Helium at zero temperature with hard-sphere and other forces

TLDR

Heliumlike systems in their ground states pose theoretical and numerical challenges, often modeled by splitting the potential into repulsive and attractive components treated perturbatively. The authors employ periodic boundary conditions to model fluid and crystal states, correct low‑lying phonon excitations for infinite‑system energy extrapolation, and use a perturbation theory that maps hard‑sphere solutions—identified via the scattering length—to smoother two‑body potentials. Numerical solutions of the Schrödinger equation now handle 256‑body hard‑sphere systems, yielding accurate energy and radial‑distribution functions; convergence for Lennard‑Jones potentials is excellent, and the resulting energy‑density curves agree with experiment within 3–10% except at high crystal densities, while the method also highlights antisymmetrization as the cause of discrepancies in He‑3.

Abstract

Various theoretical and numerical problems relating to heliumlike systems in their ground states are treated. New developments in the numerical solution of the Schr\"odinger equation permit the solution of 256-body systems with hard-sphere forces. Using periodic boundary conditions, fluid and crystal states can be described; results for the energy and radial-distribution functions are given. A new method of correcting for low-lying phonon excitations so as to extrapolate the energy of fluids to an infinite system is described. A perturbation theory relating the properties of the system with pure hard-sphere forces to those with smoother, more realistic two-body forces is introduced. As in recent work on classical systems the potential is divided into two continuous parts: One is repulsive, one attractive, the latter being treated as a perturbation. The solution for the repulsive part is taken directly from the hard-sphere problem when the radius is identified as the scattering length of the repulsive part of the smooth potential. The convergence for the Lennard-Jones potential is very good. Using our numerical results for the hard-sphere problem, with phonon corrections, together with this perturbation theory, results for energy versus density agree with experiment within our error of (3-10)% except at high crystal densities. We carry further Schiff's recent application of this perturbation theory to ${\mathrm{He}}^{3}$ and conclude that antisymmetrization by the method of Wu and Feenberg is the reason for lack of agreement with experiment in that system.