  We present a comprehensive ab initio investigation on Mg$_3$Bi$_2$, a
promising Mg-ion battery anode material with high rate capacity. Through
combined DFT (PBE, HSE06) and $G_0W_0$ electronic structure calculations, we
find that Mg$_3$Bi$_2$ is likely to be a small band gap semiconductor.
  DFT-based defect formation energies indicate that Mg vacancies are likely to
form in this material, with relativistic spin-orbit coupling significantly
lowering the defect formation energies. We show that a transition state
searching methodology based on the hybrid eigenvector-following approach can be
used effectively to search for the transition states in cases where full
spin-orbit coupling is included. Mg migration barriers found through this
hybrid eigenvector-following approach indicate that spin-orbit coupling also
lowers the migration barrier, decreasing it to a value of 0.34 eV with
spin-orbit coupling. Finally, recent experimental results on Mg diffusion are
compared to the DFT results and show good agreement. This work demonstrates
that vacancy defects and the inclusion of relativistic spin-orbit coupling in
the calculations have a profound effect in Mg diffusion in this material. It
also sheds light on the importance of relativistic spin-orbit coupling in
studying similar battery systems where heavy elements play a crucial role.
