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We study the ground state and the low-lying excitations of a trapped Bose gasin an isotropic harmonic potential for very small ($\sim 3$) to very large($\sim 10^7$) particle numbers. We use the correlated two-body basis functionsand the shape-dependent van der Waals interaction in our many-bodycalculations. We present an exhaustive study of the effect of inter-atomiccorrelations and the accuracy of the mean-field equations considering a widerange of particle numbers. We calculate the ground state energy and theone-body density for different values of the van der Waals parameter $C_{6}$.We compare our results with those of the modified Gross-Pitaevskii results, thecorrelated Hartree hypernetted-chain equations (which also utilize the two-bodycorrelated basis functions), as well as of the Diffusion Monte Carlo for hardsphere interactions. We observe the effect of the attractive tail of the vander Waals potential in the calculations of the one-body density over the trulyrepulsive zero-range potential as used in the Gross-Pitaevskii equation anddiscuss the finite-size effects. We also present the low-lying collectiveexcitations which are well described by a hydrodynamic model in the largeparticle limit.

journal of low temperature physics

Use of two-body correlated basis functions with van der Waals  interaction to study the shape-independent approximation for a large number  of trapped interacting bosons