  We present a fully ab initio based scheme to compute transport properties,
i.e. the electrical conductivity {\sigma} and thermopower S, in the presence of
electron-phonon interaction. Therefore, we explicitly investigate the
k-dependent structure of the Eliashberg spectral function , the coupling
strength, the linewidth and the relaxation time {\tau}. We obtain a
state-dependent {\tau} and show its necessity to reproduce the increased
thermopower at temperatures below the Debye temperature, without accounting for
the phonon-drag effect. Despite the detailed investigations of various k and q
dependencies, the presented scheme can be easily applied to more complicated
systems.
