Three-body correlations in liquidHe4

We report results of variational calculations of liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ with wave functions containing optimized two-body and three-body correlations. The hypernetted-chain (HNC) summation method is used, and the elementary and Abe diagrams are calculated with the scaling approximation. Comparisons with the existing Monte Carlo calculations suggest that this HNC-scaling method is almost exact. The logarithm of the three-body correlation of $f_{\mathrm{ijk}}$ contains terms having $P_l\left({\hat{r}}_{\mathrm{ij}}·{\hat{r}}_{\mathrm{ik}}\right)$, $l=0, 1, \mathrm{and} 2$. As expected on theoretical grounds, the $l=1\mathrm{}$ term of $\ln f_{\mathrm{ijk}}$ dominates, while the $l=0\mathrm{} \mathrm{and} 2$ terms give rather small changes in the binding energy. The $f_{\mathrm{ijk}}$ makes up ∼85% of the difference between the Jastrow and presumably exact Green's-function Monte Carlo (GFMC) energies. The best variational energies obtained with the HFDHE2 potential of Aziz et al. are within (2 ± 1)% of the GFMC and experimental results. The liquid structure function $S\left(k\right)\mathrm{}$ is also well explained by the variational wave function.