  For the technologically relevant spin Hall effect most theoretical approaches
rely on the evaluation of the spin-conductivity tensor. In contrast, for most
experimental configurations the generation of spin accumulation at interfaces
and surfaces is the relevant quantity. Here, we directly calculate the
accumulation of spins due to the spin Hall effect at the surface of a thin
metallic layer, making quantitative predictions for different materials. Two
distinct limits are considered, both relying on a fully relativistic
Korringa-Kohn-Rostoker density functional theory method. In the semiclassical
approach, we use the Boltzmann transport formalism and compare it directly to a
fully quantum mechanical non-equilibrium Keldysh formalism. Restricting the
calculations to the spin Hall induced, odd in spatial inversion, contribution
in the limit of the relaxation time approximation we find good agreement
between both methods, where deviations can be attributed to the complexity of
Fermi surfaces. Finally, we compare our results to experimental values of the
spin accumulation at surfaces as well as the Hall angle and find good agreement
for the trend across the considered elements.
