Scaling approximation for the elementary diagrams in hypernetted-chain calculations

A simple method to sum elementary diagrams in the calculation of two-body and three-body distribution functions by the hypernetted-chain method is proposed. The method is based on the observation that the sum of the elementary diagrams has approximately the same spacial behavior as that of the lowest-order four-body elementary diagram. Thus the sums of two-point and three-point elementary diagrams may be approximated by a scaling constant times the lowest-order contribution. The scaling factor is determined by equating energies calculated with the Jackson-Feenberg and Pandharipande-Bethe expressions. Results of calculations of the energies and distribution functions of liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ with the use of this method are reported. The results obtained with the McMillan correlation function are in almost exact agreement with the Monte Carlo results. Calculations with optimized correlation function having $r^{-2\mathrm{}}$ tails are also reported.