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

Harry Yserentant

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (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 Mathematics 2000, 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 - Modélisation Mathématique et Analyse Numérique 45.5 (2011): 803-824. <http://eudml.org/doc/273162>.

@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 Mathematics 2000, 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 - Modélisation Mathématique et Analyse Numérique},
keywords = {schrödinger equation; regularity; mixed derivatives; correlation factor; complexity; Schrödinger equation},
language = {eng},
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/273162},
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 - Modélisation Mathématique et Analyse Numérique
PY - 2011
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 Mathematics 2000, 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; Schrödinger equation
UR - http://eudml.org/doc/273162
ER -

References

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