Volume 45, Number 5, September-October 2011
|Page(s)||803 - 824|
|Published online||23 February 2011|
Institut für Mathematik,
Technische Universität Berlin,
Straße des 17. Juni 136,
Revised: 11 October 2010
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.
Mathematics Subject Classification: 35J10 / 35B65 / 41A25 / 41A63
Key words: Schrödinger equation / regularity / mixed derivatives / correlation factor / complexity
In the meantime the author became aware of the closely related work of M. Bachmayr: Hyperbolic wavelet discretization of the two-electron Schrödinger equation in an explicitly correlated formulation, Preprint AICES-2010/06-2, RWTH Aachen, in which the mixed regularity of correspondingly modified wave functions for the two-electron case is analyzed.
© EDP Sciences, SMAI, 2011
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