  We probe the accuracy limit of {\it ab initio} calculations of carrier
mobilities in semiconductors, within the framework of the Boltzmann transport
equation. By focusing on the paradigmatic case of silicon, we show that fully
predictive calculations of electron and hole mobilities require many-body
quasiparticle corrections to band structures and electron-phonon matrix
elements, the inclusion of spin-orbit coupling, and an extremely fine sampling
of inelastic scattering processes in momentum space. By considering all these
factors we obtain excellent agreement with experiment, and we identify the band
effective masses as the most critical parameters to achieve predictive
accuracy. Our findings set a blueprint for future calculations of carrier
mobilities, and pave the way to engineering transport properties in
semiconductors by design.
