Electronic Structure: Basic Theory and Practical Methods, 2nd Edition

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