Transport calculations in complex materials: A comparison of the Kubo formula, the Kubo-Greenwood formula and the microscopic response method

Recently we have introduced the microscopic response method (MRM) to compute the conductivity and Hall mobility for complex system with topological and thermal disorder, which is more convenient than the Kubo formula. We prove that for a canonical ensemble the MRM leads to the same expression as the Kubo formula. When the gradient of carrier density is small, the MRM reduces to the widely used Kubo-Greenwood formula.