birdsEyeView_DFT

birdsEyeView_DFT - arXiv:cond-mat/0211443v5...

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arXiv:cond-mat/0211443v5 [cond-mat.mtrl-sci] 18 Nov 2006 A Bird’s-Eye View of Density-Functional Theory Klaus Capelle Departamento de F´ ısica e Inform´atica Instituto de F´ ısica de S˜ao Carlos Universidade de S˜ao Paulo Caixa Postal 369, S˜ao Carlos, 13560-970 SP, Brazil keywords: density-functional theory, electronic-structure theory, electron cor- relation, many-body theory, local-density approximation Abstract This paper is the outgrowth of lectures the author gave at the Physics Institute and the Chemistry Institute of the University of S˜ao Paulo at S˜ao Carlos, Brazil, and at the VIII’th Summer School on Electronic Structure of the Brazilian Physical Society. It is an attempt to introduce density-functional theory (DFT) in a language accessi- ble for students entering the ±eld or researchers from other ±elds. It is not meant to be a scholarly review of DFT, but rather an infor- mal guide to its conceptual basis and some recent developments and advances. The Hohenberg-Kohn theorem and the Kohn-Sham equa- tions are discussed in some detail. Approximate density functionals, selected aspects of applications of DFT, and a variety of extensions of standard DFT are also discussed, albeit in less detail. Through- out it is attempted to provide a balanced treatment of aspects that are relevant for chemistry and aspects relevant for physics, but with a strong bias towards conceptual foundations. The paper is intended to be read before (or in parallel with) one of the many excellent more technical reviews available in the literature.
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Contents 1 Preface 3 2 What is density-functional theory? 4 3 DFT as a many-body theory 8 3.1 Functionals and their derivatives . . . . . . . . . . . . . . . . . 8 3.2 The Hohenberg-Kohn theorem . . . . . . . . . . . . . . . . . . 10 3.3 Complications: N and v -representability of densities, and nonunique- ness of potentials . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.4 A preview of practical DFT . . . . . . . . . . . . . . . . . . . 16 3.5 From wave functions to density functionals via Green’s func- tions and density matrices . . . . . . . . . . . . . . . . . . . . 19 3.5.1 Green’s functions . . . . . . . . . . . . . . . . . . . . . 19 3.5.2 Density matrices . . . . . . . . . . . . . . . . . . . . . 21 4 DFT as an e±ective single-body theory: The Kohn-Sham equations 25 4.1 Exchange-correlation energy: de±nition, interpretation and exact properties . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.1.1 Exchange-correlation energy . . . . . . . . . . . . . . . 26 4.1.2 Di²erent perspectives on the correlation energy . . . . 28 4.1.3 Exact properties . . . . . . . . . . . . . . . . . . . . . 30 4.2 Kohn-Sham equations . . . . . . . . . . . . . . . . . . . . . . 32 4.2.1 Derivation of the Kohn-Sham equations . . . . . . . . . 32 4.2.2 The eigenvalues of the Kohn-Sham equation . . . . . . 35 4.2.3 Hartree, Hartree-Fock and Dyson equations . . . . . . 37 4.3 Basis functions . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5 Making DFT practical: Approximations 41 5.1 Local functionals: LDA . . . . . . . . . . . . . . . . . . . . . . 43 5.2 Semilocal functionals: GEA, GGA and beyond . . . . . . . . . 45 5.3 Orbital functionals and other nonlocal approximations: hy- brids, Meta-GGA, SIC, OEP, etc. . . . . . . . . . . . . . . . . 49 6 Extensions of DFT: New frontiers and old problems 53 References 59 2
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1 Preface This paper is the outgrowth of lectures the author gave at the Physics Institute and the Chemistry Institute of the University of S˜ao Paulo at S˜ao Carlos, Brazil, and at the VIII’th Summer School on Electronic Structure of the Brazilian Physical Society [1]. The main text is a description of density-
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birdsEyeView_DFT - arXiv:cond-mat/0211443v5...

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