Helium at zero temperature with hard-sphere and other forces

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.