We present multiscale calculations to describe the spin transport behavior of the Co/Cu bilayer structure including the effect of the interface. The multiscale approach introduces the connection between the ab initio calculation used to describe the electronic structure of the system and the generalized spin accumulation model employed to describe the spin transport behavior. We have applied our model to atomically smooth and diffuse interfaces. The results demonstrate the huge importance of the use of first principle calculations, not only due to the interfacial coordinates optimization but also the magnetic and electronic properties obtained through the electronic structure. The system including the effect of interface with and without the charge fluctuation are studied. The results indicate that changes of electronic structure at the Co/Cu interface give rise to an interfacial resistance distributed over several atomic planes, similar to the effect of interface diffusion. We argue that even atomically smooth Co/Cu interfaces have properties analogous to a diffuse interface due to the variation of electronic structure at the interface.