WIAS Preprint No. 731, (2002)

Wavelet approximation of correlated wavefunctions. II. Hyperbolic wavelets and adaptive approximation schemes


  • Luo, Hongjun
  • Kolb, Dietmar
  • Flad, Heinz-Jürgen
  • Hackbusch, Wolfgang
  • Koprucki, Thomas
    ORCID: 0000-0001-6235-9412

2010 Mathematics Subject Classification

  • 65T60 81V70


  • wavelets, adaptive schemes, correlated wavefunctions, Jastrow factor, electron-electron cusp, groundstate




We have studied various aspects concerning the use of hyperbolic wavelets and adaptive approximation schemes for wavelet expansions of correlated wavefunctions. In order to analyze the consequences of reduced regularity of the wavefunction at the electron-electron cusp, we first considered a realistic exactly solvable many-particle model in one dimension. Convergence rates of wavelet expansions, with respect to L2 and H1 norms and the energy, were established for this model. We compare the performance of hyperbolic wavelets and their extensions through adaptive refinement in the cusp region, to a fully adaptive treatment based on the energy contribution of individual wavelets. Although hyperbolic wavelets show an inferior convergence behavior, they can be easily refined in the cusp region yielding an optimal convergence rate for the energy. Preliminary results for the helium atom are presented, which demonstrate the transferability of our observations to more realistic systems. We propose a contraction scheme for wavelets in the cusp region, which reduces the number of degrees of freedom and yields a favorable cost to benefit ratio for the evaluation of matrix elements.

Appeared in

  • Journal of Chemical Physics 117 (2002), pp. 3625-3638

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