Theory of Quantum Crystals. II. Three-Body Term in the Cluster Expansion

Abstract

In a previous paper, denoted as I, the ground-state properties of crystalline helium were studied by a variational method which used a cluster expansion evaluation of the energy, $E_0$. The basic approximation of that work was the truncation of the cluster expansion after the one- and two-particle terms, $E_{01}+E_{02}$. We have tested this approximation by numerical computation of the three-body term, $E_{03}$, of the expansion. Using the analytic form of the trial wave function given in I, we find, for bcc ${}_{}{}^3{}_{}{}^{}\mathrm{He}$, that $E_{03}\approx {10}^{-2\mathrm{}}{E}_{02}$ at the minimum in $E_{01}+E_{02}$. Furthermore, when $E_{03}$ is included in the variations, $E_{03}$ remains small and the minimum energy is essentially unchanged, but the values of the wave-function parameters are improved somewhat. These results indicate that the cluster expansion is converging rapidly. The computations are performed as a function of density, and improved results for the ground-state pressure, compressibility, sound velocities, and exchange integral are also presented. Similar calculations at a single density in hcp ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ and ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ show that the close packing causes $E_{03}$ to be larger; however, it is still only $\frac{1}{20}$ the value of $E_{02}$, and the truncation of the cluster expansion is probably valid here as well. It is shown that the rate of convergence of the cluster expansion depends on the form of the trial function. A numerical example is given of a form for which the approximation of I breaks down. A critique of the expansion of Brueckner and Frohberg is given. On a basis of a numerical test, it is found that they have neglected an important term in truncating their expansion, so that the validity of their variational procedure is uncertain. Some possible improvements in the theory, such as solving a differential equation for the correlation function and including the effects of the phonon spectrum, are discussed. Details of our method of evaluating the three-body terms are given in two Appendices.