the s -wave scattering length ( a$_{s}$ ). The small r$_{ij}$ behavior of ψ$_{ij}$ will be that of the zero- energy wave function η ( r$_{ij}$ ) of the pair interacting via V ( r$_{ij}$ ). Hence we introduce η ( r$_{ij}$ ) as a short-range correlation function in the PH expansion $^{19}$. We have checked explicitly that this improves the rate of convergence dramatically. In order that the small r$_{ij}$ behavior of ψ$_{ij}$ ( r$_{ij}$, r ) corresponds to the correct two-body interaction appropriate for the experimental a$_{s}$ , we adjust the very short-range behavior of the interatomic interaction (in the case of the van der Waals potential used by us, it will be the hard-core radius r$_{c}$ ) so that η ( r$_{ij}$ ) asymptotically becomes proportional to (1 − a$_{s}$ r$_{ij}$ ) $^{23}$. Substitution of this expansion of ψ in Eq. (2) and projection on a particular PH gives rise to a set of coupled differential equations in r 19 where L = l + (3 A − 6) / 2, U$_{Kl}$ ( r ) = f$_{Kl}$u l $_{K}$( r ), α = 3 A − 8 2 and β = l + 1 $_{2}$. Here, l is the orbital angular momentum of the system (assumed to be contributed by the interacting pair only) and u l $_{K}$( r ) is the coefficient of expansion of ψ$_{ij}$ ( r$_{ij}$, r ) in the correlated PH basis, while f$_{Kl}$ represents the overlap of the PH corresponding to the ( ij )-partition with the sum of PHs of all partitions $^{18}$. Expressions for the potential matrix element V$_{KK}$ ′ ( r ) and f$_{Kl}$ can be found in Ref. $^{18,19}$. Eq. (5) is solved by hyperspherical adiabatic approximation (HAA) $^{24}$, for which we assume that the hyperangular motion is much faster than the hyperradial motion. Consequently, the former can be separated adiabatically and solved to obtain an effective potential as a parametric function of r , in which the hyperradial motion takes place. The hyperangular motion is effectively solved by diagonalizing the potential matrix V$_{KK}$ ′ ( r ) together with the hyper-centrifugal potential of Eq. (5) to get the lowest eigenvalue ω$_{0}$ ( r ) [corresponding eigen column vector being χ$_{K}$$_{0}$ ( r )], which is the effective potential for the hyperradial motion. Finally, the adiabatically separated hyperradial equation [ − ℏ 2 m d 2 dr 2 + ω$_{0}$ ( r ) + ∑ K | χ$_{K}$$_{0}$ ( r ) dr | 2 − E$_{R}$ ] ζ$_{0}$ ( r ) = 0 . (6) [ − ℏ 2 m d 2 dr 2 + ℏ 2 mr 2 {L ( L + 1) + 4 K ( K + α + β + 1) } + V$_{trap}$ ( r ) − E$_{R}$ ] U$_{Kl}$ ( r ) (5) + ∑ K ′ f$_{Kl}$V$_{KK}$ ′ ( r ) f$_{K}$ ′ $_{l}$U$_{K}$ ′ $_{l}$( r ) = 0 , 5