Linear scaling density matrix search based on <i>sign</i> matrices

Németh, Károly; Scuseria, Gustavo E.

doi:10.1063/1.1308546

Cited by 45 publications

(41 citation statements)

References 15 publications

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“…Implementations of Car-Parrinello molecular dynamics have also been described [218,219], both for a simple fictitious electronic mass [218] and a variable fictitious mass to optimise convergence [219]. A closely related method to the DMM uses the sign matrix [220,221], which is equivalent to expanding the Fermi matrix (as in the Fermi Operator Expansion described in Sec. 3.2.3); another approach including implicit purification for finite temperatures has been derived [222].…”

Section: Direct and Iterative Approachesmentioning

confidence: 99%

\mathcal{O}(N) methods in electronic structure calculations

2012

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Abstract. Linear scaling methods, or O(N ) methods, have computational and memory requirements which scale linearly with the number of atoms in the system, N , in contrast to standard approaches which scale with the cube of the number of atoms. These methods, which rely on the short-ranged nature of electronic structure, will allow accurate, ab initio simulations of systems of unprecedented size. The theory behind the locality of electronic structure is described and related to physical properties of systems to be modelled, along with a survey of recent developments in real-space methods which are important for efficient use of high performance computers. The linear scaling methods proposed to date can be divided into seven different areas, and the applicability, efficiency and advantages of the methods proposed in these areas is then discussed. The applications of linear scaling methods, as well as the implementations available as computer programs, are considered. Finally, the prospects for and the challenges facing linear scaling methods are discussed.

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Section: Direct and Iterative Approachesmentioning

confidence: 99%

\mathcal{O}(N) methods in electronic structure calculations

2012

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“…The simplest of these iterative schemes, also known as the Newton-Schulz iteration and equivalent to the McWeeny procedure, has been employed in this and earlier work. 7,8 In the context of electronic structure calculations, higher order purification schemes have been proposed by Holas 9 and an efficient trace correcting purification method was proposed by Niklasson. 10,11 A detailed analysis of numerical error has been presented in Ref.…”

Section: Introductionmentioning

confidence: 99%

“…5 Starting from a properly scaled Hamiltonian the density matrix is constructed by a simple iterative procedure. [6][7][8][9][10][11][12] Effectively, high order polynomials (or rational functions) that approximate a step function are constructed and evaluated for the given Hamiltonian. Beylkin et al (Ref.…”

Section: Introductionmentioning

confidence: 99%

Linear Scaling Self-Consistent Field Calculations with Millions of Atoms in the Condensed Phase

VandeVondele

Borštnik

Hutter

2012

J. Chem. Theory Comput.

155

196

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In this work, the applicability and performance of a linear scaling algorithm is investigated for three-dimensional condensed phase systems. A simple but robust approach based on the matrix sign function is employed together with a thresholding matrix multiplication that does not require a prescribed sparsity pattern. Semiempirical methods and density functional theory have been tested. We demonstrate that self-consistent calculations with 1 million atoms are feasible for simple systems. With this approach, the computational cost of the calculation depends strongly on basis set quality. In the current implementation, high quality calculations for dense systems are limited to a few hundred thousand atoms. We report on the sparsities of the involved matrices as obtained at convergence and for intermediate iterations. We investigate how determining the chemical potential impacts the computational cost for very large systems. AbstractIn this work, the applicability and performance of a linear scaling algorithms is investigated for three dimensional condensed phase systems. A simple but robust approach based on the matrix sign function is employed together with a thresholding matrix multiplication that does not require a prescribed sparsity pattern. Semi-empirical methods and density functional theory have been tested. We demonstrate that self consistent calculations with a million atoms are feasible for simple systems. With this approach the computational cost of the calculation depends strongly on basis set quality. In the current implementation, high quality calculations for dense systems are limited to a few hundred thousand atoms. We report on the sparsities of the involved matrices as obtained at convergence and for intermediate iterations. We investigate on how determining the chemical potential impacts the computational cost for very large systems.

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“…Based on a recently developed density matrix perturbation theory [21], Perturbed Projection exploits the explicit relationship between the density matrix D and the effective Hamiltonian or Fockian F via spectral projection; D = θ(μI − F ), wherein θ is the Heaviside step func-THE COUPLED PERTURBED SELF-CONSISTENT-FIELD EQUATIONS tion (spectral projector) and the chemical potentialμ determines occupied states via Aufbau filling. Spectral projection can be carried out in a number of ways [22,23,24,25,26,27,28,29]. Of special interest here are recursive polynomial expansions of the projector, including the second order trace correcting (TC2) [22] and fourth order trace resetting (TRS4) [23] purification algorithms.…”

Section: Introductionmentioning

confidence: 99%

Higher-order response in O(N) by perturbed projection

Weber

Niklasson

Challacombe

2005

The Journal of Chemical Physics

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Perturbed projection for linear scaling solution of the coupled-perturbed self-consistent-field equations [Weber, Niklasson and Challacombe, Phys. Rev. Lett. 92, 193002 (2004)] is extended to the computation of higher order static response properties. Although generally applicable, perturbed projection is developed here in the context of the self-consistent first and second electric hyperpolarizabilities of three dimensional water clusters at the Hartree-Fock level of theory. Non-orthogonal, density matrix analogues of Wigner's 2n + 1 rule are given up to fourth order. Linear scaling and locality of the higher order response densities under perturbation by a global electric field are demonstrated.

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Linear scaling density matrix search based on sign matrices

Cited by 45 publications

References 15 publications

\mathcal{O}(N) methods in electronic structure calculations

\mathcal{O}(N) methods in electronic structure calculations

Linear Scaling Self-Consistent Field Calculations with Millions of Atoms in the Condensed Phase

Higher-order response in O(N) by perturbed projection

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