  A previously proposed non-canonical coupled-perturbed Kohn-Sham density
functional theory (KS-DFT)/Hartree-Fock (HF) treatment for spin-orbit coupling
is here generalized to infinite periodic systems. The scalar-relativistic
periodic KS-DFT/HF solution, obtained with a relativistic effective core
potential, is taken as the zeroth-order approximation. Explicit expressions are
given for the total energy through 3rd-order, which satisfy the 2N + 1 rule
(i.e. requiring only the 1st-order perturbed wave function for determining the
energy through 3rd-order). Expressions for additional 2nd-order corrections to
the perturbed wave function (as well as related one-electron properties) are
worked out at the uncoupled-perturbed level of theory. The approach is
implemented in the \textsc{Crystal} program and validated with calculations of
the total energy, electronic band structure, and density variables of
spin-current DFT on the tungsten dichalcogenide hexagonal bilayer series (i.e.
WSe$_2$, WTe$_2$, WPo$_2$, WLv$_2$), including 6p and 7p elements as a stress
test. The computed properties through second- or third-order match well with
those from reference two-component self-consistent field (2c-SCF) calculations.
For total energies, $E^{(3)}$ was found to consistently improve the agreement
against the 2c-SCF reference values. For electronic band structures, visible
differences w.r.t. 2c-SCF remained through second-order in only the single-most
difficult case of WLv$_2$. As for density variables of spin-current DFT, the
perturbed electron density, being vanishing in first-order, is the most
challenging for the perturbation theory approach. The visible differences in
the electron densities are, however, largest close to the core region of atoms
and smaller in the valence region. Perturbed spin-current densities, on the
other hand, are well reproduced in all tested cases.
