Copyright
Contents
Acknowledgments
Preface
List of Notation
Part I 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
2.15 Electronic Excitations and Optical Spectra
2.16 Topological Insulators
2.17 The Continuing Challenge: Electron Correlation
3 Theoretical Background
3.1 Basic Equations for Interacting Electrons and Nuclei
3.2 Coulomb Interaction in Condensed Matter
3.3 Force and Stress Theorems
3.4 Generalized Force Theorem and Coupling Constant Integration
3.5 Statistical Mechanics and the Density Matrix
3.6 Independent-Electron Approximations
3.7 Exchange and Correlation
Exercises
4 Periodic Solids and Electron Bands
4.1 Structures of Crystals: Lattice+ Basis
4.2 Reciprocal Lattice and Brillouin Zone
4.3 Excitations and the Bloch Theorem
4.4 Time-Reversal and Inversion Symmetries
4.5 Point Symmetries
4.6 Integration over the Brillouin Zone and Special Points
4.7 Density of States
Exercises
5 Uniform Electron Gas and sp-Bonded Metals
5.1 The Electron Gas
5.2 Noninteracting and Hartree–Fock Approximations
5.3 Correlation Hole and Energy
5.4 Binding insp-Bonded Metals
5.5 Excitations and theLindhard DielectricFunction
Exercises
Part II Density Functional Theory
6 Density Functional Theory: Foundations
6.1 Overview
6.2 Thomas–Fermi–Dirac Approximation
6.3 The Hohenberg–Kohn Theorems
6.4 Constrained Search Formulation of DFT
6.5 Extensions of Hohenberg–Kohn Theorems
6.6 Intricacies of Exact DensityFunctional Theory
6.7 Difficulties in Proceeding from the Density
Exercises
7 The Kohn–Sham Auxiliary System
7.1 Replacing One Problem with Another
7.2 The Kohn–Sham Variational Equations
7.3 Solution of the Self-Consistent Coupled Kohn–Sham Equations
7.4 Achieving Self-Consistency
7.5 Force and Stress
7.6 Interpretation ofthe Exchange–Correlation Potential Vxc
7.7 Meaning ofthe Eigenvalues
7.8 Intricacies of Exact Kohn–Sham Theory
7.9 Time-Dependent Density Functional Theory
7.10 Other Generalizations of the Kohn–Sham Approach
Exercises
8 Functionals for Exchange and Correlation I
8.1 Overview
8.2 Exc and the Exchange–Correlation Hole
8.3 Local (Spin) Density Approximation (LSDA)
8.4 How Can the Local Approximation Possibly Work As Well As It Does?
8.5 Generalized-Gradient Approximations (GGAs)
8.6 LDA and GGA Expressions for the Potential V σxc(r)
8.7 Average and Weighted Density Formulations: ADA and WDA
8.8 Functionals Fitted to Databases
Exercises
9 Functionals for Exchange and Correlation II
9.1 Beyond the Local Density and Generalized Gradient Approximations
9.2 Generalized Kohn–Sham and Bandgaps
9.3 Hybrid Functionals and Range Separation
9.4 Functionals ofthe KineticEnergy Density: Meta-GGAs
9.5 Optimized Effective Potential
9.6 Localized-Orbital Approaches: SIC and DFT+U
9.7 Functionals Derived from Response Functions
9.8 Nonlocal Functionals for van der Waals Dispersion Interactions
9.9 Modified Becke–Johnson Functional forVxc
9.10 Comparison of Functionals
Exercises
Part III Important Preliminaries on Atoms
10 Electronic Structure of Atoms
10.1 One-Electron Radial Schrödinger Equation
10.2 Independent-Particle Equations: Spherical Potentials
10.3 Spin–Orbit Interaction
10.4 Open-Shell Atoms: Nonspherical Potentials
10.5 Example of Atomic States: Transition Elements
10.6 Delta-SCF: Electron Addition, Removal, and Interaction Energies
10.7 Atomic Sphere Approximation inSolids
Exercises
11 Pseudopotentials
11.1 Scattering Amplitudes and Pseudopotentials
11.2 Orthogonalized Plane Waves (OPWs) and Pseudopotentials
11.3 Model Ion Potentials
11.4 Norm-Conserving Pseudopotentials (NCPPs)
11.5 Generation of l-Dependent Norm-Conserving Pseudopotentials
11.6 Unscreening and Core Corrections
11.7 Transferability and Hardness
11.8 Separable Pseudopotential Operators and Projectors
11.9 Extended Norm Conservation: Beyond the Linear Regime
11.10 Optimized Norm-Conserving Potentials
11.11 Ultrasoft Pseudopotentials
11.12 Projector Augmented Waves (PAWs): Keeping the Full Wavefunction
11.13 Additional Topics
Exercises
Part IV Determination of Electronic Structure: The Basic Methods Overview of Chapters 12–18
12 Plane Waves and Grids: Basics
12.1 The Independent-Particle Schrödinger Equation in a Plane Wave Basis
12.2 Bloch Theorem and Electron Bands
12.3 Nearly-Free-Electron Approximation
12.4 Form Factors and Structure Factors
12.5 Approximate Atomic-Like Potentials
12.6 Empirical Pseudopotential Method (EPM)
12.7 Calculation of Electron Density: Introduction of Grids
12.8 Real-Space Methods I: Finite Difference and Discontinuous Galerikin Methods
12.9 Real-Space Methods II: Multiresolution Methods
Exercises
13 Plane Waves and Real-Space Methods: Full Calculations
13.1 Ab initio Pseudopotential Method
13.2 Approach toSelf-Consistency and DielectricScreening
13.3 Projector Augmented Waves (PAWs)
13.4 Hybrid Functionals and Hartree–Fock inPlane Wave Methods
13.5 Supercells: Surfaces, Interfaces, Molecular Dynamics
13.6 Clusters and Molecules
13.7 Applications of Plane Wave and GridMethods
Exercises
14 Localized Orbitals: Tight-Binding
14.1 Localized Atom-Centered Orbitals
14.2 Matrix Elements with Atomic-Like Orbitals
14.3 Spin–Orbit Interaction
14.4 Slater–Koster Two-Center Approximation
14.5 Tight-Binding Bands: Example of a Single s Band
14.6 Two-Band Models
14.7 Graphene
14.8 Nanotubes
14.9 Square Lattice and CuO2 Planes
14.10 Semiconductors and Transition Metals
14.11 Total Energy, Force, and StressinTight-Binding
14.12 Transferability: Nonorthogonality and Environment Dependence
Exercises
15 Localized Orbitals: Full Calculations
15.1 Solution of Kohn–Sham Equations in Localized Bases
15.2 Analytic Basis Functions: Gaussians
15.3 Gaussian Methods: Ground-State and Excitation Energies
15.4 Numerical Orbitals
15.5 Localized Orbitals: Total Energy, Force, and Stress
15.6 Applications of Numerical Local Orbitals
15.7 Green’s Function and Recursion Methods
15.8 Mixed Basis
Exercises
16 Augmented Functions: APW, KKR, MTO
16.1 Augmented Plane Waves (APWs) and “Muffin Tins”
16.2 Solving APW Equations: Examples
16.3 The KKR or Multiple-Scattering Theory (MST) Method
16.4 Alloys and the Coherent Potential Approximation (CPA)
16.5 Muffin-Tin Orbitals (MTOs)
16.6 Canonical Bands
16.7 Localized “Tight-Binding,” MTO, and KKR Formulations
16.8 Total Energy, Force, and Pressurein Augmented Methods
Exercises
17 Augmented Functions: Linear Methods
17.1 Linearization of Equations and Linear Methods
17.2 Energy Derivative of the Wavefunction: ψ and ψ˙
17.3 General Form of Linearized Equations
17.4 Linearized Augmented Plane Waves (LAPWs)
17.5 Applications of the LAPW Method
17.6 Linear Muffin-Tin Orbital(LMTO) Method
17.7 Tight-Binding Formulation
17.8 Applications of theLMTO Method
17.9 Beyond Linear Methods: NMTO
17.10 Full Potential in Augmented Methods
Exercises
18 Locality and Linear-Scaling O(N) Methods
18.1 What Is the Problem?
18.2 Locality in Many-Body Quantum Systems
18.3 Building the Hamiltonian
18.4 Solution of Equations: Nonvariational Methods
18.5 Variational Density Matrix Methods
18.6 Variational (Generalized) Wannier Function Methods
18.7 Linear-Scaling Self-Consistent Density Functional Calculations
18.8 Factorized Density Matrixfor Large Basis Sets
18.9 Combining the Methods
Exercises
Part V From Electronic Structure to Properties of Matter
19 Quantum Molecular Dynamics (QMD)
19.1 Molecular Dynamics (MD): Forces from the Electrons
19.2 Born-Oppenheimer Molecular Dynamics
19.3 Car–Parrinello Unified Algorithm for Electrons and Ions
19.4 Expressions for Plane Waves
19.5 Non-self-consistent QMD Methods
19.6 Examples of Simulations
Exercises
20 Response Functions: Phonons and Magnons
20.1 Lattice Dynamics from Electronic Structure Theory
20.2 The Direct Approach: “Frozen Phonons,” Magnons
20.3 Phonons and Density Response Functions
20.4 Green’s Function Formulation
20.5 Variational Expressions
20.6 Periodic Perturbations and Phonon Dispersion Curves
20.7 Dielectric Response Functions, Effective Charges
20.8 Electron–Phonon Interactions and Superconductivity
20.9 Magnons and Spin Response Functions
Exercises
21 Excitation Spectra and Optical Properties
21.1 Overview
21.2 Time-Dependent Density Functional Theory (TDDFT)
21.3 Dielectric Response for Noninteracting Particles
21.4 Time-Dependent DFT and Linear Response
21.5 Time-Dependent Density-Functional Perturbation Theory
21.6 Explicit Real-Time Calculations
21.7 Optical Properties of Molecules and Clusters
21.8 Optical Properties of Crystals
21.9 Beyond the Adiabatic Approximation
Exercises
22 Surfaces, Interfaces, and Lower-Dimensional Systems
22.1 Overview
22.2 Potential at a Surface or Interface
22.3 Surface States: Tamm and Shockley
22.4 Shockley States on Metals: Gold (111) Surface
22.5 Surface States on Semiconductors
22.6 Interfaces: Semiconductors
22.7 Interfaces: Oxides
22.8 Layer Materials
22.9 One-Dimensional Systems
Exercises
23 Wannier Functions
23.1 Definitionand Properties
23.2 Maximally Projected Wannier Functions
23.3 Maximally Localized Wannier Functions
23.4 Nonorthogonal Localized Functions
23.5 Wannier Functions for Entangled Bands
23.6 Hybrid Wannier Functions
23.7 Applications
Exercises
24 Polarization, Localization, and Berry Phases
24.1 Overview
24.2 Polarization: The Fundamental Difficulty
24.3 Geometric Berry Phase Theory of Polarization
24.4 Relation to Centers of Wannier Functions
24.5 Calculation of Polarization in Crystals
24.6 Localization: A Rigorous Measure
24.7 The Thouless Quantized Particle Pump
24.8 Polarization Lattice
Exercises
Part VI Electronic Structure and Topology
25 Topology of the Electronic Structure of a Crystal: Introduction
25.1 Introduction
25.2 Topology of What?
25.3 Bulk-Boundary Correspondence
25.4 Berry Phase and Topology for Bloch States inthe Brillouin Zone
25.5 Berry Flux and Chern Numbers: Winding of the Berry Phase
25.6 Time-Reversal Symmetry and Topology of the Electronic System
25.7 Surface States and the Relation to the Quantum Hall Effect
25.8 Wannier Functions and Topology
25.9 Topological Quantum Chemistry
25.10 Majorana Modes
Exercises
26 Two-Band Models: Berry Phase, Winding, and Topology
26.1 General Formulation for Two Bands
26.2 Two-Band Models in One-Space Dimension
26.3 Shockley Transition in the Bulk Band Structure and Surface States
26.4 Winding of the Hamiltonian in One Dimension: Berry Phase and the Shockley Transition
26.5 Winding of the Berry Phase inTwo Dimensions: Chern Numbers and Topological Transitions
26.6 The Thouless Quantized Particle Pump
26.7 Graphene Nanoribbons and the Two-Site Model
Exercises
27 Topological Insulators I: Two Dimensions
27.1 Two Dimensions: sp2 Models
27.2 Chern Insulator and Anomalous Quantum Hall Effect
27.3 Spin–Orbit Interaction and the Diagonal Approximation
27.4 Topological Insulators and the Z2 Topological Invariant
27.5 Example of aTopological Insulator on a Square Lattice
27.6 From Chains to Planes: Example of a Topological Transition
27.7 Hg/CdTe Quantum Well Structures
27.8 Graphene and the Two-Site Model
27.9 Honeycomb Lattice Model with Large Spin–Orbit Interaction
Exercises
28 Topological Insulators II: Three Dimensions
28.1 Weak and Strong Topological Insulators in Three Dimensions: Four Topological Invariants
28.2 Tight-Binding Example in3D
28.3 Normal and Topological Insulators in Three Dimensions: Sb2Se3 and Bi2Se3
28.4 Weyl and Dirac Semimetals
28.5 Fermi Arcs
Exercises
Part VII Appendices Appendix A Functional Equations
A.1 Basic Definitions and Variational Equations
A.2 Functionals in Density Functional Theory Including Gradients
Exercises
Appendix B LSDA and GGA Functionals
B.1 Local Spin Density Approximation (LSDA)
B.2 Generalized-Gradient Approximation (GGAs)
B.3 GGAs: Explicit PBE Form
Appendix C Adiabatic Approximation
C.1 General Formulation
C.2 Electron-Phonon Interactions
Exercises
Appendix D Perturbation Theory, Response Functions, and Green’s Functions
D.1 Perturbation Theory
D.2 Static Response Functions
D.3 Response Functions in Self-Consistent Field Theories
D.4 Dynamic Response and Kramers–Kronig Relations
D.5 Green’s Functions
D.6 The “2n + 1 Theorem”
Exercises
Appendix E Dielectric Functions and Optical Properties
E.1 Electromagnetic Waves in Matter
E.2 Conductivity and Dielectric Tensors
E.3 The f Sum Rule
E.4 Scalar Longitudinal Dielectric Functions
E.5 Tensor Transverse Dielectric Functions
E.6 Lattice Contributions to Dielectric Response
Exercises
Appendix F Coulomb Interactions in Extended Systems
F.1 Basic Issues
F.2 Point Charges in a Background: Ewald Sums
F.3 Smeared Nuclei orIons
F.4 Energy Relative to Neutral Atoms
F.5 Surface and Interface Dipoles
F.6 Reducing Effects of Artificial Image Charges
Exercises
Appendix G Stress from Electronic Structure
G.1 Macroscopic Stress and Strain
G.2 Stress fromTwo-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 I Alternative Force Expressions
I.1 Variational Freedom and Forces
I.2 Energy Differences
I.3 Pressure
I.4 Force and Stress
I.5 Force inAPW-Type Methods
Exercises
Appendix J Scattering and Phase Shifts
J.1 Scattering and Phase Shifts forSpherical Potentials
Appendix K Useful Relations and Formulas
K.1 Bessel, Neumann, and Hankel Functions
K.2 Spherical Harmonics and Legendre Polynomials
K.3 Real Spherical Harmonics
K.4 Clebsch–Gordon and Gaunt Coefficients
K.5 Chebyshev Polynomials
Appendix L Numerical Methods
L.1 Numerical Integration and the Numerov Method
L.2 Steepest Descent
L.3 Conjugate Gradient
L.4 Quasi-Newton–Raphson Methods
L.5 Pulay DIIS Full-Subspace Method
L.6 Broyden Jacobian Update Methods
L.7 Moments, Maximum Entropy, Kernel Polynomial Method, and Random Vectors
Exercises
Appendix M Iterative Methods in Electronic Structure
M.1 Why Use Iterative Methods?
M.2 Simple Relaxation Algorithms
M.3 Preconditioning
M.4 Iterative (Krylov) Subspaces
M.5 The Lanczos Algorithm and Recursion
M.6 Davidson Algorithms
M.7 Residual Minimization in the Subspace –RMM–DIIS
M.8 Solution by Minimization of the Energy Functional
M.9 Comparison/Combination of Methods: Minimization of Residual or Energy
M.10 Exponential Projection in Imaginary Time
M.11 Algorithmic Complexity: Transforms and Sparse Hamiltonians
Exercises
Appendix N Two-Center Matrix Elements: Expressions for Arbitrary Angular Momentum l
Appendix O Dirac Equation and Spin–Orbit Interaction
O.1 The DiracEquation
O.2 The Spin–Orbit Interaction intheSchrödinger Equation
O.3 Relativistic Equations and Calculation of the Spin–Orbit Interaction in an Atom
Appendix P Berry Phase, Curvature, and Chern Numbers
P.1 Overview
P.2 Berry Phase and Berry Connection
P.3 Berry Flux and Curvature
P.4 Chern Number and Topology
P.5 Adiabatic Evolution
P.6 Aharonov–Bohm Effect
P.7 Dirac Magnetic Monopoles and Chern Number
Exercises
Appendix Q Quantum Hall Effect and Edge Conductivity
Q.1 Quantum Hall Effect and Topology
Q.2 Nature of the Surface States in the QHE
Appendix R Codes for Electronic Structure Calculations for Solids
References
Index