Series Expansion in the Variational Approach to Many-Boson Problems

The ground state of a many-boson system is studied within the range of the Bijl-Dingle-Jastrow-type description when the radial distribution function g(r) differs little from its asymptotic value. The treatment of the problem is based on the development of power series in α=1-g(0) for all physical quantities which depend on the particle density. The n-particle distribution functions p⁽ⁿ⁾ are evaluated to order α⁴ as functionals in the g(r) function for n=3 and 4 using the cluster-expansion procedure outlined by Abe. These results are used in connection with the improvement of the ground-state description when the wave function is not the optimum choice. Using p⁽³⁾ function obtained, the Bogoliubov-Born-Green-Kirkwood-Yvon equation is solved, also to order α⁴, for the two-particle correlation function U(r), and the first two leading corrections to the hypernetted-chain (HNC) approximation are obtained. The variational calculation along with the series expansion for U(r) yields formulas for the ground-state properties, including some corrections to known results. For a charged boson gas, numerical values of U(r), p⁽³⁾, p⁽⁴⁾, and the ground-state energy are computed using the Gaussian approximation for g(r), and the results show that the errors associated with the HNC approximation are small. A brief discussion is presented on the method of determining the general expansion coefficients of the correlation functions of p⁽ⁿ⁾ in terms of g(r).