Based on the independent-particle-approximation (i.e. the multiple scattering self-consistent-field theory), we studied the electronic structures of Rydberg series of the diatomic molecules with Si as their united atom, such as N2, CO, BF, A1H, LiNa, "BeNe" and "MgHe". The principal quantum number of the first state of the Rydberg series is determined from the limit atom of the molecular elecronic configuration. The dynamics of the excited molecules has been elucidated. Our theoretical results prove to be in fair agreement with the existing experimental measurements, thus it can provide a reliable basis for future refined calculations such as the configuation interaction methods.