Ground state energy of the f=1 spinor Bose-Einstein condensates

We calculate, in the standard Bogoliubov approximation, the ground state energy of the spinor BEC with hyperfine spin $f=1$ where the two-body repulsive hard-core and spin exchange interactions are both included. The coupling constants characterized these two competing interactions are expressed in terms of the corresponding s-wave scattering lengths using second-order perturbation methods. We show that the ultraviolet divergence arising in the ground state energy corrections can be exactly eliminated.