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The Kubo-Greenwood (KG) formula is often used in conjunction with Kohn-Sham(KS) density functional theory (DFT) to compute the optical conductivity,particularly for warm dense mater. For applying the KG formula, all KSeigenstates and eigenvalues up to an energy cutoff are required and thus theapproach becomes expensive, especially for high temperatures and large systems,scaling cubically with both system size and temperature. Here, we develop anapproach to calculate the KS conductivity within the stochastic DFT (sDFT)framework, which requires knowledge only of the KS Hamiltonian but not itseigenstates and values. We show that the computational effort associated withthe method scales linearly with system size and reduces in proportion to thetemperature unlike the cubic increase with traditional deterministicapproaches. In addition, we find that the method allows an accurate descriptionof the entire spectrum, including the high-frequency range, unlike thedeterministic method which is compelled to introduce a high-frequency cut-offdue to memory and computational time constraints. We apply the method tohelium-hydrogen mixtures in the warm dense matter regime at temperatures of\sim60\text{kK} and find that the system displays two conductivity phases,where a transition from non-metal to metal occurs when hydrogen atomsconstitute \sim0.3 of the total atoms in the system.

Transition to metallization in warm dense helium-hydrogen mixtures using stochastic density functional theory within the Kubo-Greenwood formalism