Physics 250-06 “Advanced Electronic Structure”

Lecture 1. Theoretical Background

Contents:

1. Historical Overview.

2. Basic Equations for Interacting Electrons.

Overview.

Electronic structure as a field of condensed matter physics:

1920es:

Band Theory of Independent Electrons of Felix Bloch.Insulators, Semiconductors, Metals.

Emergence of Quantitative Calculations. Works of Hartree (self-consistent electrostatic potentials) and Fock (antisymmetrized determinant) on atoms.

1930es:

Method of Wigner and Seitz (1933) and electronic states of Na metal. Augmented plane waves of Slater (1937). Pseudopotentials by Fermi.

Overview.

1950es:

First calculations of electronic states by Herman, Callaway, Slater for atoms and crystals.

1960es:

Density Functional Theory by Hohenberg, Kohn, Sham

1970es:

Linear Methods of Band Theory for solving Schroedinger’s equation by Ole Andersen.

Overview.

1980es:

First self-consistent programs for electronic structurecalculations developed. Energy bands and properties of

manymaterials have been computed.

1990es:

Discovery of High-Temperature Superconductivity:

Phonons and electron phonon interactions, importance of

correlations in electronic structure.

Simulations of more complex materials, Car Parinellomolecular dynamics

Overview.

Current Research in Electronic Structure

Quantitative theories for correlated materials.

Quantitative theories for complex systems (nano, bio).

Overview.

Fundamental variables to study ground state properties:

DensityTotal EnergyVolume Pressure

Fundamental questions:

Nature of bondingEquations of statePhase transitions under pressureTheory of ElasticityTheory of Magnetism, FerroelectricityPhonons, MagnonsSurfaces, Interfaces, Defects.

Overview.

Fundamental variables to study excitations:

One-Electron Energy BandsWave Functions and transition matrix elements

Fundamental questions:

Angle Resolve PhotoemissionOptical SpectroscopyExcitons Core Level SpectroscopyTransport PropertiesSuperconductivity

Basic Equations for Interacting Electrons

Many Body Hamiltonian and Schroedinger’s equation

Ground State and Excited States

Hellmann-Feynman Theorem

Coulomb Interactions:

Hartree approximation and self-consistent theoryExchange and Hartree-Fock approximation.Koopmans’ theoremBeyond Hartree-Fock: correlation effects

Periodic Solids and Electron Bands

Crystal structures, primitive translations and basis vectors.

Brillouin zone, high symmetry directions

Bloch theorem, band of eigenvalues

Symmetry considerations, irreducible BZ.

Integration over BZ:

Special point method. Tetrahedron method.

Uniform Electron Gas and Simple Metals.

Model of uniform electron gas, rs and density as two parameters

Hartree-Fock approximation for eigenvalues.

Dielectric screening, Friedel oscillations

Hartree-Fock potential for uniform electron gas. Slater x-Alpha method as a prerequisite to DFT