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

Published online by Cambridge University Press:  30 March 2012

Markus Bachmayr
Markus Bachmayr*
Affiliation:
RWTH Aachen, Institut für Geometrie und Praktische Mathematik, Templergraben 55, 52056 Aachen, Germany. e-mail: [email protected]
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Abstract

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

Keywords

Schrödinger equationmixed regularitytranscorrelated methodwaveletsseparable approximation
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 6 , November 2012 , pp. 1337 - 1362
Copyright
© EDP Sciences, SMAI, 2012

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References

R.A. Adams and J.J.F. Fournier, Sobolev Spaces, Pure and Applied Mathematics Series, 2nd edition. Academic Press 140 (2003).
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).
M. Bachmayr, Integration of products of Gaussians and wavelets with applications to electronic structure calculations. Preprint AICES, RWTH Aachen (2012).
Balder, R. and Zenger, C., The solution of multidimensional real Helmholtz equations on sparse grids. SIAM J. Sci. Comput. 17 (1996) 631–646. Google Scholar
Beylkin, G., On the representation of operators in bases of compactly supported wavelets. SIAM J. Numer. Anal. 29 (1992) 1716–1740. Google Scholar
Boys, S.F. and Handy, N.C., The determination of energies and wavefunctions with full electronic correlation. Proc. R. Soc. Lond. A 310 (1969) 43–61. Google Scholar
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).
H.-J. Bungartz, Dünne Gitter und deren Anwendung bei der adaptiven Lösung der dreidimensionalen Poisson-Gleichung. Ph.D. thesis, Technische Universität München (1992).
F. Chatelin, Spectral Approximation of Linear Operators, Computer Science and Applied Mathematics. Academic Press (1983).
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. Google ScholarPubMed
A. Cohen, Numerical Analysis of Wavelet Methods. Stud. Math. Appl. 32 (2003).
Dahmen, W. and Micchelli, C.A., Using the refinement equation for evaluating integrals of wavelets. SIAM J. Numer. Anal. 30 (1993) 507–537. Google Scholar
Daubechies, I., Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41 (1988) 909–996. Google Scholar
Dijkema, T.J., Schwab, C. and Stevenson, R., An adaptive wavelet method for solving high-dimensional elliptic PDEs. Constr. Approx. 30 (2009) 423–455. Google Scholar
Donovan, G., Geronimo, J. and Hardin, D., Intertwining multiresolution analyses and the construction of piecewise-polynomial wavelets. SIAM J. Math. Anal. 27 (1996) 1791–1815. Google Scholar
Donovan, G., Geronimo, J. and Hardin, D., Orthogonal polynomials and the construction of piecewise polynomial smooth wavelets. SIAM J. Math. Anal. 30 (1999) 1029–1056. Google Scholar
Flad, H.-J., Hackbusch, W., Kolb, D. and Schneider, R., Wavelet approximation of correlated wave functions. I. Basics. J. Chem. Phys. 116 (2002) 9641–9657. Google Scholar
Flad, H.-J., Hackbusch, W. and Schneider, R., Best N-term approximation in electronic structure calculations. II. Jastrow factors. ESAIM : M2AN 41 (2007) 261. Google Scholar
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.
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) 183–227. 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, Classics in Mathematics. Springer, Berlin, Heidelberg (1998).
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.
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. Also available as INS Preprint No. 0911. Google Scholar
Griebel, M. and Knapek, S., Optimized tensor-product approximation spaces. Constr. Approx. 16 (2000) 525. Google Scholar
Griebel, M. and Oswald, P., Tensor product type subspace splittings and multilevel iterative methods for anisotropic problems. Adv. Comput. Math. 4 (1995) 171–206. Google Scholar
J. Hamaekers, Tensor Product Multiscale Many-Particle Spaces with Finite-Order Weights for the Electronic Schödinger Equation. Ph.D. thesis, Universität Bonn (2009).
Harbrecht, H., Schneider, R. and Schwab, C., Multilevel frames for sparse tensor product spaces. Numer. Math. 110 (2008) 199–220. 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
Hille, E., A class of reciprocal functions. Ann. Math. 27 (1926) 427–464. Google Scholar
Hirschfelder, J.O., Removal of electron-electron poles from many-electron Hamiltonians. J. Chem. Phys. 39 (1963) 3145–3146. Google Scholar
Hylleraas, E., Über den Grundzustand des Heliumatoms. Z. Phys. 48 (1929) 469. Google Scholar
Kato, T., On the eigenfunctions of many-particle systems in quantum mechanics. Commun. Pure Appl. Math. X (1957) 151–177. Google Scholar
T. Kato, Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissenschaften, 2nd edition. Springer-Verlag, Berlin, Heidelberg, New York 132 (1976).
W. Klopper, R12 methods, Gaussian geminals, in Modern Methods and Algorithms of Quantum Chemistry, edited by J. Grotendorst (2000) 181–229.
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).
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) 3625–3638. Google Scholar
Y. Meyer and R. Coifman, Wavelets : Calderon-Zygmund and multilinear operators, Cambridge Studies in Advanced Mathematics. Cambridge University Press (1997).
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) 312–339. Google Scholar
M. Nooijen, and R.J. Bartlett, Elimination of Coulombic infinities through transformation of the Hamiltonian. J. Chem. Phys. 109 (1998).
M. Reed and B. Simon, Methods of Modern Mathematical Physics, Analysis of Operators IV. Academic Press (1978).
Schwab, C. and Todor, R.A., Sparse finite elements for stochastic elliptic problems – higher order moments. Computing 71 (2003) 43–63. Google Scholar
Stevenson, R., On the compressibility of operators in wavelet coordinates. SIAM J. Math. Anal. 35 (2004) 1110–1132. Google Scholar
Sweldens, W., The lifting scheme : a custom-design construction of biorthogonal wavelets. Appl. Comput. Harmon. Anal. 3 (1996) 186–200. Google Scholar
Tenno, S., A feasible transcorrelated method for treating electronic cusps using a frozen Gaussian geminal. Chem. Phys. Lett. 330 (2000) 169–174. Google Scholar
Tew, D.P. and Klopper, W., New correlation factors for explicitly correlated electronic wave functions. J. Chem. Phys. 123 (2005) 074101. Google ScholarPubMed
Yserentant, H., On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives. Numer. Math. 98 (2004) 731–759. Google Scholar
H. Yserentant, Regularity and Approximability of Electronic Wave Functions. Lect. Notes Math. 2000 (2010).
Yserentant, H., The mixed regularity of electronic wave functions multiplied by explicit correlation factors. ESAIM : M2AN 45 (2011) 803–824. Google Scholar
A. Zeiser, Direkte Diskretisierung der Schrödingergleichung auf dünnen Gittern. Ph.D. thesis, TU Berlin (2010).
Zeiser, A., Fast matrix-vector multiplication in the sparse-grid Galerkin method. J. Sci. Comput. 47 (2010) 328–346. Google Scholar
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.
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) 374–386. Google Scholar