Nonlinear PDE Days           Karlsruhe               January  20 – 21,  2011





Harry Yserentant

Technische Universität, Berlin



Regularity and approximability of electronic wave functions


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, and it is 
conventionally believed that a reduction to simplified models, such as those of the Hartree-Fock method or
density functional theory, is the only tenable approach. We indicate why this conventional wisdom need not 
to be ironclad: the unexpectedly high regularity of the solutions, which increases with the number of electrons,
the decay behavior of their mixed derivatives, and their 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. It is even possible to reach almost the same complexity as in the
one-electron case adding a simple regularizing factor that depends explicitly on the interelectronic distances.





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