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Electronic Structure

Book Description

The study of the electronic structure of materials is at a momentous stage, with the emergence of computational methods and theoretical approaches. Many properties of materials can now be determined directly from the fundamental equations for the electrons, providing insights into critical problems in physics, chemistry, and materials science. This book provides a unified exposition of the basic theory and methods of electronic structure, together with instructive examples of practical computational methods and real-world applications. Appropriate for both graduate students and practising scientists, this book describes the approach most widely used today, density functional theory, with emphasis upon understanding the ideas, practical methods and limitations. Many references are provided to original papers, pertinent reviews, and widely available books. Included in each chapter is a short list of the most relevant references and a set of exercises that reveal salient points and challenge the reader.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Dedication
  6. Contents
  7. Preface
  8. Acknowledgments
  9. Notation
  10. Part I: Overview and background topics
    1. 1. Introduction
      1. Summary
      2. 1.1 Quantum theory and the origins of electronic structure
      3. 1.2 Emergence of quantitative calculations
      4. 1.3 The greatest challenge: electron correlation
      5. 1.4 Recent developments
      6. Select further reading
    2. 2. Overview
      1. Summary
      2. 2.1 Electronic ground state: bonding and characteristic structures
      3. 2.2 Volume or pressure as the most fundamental variable
      4. 2.3 Elasticity: stress–strain relations
      5. 2.4 Magnetism and electron–electron interactions
      6. 2.5 Phonons and displacive phase transitions
      7. 2.6 Thermal properties: solids, liquids, and phase diagrams
      8. 2.7 Atomic motion: diffusion, reactions, and catalysis
      9. 2.8 Surfaces, interfaces, and defects
      10. 2.9 Nanomaterials: between molecules and condensed matter
      11. 2.10 Electronic excitations: bands and band gaps
      12. 2.11 Electronic excitations: heat capacity, conductivity, and optical spectra
      13. 2.12 Example of MgB2: bands, phonons, and superconductivity
      14. 2.13 The continuing challenge: electron correlation
      15. Select further reading
    3. 3. Theoretical background
      1. Summary
      2. 3.1 Basic equations for interacting electrons and nuclei
      3. 3.2 Coulomb interaction in condensed matter
      4. 3.3 Force and stress theorems
      5. 3.4 Statistical mechanics and the density matrix
      6. 3.5 Independent-electron approximations
      7. 3.6 Exchange and correlation
      8. 3.7 Perturbation theory and the “2n + 1 theorem”
      9. Select further reading
      10. Exercises
    4. 4. Periodic solids and electron bands
      1. Summary
      2. 4.1 Structures of crystals: lattice + basis
      3. 4.2 The reciprocal lattice and Brillouin zone
      4. 4.3 Excitations and the Bloch theorem
      5. 4.4 Time reversal and inversion symmetries
      6. 4.5 Point symmetries
      7. 4.6 Integration over the Brillouin zone and special points
      8. 4.7 Density of states
      9. Select further reading
      10. Exercises
    5. 5. Uniform electron gas and simple metals
      1. Summary
      2. 5.1 Non-interacting and Hartree–Fock approximations
      3. 5.2 The correlation hole and energy
      4. 5.3 Binding in sp-bonded metals
      5. 5.4 Excitations and the Lindhard dielectric function
      6. Select further reading
      7. Exercises
  11. Part II: Density functional theory
    1. 6. Density functional theory: foundations
      1. Summary
      2. 6.1 Thomas–Fermi–Dirac approximation: example of a functional
      3. 6.2 The Hohenberg–Kohn theorems
      4. 6.3 Constrained search formulation of density functional theory
      5. 6.4 Extensions of Hohenberg–Kohn theorems
      6. 6.5 Intricacies of exact density functional theory
      7. 6.6 Difficulties in proceeding from the density
      8. Select further reading
      9. Exercises
    2. 7. The Kohn-Sham auxiliary system
      1. Summary
      2. 7.1 Replacing one problem with another
      3. 7.2 The Kohn–Sham variational equations
      4. 7.3 Exc, Vxc, and the exchange–correlation hole
      5. 7.4 Meaning of the eigenvalues
      6. 7.5 Intricacies of exact Kohn–Sham theory
      7. 7.6 Time-dependent density functional theory
      8. 7.7 Other generalizations of the Kohn–Sham approach
      9. Select further reading
      10. Exercises
    3. 8. Functionals for exchange and correlation
      1. Summary
      2. 8.1 The local spin density approximation (LSDA)
      3. 8.2 Generalized-gradient approximations (GGAs)
      4. 8.3 LDA and GGA expressions for the potential Vσxc(r)
      5. 8.4 Non-collinear spin density
      6. 8.5 Non-local density formulations: ADA and WDA
      7. 8.6 Orbital-dependent functionals I: SIC and LDA + U
      8. 8.7 Orbital-dependent functionals II: OEP and EXX
      9. 8.8 Hybrid functionals
      10. 8.9 Tests of functionals
      11. Select further reading
      12. Exercises
    4. 9. Solving Kohn–Sham equations
      1. Summary
      2. 9.1 Self-consistent coupled Kohn–Sham equations
      3. 9.2 Total energy functionals
      4. 9.3 Achieving self-consistency
      5. 9.4 Force and stress
      6. Select further reading
      7. Exercises
  12. Part III: Important preliminaries on atoms
    1. 10. Electronic structure of atoms
      1. Summary
      2. 10.1 One-electron radial Schrodinger equation
      3. 10.2 Independent-particle equations: spherical potentials
      4. 10.3 Open-shell atoms: non-spherical potentials
      5. 10.4 Relativistic Dirac equation and spin–orbit interactions
      6. 10.5 Example of atomic states: transition elements
      7. 10.6 Delta-SCF: electron addition, removal, and interaction energies
      8. 10.7 Atomic sphere approximation in solids
      9. Select further reading
      10. Exercises
    2. 11. Pseudopotentials
      1. Summary
      2. 11.1 Scattering amplitudes and pseudopotentials
      3. 11.2 Orthogonalized plane waves (OPWs) and pseudopotentials
      4. 11.3 Model ion potentials
      5. 11.4 Norm-conserving pseudopotentials (NCPPs)
      6. 11.5 Generation of l-dependent norm-conserving pseudopotentials
      7. 11.6 Unscreening and core corrections
      8. 11.7 Transferability and hardness
      9. 11.8 Separable pseudopotential operators and projectors
      10. 11.9 Extended norm conservation: beyond the linear regime
      11. 11.10 Ultrasoft pseudopotentials
      12. 11.11 Projector augmented waves (PAWs): keeping the full wavefunction
      13. 11.12 Additional topics
      14. Select further reading
      15. Exercises
  13. Part IV: Determination of electronic structure: the three basic methods
    1. 12. Plane waves and grids: basics
      1. Summary
      2. 12.1 The independent-particle Schrodinger equation in a plane wave basis
      3. 12.2 The Bloch theorem and electron bands
      4. 12.3 Nearly-free-electron approximation
      5. 12.4 Form factors and structure factors
      6. 12.5 Approximate atomic-like potentials
      7. 12.6 Empirical pseudopotential method (EPM)
      8. 12.7 Calculation of density: introduction of grids
      9. 12.8 Real-space methods
      10. Select further reading
      11. Exercises
    2. 13. Plane waves and grids: full calculations
      1. Summary
      2. 13.1 “Ab initio” pseudopotential method
      3. 13.2 Projector augmented waves (PAWs)
      4. 13.3 Simple crystals: structures, bands,…
      5. 13.4 Supercells: surfaces, interfaces, phonons, defects
      6. 13.5 Clusters and molecules
      7. Select further reading
      8. Exercises
    3. 14. Localized orbitals: tight-binding
      1. Summary
      2. 14.1 Localized atom-centered orbitals
      3. 14.2 Matrix elements with atomic orbitals
      4. 14.3 Slater–Koster two-center approximation
      5. 14.4 Tight-binding bands: illustrative examples
      6. 14.5 Square lattice and CuO2 planes
      7. 14.6 Examples of bands: semiconductors and transition metals
      8. 14.7 Electronic states of nanotubes
      9. 14.8 Total energy, force, and stress in tight-binding
      10. 14.9 Transferability: non-orthogonality and environment dependence
      11. Select further reading
      12. Exercises
    4. 15. Localized orbitals: full calculations
      1. Summary
      2. 15.1 Solution of Kohn–Sham equations in localized bases
      3. 15.2 Analytic basis functions: gaussians
      4. 15.3 Gaussian methods: ground state and excitation energies
      5. 15.4 Numerical orbitals
      6. 15.5 Localized orbitals: total energy, force, and stress
      7. 15.6 Applications of numerical local orbitals
      8. 15.7 Green’s function and recursion methods
      9. 15.8 Mixed basis
      10. Select further reading
      11. Exercises
    5. 16. Augmented functions: APW, KKR, MTO
      1. Summary
      2. 16.1 Augmented plane waves (APWs) and “muffin tins”
      3. 16.2 Solving APW equations: examples
      4. 16.3 The KKR or multiple-scattering theory (MST) method
      5. 16.4 Alloys and the coherent potential approximation (CPA)
      6. 16.5 Muffin-tin orbitals (MTOs)
      7. 16.6 Canonical bands
      8. 16.7 Localized “tight-binding” MTO and KKR formulations
      9. 16.8 Total energy, force, and pressure in augmented methods
      10. Select further reading
      11. Exercises
    6. 17. Augmented functions: linear methods
      1. Summary
      2. 17.1 Energy derivative of the wavefunction: ψ and ψ
      3. 17.2 General form of linearized equations
      4. 17.3 Linearized augmented plane waves (LAPWs)
      5. 17.4 Applications of the LAPW method
      6. 17.5 Linear muffin-tin orbital (LMTO) method
      7. 17.6 “Ab initio” tight-binding
      8. 17.7 Applications of the LMTO method
      9. 17.8 Beyond linear methods: NMTO
      10. 17.9 Full potential in augmented methods
      11. Select further reading
      12. Exercises
  14. Part V: Predicting properties of matter from electronic structure – recent developments
    1. 18. Quantum molecular dynamics (QMD)
      1. Summary
      2. 18.1 Molecular dynamics (MD): forces from the electrons
      3. 18.2 Car–Parrinello unified algorithm for electrons and ions
      4. 18.3 Expressions for plane waves
      5. 18.4 Alternative approaches to density functional QMD
      6. 18.5 Non-self-consistent QMD methods
      7. 18.6 Examples of simulations
      8. Select further reading
      9. Exercises
    2. 19. Response functions: phonons, magnons,…
      1. Summary
      2. 19.1 Lattice dynamics from electronic structure theory
      3. 19.2 The direct approach: “frozen phonons,” magnons,…
      4. 19.3 Phonons and density response functions
      5. 19.4 Green’s function formulation
      6. 19.5 Variational expressions
      7. 19.6 Periodic perturbations and phonon dispersion curves
      8. 19.7 Dielectric response functions, effective charges,…
      9. 19.8 Electron–phonon interactions and superconductivity
      10. 19.9 Magnons and spin response functions
      11. Select further reading
      12. Exercises
    3. 20. Excitation spectra and optical properties
      1. Summary
      2. 20.1 Dielectric response for non-interacting particles
      3. 20.2 Time-dependent density functional theory and linear response
      4. 20.3 Variational Green’s function methods for dynamical linear response
      5. 20.4 Explicit real-time calculations
      6. 20.5 Beyond the adiabatic local approximation
      7. Select further reading
      8. Exercises
    4. 21. Wannier functions
      1. Summary
      2. 21.1 Definition and properties
      3. 21.2 “Maximally projected” Wannier functions
      4. 21.3 Maximally localized Wannier functions
      5. 21.4 Non-orthogonal localized functions
      6. 21.5 Wannier functions for “entangled bands”
      7. Select further reading
      8. Exercises
    5. 22. Polarization, localization, and Berry’s phases
      1. Summary
      2. 22.1 Polarization: the fundamental difficulty
      3. 22.2 Geometric Berry’s phase theory of polarization
      4. 22.3 Relation to centers of Wannier functions
      5. 22.4 Calculation of polarization in crystals
      6. 22.5 Localization: a rigorous measure
      7. 22.6 Geometric Berry’s phase theory of spin waves
      8. Select further reading
      9. Exercises
    6. 23. Locality and linear scaling O(N) methods
      1. Summary
      2. 23.1 Locality and linear scaling in many-particle quantum systems
      3. 23.2 Building the hamiltonian
      4. 23.3 Solution of equations: non-variational methods
      5. 23.4 Variational density matrix methods
      6. 23.5 Variational (generalized) Wannier function methods
      7. 23.6 Linear-scaling self-consistent density functional calculations
      8. 23.7 Factorized density matrix for large basis sets
      9. 23.8 Combining the methods
      10. Select further reading
      11. Exercises
    7. 24. Where to find more
  15. Appendix A: Functional equations
    1. Summary
    2. A.1 Basic definitions and variational equations
    3. A.2 Functionals in density functional theory including gradients
    4. Select further reading
    5. Exercises
  16. Appendix B: LSDA and GGA functionals
    1. Summary
    2. B.1 Local spin density approximation (LSDA)
    3. B.2 Generalized gradient approximation (GGAs)
    4. B.3 GGAs: explicit PBE form
    5. Select further reading
  17. Appendix C: Adiabatic approximation
    1. Summary
    2. C.1 General formulation
    3. C.2 Electron–phonon interactions
    4. Select further reading
    5. Exercises
  18. Appendix D: Response functions and Green’s functions
    1. Summary
    2. D.1 Static response functions
    3. D.2 Response functions in self-consistent field theories
    4. D.3 Dynamic response and Kramers–Kronig relations
    5. D.4 Green’s functions
    6. Select further reading
    7. Exercises
  19. Appendix E: Dielectric functions and optical properties
    1. Summary
    2. E.1 Electromagnetic waves in matter
    3. E.2 Conductivity and dielectric tensors
    4. E.3 The f sum rule
    5. E.4 Scalar longitudinal dielectric functions
    6. E.5 Tensor transverse dielectric functions
    7. E.6 Lattice contributions to dielectric response
    8. Select further reading
    9. Exercises
  20. Appendix F: Coulomb interactions in extended systems
    1. Summary
    2. F.1 Basic issues
    3. F.2 Point charges in a background: Ewald sums
    4. F.3 Smeared nuclei or ions
    5. F.4 Energy relative to neutral atoms
    6. F.5 Surface and interface dipoles
    7. F.6 Reducing effects of artificial image charges
    8. Select further reading
    9. Exercises
  21. Appendix G: Stress from electronic structure
    1. Summary
    2. G.1 Macroscopic stress and strain
    3. G.2 Stress from two-body pair-wise forces
    4. G.3 Expressions in Fourier components
    5. G.4 Internal strain
    6. Select further reading
    7. Exercises
  22. Appendix H: Energy and stress densities
    1. Summary
    2. H.1 Energy density
    3. H.2 Stress density
    4. H.3 Applications
    5. Select further reading
    6. Exercises
  23. Appendix I: Alternative force expressions
    1. Summary
    2. I.1 Variational freedom and forces
    3. I.2 Energy differences
    4. I.3 Pressure
    5. I.4 Force and stress
    6. I.5 Force in APW-type methods
    7. Select further reading
  24. Appendix J: Scattering and phase shifts
    1. Summary
    2. J.1 Scattering and phase shifts for spherical potentials
    3. Select further reading
  25. Appendix K: Useful relations and formulas
    1. Summary
    2. K.1 Bessel, Neumann, and Hankel functions
    3. K.2 Spherical harmonics and Legendre polynomials
    4. K.3 Real spherical harmonics
    5. K.4 Clebsch–Gordon and Gaunt coefficients
    6. K.5 Chebyshev polynomials
  26. Appendix L: Numerical methods
    1. Summary
    2. L.1 Numerical integration and the Numerov method
    3. L.2 Steepest descent
    4. L.3 Conjugate gradient
    5. L.4 Quasi-Newton–Raphson methods
    6. L.5 Pulay DIIS full-subspace method
    7. L.6 Broyden Jacobian update methods
    8. L.7 Moments, maximum entropy, kernel polynomial method, and random vectors
    9. Select further reading
    10. Exercises
  27. Appendix M: Iterative methods in electronic structure
    1. Summary
    2. M.1 Why use iterative methods?
    3. M.2 Simple relaxation algorithms
    4. M.3 Preconditioning
    5. M.4 Iterative (Krylov) subspaces
    6. M.5 The Lanczos algorithm and recursion
    7. M.6 Davidson algorithms
    8. M.7 Residual minimization in the subspace – RMM–DIIS
    9. M.8 Solution by minimization of the energy functional
    10. M.9 Comparison/combination of methods: minimization of residual or energy
    11. M.10 Exponential projection in imaginary time
    12. M.11 Algorithmic complexity: transforms and sparse hamiltonians
    13. Select further reading
    14. Exercises
  28. Appendix N: Code for empirical pseudopotential and tight-binding
    1. N.1 Calculations of eigenstates: modules common to all methods
    2. N.2 Plane wave empirical pseudopotential method (EPM)
    3. N.3 Slater–Koster tight-binding (TB) method
    4. N.4 Sample input file for TBPW
    5. N.5 Two-center matrix elements: expressions for arbitrary angular momentum l
  29. Appendix O: Units and conversion factors
  30. References
  31. Index