LiquidHe3. I. Calculation of the Binding Energy from Two-Body Correlations

Abstract

As the first step in a theoretical study of the properties of liquid $^{3}\mathrm{He}$, we have calculated the binding energy of the system by means of Brueckner theory. The method of Brueckner and Gammel is used to solve the Bethe-Goldstone equation, calculate the reaction matrix or $G$ matrix, and then get the two-body interaction energy contribution. The Brueckner-Gammel method is studied in some detail. For simplicity we use the approximation of an effective mass in a reference energy spectrum, which in principle should be fitted to self-consistent single-particle energies. The intermediate-state potential energies are, however, chosen to be equal to zero. Hence, the three-body energy contribution must be estimated by separate calculations. Various two-body wave functions, Fourier transforms of wave functions, and $G$-matrix elements are calculated. Also, the volume of the correlation hole, which gives the convergence parameter in the linked-cluster expansion, and an effective interaction, which is a representation of the $G$ matrix in coordinate space, are calculated, together with the binding energy for liquid $^{3}\mathrm{He}$. The calculations are repeated for various input parameters, i.e., for several values of the parameters which define the reference energy spectrum, and for several values of the initial relative momentum of the two interacting particles. The total or c.m. momentum is set equal to zero. The Brueckner-Gammel method is found to be a fairly rapid and convenient method when the complete Green's function in the Bethe-Goldstone equation is expressed in terms of a corresponding reference-spectrum Green's function, and the single-particle energies in the energy denominator in the Bethe-Goldstone equation are replaced by a reference energy spectrum. Third-order and higher-order energy contributions can probably be assumed to be built into this energy spectrum, or they may be estimated by separate calculations. The binding energy for liquid $^{3}\mathrm{He}$ with only two-body terms included is found to be approximately -\textonehalf{}\ifmmode^\circ\else\textdegree\fi{}K per particle, which is in general agreement with other calculations.