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Best \(N\)-term approximation in electronic structure calculations. I: One-electron reduced density matrix. (English) Zbl 1100.81050
The asymptotic behavior of best \(N\)-term approximation defines approximation spaces which can often be identified with Besov spaces and are norm-equivalent to \(l_q\) spaces of wavelet coefficients. Here the \(N\)-term approximation for one-electron wave functions is characterized. To this end, assumptions on the smoothness at the electron-nuclear cusps are made. For the Hartree-Fock solution with more than one electron/nucleon, tensor product Besov spaces enter into the analysis.

MSC:
81V70 Many-body theory; quantum Hall effect
65Z05 Applications to the sciences
41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)
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References:
[1] D. Braess , Asymptotics for the approximation of wave functions by exponential sums . J. Approx. Theory 83 ( 1995 ) 93 - 103 . Zbl 0868.41012 · Zbl 0868.41012 · doi:10.1006/jath.1995.1110
[2] H.-J. Bungartz and M. Griebel , Sparse grids . Acta Numerica 13 ( 2004 ) 147 - 269 . Zbl 1122.65405 · Zbl 1118.65388 · doi:10.1017/S0962492904000182
[3] A. Cohen , R.A. DeVore and R. Hochmuth , Restricted nonlinear approximation . Constr. Approx. 16 ( 2000 ) 85 - 113 . Zbl 0947.41006 · Zbl 0947.41006 · doi:10.1007/s003659910004
[4] R.A. DeVore , Nonlinear approximation . Acta Numerica 7 ( 1998 ) 51 - 150 . Zbl 0931.65007 · 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 · Zbl 0764.41024 · doi:10.2307/2374796
[6] R.A. DeVore , S.V. Konyagin and V.N. Temlyakov , Hyperbolic wavelet approximation . Constr. Approx. 14 ( 1998 ) 1 - 26 . Zbl 0895.41016 · Zbl 0895.41016 · doi:10.1007/s003659900060
[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 · Zbl 1088.82029 · doi:10.1016/j.jcp.2004.11.018
[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 · 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 · Zbl 1005.81095 · doi:10.1007/s002200200668
[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 · Zbl 0979.65101 · doi:10.1006/jcph.2000.6627
[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 . Article | Zbl 0812.35105 · Zbl 0812.35105 · doi:10.1007/BF02101740 · minidml.mathdoc.fr
[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 · Zbl 0985.81133 · doi:10.1007/PL00001031
[20] T. Kato , On the eigenfunctions of many-particle systems in quantum mechanics . Commun. Pure Appl. Math. 10 ( 1957 ) 151 - 177 . Zbl 0077.20904 · Zbl 0077.20904 · doi:10.1002/cpa.3160100201
[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 . Article · minidml.mathdoc.fr
[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 · Zbl 0899.65063
[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 · Zbl 1062.35100 · doi:10.1007/s00211-003-0498-1
[30] H. Yserentant , Sparse grid spaces for the numerical solution of the electronic Schrödinger equation . Numer. Math. 101 ( 2005 ) 381 - 389 . Zbl 1084.65125 · Zbl 1084.65125 · doi:10.1007/s00211-005-0581-x
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