Best N-term approximation in electronic structure calculations I. One-electron reduced density matrix
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 1, p. 49-61

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 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.

DOI : https://doi.org/10.1051/m2an:2006007
Classification:  41A50,  41A63,  65Z05,  81V70
Keywords: best N-term approximation, wavelets, Hartree-Fock method, density functional theory
@article{M2AN_2006__40_1_49_0,
     author = {Flad, Heinz-J\"urgen and Hackbusch, Wolfgang and Schneider, Reinhold},
     title = {Best $N$-term approximation in electronic structure calculations I. One-electron reduced density matrix},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {40},
     number = {1},
     year = {2006},
     pages = {49-61},
     doi = {10.1051/m2an:2006007},
     zbl = {1100.81050},
     mrnumber = {2223504},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2006__40_1_49_0}
}
Flad, Heinz-Jürgen; Hackbusch, Wolfgang; Schneider, Reinhold. Best $N$-term approximation in electronic structure calculations I. One-electron reduced density matrix. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 1, pp. 49-61. doi : 10.1051/m2an:2006007. http://www.numdam.org/item/M2AN_2006__40_1_49_0/

[1] D. Braess, Asymptotics for the approximation of wave functions by exponential sums. J. Approx. Theory 83 (1995) 93-103. | Zbl 0868.41012

[2] H.-J. Bungartz and M. Griebel, Sparse grids. Acta Numerica 13 (2004) 147-269. | Zbl 1122.65405

[3] A. Cohen, R.A. Devore and R. Hochmuth, Restricted nonlinear approximation. Constr. Approx. 16 (2000) 85-113. | Zbl 0947.41006

[4] R.A. Devore, Nonlinear approximation. Acta Numerica 7 (1998) 51-150. | Zbl 0931.65007

[5] R.A. Devore, B. Jawerth and V. Popov, Compression of wavelet decompositions. Amer. J. Math. 114 (1992) 737-785. | Zbl 0764.41024

[6] R.A. Devore, S.V. Konyagin and V.N. Temlyakov, Hyperbolic wavelet approximation. Constr. Approx. 14 (1998) 1-26. | Zbl 0895.41016

[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. | Zbl 1088.82029

[10] S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and T. Ostergaard Sorensen, On the regularity of the density of electronic wavefunctions. Contemp. Math. 307 (2002) 143-148. | Zbl 1041.81104

[11] S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and T. Ostergaard Sorensen, The electron density is smooth away from the nuclei. Commun. Math. Phys. 228 (2002) 401-415. | Zbl 1005.81095

[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. | Zbl 0979.65101

[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. A 23 (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. | Zbl 0812.35105

[19] M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and T. Ostergaard Sorensen, Electron wavefunctions and densities for atoms. Ann. Henri Poincaré 2 (2001) 77-100. | Zbl 0985.81133

[20] T. Kato, On the eigenfunctions of many-particle systems in quantum mechanics. Commun. Pure Appl. Math. 10 (1957) 151-177. | Zbl 0077.20904

[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). | MR 1623209

[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. | Zbl 1062.35100

[30] H. Yserentant, Sparse grid spaces for the numerical solution of the electronic Schrödinger equation. Numer. Math. 101 (2005) 381-389. | Zbl 1084.65125