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

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

doi:10.1051/m2an:2007016

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

Flad, Heinz-Jürgen ; Hackbusch, Wolfgang ; Schneider, Reinhold ¹

¹ Institut für Informatik Christian-Albrechts-Universität zu Kiel, Christian-Albrechts-Platz 4, 24098 Kiel, Germany. ; Christian-Albrechts-Universität Kiel, Christian-Albrechts-Platz 4, 24098 Kiel, Germany.

ESAIM: Mathematical Modelling and Numerical Analysis , Special issue on Molecular Modelling, Tome 41 (2007) no. 2, pp. 261-279

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Résumé

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 $ℱ^{(2)} \in A_{q}^{α} (H^{1})$ for $q > 1$ and $α = \frac{1}{q} - \frac{1}{2}$ 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.

MR Zbl

DOI : 10.1051/m2an:2007016

Classification : 41A50, 41A63, 65Z05, 81V70
Keywords: best N-term approximation, wavelets, electron correlations, Jastrow factor

Affiliations des auteurs :

Flad, Heinz-Jürgen ; Hackbusch, Wolfgang ; Schneider, Reinhold ¹

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     author = {Flad, Heinz-J\"urgen and Hackbusch, Wolfgang and Schneider, Reinhold},
     title = {Best $N$-term approximation in electronic structure calculations. {II.} {Jastrow} factors},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {261--279},
     publisher = {EDP-Sciences},
     volume = {41},
     number = {2},
     year = {2007},
     doi = {10.1051/m2an:2007016},
     mrnumber = {2339628},
     zbl = {1135.81029},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an:2007016/}
}

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Flad, Heinz-Jürgen; Hackbusch, Wolfgang; Schneider, Reinhold. Best $N$-term approximation in electronic structure calculations. II. Jastrow factors. ESAIM: Mathematical Modelling and Numerical Analysis , Special issue on Molecular Modelling, Tome 41 (2007) no. 2, pp. 261-279. doi: 10.1051/m2an:2007016

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