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The Journal of Chemical Physics

Quantum Monte Carlo investigation of small He4 clusters with a He3 impurity

Dario Bressanini, Matteo Zavaglia, Massimo Mella, Gabriele Morosi

Published Jan 8, 2000• 50 citations
Publisher

Abstract

Small helium (4He) clusters containing the lighter isotope He3 are studied by means of quantum Monte Carlo methods. Accurate ground state energies and structural properties are obtained using accurate trial wave functions and the Tang–Tonnies–Yiu (TTY) helium–helium pair potential. The dimer He4–3He is not bound; as well as the trimer He4He23. The smallest cluster containing He3 is He24He3 with a nonrigid structure having a marked linear contribution. Interestingly, this weakly bound system, with an energy one order of magnitude less than the He34 trimer, is able to bind another He3 atom, forming the tetramer He24He23, which shows the odd feature of having five out of six unbound pairs. In general, the substitution of a single He4 atom in a pure cluster with a He3 atom leads to an energetic destabilization, as the pair He4–3He is not bound. The isotopic impurity is found to perturb only weakly the distributions of the remaining He4 atoms, which retain the high floppiness already found in the pure clusters. As the number of atoms increases the isotopic impurity has the marked tendency to stay on the surface of the cluster. This behavior is consistent with the formation of the so-called “Andreev states” of a single He3 in liquid He4 helium and droplets, where the impurity tends to form single-particle states on the surface of the pure He4.

DOI
10.1063/1.480604

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Small helium (4He) clusters containing the lighter isotope He3 are studied by means of quantum Monte Carlo methods. Accurate ground state energies and structural properties are obtained using accurate trial wave functions and the Tang–Tonnies–Yiu (TTY) helium–helium pair potential. The dimer He4–3He is not bound; as well as the trimer He4He23. The smallest cluster containing He3 is He24He3 with a nonrigid structure having a marked linear contribution. Interestingly, this weakly bound system, with an energy one order of magnitude less than the He34 trimer, is able to bind another He3 atom, forming the tetramer He24He23, which shows the odd feature of having five out of six unbound pairs. In general, the substitution of a single He4 atom in a pure cluster with a He3 atom leads to an energetic destabilization, as the pair He4–3He is not bound. The isotopic impurity is found to perturb only weakly the distributions of the remaining He4 atoms, which retain the high floppiness already found in the pure clusters. As the number of atoms increases the isotopic impurity has the marked tendency to stay on the surface of the cluster. This behavior is consistent with the formation of the so-called “Andreev states” of a single He3 in liquid He4 helium and droplets, where the impurity tends to form single-particle states on the surface of the pure He4.
We report accurate ground state energies and structural properties for small clusters of 4He computed with the diffusion quantum Monte Carlo (DMC) method combined with high quality trial wave functions and using the recent analytical pair potential of Tang, Toennies, and Yiu [Phys. Rev. Lett. 74, 1546 (1995)]. Calculations based on the older HFD-B(He) potential are reported for comparison. The clusters are found to be extremely floppy and to be characterized by very diffuse wave functions. The DMC results for 4He2 and 4He3 are in excellent agreement with other calculations using conventional methods. 4He3 is found to have a noticeable contribution from nearly linear geometries. The internal structure of the clusters is described by a three particle correlation function which reveals a significantly non-spherical internal cluster structure. The energies for all cluster sizes are found to be slightly higher than those obtained with the HFD-B(He) pair potential. Exploratory calculations on the helium trimer indicate that the effects of three body interactions do not exceed the uncertainty margin provided by different state of the art pair potentials.
We have studied the stability of mixed 3He/4He^{3}\mathrm{H}\mathrm{e}/^{4}\mathrm{H}\mathrm{e}3He/4He clusters in L=0L=0L=0 states by the diffusion Monte Carlo method, employing the Tang-Toennies-Yiu He-He potential. The clusters 3He4HeN^{3}\mathrm{H}\mathrm{e}^{4}\mathrm{H}\mathrm{e}_{N}3He4HeN​ and 3He24HeN^{3}\mathrm{H}\mathrm{e}_{2}^{4}\mathrm{H}\mathrm{e}_{N}3He24​HeN​ are stable for N>1N>1N>1. The lighter atoms tend to move to the surface of the cluster. The minimum number of 4He^{4}\mathrm{H}\mathrm{e}4He atoms able to bind three 3He^{3}\mathrm{H}\mathrm{e}3He atoms in a L=0L=0L=0 state is nine. Two of three fermionic helium atoms stay on the surface, while the third one penetrates into the cluster.
Abstract Small helium clusters Hen (n = 2, 3, ... , 10) in molecular beams produced by supersonic expansion of gaseous helium have been observed in several different experiments. Among these clusters the dimer, which has a single very slightly bound state, and the trimer, which could have many bound states, have especially unusual properties which QMC calculations can explain in detail. In this paper, diffusion QMC calculations were used to determine ground state properties for clusters (n = 2, ... , 10) using two accurate, but slightly different pairwise-additive potential energy expressions, a perturbation-based potential by Tang et al.a and an empirical potential (HFD-B) proposed by Aziz et al.b Trial functions for importance sampling were based on those determined earlier by Rick et alc. Expectation values and probability distributions were determined by the descendent-weighting technique to give results independent of the trial functions. As expected, a single bound state of the dimer was found, and its average internuclear distance was predicted to be 51.5 A. Energies for clusters with n = 2, 7 with the HFD-B potential were in good agreement with those of earlier calculations. Comparisons of the results for the two different potential energy expressions showed energies differing by a few percent. Tests of possible nonadditive interactions showed these to be far less important in their effects on predicted energies.
A stability chart of mixed 4He{}^{4}\mathrm{He}4He and 3He{}^{3}\mathrm{He}3He clusters has been obtained by means of the diffusion Monte Carlo method, using both the Aziz HFD-B and the Tang-Toennies-Yiu atom-atom interaction. The investigated clusters contain up to eight 4He{}^{4}\mathrm{He}4He atoms and up to 20 3He{}^{3}\mathrm{He}3He atoms. One single 4He{}^{4}\mathrm{He}4He binds 20 3He{}^{3}\mathrm{He}3He atoms, and two 4He{}^{4}\mathrm{He}4He bind 1, 2, 8, and more than 14 3He{}^{3}\mathrm{He}3He atoms. All clusters with three or more 4He{}^{4}\mathrm{He}4He atoms are bound, although the combinations 4He33He9,10,11{}^{4}{\mathrm{He}}_{3}^{3}{\mathrm{He}}_{9,10,11}4He33​He9,10,11​ and 4He43He9{}^{4}{\mathrm{He}}_{4}^{3}{\mathrm{He}}_{9}4He43​He9​ are metastable. Clusters with 2, 8, and 20 3He{}^{3}\mathrm{He}3He atoms are particularly stable and define magic 3He{}^{3}\mathrm{He}3He numbers.
Optimized variational calculations have been carried out for clusters of He4 between N=20 and N=1000 atoms. For small cluster sizes with less or equal to 112 particles, where comparisons with existing diffusion Monte Carlo results are possible, we find good agreement for the ground state energies and densities. Using a somewhat simpler approximation, we also calculate the bound state energies of He3 atoms attached to these clusters. We then calculate excitations and the dynamic structure function. The complex and nonlocal self-energy introduced for that purpose gives access to the calculation of both elastic and inelastic scattering processes for He4 and He3 atoms impinging on the clusters.