Linear scaling methods in density functional theory must address first the assembly of the effective Hamiltonian (and its derivatives for forces) for a given density matrix, and second the problem of varying the density matrix towards the minimum energy. This talk will provide a self-contained introduction to the fast methods commonly employed for the first problem. For the Coulomb problem, we shall discuss the current adaptations of the fast multipole method for the far-field, together with special purpose methods for rapid evaluation of the near-field contributions. We shall also discuss the corresponding problem of exact exchange interactions associated with hybrid density functional theory, for which conditionally linear scaling approaches are available. All of these methods are formulated for both energies and energy gradients with respect to atomic centers (atomic forces).
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