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BK

Second edition

Cambridge (UK) : Cambridge University Press, 2020

xxviii, 762 stran : ilustrace ; 25 cm

ISBN 978-1-108-42990-0 (vázáno)

Obsahuje bibliografie, bibliografické odkazy a rejstřík

001653339

Contents // Preface // Acknowledgments // List of Notation // xix // xxiv // xxvi // Part 1 Overview and Background Topics // 1 Introduction // 1.1 Quantum Theory and the Origins of Electronic Structure // 1.2 Why Is the Independent-Electron Picture So Successful? // 1.3 Emergence of Quantitative Calculations // 1.4 The Greatest Challenge: Electron Interaction and Correlation // 1.5 Density Functional Theory // 1.6 Electronic Structure Is Now an Essential Part of Research // 1.7 Materials by Design // 1.8 Topology of Electronic Structure // 2 Overview // 2.1 Electronic Structure and the Properties of Matter // 2.2 Electronic Ground State: Bonding and Characteristic Structures // 2.3 Volume or Pressure As the Most Fundamental Variable // 2.4 How Good Is DFT for Calculation of Structures? // 2.5 Phase Transitions under Pressure // 2.6 Structure Prediction: Nitrogen Solids and Hydrogen Sulfide // Superconductors at High Pressure // 2.7 Magnetism and Electron-Electron Interactions // 2.8 Elasticity: Stress-Strain Relations // 2.9 Phonons and Displacive Phase Transitions // 2.10 Thermal Properties: Solids, Liquids, and Phase Diagrams // 2.11 Surfaces and Interfaces // 2.12 Low-Dimensional Materials and van der Waals Heterostructures // 2.13 Nanomaterials: Between Molecules and Condensed Matter // 2.14 Electronic Excitations: Bands and Bandgaps // 1 // 2 // 3 // 7 // 10 // 11 // 11 // 12 // 13 // 15 // 15 // 17 // 19 // 21 // 23 // 26 // 31 // 33 // 35 // 38 // 44 // 47 // 48 // 50

// Contents // •nm // Vili // 2.15 Electronic Excitations and Optical Spectra 54 // 2.16 Topological Insulators 57 // 2.17 The Continuing Challenge: Electron Correlation 57 // 3 Theoretical Background 60 // 3.1 Basic Equations for Interacting Electrons and Nuclei 60 // 3.2 Coulomb Interaction in Condensed Matter 64 // 3.3 Force and Stress Theorems 65 // 3.4 Generalized Force Theorem and Coupling Constant Integration 67 // 3.5 Statistical Mechanics and the Density Matrix 68 // 3.6 Independent-Electron Approximations 69 // 3.7 Exchange and Correlation 74 // Exercises 78 // 4 Periodic Solids and Electron Bands 81 // 4.1 Structures of Crystals: Lattice - Basis 81 // 4.2 Reciprocal Lattice and Brillouin Zone 90 // 4.3 Excitations and the Bloch Theorem 94 // 4.4 Time-Reversal and Inversion Symmetries 98 // 4.5 Point Symmetries 100 // 4.6 Integration over the Brillouin Zone and Special Points 101 // 4.7 Density of States 105 // Exercises 106 // 5 Uniform Electron Gas and sp-Bonded Metals . 109 // 5.1 The Electron Gas 109 // 5.2 Noninteracting and Hartree-Fock Approximations 111 // 5.3 Correlation Hole and Energy 117 // 5.4 Binding in sp-Bonded Metals 121 // 5.5 Excitations and the Lindhard Dielectric Function 122 // Exercises 126 // Part II Density Functional Theory // 6 Density Functional Theory: Foundations 129 // 6.1 Overview 129 // 6.2 Thomas-Fermi-Dirac Approximation 130 // 6.3 The Hohenberg-Kohn Theorems 131 // 6.4 Constrained Search Formulation of DFT 135 // 6.5 Extensions

of Hohenberg-Kohn Theorems 137 // 6.6 Intricacies of Exact Density Functional Theory 139 // 6.7 Difficulties in Proceeding from the Density 141 // Exercises 143 // Contents // ix // 7 The Kohn-Sham Auxiliary System 145 // 7.1 Replacing One Problem with Another 145 // 7.2 The Kohn-Sham Variational Equations 148 // 7.3 Solution of the Self-Consistent Coupled Kohn-Sham Equations 150 // 7.4 Achieving Self-Consistency 157 // 7.5 Force and Stress 160 // 7.6 Interpretation of the Exchange-Correlation Potential Vxc 161 // 7.7 Meaning of the Eigenvalues 162 // 7.8 Intricacies of Exact Kohn-Sham Theory 163 // 7.9 Time-Dependent Density Functional Theory 166 // 7.10 Other Generalizations of the Kohn-Sham Approach 167 // Exercises 168 // 8 Functionals for Exchange and Correlation I 171 // 8.1 Overview 171 // 8.2 Exc and the Exchange-Correlation Hole 172 // 8.3 Local (Spin) Density Approximation (LSDA) 174 // 8.4 How Can the Local Approximation Possibly Work As Well As It Does? 175 // 8.5 Generalized-Gradient Approximations (GGAs) 179 // 8.6 LDA and GGA Expressions for the Potential V Xc(r) 183 // 8.7 Average and Weighted Density Formulations: ADA and WDA 185 // 8.8 Functionals Fitted to Databases 185 // Exercises 186 // 9 Functionals for Exchange and Correlation II 188 // 9.1 Beyond the Local Density and Generalized Gradient Approximations 188 // 9.2 Generalized Kohn-Sham and Bandgaps 189 // 9.3 Hybrid Functionals and Range Separation 191 // 9.4 Functionals of the Kinetic Energy Density:

Meta-GGAs 195 // 9.5 Optimized Effective Potential 197 // 9.6 Localized-Orbital Approaches: SIC and DFT+U 199 // 9.7 Functionals Derived from Response Functions 203 // 9.8 Nonlocal Functionals for van der Waals Dispersion Interactions 205 // 9.9 Modified Becke-Johnson Functional for Vxc 209 // 9.10 Comparison of Functionals 209 // Exercises 213 // Part III // Important Preliminaries on Atoms // 10 Electronic Structure of Atoms 215 // 10.1 One-Electron Radial Schrödinger Equation 215 // 10.2 Independent-Particle Equations: Spherical Potentials 217 // 10.3 Spin-Orbit Interaction 219 // 10.4 Open-Shell Atoms: Nonspherical Potentials 219 // X // Contents // 10.5 Example of Atomic States: Transition Elements 221 // 10.6 Delta-SCF: Electron Addition, Removal, and Interaction Energies 224 // 10.7 Atomic Sphere Approximation in Solids 225 // Exercises 228 // 11 Pseudopotentials 230 // 11.1 Scattering Amplitudes and Pseudopotentials 230 // 11.2 Orthogonalized Plane Waves (OPWs) and Pseudopotentials 233 // 11.3 Model Ion Potentials 237 // 11.4 Norm-Conserving Pseudopotentials (NCPPs) 238 // 11.5 Generation of /-Dependent Norm-Conserving Pseudopotentials 241 // 11.6 Unscreening and Core Corrections 245 // 11.7 Transferability and Hardness 246 // 11.8 Separable Pseudopotential Operators and Projectors 247 // 11.9 Extended Norm Conservation: Beyond the Linear Regime 248 // 11.10 Optimized Norm-Conserving Potentials 249 // 11.11 Ultrasoft Pseudopotentials 250 // 11.12 Projector Augmented Waves

(PAWs): Keeping the Full Wavefunction 252 // 11.13 Additional Topics 255 // Exercises 256 // Part IV Determination of Electronic Structure: The Basic Methods // Overview of Chapters 12-18 259 // 12 Plane Waves and Grids: Basics 262 // 12.1 The Independent-Particle Schrodinger Equation in a Plane Wave Basis 262 // 12.2 Bloch Theorem and Electron Bands 264 // 12.3 Nearly-Free-Electron Approximation 265 // 12.4 Form Factors and Structure Factors 267 // 12.5 Approximate Atomic-Like Potentials 269 // 12.6 Empirical Pseudopotential Method (EPM) 270 // 12.7 Calculation of Electron Density: Introduction of Grids 272 // 12.8 Real-Space Methods I: Finite Difference and Discontinuous // Galerikin Methods 274 // 12.9 Real-Space Methods II: Multiresolution Methods 277 // Exercises 280 // 13 Plane Waves and Real-Space Methods: Full Calculations 283 // 13.1 Ab initio Pseudopotential Method 284 // 13.2 Approach to Self-Consistency and Dielectric Screening 286 // 13.3 Projector Augmented Waves (PAWs) 287 // 13.4 Hybrid Functionals and Hartree-Fock in Plane Wave Methods 288 // 13.5 Supercells: Surfaces, Interfaces, Molecular Dynamics 289 // 13.6 Clusters and Molecules 292 // Contents xi // 13.7 Applications of Plane Wave and Grid Methods 292 // Exercises 293 // 14 Localized Orbitals: Tight-Binding 295 // 14.1 Localized Atom-Centered Orbitals 296 // 14.2 Matrix Elements with Atomic-Like Orbitals 297 // 14.3 Spin-Orbit Interaction 301 // 14.4 Slater-Koster Two-Center Approximation 302 // 14.5 Tight-Binding

Bands: Example of a Single s Band 303 // 14.6 Two-Band Models 305 // 14.7 Graphene 306 // 14.8 Nanotubes 308 // 14.9 Square Lattice and Cu02 Planes 310 // 14.10 Semiconductors and Transition Metals 311 // 14.11 Total Energy, Force, and Stress in Tight-Binding 312 // 14.12 Transferability: Nonorthogonality and Environment Dependence 315 // Exercises 317 // 15 Localized Orbitals: Full Calculations 320 // 15.1 Solution of Kohn-Sham Equations in Localized Bases 320 // 15.2 Analytic Basis Functions: Gaussians 322 // 15.3 Gaussian Methods: Ground-State and Excitation Energies 324 // 15.4 Numerical Orbitals 324 // 15.5 Localized Orbitals: Total Energy, Force, and Stress 327 // 15.6 Applications of Numerical Local Orbitals 329 // 15.7 Green’s Function and Recursion Methods 329 // 15.8 Mixed Basis 330 // Exercises 331 // 16 Augmented Functions: APW, KKR, MTO 332 // 16.1 Augmented Plane Waves (APWs) and "Muffin Tins” 332 // 16.2 Solving APW Equations: Examples 337 // 16.3 The KKR or Multiple-Scattering Theory (MST) Method 342 // 16.4 Alloys and the Coherent Potential Approximation (CPA) 349 // 16.5 Muffm-Tin Orbitals (MTOs) 350 // 16.6 Canonical Bands 352 // 16.7 Localized "Tight-Binding,” MTO, and KKR Formulations 358 // 16.8 Total Energy, Force, and Pressure in Augmented Methods 360 // Exercises 362 // 17 Augmented Functions: Linear Methods 365 // 17.1 Linearization of Equations and Linear Methods 365 // 17.2 Energy Derivative of the Wavefunction: y and y 366 // 17.3

General Form of Linearized Equations 368 // 17.4 Linearized Augmented Plane Waves (LAPWs) 370 // Contents // • • // XII // 17.5 Applications of the LAPW Method 372 // 17.6 Linear Muffin-Tin Orbital (LMTO) Method 375 // 17.7 Tight-Binding Formulation 379 // 17.8 Applications of the LMTO Method 379 // 17.9 Beyond Linear Methods: NMTO 381 // 17.10 Full Potential in Augmented Methods 383 // Exercises 385 // 18 Locality and Linear-Scaling 0(A) Methods 386 // 18.1 What Is the Problem? 386 // 18.2 Locality in Many-Body Quantum Systems 388 // 18.3 Building the Hamiltonian 390 // 18.4 Solution of Equations: Nonvariational Methods 391 // 18.5 Variational Density Matrix Methods 400 // 18.6 Variational (Generalized) Wannier Function Methods 402 // 18.7 Linear-Scaling Self-Consistent Density Functional Calculations 405 // 18.8 Factorized Density Matrix for Large Basis Sets 406 // 18.9 Combining the Methods 407 // Exercises 408 // Part V From Electronic Structure to Properties of Matter // 19 Quantum Molecular Dynamics (QMD) 411 // 19.1 Molecular Dynamics (MD): Forces from the Electrons 411 // 19.2 Born-Oppenheimer Molecular Dynamics 413 // 19.3 Car-Parrinello Unified Algorithm for Electrons and Ions 414 // 19.4 Expressions for Plane Waves 418 // 19.5 Non-self-consistent QMD Methods 419 // 19.6 Examples of Simulations 419 // Exercises 424 // 20 Response Functions: Phonons and Magnons 427 // 20.1 Lattice Dynamics from Electronic Structure Theory 427 // 20.2 The Direct Approach: "Frozen

Phonons,” Magnons 430 // 20.3 Phonons and Density Response Functions 433 // 20.4 Green’s Function Formulation 435 // 20.5 Variational Expressions 436 // 20.6 Periodic Perturbations and Phonon Dispersion Curves 438 // 20.7 Dielectric Response Functions, Effective Charges 439 // 20.8 Electron-Phonon Interactions and Superconductivity 441 // 20.9 Magnons and Spin Response Functions 442 // Exercises 444 // 21 Excitation Spectra and Optical Properties 446 // 21.1 Overview 446 // 21.2 Time-Dependent Density Functional Theory (TDDFT) 447 // Contents // w // XIII // 21.3 Dielectric Response for Noninteracting Particles 448 // 21.4 Time-Dependent DFT and Linear Response 450 // 21.5 Time-Dependent Density-Functional Perturbation Theory 451 // 21.6 Explicit Real-Time Calculations 452 // 21.7 Optical Properties of Molecules and Clusters 454 // 21.8 Optical Properties of Crystals 459 // 21.9 Beyond the Adiabatic Approximation 463 // Exercises 464 // 22 Surfaces, Interfaces, and Lower-Dimensional Systems 465 // 22.1 Overview 465 // 22.2 Potential at a Surface or Interface 466 // 22.3 Surface States: Tamm and Shockley 467 // 22.4 Shockley States on Metals: Gold (111) Surface 470 // 22.5 Surface States on Semiconductors 471 // 22.6 Interfaces: Semiconductors 472 // 22.7 Interfaces: Oxides 474 // 22.8 Layer Materials 477 // 22.9 One-Dimensional Systems 478 // Exercises 479 // 23 Wannier Functions 481 // 23.1 Definition and Properties 481 // 23.2 Maximally Projected Wannier Functions

485 // 23.3 Maximally Localized Wannier Functions 487 // 23.4 Nonorthogonal Localized Functions 491 // 23.5 Wannier Functions for Entangled Bands 492 // 23.6 Hybrid Wannier Functions 494 // 23.7 Applications 495 // Exercises 496 // 24 Polarization, Localization, and Berry Phases 499 // 24.1 Overview 499 // 24.2 Polarization: The Fundamental Difficulty 501 // 24.3 Geometric Berry Phase Theory of Polarization 505 // 24.4 Relation to Centers of Wannier Functions 508 // 24.5 Calculation of Polarization in Crystals 509 // 24.6 Localization: A Rigorous Measure 510 // 24.7 The Thouless Quantized Particle Pump 512 // 24.8 Polarization Lattice . 513 // Exercises 514 // xiv // Contents // Part VI Electronic Structure and Topology // • if // 25 Topology of the Electronic Structure of a Crystal: Introduction 517 // 25.1 Introduction 517 // 25.2 Topology of What? 519 // 25.3 Bulk-Boundary Correspondence 520 // 25.4 Berry Phase and Topology for Bloch States in the Brillouin Zone 521 // 25.5 Berry Flux and Chern Numbers: Winding of the Berry Phase 524 // 25.6 Time-Reversal Symmetry and Topology of the Electronic System 526 // 25.7 Surface States and the Relation to the Quantum Hall Effect 527 // 25.8 Wannier Functions and Topology 528 // 25.9 Topological Quantum Chemistry 529 // 25.10 Majorana Modes 529 // Exercises 530 // ’ s, . gieguoisig olloos2 I*_ // 26 Two-Band Models: Berry Phase, Winding, and Topology 531 // 26.1 General Formulation for Two Bands 531 // 26.2 Two-Band Models in

One-Space Dimension 533 // 26.3 Shockley Transition in the Bulk Band Structure and Surface States 535 // 26.4 Winding of the Hamiltonian in One Dimension: Berry Phase // and the Shockley Transition 537 // 26.5 Winding of the Berry Phase in Two Dimensions: Chern Numbers // and Topological Transitions 539 // 26.6 The Thouless Quantized Particle Pump 541 // 26.7 Graphene Nanoribbons and the Two-Site Model 543 // Exercises 545 // 27 Topological Insulators I: Two Dimensions 547 // 27.1 Two Dimensions: sp2 Models 548 // 27.2 Chern Insulator and Anomalous Quantum Hall Effect 550 // 27.3 Spin-Orbit Interaction and the Diagonal Approximation 552 // 27.4 Topological Insulators and the Z2 Topological Invariant 554 // 27.5 Example of a Topological Insulator on a Square Lattice 557 // 27.6 From Chains to Planes: Example of a Topological Transition 560 // 27.7 Hg/CdTe Quantum Well Structures 561 // 27.8 Graphene and the Two-Site Model 563 // 27.9 Honeycomb Lattice Model with Large Spin-Orbit Interaction 567 // Exercises 567 // 28 Topological Insulators II: Three Dimensions 569 // 28.1 Weak and Strong Topological Insulators in Three Dimensions: // Four Topological Invariants 569 // 28.2 Tight-Binding Example in 3D 572 // 28.3 Normal and Topological Insulators in Three Dimensions: // Sb2Se3 and BÍ2Se3 573 // Contents // xv // 28.4 Weyl and Dirac Semimetals 575 // 28.5 Fermi Arcs 578 // Exercises 580 // Part VII Appendices // Appendix A Functional Equations 581 // A.I Basic Definitions and Variational

Equations 581 // A.2 Functionals in Density Functional Theory Including Gradients 582 // Exercises 583 // Appendix B LSDA and GGA Functionals 584 // B.1 Local Spin Density Approximation (LSDA) 584 // B.2 Generalized-Gradient Approximation (GGAs) 585 // B.3 GGAs: Explicit PBE Form 585 // Appendix C Adiabatic Approximation 587 // C.I General Formulation 587 // C.2 Electron-Phonon Interactions 589 // Exercises 589 // Appendix D Perturbation Theory, Response Functions, and Green’s Functions 590 // D.l Perturbation Theory 590 // D.2 Static Response Functions 591 // D.3 Response Functions in Self-Consistent Field Theories 592 // D.4 Dynamic Response and Kramers-Kronig Relations 593 // D.5 Green’s Functions 596 // D.6 The "2n 4-1 Theorem” 597 // Exercises 599 // Appendix E Dielectric Functions and Optical Properties 600 // E. 1 Electromagnetic Waves in Matter 600 // E.2 Conductivity and Dielectric Tensors 602 // E.3 The/Sum Rule 602 // E.4 Scalar Longitudinal Dielectric Functions 603 // E.5 Tensor Transverse Dielectric Functions 604 // E.6 Lattice Contributions to Dielectric Response 605 // Exercises 606 // Appendix F Coulomb Interactions in Extended Systems 607 // F.1 Basic Issues 607 // F.2 Point Charges in a Background: Ewald Sums 609 // F.3 Smeared Nuclei or Ions 613 // F.4 Energy Relative to Neutral Atoms 614 // F.5 Surface and Interface Dipoles 615 // xvi // Contents // F.6 Reducing Effects of Artificial Image Charges 616 // Exercises // Appendix G Stress from

Electronic Structure // G.l Macroscopic Stress and Strain // G.2 Stress from Two-Body Pair-Wise Forces // G.3 Expressions in Fourier Components // G.4 Internal Strain // Exercises // Appendix H Energy and Stress Densities // H. 1 Energy Density // H.2 Stress Density // H.3 Integrated Quantities // H.4 Electron Localization Function (ELF) // Exercises // Appendix 1 Alternative Force Expressions // 1.1 Variational Freedom and Forces // 1.2 Energy Differences // 1.3 Pressure // 1.4 Force and Stress // 1.5 Force in APW-Type Methods // Exercises // Appendix J Scattering and Phase Shifts // J. 1 Scattering and Phase Shifts for Spherical Potentials // 619 // 620 // 620 // 623 // 623 // 625 // 626 // 627 // 628 // 632 // 633 // 634 // 636 // 637 // 638 // 640 // 640 // 641 // 642 // 643 // 644 // 644 // Appendix K Useful Relations and Formulas 647 // K. 1 Bessel, Neumann, and Hankel Functions 647 // K.2 Spherical Harmonics and Legendre Polynomials 648 // K.3 Real Spherical Harmonics 649 // K.4 Clebsch-Gordon and Gaunt Coefficients 649 // K.5 Chebyshev Polynomials 650 // Appendix L Numerical Methods 651 // L.I Numerical Integration and the Numerov Method 651 // L .2 Steepest Descent 652 // L .3 Conjugate Gradient 653 // L .4 Quasi-Newton-Raphson Methods 655 // L.5 Pulay DIIS Full-Subspace Method 655 // L.6 Broyden Jacobian Update Methods 656 // L.7 Moments, Maximum Entropy, Kernel Polynomial Method, // and Random Vectors 657 // Exercises 659 // Contents XV // Appendix M Iterative Methods

in Electronic Structure 661 // M.l Why Use Iterative Methods? 661 // M.2 Simple Relaxation Algorithms 662 // M.3 Preconditioning 663 // M.4 Iterative (Krylov) Subspaces 664 // M.5 The Lanczos Algorithm and Recursion 665 // M.6 Davidson Algorithms 667 // M.7 Residual Minimization in the Subspace - RMM-DIIS 667 // M.8 Solution by Minimization of the Energy Functional 668 // M.9 Comparison/Combination of Methods: Minimization of // Residual or Energy 672 // M.10 Exponential Projection in Imaginary Time 672 // M.l l Algorithmic Complexity: Transforms and Sparse Hamiltonians 672 // Exercises 676 // Appendix N Two-Center Matrix Elements: Expressions for Arbitrary // Angular Momentum 1 677 // Appendix O Dirac Equation and Spin-Orbit Interaction 679 // O.1 The Dirac Equation 680 // 0.2 The Spin-Orbit Interaction in the Schrodinger Equation 681 // 0.3 Relativistic Equations and Calculation of the Spin-Orbit Interaction // in an Atom 683 // Appendix P Berry Phase, Curvature, and Chern Numbers 686 // P. 1 Overview 686 // P.2 Berry Phase and Berry Connection 687 // P.3 Berry Flux and Curvature 689 // P.4 Chern Number and Topology 691 // P.5 Adiabatic Evolution 692 // P.6 Aharonov-Bohm Effect 692 // P.7 Dirac Magnetic Monopoles and Chern Number 694 // Exercises 696 // Appendix Q Quantum Hall Effect and Edge Conductivity 697 // Q.1 Quantum Hall Effect and Topology 697 // Q.2 Nature of the Surface States in the QHE 698 // Appendix R Codes for Electronic Structure Calculations for Solids 701

// References 704 // Index 756