ABC_of_DFT

ABC_of_DFT - 1 http://chem.ps.uci.edu/kieron/dft/book/ The...

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Unformatted text preview: 1 http://chem.ps.uci.edu/kieron/dft/book/ The ABC of DFT Kieron Burke and friends Department of Chemistry, University of California, Irvine, CA 92697 April 10, 2007 2 Contents I Background 13 1 Introduction 15 1.1 Importance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2 What is a Kohn-Sham calculation? . . . . . . . . . . . . . . . . . . . . . . 16 1.3 Reinterpreting molecular orbitals . . . . . . . . . . . . . . . . . . . . . . . . 19 1.4 Sampler of applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.5 Particle in a box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.6 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2 Functionals 27 2.1 What is a functional? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 Functional derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3 Euler-Lagrange equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.4 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3 One electron 33 3.1 Variational principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 Trial wavefunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3 Three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.4 Di ff erential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.5 Virial theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.6 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4 Two electrons 39 4.1 Antisymmetry and spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.2 Hartree-Fock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.3 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.4 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5 Many electrons 47 5.1 Ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.2 Hartree-Fock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3 4 CONTENTS 5.3 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.4 Atomic configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.5 Atomic densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.6 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 II Basics 55 6 Density functional theory 57 6.1 One electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.2 Hohenberg-Kohn theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.3 Thomas-Fermi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.4 Particles in boxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6.5 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Questions ....
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