An extension of the Kubo-Greenwood formula for use in molecular simulations

We discuss the foundations and extend the range of applicability of the widely used Kubo-Greenwood formula (KGF) for the electronic conductivity. The conductivity is derived from the current density, and only the probability amplitude rather than the transition probability is used. It is shown that the contribution to the conductivity from degenerate states in a low or zero frequency external electric field and the contribution from states near resonance with a finite frequency external field are finite. The improved conductivity expression does not include the familiar "energy conserving" delta function, and no artificial broadening parameter for delta function is required for the DC limit. We explored two methods of computing current density. We discuss the role of many-electron statistics in computing the conductivity in single-particle approximations, and we show that the conventional KGF is due to the contribution from single-particle excited states.