Ground-state energetics of helium and deuterium fermion fluids

Abstract

The method of correlated basis functions (CBF) is applied to the evaluation of the ground-state energy of atomic fermion fluids as a function of density. As a first step, liquid $^{3}\mathrm{He}$ in both unpolarized and fully polarized spin configurations is considered variationally, using Slater-Jastrow trial wave functions. Results are reported for a conventional analytic choice of the state-independent two-body correlation function $f(r)$ and for the optimal $f(r)$ determined by the solution of a suitable Euler equation. The Jastrow treatment is found to be inadequate in that (i) the energy expectation value lies above the experimental equilibrium energy by some 1.5 K, and (ii) the polarized phase is predicted to be more stable than the unpolarized one. For a given polarization, a correlated basis is formed by application of the assumed Jastrow correlation factor to the elements of a complete set of noninteracting-Fermi-gas Slater determinants. The exact ground-state energy may be developed in a perturbation expansion in the correlated basis, the leading term being the Jastrow energy expectation value. Considerable improvement on the Jastrow description of the unpolarized phase is achieved upon inclusion of the correlated two-particle\char22{}two-hole component of the second-order CBF perturbation correction. At the experimental equilibrium density, this contribution, which incorporates important momentum- and spin-dependent correlations, can amount to some 0.6-1.1 K [depending on the choice of $f(r)$]. The required correlated-basis matrix elements are calculated by Fermi hypernetted-chain (FHNC) techniques, crucial Pauli effects of the elementary diagrams being introduced through the FHNC/C algorithm. The Euler equation is approximated within the same framework. The momentum-space integrations in the second-order perturbation correction are evaluated by a Monte Carlo procedure. One may reasonably expect that further refinements of the CBF method will lead to an accurate microscopic description of the ground-state energetics of liquid $^{3}\mathrm{He}$. Bulk atomic deuterium with all electronic spins aligned is treated at the same level of approximation as applied to helium. Three choices of nuclear-spin distribution are examined, with a single spin state present, or two or three equally populated nuclear spin states. The finite-density energy minimum is found to lie very close to zero energy in all three examples; a very precise many-body calculation will thus be needed to decide their liquid or gaseous nature at zero temperature under zero external pressure.