Applied Numerical Mathematics 56 (2006) 1383–1396 www.elsevier.com/locate/apnum Wavelets for density matrix computation in electronic structure calculation ✩ Reinhold Schneider ∗ , Toralf Weber Institute of Computer Science and Applied Mathematics, Christian-Albrechts-University of Kiel, Christian-Albrechts-Platz 4, 24098 Kiel, Germany Available online 9 June 2006 Abstract This paper is concerned with demanding calculations of electronic structures. We give a brief introduction to the basics of electronic structure calculation based on the electronic multi-particle Schrödinger equation. We describe the structures of Hartree– Fock, Kohn–Sham and hybrid models for closed shell systems, the aufbau principle and the self consistent field iteration. While traditional methods for computing the orbitals are scaling cubically w.r.t. the number of electrons, the computation of the density matrix offers the opportunity to achieve linear complexity. We describe several iteration schemes for the computation of the density matrix. We briefly present the concept of best n-term approximation and summarize recent regularity results obtained by the authors. They show that the density matrix is in mixed Besov spaces B s τ,τ . Adaptive sparse grid approximation will reduce the complexity by several magnitudes. We propose fast methods for matrix computations as e.g. wavelet matrix compression. Finally, first numerical experiments demonstrate the behavior of the described iteration schemes for computing the density matrix.  2006 IMACS. Published by Elsevier B.V. All rights reserved. MSC: 65T60 Keywords: Electronic Schrödinger equation; Hartree–Fock; Density functional theory; Density matrices; Linear scaling; Wavelets; Sparse grids 1. Introduction Modern developments in technology and sciences are demanding for numerical simulation of molecular structures. Among the most prominent and modern developments there are molecular biology, nano-science and design of semi- conductor devices. Even for macroscopic models in continuum mechanics unknown parameters are determined by numerical simulation on a microscopic scale. A multi-scale modeling comes up with a hierarchy of models, where the parameters for the macroscopic models are calibrated by a computation on a microscopic scale. On a molecular or atomic scale, physical phenomena are governed by the laws of quantum mechanics. Therefore any determination of molecular parameters should be based on a reliable computational tool simulating the quantum mechanical phenom- ena accurately. This is the ambitious principle of ab initio methods. The model equations should be based only on the ✩ The work of these authors has been supported by the DFG SPP 1145 “Moderne und universelle first-principles Methoden für Mehrelektronensysteme in Chemie und Physik”. * Corresponding author. E-mail address: rs@numerik.uni-kiel.de (R. Schneider). 0168-9274/$30.00  2006 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.apnum.2006.03.020