Density Functional Theory

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Density Functional Theory

Instructors: Francesco Sottile

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3 modules

Gain insight into a topic and learn the fundamentals.

Advanced level

Recommended experience

3 weeks to complete

at 10 hours a week

Flexible schedule

Learn at your own pace

3 modules

Gain insight into a topic and learn the fundamentals.

Advanced level

Recommended experience

3 weeks to complete

at 10 hours a week

Flexible schedule

Learn at your own pace

What you'll learn

Foundation (mathematical and historical) of DFT, approximation strategies.
Quality and accuracy of different approximations, practical procedure to solve the equations.
Ready to be operative and use DFT for your own research

Skills you'll gain

Details to know

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Assessments

12 assignments¹

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Taught in English

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There are 3 modules in 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.

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.

What's included

13 videos7 readings7 assignments

13 videosTotal 99 minutes

Introduction3 minutes
The many-body problem7 minutes
First objective: observables8 minutes
Examples of observables5 minutes
Observables in terms of compact quantities8 minutes
Introduction to functionals and functionals of the density7 minutes
Functional derivatives11 minutes
Define the system: the external potential8 minutes
Demonstration of the Hohenberg-Kohn theorem9 minutes
HK theorem: some important aspects10 minutes
The Thomas(-Fermi) approach9 minutes
Thomas-Fermi as density functional7 minutes
Beyond Thomas-Fermi7 minutes

7 readingsTotal 203 minutes

Reading and References15 minutes
on Notations3 minutes
Recent literature on the many-body problem15 minutes
Lecture Notes :: Functional derivatives90 minutes
A simple matrix (degenerate eigenvalue)10 minutes
Extra references10 minutes
Lecture Notes :: Thomas-Fermi approximation60 minutes

7 assignmentsTotal 230 minutes

Which strategy ?5 minutes
around Hohenberg-Kohn10 minutes
Thomas-Fermi density functional60 minutes
The many-body wavefunction: a big beast30 minutes
Functional strategy5 minutes
Exercises on Derivatives60 minutes
Beyond Thomas-Fermi60 minutes

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.

What's included

10 videos5 readings4 assignments1 peer review

10 videosTotal 90 minutes

Functional of the density :: a summary14 minutes
Auxiliary system 2.3 part19 minutes
Auxiliary systems 2.3 part25 minutes
The exchange-correlation term7 minutes
The Kohn-Sham equations10 minutes
Thoughts on the Kohn-Sham equations11 minutes
The shape of the Kohn-Sham potential6 minutes
How real is the Kohn-Sham world ?5 minutes
The Hartree equations10 minutes
From Thomas and Hartree to Walter Kohn13 minutes

5 readingsTotal 210 minutes

on the functional of the density10 minutes
References :: auxiliary systems60 minutes
Lecture Notes for functional of the density10 minutes
Lecture notes for Kohn-Sham derivation70 minutes
The Hartree equations60 minutes

4 assignmentsTotal 135 minutes

Simple auxiliary system45 minutes
the Koopmans' theorem30 minutes
Hartree's other interests30 minutes
the exchange-correlation potential30 minutes

1 peer reviewTotal 150 minutes

Kohn-Sham equations :: derivation and total energy150 minutes

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.

What's included

11 videos8 readings1 assignment

11 videosTotal 99 minutes

3.1 The universal functional4 minutes
3.2 Strategies for approximations6 minutes
The local density approximation (LDA)7 minutes
Exact constraints and GGA10 minutes
Meta-GGAs10 minutes
Hybrids functionals11 minutes
What is a band gap ?11 minutes
Band gap? What about hybrids?8 minutes
Solving the KS equation9 minutes
Summary of the Mooc10 minutes
Extra12 minutes

8 readingsTotal 280 minutes

more about the Universal Functional60 minutes
Extra references30 minutes
Coupling constant integration and PBE60 minutes
Lecture Notes for explicit and implicit functionals10 minutes
Extra References10 minutes
Lecture Notes for locality and non-locality20 minutes
Lecture Notes for the derivative discontinuity30 minutes
Solving KS equations (and another bit of history)60 minutes

1 assignmentTotal 30 minutes

the SCF scheme30 minutes

Instructors

Francesco Sottile

École Polytechnique

1 Course21,982 learners

Offered by

École Polytechnique

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