Spectra of helium clusters with up to six atoms using soft-core potentials

In this paper, we investigate small clusters of helium atoms using the hyperspherical harmonic basis. We consider systems with $A=2,3,4,5,6$ 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 $(A-1)$-atom system. Universal relations between the energies of the excited state of the $A$-atom system and the ground-state energy of the $(A-1)$-atom system are extracted, as well as the ratio between the ground state of the $A$-atom system and the ground-state energy of the trimer.

Figure 1

Ground- and excited-state energies of the $A=2,3,4,5,6$ systems as a function of ${\rho }_0$. In panel (a), we report the ground- and excited-state energy of the $A=3$ system together with the ground-state energy of $A=2$; for all of the values of ${\rho }_0$ we have considered, the excited $A=3$ state is bounded. In panel (b), we report the ground- and excited-state energy of the $A=4$ system together with the ground-state energy of $A=3$; the excited $A=4$ state is bounded for ${\rho }_0>7$. In panel (c), we report the ground- and excited-state energy of the $A=5$ system together with the ground-state energy of $A=4$; the excited $A=5$ state is bounded for ${\rho }_0>12$. In panel (d), we report the ground- and excited-state energy of the $A=6$ system together with the ground-state energy of $A=5$; the excited $A=6$ state is bounded for ${\rho }_0>10$.

Figure 2

Hyperspherical tree corresponding to Eq. (A4).