Continuous Kubo-Greenwood formula: Theory and numerical implementation

In this paper, we present the so-called continuous Kubo-Greenwood formula intended for the numerical calculation of the dynamic Onsager coefficients and, in particular, the real part of dynamic electrical conductivity. In contrast to the usual Kubo-Greenwood formula, which contains the summation over a discrete set of transitions between electron energy levels, the continuous one is formulated as an integral over the whole energy range. This integral includes the continuous functions: the smoothed squares of matrix elements, $D(\ensuremath{\varepsilon},\ensuremath{\varepsilon}+\ensuremath{\hbar}\ensuremath{\omega})$, the densities of state, $g(\ensuremath{\varepsilon})g(\ensuremath{\varepsilon}+\ensuremath{\hbar}\ensuremath{\omega})$, and the difference of the Fermi weights, $[f(\ensuremath{\varepsilon})\ensuremath{-}f(\ensuremath{\varepsilon}+\ensuremath{\hbar}\ensuremath{\omega})]/(\ensuremath{\hbar}\ensuremath{\omega})$. The function $D(\ensuremath{\varepsilon},\ensuremath{\varepsilon}+\ensuremath{\hbar}\ensuremath{\omega})$ is obtained via the specially developed smoothing procedure. From the theoretical point of view, the continuous formula is an alternative to the usual one. Both can be used to calculate matter properties and produce close results. However, the continuous formula includes the smooth functions that can be plotted and examined. Thus, we can analyze the contributions of various parts of the electron spectrum to the obtained properties. The possibility of such analysis is the main advantage of the continuous formula. The continuous Kubo-Greenwood formula is implemented in the parallel code cubogram. Using the code we demonstrate the influence of technical parameters on the simulation results for liquid aluminum. We also analyze various methods of matrix elements computation and their effect on dynamic electrical conductivity.