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.
Among the papers in the second group, we mention @cite_67 @cite_55 @cite_48 @cite_116 @cite_62 @cite_69 @cite_125 . These papers provide quantitative decay estimates for the density matrix, either based on fairly rigorous analyses of special cases, or on not fully rigorous discussions of general situations. Large use is made of approximations, asymptotics, heuristics and physically motivated assumptions, and the results are often validated by numerical calculations. Also, it is occasionally stated that while the results were derived in the case of simplified models, the conclusions should be valid in general. Several of these authors emphasize the difficulty of obtaining rigorous results for general systems in arbitrary dimension. In spite of not being fully rigorous from a mathematical point of view, these contributions are certainly very valuable and seem to have been broadly accepted by physicists and chemists. We note, however, that the results in these papers usually take the form of order-of-magnitude estimates for the density matrix @math in real space, valid for sufficiently large separations @math , rather than strict upper bounds. As said before of Kohn's results, this type of estimates may be difficult to use for computational purposes.
@cite_67 @cite_62 @cite_69 @cite_48 @cite_55 @cite_125 @cite_116
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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 . Abstract We study the recovery of Hermitian low rank matrices X ∈ C n × n from undersampled measurements via nuclear norm minimization. We consider the particular scenario where the measurements are Frobenius inner products with random rank-one matrices of the form a j a j ⁎ for some measurement vectors a 1 , … , a m , i.e., the measurements are given by b j = tr ( X a j a j ⁎ ) . The case where the matrix X = x x ⁎ to be recovered is of rank one reduces to the problem of phaseless estimation (from measurements b j = | 〈 x , a j 〉 | 2 ) via the PhaseLift approach, which has been introduced recently. We derive bounds for the number m of measurements that guarantee successful uniform recovery of Hermitian rank r matrices, either for the vectors a j , j = 1 , … , m , being chosen independently at random according to a standard Gaussian distribution, or a j being sampled independently from an (approximate) complex projective t-design with t = 4 . In the Gaussian case, we require m ≥ C r n measurements, while in the case of 4-designs we need m ≥ Cr n log ⁡ ( n ) . Our results are uniform in the sense that one random choice of the measurement vectors a j guarantees recovery of all rank r-matrices simultaneously with high probability. Moreover, we prove robustness of recovery under perturbation of the measurements by noise. The result for approximate 4-designs generalizes and improves a recent bound on phase retrieval due to Gross, Krahmer and Kueng. In addition, it has applications in quantum state tomography. Our proofs employ the so-called bowling scheme which is based on recent ideas by Mendelson and Koltchinskii. Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253 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.