Spin-orbit coupling in the spin-current-density-functional theory

Starting from the spin-current-density-functional theory for electronic systems, we extend the formulation to include spin-orbit coupling. Particular attention is devoted to the symmetry of the problem. Here we show that the exchange-correlation energy functional is invariant by the U(1)em x SU(2)spin gauge transformations. We give the transformation laws of the paramagnetic current and also the paramagnetic spin current density by the U(1)em x SU(2)spin gauge transformations. For the case where the spin-orbit coupling is taken into account, we generalize the equations of continuity satisfied by the current density and the spin current density, derived by Vignale and Rasolt