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Physical Review Letters

Efimov States for H4e Trimers?

Tomás González‐Lezana, Jesús Rubayo‐Soneira, Salvador Miret-Artés, Franco Antonio Gianturco, Gerardo Delgado-Barrio, Pablo Villarreal

Published Feb 22, 1999• 72 citations
Publisher

Abstract

Helium trimer bound states are calculated with a variational method described in terms of pair atom coordinates and distributed Gaussian functions by assuming zero total angular momentum. With the two-body interaction potential used, a weakly bound state for the dimer and only two bound states for the trimer are found. Although strictly speaking the first excited trimer state is not an Efimov state, we show that this state presents several characteristics of the Efimov behavior which are extensively discussed in the present work. Special emphasis is placed on the main geometrical configurations contributing to the two bound trimer states, and suggestions are made on the possible ways of experimentally detecting the Efimov-type state.

DOI
10.1103/physrevlett.82.1648

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Helium trimer bound states are calculated with a variational method described in terms of pair atom coordinates and distributed Gaussian functions by assuming zero total angular momentum. With the two-body interaction potential used, a weakly bound state for the dimer and only two bound states for the trimer are found. Although strictly speaking the first excited trimer state is not an Efimov state, we show that this state presents several characteristics of the Efimov behavior which are extensively discussed in the present work. Special emphasis is placed on the main geometrical configurations contributing to the two bound trimer states, and suggestions are made on the possible ways of experimentally detecting the Efimov-type state.
Helium trimer bound states are calculated by means of a variational method described in terms of atom pair coordinates and distributed Gaussian basis functions for zero total angular momentum. To show the feasibility of this method, we also apply it to the calculation of the first vibrational levels of the Ar3 and Ne3 clusters. Special emphasis is made on the study of the possible Efimov behavior of the first excited state found in the He34 trimer. Geometrical configurations of the ground and first excited states of these rare gas trimers have been exhaustively studied owing to the proper symmetry of the coordinates chosen.
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.
The weakly bound diatomic systems He24 and HeH−4 have been found able to support only one bound state for J=0, although the latter also supports an additional bound state for J=1. In the present paper we, therefore, study the structure of the bound states which might exist for the weakly bound triatomic He24H−, in its J=0 state, by describing the full potential as a simple addition of two-body (2B) interactions. We carry out bound state calculations using both Jacobi coordinates within a discrete variable representation (DVR) and pair coordinates with a distributed Gaussian function (DGF) expansion. The system is shown to possess two bound states with respect to its lower dissociation threshold and two further “ghost” states before the complete break-up threshold. The spatial structures of such states and of the floppy complex are analyzed in detail, as is the possibility of detecting Efimov-type states in such a weakly bound aggregate. Finally, the inclusion of three-body (3B) forces in the description of the full interaction and its effect on the number of possible bound states is also discussed.
The possible presence of Efimov-type states in isotopic variants of triatomic molecules composed of one Li and two boson He atoms is investigated. Firstly, through the study of two-body interactions in the frame of the effective range theory, it is shown that, at least with the present choice of dimer potentials, the system containing 7 Li unlikely presents any Efimov state, while the 6 Li isotopic variant would be a better candidate in this regard. Then, triatomic bound states are searched through variational calculations in terms of pair inter-atomic distances and distributed Gaussian functions (DGF). The calculations were performed at several values of a parameter λ controlling the strength of the potential. Each system presents two bound levels at the “physical” situation λ = 1. By increasing lambda, the excited levels become “ghost” states as their energies are above the helium dimer energy and present, therefore, an Efimov character. Hence, we conclude that three-body (3B) predictions coming only from two-body (2B) calculations do not have, at least for the systems studied here, any great relevance. Such predictions are not in fact fulfilled for LiHe 2 , as shown by the greatly reduced relevance of isotopic effects in our actual 3B calculations.
We have searched for bound states of helium trimers with nonzero angular momenta. Including only pair interactions between helium atoms, we solved the Schrödinger equation in hyperspherical coordinates in the adiabatic approximation. From the resulting potential curves - which are mostly repulsive for nonzero angular momenta - we conclude that there are no bound rotational excited states in any of the isotopes of the helium trimer. The symmetry properties of the trimer wavefunctions in the body-fixed frame are also analysed.