s∗-compressibility of the discrete Hartree-Fock equation

Heinz-Jürgen Flad; Reinhold Schneider

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2012)

  • Volume: 46, Issue: 5, page 1055-1080
  • ISSN: 0764-583X

Abstract

top
The Hartree-Fock equation is widely accepted as the basic model of electronic structure calculation which serves as a canonical starting point for more sophisticated many-particle models. We have studied the s∗-compressibility for Galerkin discretizations of the Hartree-Fock equation in wavelet bases. Our focus is on the compression of Galerkin matrices from nuclear Coulomb potentials and nonlinear terms in the Fock operator which hitherto has not been discussed in the literature. It can be shown that the s∗-compressibility is in accordance with convergence rates obtained from best N-term approximation for solutions of the Hartree-Fock equation. This is a necessary requirement in order to achieve numerical solutions for these equations with optimal complexity using the recently developed adaptive wavelet algorithms of Cohen, Dahmen and DeVore.

How to cite

top

Flad, Heinz-Jürgen, and Schneider, Reinhold. "s∗-compressibility of the discrete Hartree-Fock equation." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 46.5 (2012): 1055-1080. <http://eudml.org/doc/273274>.

@article{Flad2012,
abstract = {The Hartree-Fock equation is widely accepted as the basic model of electronic structure calculation which serves as a canonical starting point for more sophisticated many-particle models. We have studied the s∗-compressibility for Galerkin discretizations of the Hartree-Fock equation in wavelet bases. Our focus is on the compression of Galerkin matrices from nuclear Coulomb potentials and nonlinear terms in the Fock operator which hitherto has not been discussed in the literature. It can be shown that the s∗-compressibility is in accordance with convergence rates obtained from best N-term approximation for solutions of the Hartree-Fock equation. This is a necessary requirement in order to achieve numerical solutions for these equations with optimal complexity using the recently developed adaptive wavelet algorithms of Cohen, Dahmen and DeVore.},
author = {Flad, Heinz-Jürgen, Schneider, Reinhold},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {Hartree-Fock equation; matrix compression; bestn-term approximation; wavelet bases; Galerkin matrices; Coulomb potentials; best -term approximation; numerical results},
language = {eng},
number = {5},
pages = {1055-1080},
publisher = {EDP-Sciences},
title = {s∗-compressibility of the discrete Hartree-Fock equation},
url = {http://eudml.org/doc/273274},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Flad, Heinz-Jürgen
AU - Schneider, Reinhold
TI - s∗-compressibility of the discrete Hartree-Fock equation
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2012
PB - EDP-Sciences
VL - 46
IS - 5
SP - 1055
EP - 1080
AB - The Hartree-Fock equation is widely accepted as the basic model of electronic structure calculation which serves as a canonical starting point for more sophisticated many-particle models. We have studied the s∗-compressibility for Galerkin discretizations of the Hartree-Fock equation in wavelet bases. Our focus is on the compression of Galerkin matrices from nuclear Coulomb potentials and nonlinear terms in the Fock operator which hitherto has not been discussed in the literature. It can be shown that the s∗-compressibility is in accordance with convergence rates obtained from best N-term approximation for solutions of the Hartree-Fock equation. This is a necessary requirement in order to achieve numerical solutions for these equations with optimal complexity using the recently developed adaptive wavelet algorithms of Cohen, Dahmen and DeVore.
LA - eng
KW - Hartree-Fock equation; matrix compression; bestn-term approximation; wavelet bases; Galerkin matrices; Coulomb potentials; best -term approximation; numerical results
UR - http://eudml.org/doc/273274
ER -

References

top
  1. [1] D. Andrae, Numerical self-consistent field method for polyatomic molecules. Mol. Phys.99 (2001) 327–334. 
  2. [2] T.A. Arias, Multiresolution analysis of electronic structure : Semicardinal and wavelet bases. Rev. Mod. Phys.71 (1999) 267–312. 
  3. [3] O. Beck, D. Heinemann and D. Kolb, Fast and accurate molecular Hartree-Fock with a finite-element multigrid method. arXiv:physics/0307108 (2003). 
  4. [4] F.A. Bischoff and E.F. Valeev, Low-order tensor approximations for electronic wave functions : Hartree-Fock method with guaranteed precision. J. Chem. Phys. 134 (2011) 104104. 
  5. [5] D. Braess, Asymptotics for the approximation of wave functions by exponential sums. J. Approx. Theory83 (1995) 93–103. Zbl0868.41012MR1354964
  6. [6] S.C. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods. Springer, New York (2008). Zbl1135.65042MR2373954
  7. [7] H.-J. Bungartz and M. Griebel, Sparse grids. Acta Numer.13 (2004) 147–269. Zbl1118.65388MR2249147
  8. [8] E. Cancès, SCF algorithms for HF electronic calculations, in Mathematical Models and Methods for Ab Initio Quantum Chemistry, edited by M. Defranceschi and C. Le Bris, Springer, Berlin. Lect. Notes Chem. 74 (2000) 17–43. Zbl0992.81103MR1855573
  9. [9] E. Cancès and C. Le Bris, On the convergence of SCF algorithms for the Hartree-Fock equations. ESAIM : M2AN 34 (2000) 749–774. Zbl1090.65548MR1784484
  10. [10] A. Cohen, W. Dahmen and R.A. DeVore, Adaptive wavelet methods for elliptic operator equations, convergence rates. Math. Comp.70 (2001) 27–75. Zbl0980.65130MR1803124
  11. [11] W. Dahmen, T. Rohwedder, R. Schneider and A. Zeiser, Adaptive eigenvalue computation : complexity estimates. Numer. Math.110 (2008) 277–312. Zbl1157.65029MR2430982
  12. [12] R.A. DeVore, Nonlinear approximation. Acta Numer.7 (1998) 51–150. Zbl0931.65007MR1689432
  13. [13] Y.V. Egorov and B.-W. Schulze, Pseudo-Differential Operators, Singularities, Applications. Birkhäuser, Basel (1997). Zbl0877.35141MR1443430
  14. [14] T.D. Engeness and T.A. Arias, Multiresolution analysis for efficient, high precision all-electron density-functional calculations. Phys. Rev. B 65 (2002) 165106. 
  15. [15] H.-J. Flad, W. Hackbusch and R. Schneider, Best N-term approximation in electronic structure calculation. I. One-electron reduced density matrix. ESAIM : M2AN 40 (2006) 49–61. Zbl1100.81050MR2223504
  16. [16] H.-J. Flad, R. Schneider and B.-W. Schulze, Asymptotic regularity of solutions of Hartree-Fock equations with Coulomb potential. Math. Methods Appl. Sci.31 (2008) 2172–2201. Zbl1158.35435MR2467184
  17. [17] H. Flanders, Differentiation under the integral sign. Amer. Math. Monthly80 (1973) 615–627. Zbl0266.26010MR340514
  18. [18] L. Genovese, T. Deutsch, A. Neelov, S. Goedecker and G. Beylkin, Efficient solution of Poisson’s equation with free boundary conditions. J. Chem. Phys. 125 (2006) 074105. 
  19. [19] L. Genovese, A. Neelov, S. Goedecker, T. Deutsch, S.A. Ghasemi, A. Willand, D. Caliste, O. Zilberberg, M. Rayson, A. Bergman and R. Schneider, Daubechies wavelets as a basis set for density functional pseudopotential calculations. J. Chem. Phys. 129 (2008) 014109. 
  20. [20] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1998). Zbl1042.35002
  21. [21] M. Griebel and J. Hamaekers, Tensor product multiscale many-particle spaces with finite-order weights for the electronic Schrödinger equation. Z. Phys. Chem.224 (2010) 527–543. 
  22. [22] P. Hajłasz, P. Koskela and H. Tuominen, Sobolev embeddings, extensions and measure density condition. J. Funct. Anal.254 (2008) 1217–1234. Zbl1136.46029MR2386936
  23. [23] 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. 
  24. [24] D. Heinemann, A. Rosén and B. Fricke, Solution of the Hartree-Fock equations for atoms and diatomic molecules with the finite element method. Phys. Scr.42 (1990) 692–696. 
  25. [25] T. Helgaker, P. Jørgensen and J. Olsen, Molecular Electronic-Structure Theory. Wiley, New York (1999). 
  26. [26] W. Klopper, F.R. Manby, S. Ten-No and E.F. Valeev, R12 methods in explicitly correlated molecular electronic structure theory. Int. Rev. Phys. Chem.25 (2006) 427–468. 
  27. [27] J. Kobus, L. Laaksonen and D. Sundholm, A numerical Hartree-Fock program for diatomic molecules. Comput. Phys. Commun.98 (1996) 346–358. 
  28. [28] W. Kutzelnigg, Theory of the expansion of wave functions in a Gaussian basis. Int. J. Quantum Chem.51 (1994) 447–463. 
  29. [29] E.H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems. Commun. Math. Phys.53 (1977) 185–194. MR452286
  30. [30] P.L. Lions, Solutions of Hartree-Fock equations for Coulomb systems. Commun. Math. Phys.109 (1987) 33–97. Zbl0618.35111MR879032
  31. [31] A.I. Neelov and S. Goedecker, An efficient numerical quadrature for the calculation of the potential energy of wavefunctions expressed in the Daubechies wavelet basis. J. Comp. Phys.217 (2006) 312–339. Zbl1102.65110MR2260604
  32. [32] T. Rohwedder, R. Schneider and A. Zeiser, Perturbed preconditioned inverse iteration for operator eigenvalue problems with applications to adaptive wavelet discretization. Adv. Comput. Math.34 (2011) 43–66. Zbl1208.65160MR2783301
  33. [33] R. Schneider, Multiskalen-und Wavelet-Matrixkompression. Teubner, Stuttgart (1998). MR1623209
  34. [34] C. Schwab and R. Stevenson, Adaptive wavelet algorithms for elliptic PDE’s on product domains. Math. Comp.77 (2008) 71–92. Zbl1127.41009MR2353944
  35. [35] O. Sinanoğlu, Perturbation theory of many-electron atoms and molecules. Phys. Rev.122 (1961) 493–499. Zbl0105.43006MR122423
  36. [36] O. Sinanoğlu, Theory of electron correlation in atoms and molecules. Proc. R. Soc. Lond., Ser. A 260 (1961) 379–392. Zbl0098.23101MR119881
  37. [37] R. Stevenson, On the compressibility of operators in wavelet coordinates. SIAM J. Math. Anal.35 (2004) 1110–1132. Zbl1087.47012MR2050194
  38. [38] 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. MR2103733
  39. [39] H. Yserentant, On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives. Numer. Math.98 (2004) 731–759. Zbl1062.35100MR2099319

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.