  Spin-current density functional theory (SCDFT) is a formally exact framework
designed to handle the treatment of interacting many-electron systems including
spin-orbit coupling at the level of the Pauli equation. In practice, robust and
accurate calculations of the electronic structure of these systems call for
functional approximations that depend not only on the densities, but also on
spin-orbitals. Here we show that the call can be answered by resorting to an
extension of the Kohn-Sham formalism, which admits the use of non-local
effective potentials, yet it is firmly rooted in SCDFT. The power of the
extended formalism is demonstrated by calculating the spin-orbit-induced
band-splittings of inversion-asymmetric MoSe$_2$ monolayer and
inversion-symmetric bulk $\alpha$-MoTe$_2$. We show that quantitative agreement
with experimental data is obtainable via global hybrid approximations by
setting the fraction of Fock exchange at the same level which yields accurate
values of the band gap. Key to these results is the ability of the method to
self-consistently account for the spin currents induced by the spin-orbit
interaction. The widely used method of refining spin-density functional theory
by a second-variational treatment of spin-orbit coupling is unable to match our
SCDFT results.
