In this paper, the bound-state energy eigenvalues for several diatomic molecules (O2, I2, N2, H2, CO, NO, LiH, HCl) are computed for various quantum numbers using the shifted 1/N expansion method with the Kratzer, Morse and Modified Morse potentials. Numerical results of the energy eigenvalues for the selected diatomic molecules are discussed. Our results for energy eigenvalues agree perfectly with the results obtained in the literature. A comparative study is performed for four diatomic molecules (H2, N2, CO and HCl) in their ground states using the three potentials.