Notes on Fundamentals and Applications of Density Functional Theory (DFT)

Introduction

Speaker: Astrid Markinson, PhD candidate in Material Science and Engineering.
Topic: Fundamentals and applications of Density Functional Theory (DFT).
Structure:
- Part 1: Introduction to fundamentals of DFT.
- Part 2: How to perform DFT calculations using software (VASP).

Computational material science utilizes computer simulations to explore material properties.
DFT is used for first-principle calculations without empirical parameters, based entirely on quantum mechanics.

Wave Function: Represents the quantum state of particles in a system.
Operator: A mathematical operation on variables, in quantum mechanics, operators correspond to observables.
Ground State: The most stable state of a system with the lowest energy.

Goal: Find the ground state of a set of particles by solving the many-body Schrödinger equation.
Born-Oppenheimer Approximation: Decouples the dynamics of slow nuclei from fast electrons, simplifying the problem.
The many-body problem becomes complex as the number of particles increases, necessitating DFT.

Focuses on electron density rather than wave function, reducing the problem to three spatial dimensions.
Hartree Theorem: States that the ground state energy is a unique functional of the electron density.
Energy Functional: Divided into known and unknown parts; the unknown part accounts for electron interaction (exchange-correlation functional).

Two fundamental theorems provide a basis for obtaining ground state electron density and energy functional.
Self-consistency in calculations is achieved by iterating density and energy calculations until convergence.

Crystal Definition: A periodic arrangement of atoms, with unique potentials affecting electrons.
Bloch Theorem: Describes electron wave functions in crystals, resulting in Bloch waves.
Key Concepts:
- Cutoff Energy: Defines the maximum kinetic energy of plane wave basis sets used in calculations.
- K-point Sampling: Sampling in reciprocal space is necessary for calculating properties; only the first Brillouin zone is considered.
- Pseudopotentials: Simplifies calculations by considering only valence electrons, treating inner electrons as part of an effective potential.
- Periodic Boundary Conditions: Used to model crystal structures, ensuring that calculations are representative of infinite systems.

VASP (Vienna Ab initio Simulation Package): A commercial software package widely used for DFT calculations.
Capable of modeling systems with periodic boundary conditions and utilizing pseudopotential methods.

INCAR: Input parameters for the calculation (e.g., energy cutoffs, convergence criteria).
POSCAR: Contains the geometry and arrangement of atoms in the simulation cell.
POTCAR: Information on the pseudopotentials and exchange-correlation functionals.
KPOINTS: Defines the mesh of k-points for sampling in the Brillouin zone.

Example: Barium Titanate (perovskite structure).
Define input files and parameters.
Conduct convergence testing for cutoff energy and k-points to ensure reliable results.

CONTCAR: Contains the relaxed structure after calculations.
OUTCAR: Comprehensive output file with details of electronic steps, forces, and energies.

DFT serves as a powerful tool for predicting material properties and investigating systems at an atomic level.
Calculating and understanding DFT is manageable with the proper tools and methodologies.
DFT should complement experimental approaches rather than completely replace them.

Suggested reading materials and VASP manuals for further understanding and reference.

These notes summarize the key points from the seminar on Density Functional Theory (DFT) presented by Astrid Markinson.