Density Functional Theory

Instructors: Francesco Sottile

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3 modules

Gain insight into a topic and learn the fundamentals.

251 reviews

Advanced level

Recommended experience

Flexible schedule

3 weeks at 10 hours a week

Learn at your own pace

96%

Most learners liked this course

3 modules

Gain insight into a topic and learn the fundamentals.

251 reviews

Advanced level

Recommended experience

Flexible schedule

3 weeks at 10 hours a week

Learn at your own pace

96%

Most learners liked this course

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 videos Total 99 minutes

Introduction 3 minutes
The many-body problem 7 minutes
First objective: observables 8 minutes
Examples of observables 5 minutes
Observables in terms of compact quantities 8 minutes
Introduction to functionals and functionals of the density 7 minutes
Functional derivatives 11 minutes
Define the system: the external potential 8 minutes
Demonstration of the Hohenberg-Kohn theorem 9 minutes
HK theorem: some important aspects 10 minutes
The Thomas(-Fermi) approach 9 minutes
Thomas-Fermi as density functional 7 minutes
Beyond Thomas-Fermi 7 minutes

7 readings Total 203 minutes

Reading and References 15 minutes
on Notations 3 minutes
Recent literature on the many-body problem 15 minutes
Lecture Notes :: Functional derivatives 90 minutes
A simple matrix (degenerate eigenvalue) 10 minutes
Extra references 10 minutes
Lecture Notes :: Thomas-Fermi approximation 60 minutes

7 assignments Total 230 minutes

Which strategy ? 5 minutes
around Hohenberg-Kohn 10 minutes
Thomas-Fermi density functional 60 minutes
The many-body wavefunction: a big beast 30 minutes
Functional strategy 5 minutes
Exercises on Derivatives 60 minutes
Beyond Thomas-Fermi 60 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 videos Total 90 minutes

Functional of the density :: a summary 14 minutes
Auxiliary system 2.3 part1 9 minutes
Auxiliary systems 2.3 part2 5 minutes
The exchange-correlation term 7 minutes
The Kohn-Sham equations 10 minutes
Thoughts on the Kohn-Sham equations 11 minutes
The shape of the Kohn-Sham potential 6 minutes
How real is the Kohn-Sham world ? 5 minutes
The Hartree equations 10 minutes
From Thomas and Hartree to Walter Kohn 13 minutes

5 readings Total 210 minutes

on the functional of the density 10 minutes
References :: auxiliary systems 60 minutes
Lecture Notes for functional of the density 10 minutes
Lecture notes for Kohn-Sham derivation 70 minutes
The Hartree equations 60 minutes

4 assignments Total 135 minutes

Simple auxiliary system 45 minutes
the Koopmans' theorem 30 minutes
Hartree's other interests 30 minutes
the exchange-correlation potential 30 minutes

1 peer review Total 150 minutes

Kohn-Sham equations :: derivation and total energy 150 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.