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The Journal of Chemical Physics

Trimer binding in two dimensions

F. Cabral, Ludwig W. Bruch

Published May 15, 1979• 24 citations
Publisher

Abstract

The ground state energies of loosely bound boson and fermion trimers interacting via Lennard-Jones (12,6) pair potentials in two dimensions are evaluated with variational calculations. Threshold coupling constants for self-binding are determined by extrapolation to zero binding energy. The results are used to give estimates of the binding energy of the 4He trimer and in a discussion of the ordering of dimer and trimer thresholds in two dimensions.

DOI
10.1063/1.437251

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•Jan 1, 1979•0 citations
The ground state energies of loosely bound boson and fermion trimers interacting via Lennard-Jones (12,6) pair potentials in two dimensions are evaluated with variational calculations. Threshold coupling constants for self-binding are determined by extrapolation to zero binding energy. The results are used to give estimates of the binding energy of the 4He trimer and in a discussion of the ordering of dimer and trimer thresholds in two dimensions.
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.
Qualitative features are discussed for the binding of three identical bosons interacting through pair potentials in two dimensions. Two special cases, known to yield pathologies in three dimensions, are examined using the Faddeev equation for the bound states. The Thomas effect does not occur: in the model which is treated, the trimer binding energy is finite for a zero-range force with a finite dimer energy. The Efimov effect can only occur under more restrictive conditions than in three dimensions: the number of bound trimer states is finite at the dimer threshold for a range of potential models. The trimer ground-state energy is determined as a function of the coupling constant for a simple model, and variational results for loosely bound Lennard-Jones trimers are shown to reflect a general trend found for the model.
The ground state of molecular trimers is studied in three dimensions for these pair potential models: the square well, the exponential, the Yukawa, the Gaussian, and the Lennard-Jones 12–6. Three spin cases are considered: three spin-zero bosons and the spin- (1/2) and spin- (3/2) states of three spin- (1/2) fermions. Variational wavefunctions are constructed which satisfy the exchange symmetry requirements for these cases. Bounds are obtained for the threshold coupling constants at which self-bound trimers occur. Consequences for possible self-bound trimers of 3He are discussed.
The results of accurate variational calculations for the ground-state binding energies of several NNN-boson systems in two and three dimensions are reported; the systems specifically considered are small (N=2,3,4,and5N=2, 3, 4, \mathrm{and} 5N=2,3,4,and5) clusters of 4He^{4}\mathrm{He}4He atoms interacting through pairwise Lennard-Jones potentials. Evidence is found strongly suggesting each set of systems have a common functional relationship linking the binding energies and the De Boer quantum constant near threshold for self-binding. These calculations also confirm the existence of a unique threshold value of the interaction strength for all NNN-boson systems in 2D2D2D. In addition, the linear relation between the binding energies of helium clusters present in 3D3D3D holds ture in 2D2D2D as well.