Motivated by applications in quantum chemistry and solid state physics, we apply general results from approximation theory and matrix analysis to the study of the decay properties of spectral projectors associated with large and sparse Hermitian matrices. Our theory leads to a rigorous proof of the exponential off-diagonal decay ("nearsightedness") for the density matrix of gapped systems at zero electronic temperature in both orthogonal and non-orthogonal representations, thus providing a firm theoretical basis for the possibility of linear scaling methods in electronic structure calculations for non-met allic systems. We further discuss the case of density matrices for met allic systems at positive electronic temperature. A few other possible applications are also discussed.
It is interesting to compare these statements with two earlier ones by Stefan Goedecker. In @cite_9 he wrote (page 261):
@cite_9
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Electronic structure calculations which are based on Wannier, like localized orbitals or the related density matrix, are an alternative to conventional calculations based on extended orbitals. For large systems this approach is potentially faster since it offers O(N) scaling with respect to the number of atoms in the system. We derive a class of algorithms based on projection to calculate either the localized orbitals or the density matrix.