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s∗-compressibility of the discrete Hartree-Fock equation

Published online by Cambridge University Press:  13 February 2012

Heinz-Jürgen Flad  and
Reinhold Schneider
Heinz-Jürgen Flad
Affiliation:
Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany. flad@math.tu-berlin.de
Reinhold Schneider
Affiliation:
Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany. flad@math.tu-berlin.de
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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.

Keywords

Hartree-Fock equationmatrix compressionbestN-term approximation
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 5 , September 2012 , pp. 1055 - 1080
Copyright
© EDP Sciences, SMAI, 2012

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References

Andrae, D., Numerical self-consistent field method for polyatomic molecules. Mol. Phys. 99 (2001) 327–334. Google Scholar
Arias, T.A., Multiresolution analysis of electronic structure : Semicardinal and wavelet bases. Rev. Mod. Phys. 71 (1999) 267–312. Google Scholar
O. Beck, D. Heinemann and D. Kolb, Fast and accurate molecular Hartree-Fock with a finite-element multigrid method. arXiv:physics/0307108 (2003).
Bischoff, F.A. and Valeev, E.F., Low-order tensor approximations for electronic wave functions : Hartree-Fock method with guaranteed precision. J. Chem. Phys. 134 (2011) 104104. Google Scholar
Braess, D., Asymptotics for the approximation of wave functions by exponential sums. J. Approx. Theory 83 (1995) 93–103. Google Scholar
S.C. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods. Springer, New York (2008).
Bungartz, H.-J. and Griebel, M., Sparse grids. Acta Numer. 13 (2004) 147–269. Google Scholar
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.
Cancès, E. and Le Bris, C., On the convergence of SCF algorithms for the Hartree-Fock equations. ESAIM : M2AN 34 (2000) 749–774. Google Scholar
Cohen, A., Dahmen, W. and DeVore, R.A., Adaptive wavelet methods for elliptic operator equations, convergence rates. Math. Comp. 70 (2001) 27–75. Google Scholar
Dahmen, W., Rohwedder, T., Schneider, R. and Zeiser, A., Adaptive eigenvalue computation : complexity estimates. Numer. Math. 110 (2008) 277–312. Google Scholar
DeVore, R.A., Nonlinear approximation. Acta Numer. 7 (1998) 51–150. Google Scholar
Y.V. Egorov and B.-W. Schulze, Pseudo-Differential Operators, Singularities, Applications. Birkhäuser, Basel (1997).
Engeness, T.D. and Arias, T.A., Multiresolution analysis for efficient, high precision all-electron density-functional calculations. Phys. Rev. B 65 (2002) 165106. Google Scholar
Flad, H.-J., Hackbusch, W. and Schneider, R., Best N-term approximation in electronic structure calculation. I. One-electron reduced density matrix. ESAIM : M2AN 40 (2006) 49–61. Google Scholar
Flad, H.-J., Schneider, R. and Schulze, B.-W., Asymptotic regularity of solutions of Hartree-Fock equations with Coulomb potential. Math. Methods Appl. Sci. 31 (2008) 2172–2201. Google Scholar
Flanders, H., Differentiation under the integral sign. Amer. Math. Monthly 80 (1973) 615–627. Google Scholar
Genovese, L., Deutsch, T., Neelov, A., Goedecker, S. and Beylkin, G., Efficient solution of Poisson’s equation with free boundary conditions. J. Chem. Phys. 125 (2006) 074105. Google ScholarPubMed
Genovese, L., Neelov, A., Goedecker, S., Deutsch, T., Ghasemi, S.A., Willand, A., Caliste, D., Zilberberg, O., Rayson, M., Bergman, A. and Schneider, R., Daubechies wavelets as a basis set for density functional pseudopotential calculations. J. Chem. Phys. 129 (2008) 014109. Google ScholarPubMed
D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1998).
Griebel, M. and Hamaekers, J., Tensor product multiscale many-particle spaces with finite-order weights for the electronic Schrödinger equation. Z. Phys. Chem. 224 (2010) 527–543. Google Scholar
Hajłasz, P., Koskela, P. and Tuominen, H., Sobolev embeddings, extensions and measure density condition. J. Funct. Anal. 254 (2008) 1217–1234. Google Scholar
Harrison, R.J., Fann, G.I., Yanai, T., Gan, Z. and Beylkin, G., Multiresolution quantum chemistry : Basic theory and initial applications. J. Chem. Phys. 121 (2004) 11587–11598. Google ScholarPubMed
Heinemann, D., Rosén, A. and Fricke, B., Solution of the Hartree-Fock equations for atoms and diatomic molecules with the finite element method. Phys. Scr. 42 (1990) 692–696. Google Scholar
T. Helgaker, P. Jørgensen and J. Olsen, Molecular Electronic-Structure Theory. Wiley, New York (1999).
Klopper, W., Manby, F.R., Ten-No, S. and Valeev, E.F., R12 methods in explicitly correlated molecular electronic structure theory. Int. Rev. Phys. Chem. 25 (2006) 427–468. Google Scholar
Kobus, J., Laaksonen, L. and Sundholm, D., A numerical Hartree-Fock program for diatomic molecules. Comput. Phys. Commun. 98 (1996) 346–358. Google Scholar
Kutzelnigg, W., Theory of the expansion of wave functions in a Gaussian basis. Int. J. Quantum Chem. 51 (1994) 447–463. Google Scholar
Lieb, E.H. and Simon, B., The Hartree-Fock theory for Coulomb systems. Commun. Math. Phys. 53 (1977) 185–194. Google Scholar
Lions, P.L., Solutions of Hartree-Fock equations for Coulomb systems. Commun. Math. Phys. 109 (1987) 33–97. Google Scholar
Neelov, A.I. and Goedecker, S., 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. Google Scholar
Rohwedder, T., Schneider, R. and Zeiser, A., Perturbed preconditioned inverse iteration for operator eigenvalue problems with applications to adaptive wavelet discretization. Adv. Comput. Math. 34 (2011) 43–66. Google Scholar
R. Schneider, Multiskalen-und Wavelet-Matrixkompression. Teubner, Stuttgart (1998).
Schwab, C. and Stevenson, R., Adaptive wavelet algorithms for elliptic PDE’s on product domains. Math. Comp. 77 (2008) 71–92. Google Scholar
Sinanoğlu, O., Perturbation theory of many-electron atoms and molecules. Phys. Rev. 122 (1961) 493–499. Google Scholar
Sinanoğlu, O., Theory of electron correlation in atoms and molecules. Proc. R. Soc. Lond., Ser. A 260 (1961) 379–392. Google Scholar
Stevenson, R., On the compressibility of operators in wavelet coordinates. SIAM J. Math. Anal. 35 (2004) 1110–1132. Google Scholar
Yanai, T., Fann, G.I., Gan, Z., Harrison, R.J. and Beylkin, G., Multiresolution quantum chemistry in multiwavelet basis : Hartree-Fock exchange. J. Chem. Phys. 121 (2004) 6680–6688. Google Scholar
Yserentant, H., On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives. Numer. Math. 98 (2004) 731–759.Google Scholar