Order-N tight-binding methods for electronic-structure and molecular dynamics

Abstract During the last few years, there has been an intense effort in the development of the so-called Order- N methods to solve the electronic-structure problem, for which the numerical efforts scale only linearly with the size of the system. The combination of these algorithms with total energy schemes has expanded our capability of performing electronic-structure based molecular dynamics (MD) simulations for systems of unprecedented size. In general, Order- N methods yield approximate solutions, based on physically motivated approximations. The central idea is, in most cases, the concept of localization (or the dependence of the relevant physical quantities on only the local environment). Therefore, the Tight-Binding (TB) formulation (or, more generally, the use of some kind of localized basis set), either from first principles or in an empirical form, is a natural framework to develop and apply Order- N schemes. In this paper we analyze the main ideas involved in these methods and their different implementations. We will focus on schemes to compute total energies and forces, therefore suited for MD simulations, and also on approaches to study the spectral properties like the density of states and eigenvalue information. These two classes of methods provided valuable complementary information and are often based on very similar assumptions and formalisms.

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