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.
Finally, in @cite_32 the authors present the results of numerical calculations for various insulators in order to gain some insight on the dependence of the decay length on the gap. Their experiments confirm that the decay behavior of @math can be strongly anisotropic, and that different rates of decay may occur in different directions; this is consistent with the analytical results in @cite_116 .
@cite_116 @cite_32
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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. We compute the single-particle density matrix in large (500-, 512-, and 1000-atom) models of fcc aluminum and crystalline (diamond) and amorphous silicon and carbon. We use an approximate density functional Hamiltonian in the local density approximation. The density matrix for fcc aluminum is found to closely approximate the results for jellium, and the crystalline and amorphous insulators exhibit exponential decay albeit with pronounced anisotropy. We compare the computed decays to existing predictions of the fall off of the density matrix in insulators and find that the open_quotes tight-binding close_quotes prediction of Kohn [W. Kohn, Phys. Rev. >115, 809 (1959)] provides the best overall fit to our calculations for Si and C.