide further support to a universal scaling function relating the binding energies of two successive tetramer states. The correlated scaling between the tetramer energies comes from the sensitivity of the four-boson system to a short-range four-body scale. Each excited NNNth tetramer energy B4(N){B}_{4}^{(N)}B4(N)​ moves as the short-range four-body scale changes, while the trimer properties are kept fixed, with the next excited tetramer B4(N+1){B}_{4}^{(N+1)}B4(N+1)​ emerging from the atom-trimer threshold for a universal ratio ${B}_{4}^{(N)}/{B}_{3}={B}_{4}^{(N)}/{B}_{4}^{(N+1)}\ensuremath{\simeq}4.6$, which does not depend on NNN. We show that both channels of the FY decomposition [atom-trimer (KKK type) and dimer-dimer (HHH type)] present high-momentum tails that reflect the short-range four-body scale. We also found that the HHH channel is favored over the KKK channel at low momentum, when the four-body momentum scale largely overcomes the three-body scale.