Best N-term approximation in electronic structure calculations. II. Jastrow factors

Heinz-Jürgen Flad; Wolfgang Hackbusch; Reinhold Schneider

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

  • Volume: 41, Issue: 2, page 261-279
  • ISSN: 0764-583X

Abstract

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We present a novel application of best N-term approximation theory in the framework of electronic structure calculations. The paper focusses on the description of electron correlations within a Jastrow-type ansatz for the wavefunction. As a starting point we discuss certain natural assumptions on the asymptotic behaviour of two-particle correlation functions ℱ ( 2 ) near electron-electron and electron-nuclear cusps. Based on Nitsche's characterization of best N-term approximation spaces A q α ( H 1 ) , we prove that ℱ ( 2 ) ∈ A q α ( H 1 ) for q>1 and α = 1 q - 1 2 with respect to a certain class of anisotropic wavelet tensor product bases. Computational arguments are given in favour of this specific class compared to other possible tensor product bases. Finally, we compare the approximation properties of wavelet bases with standard Gaussian-type basis sets frequently used in quantum chemistry.


How to cite

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Flad, Heinz-Jürgen, Hackbusch, Wolfgang, and Schneider, Reinhold. "Best N-term approximation in electronic structure calculations. II. Jastrow factors." ESAIM: Mathematical Modelling and Numerical Analysis 41.2 (2007): 261-279. <http://eudml.org/doc/250137>.

@article{Flad2007,
abstract = { We present a novel application of best N-term approximation theory in the framework of electronic structure calculations. The paper focusses on the description of electron correlations within a Jastrow-type ansatz for the wavefunction. As a starting point we discuss certain natural assumptions on the asymptotic behaviour of two-particle correlation functions $\mathcal\{F\}^\{(2)\}$ near electron-electron and electron-nuclear cusps. Based on Nitsche's characterization of best N-term approximation spaces $A_\{q\}^\{\alpha\}(H^\{1\})$, we prove that $\left. \mathcal\{F\}^\{(2)\}\in A_\{q\}^\{\alpha\}(H^\{1\})\right. $ for q>1 and $\alpha=\frac\{1\}\{q\}-\frac\{1\}\{2\}$ with respect to a certain class of anisotropic wavelet tensor product bases. Computational arguments are given in favour of this specific class compared to other possible tensor product bases. Finally, we compare the approximation properties of wavelet bases with standard Gaussian-type basis sets frequently used in quantum chemistry.
},
author = {Flad, Heinz-Jürgen, Hackbusch, Wolfgang, Schneider, Reinhold},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Best N-term approximation; wavelets; electron correlations; Jastrow factor.},
language = {eng},
month = {6},
number = {2},
pages = {261-279},
publisher = {EDP Sciences},
title = {Best N-term approximation in electronic structure calculations. II. Jastrow factors},
url = {http://eudml.org/doc/250137},
volume = {41},
year = {2007},
}

TY - JOUR
AU - Flad, Heinz-Jürgen
AU - Hackbusch, Wolfgang
AU - Schneider, Reinhold
TI - Best N-term approximation in electronic structure calculations. II. Jastrow factors
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2007/6//
PB - EDP Sciences
VL - 41
IS - 2
SP - 261
EP - 279
AB - We present a novel application of best N-term approximation theory in the framework of electronic structure calculations. The paper focusses on the description of electron correlations within a Jastrow-type ansatz for the wavefunction. As a starting point we discuss certain natural assumptions on the asymptotic behaviour of two-particle correlation functions $\mathcal{F}^{(2)}$ near electron-electron and electron-nuclear cusps. Based on Nitsche's characterization of best N-term approximation spaces $A_{q}^{\alpha}(H^{1})$, we prove that $\left. \mathcal{F}^{(2)}\in A_{q}^{\alpha}(H^{1})\right. $ for q>1 and $\alpha=\frac{1}{q}-\frac{1}{2}$ with respect to a certain class of anisotropic wavelet tensor product bases. Computational arguments are given in favour of this specific class compared to other possible tensor product bases. Finally, we compare the approximation properties of wavelet bases with standard Gaussian-type basis sets frequently used in quantum chemistry.

LA - eng
KW - Best N-term approximation; wavelets; electron correlations; Jastrow factor.
UR - http://eudml.org/doc/250137
ER -

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Citations in EuDML Documents

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  1. Markus Bachmayr, Hyperbolic wavelet discretization of the two-electron Schrödinger equation in an explicitly correlated formulation
  2. Markus Bachmayr, Hyperbolic wavelet discretization of the two-electron Schrödinger equation in an explicitly correlated formulation
  3. Harry Yserentant, The mixed regularity of electronic wave functions multiplied by explicit correlation factors
  4. Markus Bachmayr, Hyperbolic wavelet discretization of the two-electron Schrödinger equation in an explicitly correlated formulation
  5. Markus Bachmayr, Hyperbolic wavelet discretization of the two-electron Schrödinger equation in an explicitly correlated formulation
  6. Harry Yserentant, The mixed regularity of electronic wave functions multiplied by explicit correlation factors

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