Abstract
This paper presents a numerical convergence study of a hyperspherical-harmonics expansion for binding energies of a system of 4≤N≤728 helium atoms using a phenomenological soft attractive two-body He-He potential and a repulsive three-body force aimed at compensating for the absence of the two-body repulsive core. Earlier calculations with such a potential have shown an improved convergence when N increases from four to six. The present study reveals that the improved convergence occurs only for a limited range of N determined by the range of the three-body repulsion. For a soft repulsive three-body force, the convergence is fast for N≤20, while for a short-range three-body repulsion it deteriorates at N≥10. The reasons for this deterioration are discussed. The range of the three-body force also determines the binding energy behavior with N, and it is also responsible for binding the excited states. The long-range force binds all first excited 0+ states but strongly underbinds the systems of N helium atoms at large N. The short-range force does not bind the first 0+ states for A≤7 but gives better predictions of binding energies as compared to the calculations of other authors though overestimating them. Some options to improve both the description of the binding energies and the convergence of the hyperspherical-harmonics expansion using phenomenological forces are discussed. It is pointed out that a fast convergence is very much needed for the reliable predictions of states with nonzero angular momentum, examples of which are also given.