  In collinear magnets lacking inversion symmetry application of electric
currents induces torques on the magnetization and conversely magnetization
dynamics induces electric currents. The two effects, which both rely on
spin-orbit interaction (SOI), are reciprocal to each other and denoted direct
spin-orbit torque (SOT) and inverse spin-orbit torque (ISOT), respectively. We
derive expressions for SOT and ISOT within the Kubo linear response formalism.
We show that expressions suitable for density-functional theory calculations
can be derived either starting from a Kohn-Sham Hamiltonian with time-dependent
exchange field or by expressing general susceptibilities in terms of the
Kohn-Sham susceptibilities. For the case of magnetic bilayer systems we derive
the general form of the ISOT current induced under ferromagnetic resonance.
Using \textit{ab initio} calculations within density-functional theory we
investigate SOT and ISOT in Co/Pt(111) magnetic bilayers. We determine the
spatial distribution of spin and charge currents as well as torques in order to
expose the mechanisms underlying SOT and ISOT and to highlight their
reciprocity on the microscopic level. We find that the spin Hall effect is
position-dependent close to interfaces.
