Introduction to Density Functional Theory

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1 1 Introduction to Density Functional Theory 21 February 2011; V172 P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 21 February 2011 Introduction to DFT

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4 4 Ab initio Computational Based Design P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 21 February 2011 Introduction to DFT

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14 14 Many Particle problem in Solids P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 21 February 2011 Introduction to DFT

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20 Hartree Method for many particle Solids 20 P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 21 February 2011 Introduction to DFT

21 Hartree Hamiltonian 21 P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 21 February 2011 Introduction to DFT

22 SCF procedure 22 P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 21 February 2011 Introduction to DFT

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25 Introduction of Exchange (Spin) : Hartree Fock Method 25 P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 21 February 2011 Introduction to DFT

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27 Failure of HF Method 27 P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 21 February 2011 Introduction to DFT

28 28 Exchange and Correlation P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 21 February 2011 Introduction to DFT

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36 How to get an approximate XC functional? 36 P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 21 February 2011 Introduction to DFT

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38 What is functional? 38 P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 21 February 2011 Introduction to DFT

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40 Density Functional Theory (DFT) is a formally exact representation of the N electron Schrödinger Equation 40 Schrödinger view DFT view Formally equivalent electron interaction external potential Hard problem to solve (scales like N 5 ) Kohn-Sham particle (non-interacting) effective potential Easy problem to solve ( scales like N 3 or better) Hohenberg and Kohn proved this (1964) Nobel prize in Chemistry for Kohn in 1998 P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 21 February 2011 Introduction to DFT

41 1998 Nobel Prize Laureates, Chemistry 41 "for his development of the density-functional theory" "for his development of computational methods in quantum chemistry" Advent of Quantitative Theory and Modeling P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 21 February 2011 Introduction to DFT

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44 44 Exchange-Correlation Potential P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 21 February 2011 Introduction to DFT

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57 Success and failure of standard DFT in solids 57 Standard LDA (GGA) gives good description of structural and electronic properties of most solids (lattice parameters within 1-2%, at least qualitatively correct bandstructure, metal-insulator, magnetism, ) Problems: localized (correlated) electrons late 3d transition metal oxides/halides metals instead of insulators (FeO, FeF 2, cuprates, ) nonmagnetic instead of anti-ferromagnetic (La 2 CuO 4, YBa 2 Cu 3 O 6 ) 4f, 5f electrons all f-states pinned at the Fermi energy, always metallic orbital moments much too small weakly correlated metals FeAl is ferromagnetic in theory, but nonmagnetic experimentally 3d-band position, exchange splitting, P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 21 February 2011 Introduction to DFT

58 58 Is LDA/GGA repairable? Ab initio methods GGA: usually improvement, but often too small. Hartree-Fock: completely neglects correlation, very poor in solids Exact exchange: imbalance between exact X and approximate (no) C Hybrid-Functionals: mix of HF + LDA; good for insulators, poor for metals GW: gaps in semiconductors, but ground state? expensive! Quantum Monte-Carlo: very expensive Not fully ab initio Self-interaction-correction: vanishes for Bloch states Orbital polarization: Hund s 2 nd rule by atomic Slater-parameter LDA+U: strong Coulomb repulsion via external Hubbard U parameter DMFT: extension of LDA+U for weakly correlated systems P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 21 February 2011 Introduction to DFT

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63 Hybrid functionals (mixing of LDA with HF) 63 P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 21 February 2011 Introduction to DFT

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67 Lecture Summary 67 Density functional theory is an exact reformulation of many-body quantum mechanics in terms of the probability density rather than the wave function The ground-state energy can be obtained by minimization of the energy functional E[n]. All we know about the functional is that it exists, however, its form is unknown. Kohn-Sham reformulation in terms of single-particle orbitals helps in the development of approximations and is the form used in density functional calculations today. P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 21 February 2011 Introduction to DFT

68 68 Summary DFT has been the mainstay of electronic structure calculations more than 25 years. Because approximate functional provide a useful balance between accuracy and computational cost. This allowed much larger systems to be treated. DFT is a completely different way of approaching any interacting problem, by mapping it exactly to a much easier-to-solve non-interacting problem. Its methodology is applied into a large variety of systems as atoms,molecules, clusters and solids. Limitations of LDA/GGA were partly overcome by Hybrid functionals. P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 21 February 2011 Introduction to DFT

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