The mixed regularity of electronic wave functions multiplied by explicit correlation factors***

Harry Yserentant

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

  • Volume: 45, Issue: 5, page 803-824
  • ISSN: 0764-583X

Abstract

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The electronic Schrödinger equation describes the motion of N electrons under Coulomb interaction forces in a field of clamped nuclei. The solutions of this equation, the electronic wave functions, depend on 3N variables, three spatial dimensions for each electron. Approximating them is thus inordinately challenging. As is shown in the author's monograph [Yserentant, Lecture Notes in Mathematics2000, Springer (2010)], the regularity of the solutions, which increases with the number of electrons, the decay behavior of their mixed derivatives, and the antisymmetry enforced by the Pauli principle contribute properties that allow these functions to be approximated with an order of complexity which comes arbitrarily close to that for a system of two electrons. The present paper complements this work. It is shown that one can reach almost the same complexity as in the one-electron case adding a simple regularizing factor that depends explicitly on the interelectronic distances.

How to cite

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Yserentant, Harry. "The mixed regularity of electronic wave functions multiplied by explicit correlation factors***." ESAIM: Mathematical Modelling and Numerical Analysis 45.5 (2011): 803-824. <http://eudml.org/doc/197501>.

@article{Yserentant2011,
abstract = { The electronic Schrödinger equation describes the motion of N electrons under Coulomb interaction forces in a field of clamped nuclei. The solutions of this equation, the electronic wave functions, depend on 3N variables, three spatial dimensions for each electron. Approximating them is thus inordinately challenging. As is shown in the author's monograph [Yserentant, Lecture Notes in Mathematics2000, Springer (2010)], the regularity of the solutions, which increases with the number of electrons, the decay behavior of their mixed derivatives, and the antisymmetry enforced by the Pauli principle contribute properties that allow these functions to be approximated with an order of complexity which comes arbitrarily close to that for a system of two electrons. The present paper complements this work. It is shown that one can reach almost the same complexity as in the one-electron case adding a simple regularizing factor that depends explicitly on the interelectronic distances. },
author = {Yserentant, Harry},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Schrödinger equation; regularity; mixed derivatives; correlation factor; complexity; correlation factor},
language = {eng},
month = {2},
number = {5},
pages = {803-824},
publisher = {EDP Sciences},
title = {The mixed regularity of electronic wave functions multiplied by explicit correlation factors***},
url = {http://eudml.org/doc/197501},
volume = {45},
year = {2011},
}

TY - JOUR
AU - Yserentant, Harry
TI - The mixed regularity of electronic wave functions multiplied by explicit correlation factors***
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2011/2//
PB - EDP Sciences
VL - 45
IS - 5
SP - 803
EP - 824
AB - The electronic Schrödinger equation describes the motion of N electrons under Coulomb interaction forces in a field of clamped nuclei. The solutions of this equation, the electronic wave functions, depend on 3N variables, three spatial dimensions for each electron. Approximating them is thus inordinately challenging. As is shown in the author's monograph [Yserentant, Lecture Notes in Mathematics2000, Springer (2010)], the regularity of the solutions, which increases with the number of electrons, the decay behavior of their mixed derivatives, and the antisymmetry enforced by the Pauli principle contribute properties that allow these functions to be approximated with an order of complexity which comes arbitrarily close to that for a system of two electrons. The present paper complements this work. It is shown that one can reach almost the same complexity as in the one-electron case adding a simple regularizing factor that depends explicitly on the interelectronic distances.
LA - eng
KW - Schrödinger equation; regularity; mixed derivatives; correlation factor; complexity; correlation factor
UR - http://eudml.org/doc/197501
ER -

References

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  1. E. Cancès, C. Le Bris and Y. Maday, Méthodes Mathématiques en Chimie Quantique. Springer (2006).  
  2. H.-J. Flad, W. Hackbusch and R. Schneider, Best N-term approximation in electronic structure calculations. I. One-electron reduced density matrix. ESAIM: M2AN40 (2006) 49–61.  
  3. H.-J. Flad, W. Hackbusch and R. Schneider, Best N-term approximation in electronic structure calculations. II. Jastrow factors. ESAIM: M2AN41 (2007) 261–279.  
  4. S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and T. Østergard Sørensen, Sharp regularity estimates for Coulombic many-electron wave functions. Commun. Math. Phys.255 (2005) 183–227.  
  5. S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and T. Østergard Sørensen, Analytic structure of many-body Coulombic wave functions. Commun. Math. Phys.289 (2009) 291–310.  
  6. T. Helgaker, P. Jørgensen and J. Olsen, Molecular Electronic Structure Theory. John Wiley & Sons (2000).  
  7. M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and T. Østergard Sørensen, Electron wavefunctions and densities for atoms. Ann. Henri Poincaré2 (2001) 77–100.  
  8. E.A. Hylleraas, Neue Berechnung der Energie des Heliums im Grundzustande, sowie des tiefsten Terms von Ortho-Helium. Z. Phys.54 (1929) 347–366.  
  9. W. Kohn, Nobel lecture: Electronic structure of matter-wave functions and density functionals. Rev. Mod. Phys.71 (1999) 1253–1266.  
  10. W. Kutzelnigg, r12-dependent terms in the wave function as closed sums of partial wave amplitudes for large l. Theor. Chim. Acta68 (1985) 445–469.  
  11. W. Kutzelnigg and W. Klopper, Wave functions with terms linear in the interelectronic coordinates to take care of the correlation cusp. I. General theory. J. Chem. Phys.94 (1991) 1985–2001.  
  12. C. Le Bris Ed., Handbook of Numerical Analysis, Computational ChemistryX. North Holland (2003).  
  13. C. Le Bris, Computational chemistry from the perspective of numerical analysis. Acta Numer.14 (2005) 363–444.  
  14. A.J. O'Connor, Exponential decay of bound state wave functions. Commun. Math. Phys.32 (1973) 319–340.  
  15. J. Pople, Nobel lecture: Quantum chemical models. Rev. Mod. Phys.71 (1999) 1267–1274.  
  16. J. Rychlewski Ed., Explicitly Correlated Wave Functions in Chemistry and Physics, Progress in Theoretical Chemistry and Physics13. Kluwer (2003).  
  17. H. Yserentant, On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives. Numer. Math.98 (2004) 731–759.  
  18. H. Yserentant, The hyperbolic cross space approximation of electronic wavefunctions. Numer. Math.105 (2007) 659–690.  
  19. H. Yserentant, Regularity and Approximability of Electronic Wave Functions, Lecture Notes in Mathematics2000. Springer (2010).  

Citations in EuDML Documents

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  1. Markus Bachmayr, Hyperbolic wavelet discretization of the two-electron Schrödinger equation in an explicitly correlated formulation
  2. Markus Bachmayr, Hyperbolic wavelet discretization of the two-electron Schrödinger equation in an explicitly correlated formulation
  3. Markus Bachmayr, Hyperbolic wavelet discretization of the two-electron Schrödinger equation in an explicitly correlated formulation
  4. Markus Bachmayr, Hyperbolic wavelet discretization of the two-electron Schrödinger equation in an explicitly correlated formulation

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