Decay Properties of Spectral Projectors with Applications to Electronic Structure

Benzi, Michele; Boito, Paola; Razouk, Nader

doi:10.1137/100814019

Cited by 136 publications

(198 citation statements)

References 123 publications

(251 reference statements)

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“…[20][21][22] In addition, the density matrix, ρ (r, r ′ ), itself decays exponentially with distance, |r − r ′ |, in the position representation for systems with a non-negligible HOMO-LUMO gap. 3,23,24 For electronic structure theory based on localized AO basis functions, this translates into sparsity of the density matrix in the AO representation, as well as potential localization of the molecular orbitals (MOs). This sparsity is basis-set dependent and has been studied empirically.…”

Section: Introductionmentioning

confidence: 99%

Fast, accurate evaluation of exact exchange: The occ-RI-K algorithm

Manzer

Horn

Mardirossian

et al. 2015

The Journal of Chemical Physics

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Construction of the exact exchange matrix, K, is typically the rate-determining step in hybrid density functional theory, and therefore, new approaches with increased efficiency are highly desirable. We present a framework with potential for greatly improved efficiency by computing a compressed exchange matrix that yields the exact exchange energy, gradient, and direct inversion of the iterative subspace (DIIS) error vector. The compressed exchange matrix is constructed with one index in the compact molecular orbital basis and the other index in the full atomic orbital basis. To illustrate the advantages, we present a practical algorithm that uses this framework in conjunction with the resolution of the identity (RI) approximation. We demonstrate that convergence using this method, referred to hereafter as occupied orbital RI-K (occ-RI-K), in combination with the DIIS algorithm is well-behaved, that the accuracy of computed energetics is excellent (identical to conventional RI-K), and that significant speedups can be obtained over existing integral-direct and RI-K methods. For a 4400 basis function C 68 H 22 hydrogen-terminated graphene fragment, our algorithm yields a 14× speedup over the conventional algorithm and a speedup of 3.3× over RI-K. C 2015 AIP Publishing LLC. [http://dx

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Section: Introductionmentioning

confidence: 99%

Fast, accurate evaluation of exact exchange: The occ-RI-K algorithm

Manzer

Horn

Mardirossian

et al. 2015

The Journal of Chemical Physics

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“…In the last 15 years, a number of significant results have established exponentially decaying bounds for the magnitude of the entries of holomorphic functions of sparse matrices [11,12,13]. These results have given rise to a flurry of applications in linear algebra and physics as they underly efficient approximation techniques, see for example [11,14,15].…”

Section: Decay Propertiesmentioning

confidence: 99%

“…These results have given rise to a flurry of applications in linear algebra and physics as they underly efficient approximation techniques, see for example [11,14,15]. As we shall see below, the techniques used to prove these results do not extend to the time-ordered exponential of time-dependent matrices which do not commute with themselves at different times.…”

Section: Decay Propertiesmentioning

confidence: 99%

An exact formulation of the time-ordered exponential using path-sums

Giscard

Lui

Thwaite

et al. 2015

Journal of Mathematical Physics

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We present the path-sum formulation for OE [H](t ′ , t) = T exp t ′ t H(τ ) dτ , the time-ordered exponential of a time-dependent matrix H(t). The path-sum formulation gives OE[H] as a branched continued fraction of finite depth and breadth. The terms of the path-sum have an elementary interpretation as selfavoiding walks and self-avoiding polygons on a graph. Our result is based on a representation of the time-ordered exponential as the inverse of an operator, the mapping of this inverse to sums of walks on graphs and the algebraic structure of sets of walks. We give examples demonstrating our approach. We establish a super-exponential decay bound for the magnitude of the entries of the timeordered exponential of sparse matrices. We give explicit results for matrices with commonly encountered sparse structures.Keywords: Time-ordered exponential, path-ordered exponential, path-sum, differential equation, finite continued fraction, super-exponential decay ContextThe time-ordered exponential function OE[H](t ′ , t) = T exp t ′ t H(τ ) dτ , also known as path-ordered exponential, is the unique solution of the system of differential equationssuch that OE[H](t ′ , t ′ ) = I is the identity at all times. We take H ∈ C n×n [I], n ∈ N\{0}, to be a matrix depending smoothly on the continuous variable t ∈ I, which we call time without loss of generality. In spite of the importance of the

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“…The interest for the decay behavior of matrix functions stems largely from its importance for a number of applications, including numerical analysis [6,13,16,17,22,40,46], harmonic analysis [2,26,33], quantum chemistry [5,11,37,42], signal processing [35,43], quantum information theory [14,15,23], multivariate statistics [1], queuing models [9,10], control of large-scale dynamical systems [29], quantum dynamics [25], random matrix theory [41], and others. The first case to be analyzed in detail was that of f (A) = A −1 ; see [17,18,22,34].…”

Section: Introductionmentioning

confidence: 99%

Decay Bounds for Functions of Hermitian Matrices with Banded or Kronecker Structure

Benzi¹,

Simoncini²

2015

SIAM J. Matrix Anal. & Appl.

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Abstract. We present decay bounds for completely monotonic functions of Hermitian matrices, where the matrix argument is banded or a Kronecker sum of banded matrices. This class includes the exponential, the negative fractional roots, and other functions that are important in applications. Besides being significantly tighter than previous estimates, the new bounds closely capture the actual (nonmonotonic) decay behavior of the entries of functions of matrices with Kronecker sum structure. We also discuss extensions to more general sparse matrices.Key words. completely monotonic matrix functions, banded matrices, sparse matrices, offdiagonal decay, Kronecker structure AMS subject classifications. 15A16, 65F60 DOI. 10.1137/1510061591. Introduction. The decay behavior of the entries of functions of banded and sparse matrices has attracted considerable interest over the years. It has been known for some time that if A is a banded Hermitian matrix and f is a smooth function with no singularities in a neighborhood of the spectrum of A, then the entries in f (A) usually exhibit rapid decay in magnitude away from the main diagonal. The decay rates are typically exponential, with even faster decay in the case of entire functions.The interest for the decay behavior of matrix functions stems largely from its importance for a number of applications, including numerical analysis [6,13,16,17,22,40,46] [17,18,22,34]. In these papers one can find exponential decay bounds for the entries of the inverse of banded matrices. A related but quite distinct line of research concerned the study of inverse-closed matrix algebras, where the decay behavior in the entries of a (usually infinite) matrix A is "inherited" by the entries of A −1 . Here we mention [33], where it was observed that a similar decay behavior occurs for the entries of f (A) = A −1/2 , as well as [2,3,26,27,35], among others.The study of the decay behavior for general analytic functions of banded matrices, including the important case of the matrix exponential, was initiated in [6,32] and continued for possibly nonnormal matrices and general sparsity patterns in [7]; further contributions in these directions include [4,16,38,42]. Collectively, these papers have largely elucidated the question of when one can expect exponential decay in the entries

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Decay Properties of Spectral Projectors with Applications to Electronic Structure

Cited by 136 publications

References 123 publications

Fast, accurate evaluation of exact exchange: The occ-RI-K algorithm

Fast, accurate evaluation of exact exchange: The occ-RI-K algorithm

An exact formulation of the time-ordered exponential using path-sums

Decay Bounds for Functions of Hermitian Matrices with Banded or Kronecker Structure

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