Localization of electronic structure plays a important role in designing efficient algorithms for density functional theory and for theoretical understanding of systems in the thermodynamic limit. In this lecture, we will discuss the localization from the perspective of exponential decay of density matrix for systems with band gap in the spectrum or systems at finite temperature. Using the spectral representation of density matrices, the localization originates from the decay of the Green's function of the effective Hamiltonian. We will provide a self-contained proof of the exponential decay using the method of exponential weighting. Time permitting, we will briefly explain how localization can be applied for analysis and algorithm development for DFT.
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