Francesco Sottile +1 more instructor

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Density Functional Theory

École Polytechnique

About this Course

The aim of this course is to give a thorough introduction to Density Functional Theory (DFT). DFT is today the most widely used method to study interacting electrons, and its applicability ranges from atoms to solid systems, from nuclei to quantum fluids.  
In this course, we introduce the most important concepts underlying DFT, its foundation, and basic ideas. We will in particular stress the features and reasons that lead DFT to become the dominant method for simulating quantum mechanical systems. 

The course is intended for students and researchers with knowledge of basic quantum mechanics. No experience in simulation or solid-state physics is required. We try to give a concise mathematical background when particular concepts are needed.

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Advanced Level

Approx. 25 hours to complete

English

Subtitles: English

Instructors

Francesco Sottile

Professor

Laboratoire des Solides Irradies (LSI)

41 Learners

1 Course

Lucia Reining

Professor

Theoretical Spectroscopy Group

41 Learners

1 Course

Shareable Certificate

Earn a Certificate upon completion

100% online

Start instantly and learn at your own schedule.

Flexible deadlines

Reset deadlines in accordance to your schedule.

Advanced Level

Approx. 25 hours to complete

English

Subtitles: English

Offered by

École Polytechnique

L’École polytechnique associe recherche, enseignement et innovation au meilleur niveau scientifique et technologique mondial pour répondre aux défis du XXIe siècle. En tête des écoles d’ingénieur françaises depuis plus de 200 ans, sa formation promeut une culture d’excellence scientifique pluridisciplinaire, ouverte dans une forte tradition humaniste.

Syllabus - What you will learn from this course

Week

Week 1

9 hours to complete

From the Many-Body problem to Density Functional Theory

This week will introduce the Density Functional Theory concepts. The week starts from an introduction to the many-body problem, and how things could be reformulated using the electron density. We will focus on observables, in particular those most directly related to the density. Finally we will discuss the Hohenberg-Kohn theorems. A little historical detour is taken at the end, where we will follow the footprints of the first steps to introduce a functional of the density in quantum problems.

9 hours to complete

13 videos (Total 99 min), 7 readings, 7 quizzes

13 videos

Introduction2m

The many-body problem6m

First objective: observables8m

Examples of observables4m

Observables in terms of compact quantities7m

Introduction to functionals and functionals of the density7m

Functional derivatives11m

Define the system: the external potential8m

Demonstration of the Hohenberg-Kohn theorem9m

HK theorem: some important aspects9m

The Thomas(-Fermi) approach8m

Thomas-Fermi as density functional6m

Beyond Thomas-Fermi6m

7 readings

Reading and References15m

on Notations3m

Recent literature on the many-body problem15m

Lecture Notes :: Functional derivatives1h 30m

A simple matrix (degenerate eigenvalue)10m

Extra references10m

Lecture Notes :: Thomas-Fermi approximation1h

7 practice exercises

The many-body wavefunction: a big beast30m

Functional strategy5m

Which strategy ?5m

Exercises on Derivatives1h

around Hohenberg-Kohn10m

Thomas-Fermi density functional1h

Beyond Thomas-Fermi1h

Week

Week 2

10 hours to complete

From density to the Kohn-Sham world

This second week introduces the Kohn-Sham world, a weird and very efficient world in which independent particles are governed by equations that give the exact density. At least, in principle, for the potential felt by the independent particle is unknown, as well as pathological. Again, a brief historical point of view ends the module.

10 hours to complete

10 videos (Total 90 min), 5 readings, 5 quizzes

10 videos

Functional of the density :: a summary14m

Auxiliary system 2.3 part19m

Auxiliary systems 2.3 part25m

The exchange-correlation term6m

The Kohn-Sham equations9m

Thoughts on the Kohn-Sham equations10m

The shape of the Kohn-Sham potential6m

How real is the Kohn-Sham world ?4m

The Hartree equations9m

From Thomas and Hartree to Walter Kohn13m

5 readings

on the functional of the density10m

References :: auxiliary systems1h

Lecture Notes for functional of the density10m

Lecture notes for Kohn-Sham derivation1h 10m

The Hartree equations1h

4 practice exercises

Simple auxiliary system45m

the exchange-correlation potential30m

the Koopmans' theorem30m

Hartree's other interests30m

Week

Week 3

7 hours to complete

Approximations and strategies

This third week tackles some more practical aspects of DFT. In particular the strategies for approximating the unknown exchange-correlation potential. Two other important points are the concept of band-gap and the practical scheme to solve the KS equations, via a self-consistent approach.

7 hours to complete

11 videos (Total 99 min), 8 readings, 1 quiz

11 videos

3.1 The universal functional4m

3.2 Strategies for approximations5m

The local density approximation (LDA)7m

Exact constraints and GGA9m

Meta-GGAs9m

Hybrids functionals11m

What is a band gap ?11m

Band gap? What about hybrids?7m

Solving the KS equation9m

Summary of the Mooc10m

Extra12m

8 readings

more about the Universal Functional1h

Extra references30m

Coupling constant integration and PBE1h

Lecture Notes for explicit and implicit functionals10m

Extra References10m

Lecture Notes for locality and non-locality20m

Lecture Notes for the derivative discontinuity30m

Solving KS equations (and another bit of history)1h

1 practice exercise

the SCF scheme30m

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Density Functional Theory

Density Functional Theory