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arXiv:2605.12315 · Journal of Physics B Atomic Molecular and Optical Physics

Natural and Dyson orbitals in small helium drops

N. K. Timofeyuk

Published May 15, 2025• 0 citations
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Abstract

Abstract The natural and Dyson orbitals are studied for small helium drops comprising 5 to 20 helium atoms interacting via a soft two-body Gaussian potential. The wave functions of these drops have been obtained in the hyperspherical cluster model (HCM) which provides a correct description of the single-particle behavior at large separations from the system. The natural orbitals are obtained from diagonalization of the nonlocal one-body density matrix, while Dyson orbitals are constructed by the direct overlap of the wave functions of two drops differing by one boson. This overlap converges with an increasing basis of the HCM. The shapes and occupancies of the natural orbitals, as well as their link to Dyson overlap and evolution with increasing number of atoms, are discussed. Both natural and Dyson orbitals can be used to represent the density of the system. However, the natural orbital representation is demonstrated to be superior. With increasing boson numbers the difference between Dyson and natural orbitals becomes less prominent and is expected to disappear in infinitely large systems of identical bosons.

DOI
10.1088/1361-6455/add977

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Abstract The natural and Dyson orbitals are studied for small helium drops comprising 5 to 20 helium atoms interacting via a soft two-body Gaussian potential. The wave functions of these drops have been obtained in the hyperspherical cluster model (HCM) which provides a correct description of the single-particle behavior at large separations from the system. The natural orbitals are obtained from diagonalization of the nonlocal one-body density matrix, while Dyson orbitals are constructed by the direct overlap of the wave functions of two drops differing by one boson. This overlap converges with an increasing basis of the HCM. The shapes and occupancies of the natural orbitals, as well as their link to Dyson overlap and evolution with increasing number of atoms, are discussed. Both natural and Dyson orbitals can be used to represent the density of the system. However, the natural orbital representation is demonstrated to be superior. With increasing boson numbers the difference between Dyson and natural orbitals becomes less prominent and is expected to disappear in infinitely large systems of identical bosons.
Abstract To describe long-range behaviour of one particle removed from a few- or a many-body system, a hyperspherical cluster model has been developed. It has been applied to the ground and first excited states of helium drops with five, six, eight and ten atoms interacting via a two-body soft gaussian potential. Convergence of the hyperspherical cluster harmonics expansion is studied for binding energies, root-mean-squared radii and overlaps of the wave functions of two helium drops differing by one atom. It was shown that with increasing model space the functional form of such overlaps at large distances converges to the correct asymptotic behaviour. The asymptotic normalization coefficients that quantify the overlaps’ amplitudes in this region are calculated. It was also shown that in the first excited state one helium atom stays far apart from the rest forming a two-body molecule, or a halo. The probability of finding the halo atom in the classically-forbidden region of space depends on the definition of the latter and on the valence atom binding energy. The total norm of the overlap integrals, the spectroscopic factor, represents the number of partitions of a many-body state into a chosen state of the system with one particle removed. The spectroscopic factors have been calculated and their sum rules are discussed giving a further insight into the structure of helium drops.
Drops containing 20 to 240 atoms of Bose liquid 4He^{4}\mathrm{He}4He and Fermi liquid 3He^{3}\mathrm{He}3He are studied by variational Monte Carlo methods as simple examples of correlated inhomogeneous quantum systems. In the present work we report wave functions of natural, quasiparticle, and mean-field orbitals in these drops. The wave functions of natural orbitals are determined by diagonalizing the one-particle density matrix. The condensate fraction and wave function in the Bose-liquid drops are calculated. The quasihole states are defined in the spirit of Landau's theory, and their wave functions are calculated from the overlaps between states containing N and N-1 atoms. In Bose-liquid drops the wave function of the quasihole orbital is similar to that of the condensate; however, in Fermi-liquid drops the quasihole wave functions are different from those of natural or mean-field orbitals. We find that a simple local-density approximation provides an accurate relation between the mean-field and quasihole wave functions for both Bose- and Fermi-liquid drops. The wave functions of natural orbitals are very localized and simple methods to construct them from mean-field wave functions are also discussed. The momentum distributions of atoms in the drops are calculated and compared with those of extended liquids. The deviations of the momentum distributions from mean-field distributions are also discussed.
In this work we investigate small clusters of bosons using the hyperspherical harmonic basis. We consider systems with A=2,3,4,5,6A=2,3,4,5,6A=2,3,4,5,6 particles interacting through a soft interparticle potential. In order to make contact with a real system, we use an attractive Gaussian potential that reproduces the values of the dimer binding energy and the atom-atom scattering length obtained with one of the most widely used 4{}^{4}4He-4{}^{4}4He interactions, the LM2M2 potential of Aziz and Slaman. The intensity of the potential is varied in order to explore the clusters' spectra in different regions with large positive and large negative values of the two-body scattering length. In addition, we include a repulsive three-body force to reproduce the trimer binding energy. With this model, consisting in the sum of a two- and three-body potential, we have calculated the spectrum of the four-, five-, and six-particle systems. In all the regions explored, we have found that these systems present two states, one deep and one shallow close to the $A\ensuremath{-}1$ threshold. Some universal relations between the energy levels are extracted; in particular, we have estimated the universal ratios between thresholds of the three-, four-, and five-particle continua using the two-body Gaussian potential. They agree with recent measurements and theoretical predictions.
In this paper, we investigate small clusters of helium atoms using the hyperspherical harmonic basis. We consider systems with A=2,3,4,5,6A=2,3,4,5,6A=2,3,4,5,6 atoms with an interparticle potential which does not present a strong repulsion at short distances. We use an attractive Gaussian potential that reproduces the values of the dimer binding energy, the atom-atom scattering length, and the effective range obtained with one of the widely used He-He interactions, the Aziz and Slaman potential, called LM2M2. In systems with more than two atoms, we consider a repulsive three-body force that, by construction, reproduces the trimer binding energy of the LM2M2 potential. With this model, consisting of the sum of a two- and three-body potential, we have calculated the spectrum of clusters formed by four, five, and six helium atoms. We have found that these systems present two bound states, one deep and one shallow, close to the threshold fixed by the energy of the $(A\ensuremath{-}1)$-atom system. Universal relations between the energies of the excited state of the AAA-atom system and the ground-state energy of the $(A\ensuremath{-}1)$-atom system are extracted, as well as the ratio between the ground state of the AAA-atom system and the ground-state energy of the trimer.
For a quantum system of NNN identical, harmonically interacting particles in a one-dimensional harmonic trap we calculate for the bosonic and fermionic ground state the corresponding one-particle reduced density operator ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{1}$ analytically. In case of bosons ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{1}$ is a Gibbs state for an effective harmonic oscillator. Hence the natural orbitals are Hermite functions and their occupation numbers obey a Boltzmann distribution. Intriguingly, for fermions with not too large couplings the natural orbitals coincide up to just a small error with the bosonic ones. In case of strong coupling this still holds qualitatively. Moreover, the decay of the decreasingly ordered fermionic natural occupation numbers is given by the bosonic one, but modified by an algebraic prefactor. Significant differences to bosons occur only for the largest occupation numbers. After all, the ``gap'' at the ``Fermi level'' decreases with increasing coupling strength but remains well pronounced even for strong interaction.
A formal proof is given that for NNN-boson systems in which two-body interaction potentials are described by a single Gaussian the ratio of higher-order to the lowest-order hyper-radial potentials decreases at $N\ensuremath{\rightarrow}\ensuremath{\infty}$ as ${N}^{\ensuremath{-}1/2}$ or faster. As a result, for such potentials, the convergence of binding energies for ground and several lowest excited states, obtained in expansion of the NNN-body wave function over the hyperspherical-harmonics basis, improves with increasing number of bosons. For a phenomenological three-body repulsive potential, introduced to account for the missing hard core, the ratio of higher-order to lowest-order hyper-radial potential, corresponding to this three-body potential, also decreases as ${N}^{\ensuremath{-}1/2}$ or faster when $N\ensuremath{\rightarrow}\ensuremath{\infty}$. Although adding the three-body contributions leads to increased influence from the total nondiagonal couplings around the node of the lowest hyper-radial potential, the arguments are given that this should not dramatically deteriorate the convergence if the range of repulsion is properly chosen. This means that the hyperspherical-harmonics expansion with soft two-body and repulsive three-body effective forces may become an attractive tool for studying the spectra of many-body systems. It is suggested that fine tuning of the three-body repulsion to reproduce the binding energies over a large region of NNN is possible. In particular, it has been shown that an NNN-independent choice for the three-body repulsion exists for which the ground-state binding energies of $N\ensuremath{\le}112$ atoms of helium, obtained in the lowest-order approximation of the hyperspherical-harmonics expansion, are close to the prediction of the Green's function Monte Carlo method with a hard-core He-He potential.