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##### A Tutorial on Density Functional Theory

**Authors**: F. Nogueira, A. Castro, and M.A.L. Marques

**Ref.**: in *A Primer in Density-Functional Theory*, C. Fiolhais, F. Nogueira, and M.A.L. Marques (ed.), Lecture Notes in Physics, Vol. 620, Springer, Berlin, 218-256 (2003)

**Abstract**: The success of density functional theory (DFT) is clearly demonstrated by the overwhelming amount of research articles describing results obtained within DFT that were published in the last decades. There is also a fair number of books reviewing the basics of the theory and its extensions (e.g., the present volume, [1] and [2]). These works fall mainly into three classes: those dealing with the theory (proposing extensions, new functionals, etc.), those concerned with the technical aspects of the numerical implementations, and others -- the vast majority -- presenting results. In our opinion, any scientist working in the field should have a sound knowledge of the three classes. For example, a theorist developing a new functional should be aware of the difficulties in implementing it. Or the applied scientist, performing calculations on specific systems, should know the limitations of the theory and of the numerical implementation she/he is using. The goal of this chapter is to supply the beginner with a brief pedagogical overview of DFT, combining the above-mentioned aspects. However, we will not review its foundations -- we redirect the reader to the chapter of J. Perdew and S. Kurth that opens this book. Obviously, we will not be able to provide many details, but we hope that the beginner obtains a general impression of the capabilities and limitations of DFT.

This chapter is written in the form of a tutorial, combining basic theoretical and numerical aspects with specific examples, running from the simplest hydrogen atom to more complex molecules and solids. For the examples we used only freely available codes [3], so that the reader may easily reproduce the calculations. All input and output files can be found in the web site http://www.tddft.org/DFT2001/. The chapter follows closely the outline of the practical sessions held at Caramulo, during the DFT2001 summer school. Some theoretical or numerical aspects that were required in the practical sessions were, however, not covered by any of the lectures in Caramulo (e.g., pseudo-potentials). To fill this gap we provide in this chapter a brief account of some of them. We do not intend to discuss every possible numeric implementation of DFT. In particular, we do not include any explicit example of a localized basis set DFT calculation. Neither do we intend to present an extensive survey of the numerical aspects of each technique. We expect, however, that the technical details given are sufficient to enable the reader to perform himself the simulations presented herein.

The outline of the chapter is the following: We start, in Sect. 6.2, by giving a technical overview on how to solve the Kohn-Sham equations. The next section is devoted to pseudo-potentials, an essential ingredient of many DFT calculations. In Sect. 6.4 we present our first test case, namely atoms, before we proceed to some plane-wave calculations in Sect. 6.5. The final example, methane calculated using a real-space implementation, is presented in Sect. 6.6. We will use atomic units throughout this chapter, except when explicitly stated otherwise.

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