Validity of the on-site spin-orbit coupling approximation

Spin-orbit coupling (SOC) is generally understood as a highly localized interaction within each atom, whereby core electrons holding large $J$ splittings transfer the SOC to the valence electrons of the same atom, while their direct impact on neighbor valence orbitals is usually small. Seivane and Ferrer [Phys. Rev. Lett. 99, 183401 (2007)] proposed an approach within a tight-binding type ab initio framework assuming that the transfer of SOC from core to valence orbitals only takes place when both are on the same atom, leading to the so-called on-site approximation, which then has been successfully applied to a variety of systems. In this work we thoroughly test its general validity by confronting SOC related properties such as spin splittings, spin textures, or magnetic anisotropies calculated under the on-site approximation versus the more general approach where all the contributions to the SOC, including three-center integrals, are explicitly included. After considering a variety of systems with different dimensionalities, all presenting a strong SOC, we conclude that although the on-site approximation often provides accurate results, it breaks down in some systems where $5d$ electrons are close to the Fermi level due to their strong SOC and moderately large spatial extension. Furthermore, there are a few examples where subtle inaccuracies lead to qualitatively wrong conclusions, the most clear case being the doping of the topological surface state in ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$(0001). Finally, magnetic anisotropy energies calculated under this approximation tend to be underestimated.