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, as representatives of the third group of papers we select @cite_78 and @cite_32 . The authors of @cite_78 use the Fermi--Dirac approximation of the density matrix and consider its expansion in the Chebyshev basis. From an estimate of the rate of decay of the coefficients of the Chebyshev expansion of @math , they obtain estimates for the number of terms needed to satisfy a prescribed error in the approximation of the density matrix. In turn, this yields estimates for the rate of decay as a function of the extreme eigenvalues and spectral gap of the discrete Hamiltonian. Because of some ad hoc assumptions and the many approximations used the arguments in this paper cannot be considered mathematically rigorous, and the estimates thus obtained are not always accurate. Nevertheless, the idea of using a polynomial approximation for the Fermi--Dirac function and the observation that exponential decay of the expansion coefficients implies exponential decay in the (approximate) density matrix is quite valuable and, as we show in this paper, can be made fully rigorous.
@cite_32 @cite_78
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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. The range and sparsity of the one-electron density matrix (DM) in density functional theory is studied for large systems using the analytical properties of its Chebyshev expansion. General estimates of the range of the DM are derived, showing that the range is inversely proportional to the square root of an insulator band gap and inversely proportional to the square root of the temperature. These findings support open_quotes principle of nearsightedness close_quotes introduced recently by W. Kohn [Phys.Rev.Lett. bold 76 , 3168 (1996)]. These estimates are used to study the complexity of several linear system-size scaling electronic structure algorithms which differ in their dependence on the geometric dimensionality of the system. copyright ital 1997 ital The American Physical Society