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.
To put our work further into perspective, we quote from two prominent researchers in the field of electronic structure, one a mathematician, the other a physicist. In his excellent survey @cite_90 Claude Le Bris, discussing the basis for linear scaling algorithms, i.e., the assumed sparsity of the density matrix, wrote (pages 402 and 404):
@cite_90
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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 We give two different and simple constructions for dimensionality reduction in e2 via linear mappings that are sparse: only an O(e)-fraction of entries in each column of our embedding matrices are non-zero to achieve distortion 1 + e with high probability, while still achieving the asymptotically optimal number of rows. These are the first constructions to provide subconstant sparsity for all values of parameters, improving upon previous works of Achlioptas [2003] and [2010]. Such distributions can be used to speed up applications where e2 dimensionality reduction is used. The subject of sparse matrices has its roots in such diverse fields as management science, power systems analysis, surveying, circuit theory, and structural analysis. Mathematical models in all these areas give rise to very large systems of linear equations which could not be solved were it not for the fact that the matrices contain relatively few non-zero entries. Only comparatively recently, in the last fifteen years or so, has it become apparent that the equations can be solved even when the pattern is irregular, and it is primarily the solution of such problems that is considered in this book. The subject is intensely practical and this book is written with practicalities ever in mind. Whenever two methods are applicable, their rival merits are considered, and conclusions are based on practical experience where theoretical comparison is not possible. Non-numeric computing techniques have been included, as well as frequent illustrations, in an attempt to bridge the usually wide gap between the printed page and the working computer code. Despite this practical bias, it is recognized that many aspects of the subject are of interest in their own right, and the book aims to be suitable also for a student course, probably at MSc level. Exercises have been included to illustrate and strengthen understanding of the material, as well as to extend it. Efficient use of sparsity is a key to solving large problems in many fields. This book will supply both insight and answers for those attempting to solve these problems. 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.