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Hyperbolic wavelet discretization of the two-electron Schrödinger equation in an explicitly correlated formulation

  • Markus Bachmayr (a1)

Abstract

In the framework of an explicitly correlated formulation of the electronic Schrödinger equation known as the transcorrelated method, this work addresses some fundamental issues concerning the feasibility of eigenfunction approximation by hyperbolic wavelet bases. Focusing on the two-electron case, the integrability of mixed weak derivatives of eigenfunctions of the modified problem and the improvement compared to the standard formulation are discussed. Elements of a discretization of the eigenvalue problem based on orthogonal wavelets are described, and possible choices of tensor product bases are compared especially from an algorithmic point of view. The use of separable approximations of potential terms for applying operators efficiently is studied in detail, and estimates for the error due to this further approximation are given.

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[1] R.A. Adams and J.J.F. Fournier, Sobolev Spaces, Pure and Applied Mathematics Series, 2nd edition. Academic Press 140 (2003).
[2] S. Agmon, Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations : Bounds on Eigenfunctions of N-Body Schrödinger Operators, Mathematical Notes. Princeton University Press (1982).
[3] M. Bachmayr, Integration of products of Gaussians and wavelets with applications to electronic structure calculations. Preprint AICES, RWTH Aachen (2012).
[4] Balder, R. and Zenger, C., The solution of multidimensional real Helmholtz equations on sparse grids. SIAM J. Sci. Comput. 17 (1996) 631646.
[5] Beylkin, G., On the representation of operators in bases of compactly supported wavelets. SIAM J. Numer. Anal. 29 (1992) 17161740.
[6] Boys, S.F. and Handy, N.C., The determination of energies and wavefunctions with full electronic correlation. Proc. R. Soc. Lond. A 310 (1969) 4361.
[7] D. Braess and W. Hackbusch, On the efficient computation of high-dimensional integrals and the approximation by exponential sums, in Multiscale, Nonlinear and Adaptive Approximation, edited by R. DeVore and A. Kunoth. Springer, Berlin, Heidelberg (2009).
[8] H.-J. Bungartz, Dünne Gitter und deren Anwendung bei der adaptiven Lösung der dreidimensionalen Poisson-Gleichung. Ph.D. thesis, Technische Universität München (1992).
[9] F. Chatelin, Spectral Approximation of Linear Operators, Computer Science and Applied Mathematics. Academic Press (1983).
[10] Chinnamsetty, S.R., Espig, M., Khoromskij, B.N., Hackbusch, W. and Flad, H.-J., Tensor product approximation with optimal rank in quantum chemistry. J. Chem. Phys. 127 (2007) 084110.
[11] A. Cohen, Numerical Analysis of Wavelet Methods. Stud. Math. Appl. 32 (2003).
[12] Dahmen, W. and Micchelli, C.A., Using the refinement equation for evaluating integrals of wavelets. SIAM J. Numer. Anal. 30 (1993) 507537.
[13] Daubechies, I., Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41 (1988) 909996.
[14] Dijkema, T.J., Schwab, C. and Stevenson, R., An adaptive wavelet method for solving high-dimensional elliptic PDEs. Constr. Approx. 30 (2009) 423455.
[15] Donovan, G., Geronimo, J. and Hardin, D., Intertwining multiresolution analyses and the construction of piecewise-polynomial wavelets. SIAM J. Math. Anal. 27 (1996) 17911815.
[16] Donovan, G., Geronimo, J. and Hardin, D., Orthogonal polynomials and the construction of piecewise polynomial smooth wavelets. SIAM J. Math. Anal. 30 (1999) 10291056.
[17] Flad, H.-J., Hackbusch, W., Kolb, D. and Schneider, R., Wavelet approximation of correlated wave functions. I. Basics. J. Chem. Phys. 116 (2002) 96419657.
[18] Flad, H.-J., Hackbusch, W. and Schneider, R., Best N-term approximation in electronic structure calculations. II. Jastrow factors. ESAIM : M2AN 41 (2007) 261.
[19] H.-J. Flad, W. Hackbusch, B.N. Khoromskij and R. Schneider, Matrix Methods : Theory, Algorithms and Applications, in Concepts of Data-Sparse Tensor-Product Approximation in Many-Particle Modelling. World Scientific (2010) 313–347.
[20] Fournais, S., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T. and Østergaard Sørensen, T., Sharp regularity results for many-electron wave functions. Commun. Math. Phys. 255 (2005) 183227.
[21] 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.
[22] 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.
[23] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer, Berlin, Heidelberg (1998).
[24] M. Griebel and J. Hamaekers, A wavelet based sparse grid method for the electronic Schrödinger equation, in Proc. of the International Congress of Mathematicians, edited by M. Sanz-Solé, J. Soria, J. Varona and J. Verdera III (2006) 1473–1506.
[25] 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) 527543. Also available as INS Preprint No. 0911.
[26] Griebel, M. and Knapek, S., Optimized tensor-product approximation spaces. Constr. Approx. 16 (2000) 525.
[27] Griebel, M. and Oswald, P., Tensor product type subspace splittings and multilevel iterative methods for anisotropic problems. Adv. Comput. Math. 4 (1995) 171206.
[28] J. Hamaekers, Tensor Product Multiscale Many-Particle Spaces with Finite-Order Weights for the Electronic Schödinger Equation. Ph.D. thesis, Universität Bonn (2009).
[29] Harbrecht, H., Schneider, R. and Schwab, C., Multilevel frames for sparse tensor product spaces. Numer. Math. 110 (2008) 199220.
[30] 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) 1158711598.
[31] Hille, E., A class of reciprocal functions. Ann. Math. 27 (1926) 427464.
[32] Hirschfelder, J.O., Removal of electron-electron poles from many-electron Hamiltonians. J. Chem. Phys. 39 (1963) 31453146.
[33] Hylleraas, E., Über den Grundzustand des Heliumatoms. Z. Phys. 48 (1929) 469.
[34] Kato, T., On the eigenfunctions of many-particle systems in quantum mechanics. Commun. Pure Appl. Math. X (1957) 151177.
[35] T. Kato, Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissenschaften, 2nd edition. Springer-Verlag, Berlin, Heidelberg, New York 132 (1976).
[36] W. Klopper, R12 methods, Gaussian geminals, in Modern Methods and Algorithms of Quantum Chemistry, edited by J. Grotendorst (2000) 181–229.
[37] H.-C. Kreusler and H. Yserentant, The mixed regularity of electronic wave functions in fractional order and weighted Sobolev spaces. Preprint 94, DFG SPP 1324 (2011).
[38] Luo, H., Kolb, D., Flad, H.-J., Hackbusch, W. and Koprucki, T., Wavelet approximation of correlated wave functions. II. Hyperbolic wavelets and adaptive approximation schemes. J. Chem. Phys. 117 (2002) 36253638.
[39] Y. Meyer and R. Coifman, Wavelets : Calderon-Zygmund and multilinear operators, Cambridge Studies in Advanced Mathematics. Cambridge University Press (1997).
[40] Neelov, A. and Goedecker, S., An efficient numerical quadrature for the calculation of the potential energy of wavefunctions expressed in the Daubechies wavelet basis. J. Comput. Phys. 217 (2006) 312339.
[41] M. Nooijen, and R.J. Bartlett, Elimination of Coulombic infinities through transformation of the Hamiltonian. J. Chem. Phys. 109 (1998).
[42] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Analysis of Operators IV. Academic Press (1978).
[43] Schwab, C. and Todor, R.A., Sparse finite elements for stochastic elliptic problems – higher order moments. Computing 71 (2003) 4363.
[44] Stevenson, R., On the compressibility of operators in wavelet coordinates. SIAM J. Math. Anal. 35 (2004) 11101132.
[45] Sweldens, W., The lifting scheme : a custom-design construction of biorthogonal wavelets. Appl. Comput. Harmon. Anal. 3 (1996) 186200.
[46] Tenno, S., A feasible transcorrelated method for treating electronic cusps using a frozen Gaussian geminal. Chem. Phys. Lett. 330 (2000) 169174.
[47] Tew, D.P. and Klopper, W., New correlation factors for explicitly correlated electronic wave functions. J. Chem. Phys. 123 (2005) 074101.
[48] Yserentant, H., On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives. Numer. Math. 98 (2004) 731759.
[49] H. Yserentant, Regularity and Approximability of Electronic Wave Functions. Lect. Notes Math. 2000 (2010).
[50] Yserentant, H., The mixed regularity of electronic wave functions multiplied by explicit correlation factors. ESAIM : M2AN 45 (2011) 803824.
[51] A. Zeiser, Direkte Diskretisierung der Schrödingergleichung auf dünnen Gittern. Ph.D. thesis, TU Berlin (2010).
[52] Zeiser, A., Fast matrix-vector multiplication in the sparse-grid Galerkin method. J. Sci. Comput. 47 (2010) 328346.
[53] A. Zeiser, Wavelet approximation in weighted Sobolev spaces of mixed order with applications to the electronic Schrödinger equation. To appear in Constr. Approx. (2011) DOI : 10.1007/s00365-011-9138-7.
[54] Zweistra, H.J.A., Samson, C.C.M. and Klopper, W., Similarity-transformed Hamiltonians by means of Gaussian-damped interelectronic distances. Collect. Czech. Chem. Commun. 68 (2003) 374386.

Keywords

Hyperbolic wavelet discretization of the two-electron Schrödinger equation in an explicitly correlated formulation

  • Markus Bachmayr (a1)

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