Scaling of the Computational Effort with Respect to the Size of the System

Methods exhibiting linear scaling with respect to the size of the system, so called O(N) methods, are an essential tool for the calculation of the electronic structure of large systems containing many atoms. They are based on algorithms which take advantage of the decay properties of the density matrix. In this article the physical decay properties of the density matrix will therefore first be studied for both metals and insulators. Several approaches to construct O(N) algorithms will then be presented and critically examined under various aspects. Finally some issues which are relevant only for self-consistent O(N) methods such as the calculation of the Hartree potential and mixing issues will be discussed.