s * -compressibility of the discrete Hartree-Fock equation
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 5, pp. 1055-1080.

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.

DOI: 10.1051/m2an/2011077
Classification: 65Z05,  35Q40,  35C20,  35J10
Keywords: Hartree-Fock equation, matrix compression, bestn-term approximation
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Flad, Heinz-Jürgen; Schneider, Reinhold. $s^\ast $-compressibility of the discrete Hartree-Fock equation. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 5, pp. 1055-1080. doi : 10.1051/m2an/2011077. http://archive.numdam.org/articles/10.1051/m2an/2011077/

[1] D. Andrae, Numerical self-consistent field method for polyatomic molecules. Mol. Phys. 99 (2001) 327-334.

[2] T.A. Arias, Multiresolution analysis of electronic structure : Semicardinal and wavelet bases. Rev. Mod. Phys. 71 (1999) 267-312.

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

[6] S.C. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods. Springer, New York (2008). | MR | Zbl

[7] H.-J. Bungartz and M. Griebel, Sparse grids. Acta Numer. 13 (2004) 147-269. | MR | Zbl

[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. | MR | Zbl

[9] E. Cancès and C. Le Bris, On the convergence of SCF algorithms for the Hartree-Fock equations. ESAIM : M2AN 34 (2000) 749-774. | Numdam | MR | Zbl

[10] A. Cohen, W. Dahmen and R.A. Devore, Adaptive wavelet methods for elliptic operator equations, convergence rates. Math. Comp. 70 (2001) 27-75. | MR | Zbl

[11] W. Dahmen, T. Rohwedder, R. Schneider and A. Zeiser, Adaptive eigenvalue computation : complexity estimates. Numer. Math. 110 (2008) 277-312. | MR | Zbl

[12] R.A. Devore, Nonlinear approximation. Acta Numer. 7 (1998) 51-150. | MR | Zbl

[13] Y.V. Egorov and B.-W. Schulze, Pseudo-Differential Operators, Singularities, Applications. Birkhäuser, Basel (1997). | MR | Zbl

[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] 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. | Numdam | MR | Zbl

[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. | MR | Zbl

[17] H. Flanders, Differentiation under the integral sign. Amer. Math. Monthly 80 (1973) 615-627. | MR | Zbl

[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] 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] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1998). | Zbl

[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] P. Hajłasz, P. Koskela and H. Tuominen, Sobolev embeddings, extensions and measure density condition. J. Funct. Anal. 254 (2008) 1217-1234. | MR | Zbl

[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] 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] T. Helgaker, P. Jørgensen and J. Olsen, Molecular Electronic-Structure Theory. Wiley, New York (1999).

[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] J. Kobus, L. Laaksonen and D. Sundholm, A numerical Hartree-Fock program for diatomic molecules. Comput. Phys. Commun. 98 (1996) 346-358.

[28] W. Kutzelnigg, Theory of the expansion of wave functions in a Gaussian basis. Int. J. Quantum Chem. 51 (1994) 447-463.

[29] E.H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems. Commun. Math. Phys. 53 (1977) 185-194. | MR

[30] P.L. Lions, Solutions of Hartree-Fock equations for Coulomb systems. Commun. Math. Phys. 109 (1987) 33-97. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[33] R. Schneider, Multiskalen-und Wavelet-Matrixkompression. Teubner, Stuttgart (1998). | MR

[34] C. Schwab and R. Stevenson, Adaptive wavelet algorithms for elliptic PDE's on product domains. Math. Comp. 77 (2008) 71-92. | MR | Zbl

[35] O. Sinanoğlu, Perturbation theory of many-electron atoms and molecules. Phys. Rev. 122 (1961) 493-499. | MR | Zbl

[36] O. Sinanoğlu, Theory of electron correlation in atoms and molecules. Proc. R. Soc. Lond., Ser. A 260 (1961) 379-392. | MR | Zbl

[37] R. Stevenson, On the compressibility of operators in wavelet coordinates. SIAM J. Math. Anal. 35 (2004) 1110-1132. | MR | Zbl

[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. | MR

[39] H. Yserentant, On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives. Numer. Math. 98 (2004) 731-759. | MR | Zbl

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