Electric conductance tensor from an <i>ab initio</i> approach that includes the energy dissipation term

We previously considered the effect of the energy dissipation term in time-dependent current-density-functional theory (ed-tdCDFT). Here, we expand the ed-tdCDFT by using energy bands composed of magnetic Bloch states. To derive the electric conductance tensor representing the conductivity parallel and perpendicular to the applied electric field, we apply perturbation theory up to the second order to the Kohn-Sham equation of the present ed-tdCDFT. The nondiagonal part of the electric conductance tensor consists of two terms, with the first term providing the integer quantum Hall effect (IQHE). The second term of the nondiagonal part corresponds to the Hall effect (HE). The diagonal part provides a quantum-mechanical expression of Ohm's conductivity. It is shown that the second term of the nondiagonal part is reduced to the HE in the classical limit, and that the diagonal part is reduced to Ohm's conductivity in the Drude model. This result indicates that the energy dissipation term is appropriately incorporated into the present ed-tdCDFT. A key contribution is the unified explanation of Ohm's law, the HE, and the IQHE, using a single theoretical framework.