Agenda

DFT and memory systems

Why collaborate with a DFT specialist?

Higher order corrections

Geometry optimization

AFLOW project

What is the dispersion of electrons?

Software packages for DFT

Density Functional Theory of clusters

Conclusion

DFT improves on Hartree-Fock

Equations and Definitions

The Schödinger functional

Hierarchy of DFTs

Local spin density approximation (LSDA)

Hamiltonian as a function of perturbation strength

B3LYP Functional

Hartree-Fock exchange at long range

Grid sensitivity

DFT is not good for metals with partially filled orbitals

How Quantum Mechanics Works

Multi-Dimensional Schrodinger Equation

Why do we need this much information?

Configuration space

What is DFT?

Non-interacting electrons

How to support this show

Could a black hole destroy the Moon?

Matryoshka Brain

Intro

Operators

Approximation: electron as point charge in field of other electrons

Self-consistency scheme

Reciprocal space and Fourier series

The Brillouin zone

Simulation of defects

Perovskite example

Relative energies

Conclusion

Density Functional Theory: Introduction and Applications

Density Functional Theory: Introduction and Applications

Overview

Computational Material Science

Microscopic Scale: Quantum Mechanics

Microscopic Scale: Quantum Mechanics

Microscopic Scale: Quantum Mechanics

Microscopic Scale: Quantum Mechanics

Overview

Density Functional Theory: Formulation and Implementation

Question: Have we made an approximation yet?

Density Functional Theory: Formulation and Implementation

Question: Have we made an approximation yet?

Density Functional Theory: Formulation and Implementation

Overview

Density Functional Theory: Applications

Density Functional Theory: Applications

Example I: Total-energy calculations and convergence

Example II: Bulk modulus

Example III: Electronic band structure

Example III: Electronic band structure

Summary

Intro

Aim of this lecture series

Schrodinger Equation

The Hadamard product

Electrons are identical particles

What is a functional?

Hamiltonian equation

The Gibbs equation

Single-electron Wavefunction

Conclusion

Introduction

Time series data from sound recordings

Julia notebook: Playing with sound - WAV files

Drawing waveforms

Combining (superposing) different frequencies

Julia: FFT function

Discrete Fourier Transform (DFT) vs Fast Fourier Transform (FFT)

Musical overtones: Magnitude of the FFT

Analyzing a sound file using the FFT

Defining the DFT mathematically

First term of the DFT

Equally-spaced points on unit circle in the complex plane

Idea of Fourier transform of a signal: walking around a circle

Adding complex numbers as adding vectors

Angle of DFT gives information about phase

Interpreting the second term of the DFT

General formula for DFT

Implementing the DFT in Julia

Julia: Array comprehension

Comparison of DFT with FFT results

Julia: isapprox for testing approximate equality

Pre-computing an array of powers

Julia: Modulo (%)

Julia: OffsetArray for zero-based indexing

DFT as polynomials

Introduction

Periodic Discrete Complex Exponential (The building block of dft)

The Main Idea of Discrete Fourier Transform

Discrete Fourier Transform (DFT) (Analysis Formula)

Inverse Discrete Fourier Transform (IDFT) (Synthesis Formula)

Conclusion

Lecture outline

Hamiltonian and wavefunction

The Hartree-Fock approximation

DFT approximated voltages

Why is DFT a 0K technique?

Correlation Factors and Transport Coefficients

Octahedral motifs

Cluster expansion and training data

Charge densities

Conclusion

Intro

Wavefunction in Hilbert space

Coulombic interaction between two electrons

Interaction between planets around the Sun

Wavefunction as a function of 6 variables

The variational principle

Fermion statistics

Coulomb functional

Self-consistent calculation

Conclusion