The thermoelectric properties of 1.6-nm-thick Si square nanowires with [100] crystalline orientation are calculated over a wide temperature range from 0 K to 1000 K, taking into account atomistic electron-phonon interaction. In our model, the [010] and [001] facets are passivated by hydrogen and there are Si-Si dimers on the nanowire surface. The electronic structure was calculated by using the sp3 spin-orbit-coupled atomistic second-nearest-neighbor tight-binding model. The phonon dispersion was calculated from a valence force field model of the Brenner type. A scheme for calculating electron-phonon matrix elements from a second-nearest-neighbor tight-binding model is presented. Based on Fermi's golden rule, the electron-phonon transition rate was obtained by combining the electron and phonon eigenstates. Both elastic and inelastic scattering processes are taken into consideration. The temperature dependence of transport characteristics was calculated by using a solution of the linearized Boltzmann transport equation obtained by means of the iterative orthomin method. At room temperature, the electron mobility is 195 cm2 V-1 s-1 and increases with temperature, while a figure of merit ZT=0.38 is reached for n-type doping with a concentration of n=1019 cm-3.