Numerische Mathematik manuscript No. (will be inserted by the editor) Harry Yserentant The hyperbolic cross space approximation of electronic wavefunctions Revised version of September 13, 2006; to appear in Numerische Mathematik Abstract The electronic Schr¨ odinger equation describes the motion of electrons under Coulomb interaction forces in the field of clamped nuclei and forms the ba- sis of quantum chemistry. The present article continues the author’s recent work [Numer. Math. 98, 731-759 (2004)] on the regularity of its solutions, the electronic wavefunctions. It was shown in the mentioned article that these wavefunctions possess square integrable high-order mixed weak derivatives as they are needed to justify and underpin sparse grid and hyperbolic cross space approximation tech- niques theoretically. How fast the norms of these derivatives can increase with the number of electrons is studied in the present article. Provided that all quanti- ties are properly related to a characteristic lengthscale of the considered atomar or molecular system, the norms of the derivatives remain bounded independent of the number of electrons and the number, the positions, and the charges of the nuclei. Mathematics Subject Classification (2000) 35J10, 35B65, 41A63, 65D99 1 Introduction The basis of almost all quantum chemistry is the Schr¨ odinger equation. It de- scribes a system of electrons and nuclei that interact by Coulomb attraction and repulsion forces. The much slower motion of the nuclei is usually separated from that of the electrons. This results in the electronic Schr ¨ odinger equation, the prob- lem to find the eigenvalues and eigenfunctions of the electronic Hamilton operator H = − 1 2 N ∑ i=1 Δ i − N ∑ i=1 K ∑ ν =1 Z ν |x i − a ν | + 1 2 N ∑ i, j=1 i= j 1 |x i − x j | , (1.1) H. Yserentant Institut f ¨ ur Mathematik, TU Berlin, 10623 Berlin, Germany E-mail: yserentant@math.tu-berlin.de