Ground state of liquid helium in the Kirkwood superposition approximation

We calculate the ground-state energies and pressures for ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ and (normal) ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ using radial distribution functions which are solutions of the Born-Bogoliubov-Green-Kirkwood-Yvon (BBGKY) integral equation with the Kirkwood superposition approximation (KSA) used for the three-particle distribution function. We compare the BBGKY-KSA results with those obtained from the hypernetted chain equation, molecular dynamics, and experiment. We conclude that the BBGKY-KSA yields poor results for ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ but at the lower ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ densities offers a reasonable approximation to the molecular-dynamics results. The ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ energies are obtained by means of the statistical cluster expansion of Wu and Feenberg and we discuss the convergence of the series. Further, we show that the familiar parametrized power-law form used for the pair function yields pressures consistent with the virial theorem.