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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 Schrodinger 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 Bτ, τs. 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. |
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