Elsevier

Chemical Physics Letters

Volume 335, Issues 1–2, 16 February 2001, Pages 105-110
Chemical Physics Letters

Rotation–vibration interaction in 4He trimers

https://doi.org/10.1016/S0009-2614(00)01423-8Get rights and content

Abstract

An approximate treatment of the rotation–vibration interaction is considered in the helium trimer system to study the existence of bound states with non-zero (J=1) total angular momentum and to elucidate its Efimov character. With the interaction used here, at least one bound state is found. In terms of a λ-parameter modifying the strength of the two-body potential, the radial part of the wave function changes dramatically with tiny variations around the physical value λ=1. High-resolution rotational Raman spectroscopy in a supersonic beam expansion is proposed as a feasible way of observing the bound state predicted.

Introduction

Quantum liquids and clusters are currently being very actively investigated. In particular, helium clusters provide prototype systems which display unusual behaviors manifested in different phenomena such as, for example, atomic optics [1], cold atom scattering [2], [3] Bose–Einstein condensation and superfluidity [4]. Furthermore, in the kinetics of formation of small clusters from 4He beams, dimers and trimers play a very important role in the different recombination mechanisms envisaged to lead to clusters of increasing size. Thus, a very precise knowledge of the partition functions for these small clusters is essential for a correct statistical description of formation dynamics at low temperatures since the addition of a new bound state can represent a marked change in the partition function. As a consequence, the existence of possible excited states of these clusters has recently lead to a very fruitful and enthusiastic debate in which the so-called Efimov states [5], [6] are at the center of the dicussion.

Very recently [7], [8] helium trimer bound states have been calculated by us using a variational method described in terms of pair–atom coordinates and where the nuclear bound states are expanded over symmetrized products of distributed Gaussian functions for a total angular momentum J=0. With the two-body interaction potential that we used [9], only two bound states for the trimer with energies E(v=0)vib=−0.1523 cm−1 and E(v=1)vib=−1.24×10−3 cm−1 have been found. As we have further shown [7], [8], the corresponding excited state of the boson system displays several characteristics of Efimov behavior. The different triangular configurations contributing to both bound trimer states have also been extracted from our calculations where, due to the great floppiness of this system, they could be assessed in terms of the rms bond-length fluctuations (δrms(v=0)=0.63 and δrms(v=1)=0.56) and classified as a melted or liquid-like, non-rigid cluster according to the Lindemann criterium (δrms>0.1) [10]. Strong structural deformations are therefore expected to occur upon rotations which will inevitably increase its liquid-like character.

Recent helium trimer calculations with non-zero total angular momentum [11], [12] have not shown conclusive results about the possible existence of rotational states in such a special system. In this work, and due to obvious inherent difficulties of this system to tackle a proper and direct solution of the problem, our strategy in carrying out approximate J=1−rotational calculations can be summarized in four steps. First, the rotation–vibration interaction has been considered, as a good starting point, through a Hamiltonian expressed as a sum of the exact Hamiltonian at J=0 and the rotational, rigid rotor-like Hamiltonian, in a similar vein to that followed by the approximate Van Vleck procedure [13]. Second, as rotational basis we have chosen to work with the prolate symmetric top basis set |JKM〉 (M and K are the projections of J on the space- and body-fixed z-axis, respectively). On the other hand, the energy levels of an asymmetric top rotor as a function of the so-called Ray's asymmetry parameter are correlated, in the limits ±1, to oblate and prolate symmetric top energy levels, respectively [13]. Thus, the lowest rotational energy value has been taken as that which correlates in the prolate limit with J=1 and Kp=0, and in the oblate limit, with J=1 and Ko=1. For all the cases treated here the corresponding rotational eigenfunction is |10M〉. Third, as a convenient basis set to describe the non-rigid behavior of the whole cluster, a product of each J=0 vibrational eigenfunction times the prolate symmetric top basis is taken to represent the full Hamiltonian. Fourth, from the diagonalization of the corresponding 2×2 Hamiltonian matrix which amounts to all orders of perturbation theory in the |v,10M〉-block, we estimate an upper limit to the actual energy of the lowest J=1−rotational state, as long as the following points are fulfilled: (1) the two bound J=0 states are considered to be accurate enough; and (2) the starting Hamiltonian accounted for here can adequately describe the system. This procedure has been successfully tested on the Ar3 system at J=1, the binding energy of the deepest state being underestimated only by 0.01%. For He3 clusters, the key point in the last step consists of the fact that the two vibrational wave functions have a completely different nature since one represents an Efimov-type state, with very large spatial extension, whereas the other corresponds to a standard ground state. As we shall see, this unusual diagonalization analyzed as a function of the fluctuations of the interaction potential, by means of a λ-parameter modifying the strength of the two-body potential, turns out to be critical in settling the discussion on the existence of rotational states as well as their possible Efimov-type features. Finally, a plausible way of detecting the predicted bound state by means of high-resolution rotational Raman spectroscopy is proposed.

Section snippets

Theory and discussion of results

One of the advantages of our variational treatment is that we expand the eigenfunctions of the total Hamiltonian for J=0 in terms of φi basis functions, easily recognizable as triangular configurations, and written as [7], [8]Φv(R1,R2,R3)=∑ia(v)iφi(R1,R2,R3)=Nlmn−1/2ia(v)iP∈S3P[ϕl(R1)ϕm(R2)ϕn(R3)],where i denotes a collective index, i=(lmn) and the φi functions, in turn, are built as symmetrized products of pair Gaussian functions, ϕ(R). Basically, each φi function describes the six

Acknowledgements

This work has been supported in part by DGICYT (Spain) under contract PB95-0071, by an EC Research Network with Contract Number HPRN-CT-1999-00005, by the Italian National Research Council (CNR) and Ministry for University and Research (MURST) and by a joint Italian–Spanish Project with No. HI1999-0157. We would like to thank S. Montero for many interesting and fruitful discussions.

References (16)

  • V. Efimov

    Phys. Lett. B

    (1970)
  • V. Efimov

    Nucl. Phys. A

    (1973)
  • W. Schöllkopf et al.

    Science

    (1994)
  • S. Inouye et al.

    Nature

    (1998)
  • P.O. Fedichev et al.

    Phys. Rev. Lett.

    (1996)
  • S.Y. Larsen

    Phys. Rev.

    (1963)
  • T. González-Lezana et al.

    Phys. Rev. Lett.

    (1999)
  • T. González-Lezana et al.

    J. Chem. Phys.

    (1999)
There are more references available in the full text version of this article.

Cited by (0)

View full text