Note on the Two-Body Correlation Approximation in Liquid Helium-3

In an earlier paper we proposed the calculation of the low-temperature properties of liquid ${\mathrm{He}}^3$ in the thermodynamically consistent "$T$-matrix" (TCTM) approximation, and presented a preliminary, zero-temperature calculation employing a separable potential and noninteracting spectral functions. This paper continues the program by calculating, with the local Phillipson-Morse-Dalgarno (PMD) potential of Bernstein and Morse, the zero-temperature properties at a single Fermi momentum. A calculation employing noninteracting spectral functions is performed as the zeroth-order solution of the complete TCTM approximation. The calculated properties differ from those obtained in the earlier calculation, and the difference is understood by comparing de Boer's 6-12 potential to which the separable potential was fitted with the PMD potential. Using the zeroth-order solution as input, an interative solution of the coupled integral equations of the TCTM approximation is obtained. The convergence of the iterative procedure is rapid, and a spectral function for the interacting system is obtained. The calculated properties do not agree with the experimentally determined properties, but a TCTM approximation using an interatomic potential in better agreement with the second-virial-coefficient data for helium than the PMD potential is expected to produce results in better agreement with experiment.