Strong Coulomb Correlations in Electronic Structure Calculations Advances in Condensed Matter Science A series edited by D.D. Sarma, G. Kotliar and Y. Tokura Volume 1 Strong Coulomb Correlations in Electronic Structure Calculations: Beyond the Local Density Approximation, edited by Vladimir I. Anisimov Forthcoming titles in the series Colossal Magnetoresistive Oxides, edited by Y. Tokura Introduction to DV-Xa Quantum Chemistry, by J. Kawai and H. Adachi Advances in Amorphous Semiconductors, by J. Singh and K. Shimakawa Electronic Structure of Non-periodic Solids: Alloys, Suifaces and Clusters, by A. Mookerjee and D.D. Sarma This book is part of a series. The publisher will accept continuation orders which may be cancelled at any time and which provide for automatic billing and shipping of each title in the series upon publication. Please write for details. 2000 Publisher’s Note The publisher has gone to great lengths to ensure the quality of this book but points out that some imperfections from the original may be apparent. Contents Contributors ix Introduction xi Chapter 1. The GW Approximation and Vertex Corrections F erdi Aryasetiawan 1 Introduction 1 2 Theory 4 2.1 The Green function and the self-energy 4 2.2 The response and polarization function 9 2.3 The Hedin equations 10 2.4 The self-energy expansion 10 2.5 Spectral representations 11 2.6 The GW approximation 14 3 Computational Method 21 3.1 Basis functions 21 3.2 Special directions 24 3.3 Evaluation of the Coulomb Matrix 27 3.4 Evaluation of the polarization P 29 3.5 Evaluation of the self-energy I: 32 4 Applications 34 4.1 Simple metals and semiconductors 35 4.2 Transition metals 38 4.3 f system: Gd 54 5 Vertex Corrections: Beyond GW 55 5.1 The cumulant expansion 59 5.2 T -matrix approximation 68 6 Self-consistency and Total Energies 80 6.1 Self-consistency 80 6.2 Total energies 85 References 87 v vi CONTENTS Chapter 2. LDA+U Method: Screened Coulomb Interaction in the Mean-field Approximation Vladimir I. Anisimov and A.l. Lichtenstein 1 Introduction 97 2 LDA+U Method: Formulation and Practical Realization 98 ·3 Relationship Between LDA+U and GW Approximation 101 4 Localized States: 3d and 4f Orbitals 109 4.1 Rare-earth metals: Gd 109 4.2 Transition metal impurities in alkaline metals 111 4.3 Rare-earth elements compounds: CeSb 111 4.4 PrBaCu0 114 2 3 7 5 Mott-Hubbard Insulators 116 5.1 3d-transition metal oxides 116 5.2 Transition metal impurities in insulators: Fe-impurity in MgO 121 5.3 Linear chain (MX) compounds 123 6 Electron-lattice Interactions: Jahn-Teller Distortions and Polarons 123 6.1 Cooperative Jahn-Teller distortions in transition metal compounds: KCuF 123 3 6.2 Polarons in doped Mott insulators: L~xSrxCu04 and La_xSrxNiO 127 2 4 7 Metal-insulator Transition 130 7.1 FeSi 130 7.2 LaCo0 133 3 8 Charge and Orbital Ordering 134 8.1 Magnetite Fe0 134 3 4 8.2 Doped manganite Pr_xCaxMn0 138 1 3 9 Beyond the Mean-field Approximation: Spectral Properties and Quasiparticle Bands 144 9.1 Spectral properties from the Anderson impurity model 144 9.2 Quasiparticle bands in FLEX approximation 150 9.3 Dynamical mean-field theory 152 10 Conclusion 159 References 161 CONTENTS vii Chapter 3. LSDA and Self-Interaction Correction Takeo Fujiwara, Masao Arai and Yasushi Ishii 1 Introduction 167 2 Total Energy and Self-Interaction Correction 168 2.1 LSDA total energy 168 2.2 LSDA self-interaction correction 170 2.3 Canonical orbitals 176 2.4 Self-interaction correction in metallic systems 179 2.5 Canonical orbitals and many-electron wavefunctions 180 3 SIC in Solid Hydrogen and Transition Metal Monoxides 182 3.1 Solid hydrogen 183 3.2 Transition metal compounds 185 4 SIC in Rare Earth Metals and Rare Earth Metal Compounds 194 4.1 Rare earth metals 195 4.2 Rare earth compounds - cerium pnictides 198 5 Conclusion 199 References 200 Chapter 4. Orbital Functionals in Density Functional Theory: The Optimized Effective Potential Method T. Grabo, T. Kreibich, S. Kurth and E.K. U. Gross 1 Introduction 203 2 The OEP Method, Basic Formalism 209 2.1 Derivation of the OEP equations 209 2.2 Approximation of Krieger, Li and Iafrate 212 2.3 Rigorous properties of the OEP and KLI potentials 217 2.4 Hartree-Fock versus x-only OEP, a comparison 235 3 Relativistic Generalization of the OEP and KLI Methods 236 3.1 Relativistic optimized effective potential method 238 3.2 Relativistic KLI approximation 240 3.3 Relativistic OEP in the electrostatic case 244 4 Numerical Results 245 4.1 Exchange-only calculations for nonrelativistic systems 245 4.2 Comparison of nonrelativistic with relativistic results 262 viii CONTENTS 4.3 Inclusion of correlation contributions for nonrelativistic systems 270 4.4 Solids 293 5 Beyond the OEP- A Connection with Many-Body Perturbation Theory 297 Acknowledgments 306 References 307 Index 313 Contributors Vladimir I. Anisimov Institute of Metal Physics, Ekaterinburg, GSP-170, Russia Masao Arai Department of Applied Physics, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan Ferdi Aryasetiawan Department of Theoretical Physics, University of Lund, Solvegatan 14A, S-223 62 Lund, Sweden Takeo Fujiwara Department of Applied Physics, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan T. Grabo Institut ftir Theoretische Physik, UniversiUit Wiirzburg, Am Hubland, 97074 Wiirzburg, Germany E.K.U. Gross Institut ftir Theoretische Physik, UniversiUit Wiirzburg, Am Hubland, 97074 Wiirzburg, Germany Yasushi Ishii Department of Applied Physics, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan T. Kreibich Institut ftir Theoretische Physik, Universitat Wiirzburg, Am Hubland, 97074 Wtirzburg, Germany S. Kurth Department of Physics and Quantum Theory Group, Tulane University, New Orleans, LA 70118, USA A.I. Lichtenstein Institut ftir Festkorperforschung 4es Forschungszentrum Jtilich, D-52425 Jtilich, Germany ix