Best<i>N</i>-term approximation in electronic structure calculations. II. Jastrow factors

Flad, Heinz-Jürgen; Hackbusch, Wolfgang; Schneider, Reinhold

doi:10.1051/m2an:2007016

Cited by 25 publications

(21 citation statements)

References 52 publications

(102 reference statements)

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“…This convergence rate is independent of the number N of electrons and comes arbitrarily close to that for the one-electron case. We conjecture that the convergence rate can even still be improved as the studies [2,3] indicate.…”

Section: Approximation and Complexity Of The Eigenfunctionsmentioning

confidence: 85%

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The mixed regularity of electronic wave functions multiplied by explicit correlation factors

Yserentant¹

2011

ESAIM: M2AN

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Abstract. The electronic Schrödinger equation describes the motion of N electrons under Coulombinteraction forces in a field of clamped nuclei. The solutions of this equation, the electronic wave functions, depend on 3N variables, three spatial dimensions for each electron. Approximating them is thus inordinately challenging. As is shown in the author's monograph [Yserentant, Lecture Notes in Mathematics 2000, Springer (2010], the regularity of the solutions, which increases with the number of electrons, the decay behavior of their mixed derivatives, and the antisymmetry enforced by the Pauli principle contribute properties that allow these functions to be approximated with an order of complexity which comes arbitrarily close to that for a system of two electrons. The present paper complements this work. It is shown that one can reach almost the same complexity as in the one-electron case adding a simple regularizing factor that depends explicitly on the interelectronic distances.

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Section: Approximation and Complexity Of The Eigenfunctionsmentioning

confidence: 85%

“…, x N ) ∈ (R 3 ) N into parts x i ∈ R 3 that are associated with the electron positions. The components of these vectors are the real numbers x i,1 , x i,2 , and x i, 3 . Accordingly, we label partial derivatives doubly, that is, by multi-indices…”

Section: The Function Spacesmentioning

confidence: 99%

The mixed regularity of electronic wave functions multiplied by explicit correlation factors

Yserentant¹

2011

ESAIM: M2AN

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“…. , Z K , and the x j = (x j,1 , x j,2 , x j, 3 associated Laplacians so that Δ = N j=1 Δ j is the 3N -dimensional Laplacian. Let x = (x 1 , x 2 , .…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Non-Isotropic Cusp Conditions and Regularity of the Electron Density of Molecules at the Nuclei

Fournais

Sørensen

Hoffmann‐Ostenhof

et al. 2007

Ann. Henri Poincaré

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We investigate regularity properties of molecular one-electron densities ρ near the nuclei. In particular we derive a representationwith an explicit function F, only depending on the nuclear charges and the positions of the nuclei, such that μ ∈ C 1,1 (R 3 ), i.e., μ has locally essentially bounded second derivatives. An example constructed using Hydrogenic eigenfunctions shows that this regularity result is sharp. For atomic eigenfunctions which are either even or odd with respect to inversion in the origin, we prove that μ is even C 2,α (R 3 ) for all α ∈ (0, 1). Placing one nucleus at the origin we study ρ in polar coordinates x = rω and investigate ∂ ∂r ρ(r, ω) and ∂ 2 ∂r 2 ρ(r, ω) for fixed ω as r tends to zero. We prove non-isotropic cusp conditions of first and second order, which generalize Kato's classical result.

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“…Furthermore, from a theoretical point of view, such an adaptive best M -term approximation requires a new, not yet existing mixed Besov regularity theory for the electronic Schrödinger equation. Here, exponential Jastrow factors in the two-electron case have been studied in [33] to deduce some preliminary assertions on possible convergence rates. Their results yield a convergence rate of order −1/2 for the related two-particle correlation functions, compared to a rate of order −1/4 when applying a linear approximation scheme.…”

Section: Discussionmentioning

confidence: 99%

Tensor Product Multiscale Many-Particle Spaces with Finite-Order Weights for the Electronic Schrödinger Equation

Griebel¹,

Hamaekers²

2010

Zeitschrift Für Physikalische Chemie

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In this article we combine the favorable properties of efficient Gaussian type orbitals basis sets, which are applied with good success in conventional electronic structure methods, and tensor product multiscale bases, which provide guaranteed convergence rates and allow for adaptive resolution. To this end, we develop and study a new approach for the treatment of the electronic Schrödinger equation based on a modified adaptive sparse grid technique and a certain particle-wise decomposition with respect to oneparticle functions obtained by a nonlinear rank-1 approximation. Here, we employ a multiscale Gaussian frame for the sparse grid spaces and we use Gaussian type orbitals to represent the rank-1 approximation. With this approach we are able to treat small atoms and molecules with up to six electrons.2

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BestN-term approximation in electronic structure calculations. II. Jastrow factors

Cited by 25 publications

References 52 publications

The mixed regularity of electronic wave functions multiplied by explicit correlation factors

The mixed regularity of electronic wave functions multiplied by explicit correlation factors

Non-Isotropic Cusp Conditions and Regularity of the Electron Density of Molecules at the Nuclei

Tensor Product Multiscale Many-Particle Spaces with Finite-Order Weights for the Electronic Schrödinger Equation

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