Best N-term approximation in electronic structure calculations I. One-electron reduced density matrix

Heinz-Jürgen Flad; Wolfgang Hackbusch; Reinhold Schneider

ESAIM: Mathematical Modelling and Numerical Analysis (2006)

  • Volume: 40, Issue: 1, page 49-61
  • ISSN: 0764-583X

Abstract

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We discuss best N-term approximation spaces for one-electron wavefunctions φ i and reduced density matrices ρ emerging from Hartree-Fock and density functional theory. The approximation spaces A q α ( H 1 ) for anisotropic wavelet tensor product bases have been recently characterized by Nitsche in terms of tensor product Besov spaces. We have used the norm equivalence of these spaces to weighted ℓ q spaces of wavelet coefficients to proof that both φ i and ρ are in A q α ( H 1 ) for all α > 0 with α = 1 q - 1 2 . Our proof is based on the assumption that the φ i possess an asymptotic smoothness property at the electron-nuclear cusps.

How to cite

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Flad, Heinz-Jürgen, Hackbusch, Wolfgang, and Schneider, Reinhold. "Best N-term approximation in electronic structure calculations I. One-electron reduced density matrix." ESAIM: Mathematical Modelling and Numerical Analysis 40.1 (2006): 49-61. <http://eudml.org/doc/249694>.

@article{Flad2006,
abstract = { We discuss best N-term approximation spaces for one-electron wavefunctions $\phi_i$ and reduced density matrices ρ emerging from Hartree-Fock and density functional theory. The approximation spaces $A^\alpha_q(H^1)$ for anisotropic wavelet tensor product bases have been recently characterized by Nitsche in terms of tensor product Besov spaces. We have used the norm equivalence of these spaces to weighted $\ell_q$ spaces of wavelet coefficients to proof that both $\phi_i$ and ρ are in $A^\alpha_q(H^1)$ for all $\alpha > 0$ with $\alpha = \frac\{1\}\{q\} - \frac\{1\}\{2\}$. Our proof is based on the assumption that the $\phi_i$ possess an asymptotic smoothness property at the electron-nuclear cusps. },
author = {Flad, Heinz-Jürgen, Hackbusch, Wolfgang, Schneider, Reinhold},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Best N-term approximation; wavelets; Hartree-Fock method; density functional theory.; electron wave function; best -term approximation},
language = {eng},
month = {2},
number = {1},
pages = {49-61},
publisher = {EDP Sciences},
title = {Best N-term approximation in electronic structure calculations I. One-electron reduced density matrix},
url = {http://eudml.org/doc/249694},
volume = {40},
year = {2006},
}

TY - JOUR
AU - Flad, Heinz-Jürgen
AU - Hackbusch, Wolfgang
AU - Schneider, Reinhold
TI - Best N-term approximation in electronic structure calculations I. One-electron reduced density matrix
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2006/2//
PB - EDP Sciences
VL - 40
IS - 1
SP - 49
EP - 61
AB - We discuss best N-term approximation spaces for one-electron wavefunctions $\phi_i$ and reduced density matrices ρ emerging from Hartree-Fock and density functional theory. The approximation spaces $A^\alpha_q(H^1)$ for anisotropic wavelet tensor product bases have been recently characterized by Nitsche in terms of tensor product Besov spaces. We have used the norm equivalence of these spaces to weighted $\ell_q$ spaces of wavelet coefficients to proof that both $\phi_i$ and ρ are in $A^\alpha_q(H^1)$ for all $\alpha > 0$ with $\alpha = \frac{1}{q} - \frac{1}{2}$. Our proof is based on the assumption that the $\phi_i$ possess an asymptotic smoothness property at the electron-nuclear cusps.
LA - eng
KW - Best N-term approximation; wavelets; Hartree-Fock method; density functional theory.; electron wave function; best -term approximation
UR - http://eudml.org/doc/249694
ER -

References

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  1. D. Braess, Asymptotics for the approximation of wave functions by exponential sums. J. Approx. Theory83 (1995) 93–103.  
  2. H.-J. Bungartz and M. Griebel, Sparse grids. Acta Numerica13 (2004) 147–269.  
  3. A. Cohen, R.A. DeVore and R. Hochmuth, Restricted nonlinear approximation. Constr. Approx.16 (2000) 85–113.  
  4. R.A. DeVore, Nonlinear approximation. Acta Numerica7 (1998) 51–150.  
  5. R.A. DeVore, B. Jawerth and V. Popov, Compression of wavelet decompositions. Amer. J. Math.114 (1992) 737–785.  
  6. R.A. DeVore, S.V. Konyagin and V.N. Temlyakov, Hyperbolic wavelet approximation. Constr. Approx.14 (1998) 1–26.  
  7. H.-J. Flad, W. Hackbusch, D. Kolb and R. Schneider, Wavelet approximation of correlated wavefunctions. I. Basics. J. Chem. Phys.116 (2002) 9641–9657.  
  8. H.-J. Flad, W. Hackbusch, H. Luo and D. Kolb, Diagrammatic multiresolution analysis for electron correlations. Phys. Rev. B.71 (2005) 125115.  
  9. H.-J. Flad, W. Hackbusch, H. Luo and D. Kolb, Wavelet approach to quasi two-dimensional extended many-particle systems. I. supercell Hartree-Fock method. J. Comp. Phys.205 (2005) 540–566.  
  10. S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and T. Ostergaard S orensen, On the regularity of the density of electronic wavefunctions. Contemp. Math.307 (2002) 143–148.  
  11. S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and T. Ostergaard S orensen, The electron density is smooth away from the nuclei. Commun. Math. Phys.228 (2002) 401–415.  
  12. J. Garcke and M. Griebel, On the computation of the eigenproblems of hydrogen and helium in strong magnetic and electric fields with the sparse grid combination technique. J. Comp. Phys.165 (2000) 694–716.  
  13. A. Halkier, T. Helgaker, P. Jørgensen, W. Klopper and J. Olsen, Basis-set convergence of the energy in molecular Hartree-Fock calculations. Chem. Phys. Lett.302 (1999) 437–446.  
  14. R.J. Harrison, G.I. Fann, T. Yanai, Z. Gan and G. Beylkin, Multiresolution quantum chemistry: Basic theory and initial applications. J. Chem. Phys.121 (2004) 11587–11598.  
  15. T. Helgaker, P. Jørgensen and J. Olsen, Molecular Electronic-Structure Theory, Wiley, New York (1999).  
  16. R.N. Hill, Rates of convergence and error estimation formulas for the Rayleigh-Ritz variational method. J. Chem. Phys.83 (1985) 1173–1196.  
  17. M. Hoffmann-Ostenhof and R. Seiler, Cusp conditions for eigenfunctions of n-electron systems, Phys. Rev. A23 (1981) 21–23.  
  18. M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and H. Stremnitzer, Local properties of Coulombic wave functions. Commun. Math. Phys.163 (1994) 185–215.  
  19. M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and T. Ostergaard S orensen, Electron wavefunctions and densities for atoms. Ann. Henri Poincaré2 (2001) 77–100.  
  20. T. Kato, On the eigenfunctions of many-particle systems in quantum mechanics. Commun. Pure Appl. Math.10 (1957) 151–177.  
  21. W. Kutzelnigg, Theory of the expansion of wave functions in a Gaussian basis. Int. J. Quantum Chem.51 (1994) 447–463.  
  22. W. Kutzelnigg and J.D. Morgan III, Rates of convergence of the partial-wave expansions of atomic correlation energies. J. Chem. Phys.96 (1992) 4484–4508.  
  23. E.H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems. Commun. Math. Phys.53 (1977) 185–194.  
  24. H. Luo, D. Kolb, H.-J. Flad, W. Hackbusch and T. Koprucki, Wavelet approximation of correlated wavefunctions. II. Hyperbolic wavelets and adaptive approximation schemes. J. Chem. Phys.117 (2002) 3625–3638.  
  25. P.-A. Nitsche, Best N-term approximation spaces for sparse grids, Research Report No. 2003-11, Seminar für Angewandte Mathematik, ETH Zürich.  
  26. R. Schneider, Multiskalen- und Wavelet-Matrixkompression, Teubner, Stuttgart (1998).  
  27. T. Yanai, G.I. Fann, Z. Gan, R.J. Harrison and G. Beylkin, Multiresolution quantum chemistry in multiwavelet basis: Hartree-Fock exchange. J. Chem. Phys.121 (2004) 6680–6688.  
  28. T. Yanai, G.I. Fann, Z. Gan, R.J. Harrison and G. Beylkin, Multiresolution quantum chemistry in multiwavelet basis: Analytic derivatives for Hartree-Fock and density functional theory. J. Chem. Phys.121 (2004) 2866–2876.  
  29. H. Yserentant, On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives. Numer. Math.98 (2004) 731–759.  
  30. H. Yserentant, Sparse grid spaces for the numerical solution of the electronic Schrödinger equation. Numer. Math.101 (2005) 381–389.  

Citations in EuDML Documents

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  1. Harry Yserentant, The mixed regularity of electronic wave functions multiplied by explicit correlation factors
  2. Heinz-Jürgen Flad, Reinhold Schneider, s∗-compressibility of the discrete Hartree-Fock equation
  3. Heinz-Jürgen Flad, Wolfgang Hackbusch, Reinhold Schneider, Best -term approximation in electronic structure calculations. II. Jastrow factors
  4. Heinz-Jürgen Flad, Reinhold Schneider, -compressibility of the discrete Hartree-Fock equation
  5. Harry Yserentant, The mixed regularity of electronic wave functions multiplied by explicit correlation factors

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