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.
 The cost of Hartree−Fock and local correlation methods is strongly dependent on the locality of the one-particle density matrix and localized orbitals. In this paper the locality and sparsity of the one-particle density matrix is investigated numerically and theoretically, primarily at the Hartree−Fock level, for linear alkanes containing up to 320 carbon atoms. A method for the calculation of localized, atom-centered, occupied orbitals is presented and compared with the Boys' localization procedure. The atom-centered orbitals are ideally suited for use in local-correlation calculations. The connection between the size of optimally localized orbitals, the locality of the density matrix, and the onset of linear scaling is investigated. Analytic results for the asymptotic decay of the electron density matrix in insulators have been obtained in all three dimensions ( @math ) for a tight-binding model defined on a simple cubic lattice. The anisotropic decay length is shown to be dependent on the energy parameters of the model. The existence of the power-law prefactor, @math , is demonstrated.  The spatial decay properties of Wannier functions and related quantities have been investigated using analytical and numerical methods. We find that the form of the decay is a power law times an exponential, with a particular power-law exponent that is universal for each kind of quantity. In one dimension we find an exponent of @math for Wannier functions, @math for the density matrix and for energy matrix elements, and @math or @math for different constructions of nonorthonormal Wannier-like functions. Analytical results for the asymptotic spatial decay of the density matrix p(r,r') in the tight-binding model of the two-dimensional met al are presented. In various dimensions D, it is found analytically and numerically that the density matrix decays with distance according to the power law ρ(r,r')‖r-r'‖ - ( D + 1 ) 2 . We provide a tight-binding model of insulator, for which we derive an exact analytic form of the one-body density matrix and its large-distance asymptotics in dimensions @math . The system is built out of a band of single-particle orbitals in a periodic potential. Breaking of the translational symmetry of the system results in two bands, separated by a direct gap whose width is proportional to the unique energy parameter of the model. The form of the decay is a power law times an exponential. We determine the power in the power law and the correlation length in the exponential, versus the lattice direction, the direct-gap width, and the lattice dimension. In particular, the obtained exact formulae imply that in the diagonal direction of the square lattice the inverse correlation length vanishes linearly with the vanishing gap, while in non-diagonal directions, the linear scaling is replaced by the square root one. Independently of direction, for sufficiently large gaps the inverse correlation length grows logarithmically with the gap width.