At zero temperature, two-site dynamical mean field theory is applied to the dynamic Hubbard model. The dynamic Hubbard model describes the orbital relaxation that occurs when two electrons occupy the same site, by using a two-level boson field at each site. At finite boson frequency, the appearance of a Mott gap is found to be enhanced even though it shows a metallic phase with the same bare on-site interaction U in the conventional Hubbard model. The lack of electron-hole symmetry is highlighted through the quasiparticle weight and the single-particle density of states at different fillings, which qualitatively differentiates the dynamic Hubbard model from other conventional Hubbard-type models.