Real-space local polynomial basis for solid-state electronic-structure calculations: A finite-element approach

Pask, John E.; Klein, Bern; Fong, C. Y.; Sterne, P. A.

doi:10.1103/physrevb.59.12352

Cited by 106 publications

(119 citation statements)

References 44 publications

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“…Ackermann et al (1994) used a tetrahedral discretization with orders p = 1 − 5. Pask et al (1999) utilized piecewise cubic functions (termed "serendipity" elements). Yu et al (1994) employed a Lobatto-Gauss basis set with orders ranging from five to seven.…”

Section: Finite-element Basesmentioning

confidence: 99%

Real-space mesh techniques in density-functional theory

2000

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This review discusses progress in efficient solvers which have as their foundation a representation in real space, either through finite-difference or finite-element formulations. The relationship of real-space approaches to linear-scaling electrostatics and electronic structure methods is first discussed. Then the basic aspects of real-space representations are presented. Multigrid techniques for solving the discretized problems are covered; these numerical schemes allow for highly efficient solution of the grid-based equations. Applications to problems in electrostatics are discussed, in particular numerical solutions of Poisson and Poisson-Boltzmann equations. Next, methods for solving self-consistent eigenvalue problems in real space are presented; these techniques have been extensively applied to solutions of the Hartree-Fock and Kohn-Sham equations of electronic structure, and to eigenvalue problems arising in semiconductor and polymer physics. Finally, real-space methods have found recent application in computations of optical response and excited states in time-dependent density functional theory, and these computational developments are summarized. Multiscale solvers are competitive with the most efficient available plane-wave techniques in terms of the number of self-consistency steps required to reach the ground state, and they require less work in each self-consistency update on a uniform grid. Besides excellent efficiencies, the decided advantages of the real-space multiscale approach are 1) the near-locality of each function update, 2) the ability to handle global eigenfunction constraints and potential updates on coarse levels, and 3) the ability to incorporate adaptive local mesh refinements without loss of optimal multigrid efficiencies.

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Section: Finite-element Basesmentioning

confidence: 99%

Real-space mesh techniques in density-functional theory

2000

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“…The combination of adaptivity and a high order convergence rate is typically not achieved in other electronic structure programs using systematic real space methods [6]. An adaptive finite element code, using cubic polynomial shape functions [7], has a convergence rate of h 6 . Finite difference methods have sometimes low [8] h 3 or high convergence rates [9] but are not adaptive.…”

Section: Daubechies Wavelets Familymentioning

confidence: 99%

Wavelets for electronic structure calculations

Deutsch

Genovese

2011

Journées de la Neutronique

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Abstract. In 2005, the EU FP6-STREP-NEST BigDFT project funded a consortium of four laboratories, with the aim of developing a novel approach for Density Functional Theory (DFT) calculations based on Daubechies wavelets. Rather than simply building a DFT code from scratch, the objective of this three-years project was to test the potential benefit of a new formalism in the context of electronic structure calculations. Daubechies wavelets exhibit a set of properties which make them ideal for a precise and optimised DFT approach. In particular, their systematicity allows to provide a reliable basis set for high-precision results, whereas their locality (both in real and reciprocal space) is highly desired for improve the efficiency and the flexibility of the treatment. In this contribution we will provide a bird's-eye view on the computational methods in DFT, and we then focus on DFT approaches and on the way they are implemented in the BigDFT code, to explain how we can take benefit from the peculiarities of such basis set in the context of electronic structure calculations.In the recent years, the development of efficient and reliable methods for studying matter at atomistic level has become an asset for important advancements in the context of material science. Both modern technological evolution and the need for new conception of materials and nanoscaled devices require a deep understanding of the properties of systems of many atoms from a fundamental viewpoint. To this aim, the support of computer simulation can be of great importance. Indeed, via computer simulation scientists try to model systems with many degrees of freedom by giving a set of "rules" of general validity (under some assumptions).Once these "rules" come from first-principles laws, these simulation have the ambition to model system properties from a fundamental viewpoint. With such a tool, the properties of existing materials can be studied in deep, and new materials and molecules can be conceived, with potentially enormous scientific and technological impact. In this context, the advent of modern supercomputers represents an important resource in view of advancements in this field. In other terms, the physical properties which can be analysed via such methods are tightly connected to the computational power which can be exploited for calculation. A high-performance computing electronic structure program will make the analysis of more complex systems and environments possible, thus opening a path towards new discoveries. It is thus important to provide reliable solutions to benefit from the enhancements of computational power in order to use these tools in more challenging systems.

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“…It is therefore from the beginning free of long range interactions between supercells, that falsify results if plane waves are used to describe non-periodic systems. Due to the convolutions we have to evaluate our method has a N log(N ) scaling instead of the ideal linear scaling, Due to its small prefactor the method is however most efficient when dealing with localized densities such as can be found for example in the context of ab initio pseudo-potential electronic structure calculations using finite differences (8) finite elements (9) or plane waves for non-periodic systems.…”

Section: Introductionmentioning

confidence: 99%

Efficient solution of Poisson’s equation with free boundary conditions

Genovese

Deutsch

Neelov

et al. 2006

The Journal of Chemical Physics

208

251

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Interpolating scaling functions give a faithful representation of a localized charge distribution by its values on a grid. For such charge distributions, using a Fast Fourier method, we obtain highly accurate electrostatic potentials for free boundary conditions at the cost of O(N logN) operations, where N is the number of grid points. Thus, with our approach, free boundary conditions are treated as efficiently as the periodic conditions via plane wave methods.

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Real-space local polynomial basis for solid-state electronic-structure calculations: A finite-element approach

Cited by 106 publications

References 44 publications

Real-space mesh techniques in density-functional theory

Real-space mesh techniques in density-functional theory

Wavelets for electronic structure calculations

Efficient solution of Poisson’s equation with free boundary conditions

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