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16 CHAPTER 1. INTRODUCTION
necessary requirements are a good background in elementary quantum mechanics, no fear
of calculus of more than one variable, and a desire to learn. The student should end up
knowing what the Kohn-Sham equations are, what functionals are, how much (or little) is
known of their exact properties, how they can be approximated, and how insight into all these
Chapter 1 things produces understanding of the errors in electronic structure calculations. Preliminary
forms of my lecture notes have been used throughout the world during 5 years its taken to
complete this book. The hope is that these notes will be used by students worldwide to gain
Introduction a better understanding of this fundamental theory. I ask only that you send me an email (to
kieron@rutchem.rutgers.edu) if you use this material. In return, I am happy to grade problems
and answer questions for all who are interested. These notes are copyright of Kieron Burke.
In which we introduce some of the basic concepts of modern density functional theory, No reproduction for purposes of sale is allowed.
including the Kohn-Sham description of a system, and give a simple but powerful example The book is laid out in the form of an undergraduate text, and requires the working of
of DFT at work. many exercises (although many of the answers are given). The idea is that the book and
exercises should be easy reading, but leave the reader with very clear concepts of modern
density functional theory. Throughout the text, there are exercises that must be performed
1.1 Importance to get full value from the book. Also, at the end of each chapter, there are questions aimed
at making you think about the material. These should be thought about and answere as you
1 Density functional theory (DFT) has long been the mainstay of electronic structure calcula-
go along, but answers don’t need to be written out as explicitly as for the problems.
tions in solid-state physics. In the 19990’s it became very popular in quantum chemistry. This
is because approximate functionals were shown to provide a useful balance between accuracy
and computational cost. This allowed much larger systems to be treated than by traditional 1.2 What is a Kohn-Sham calculation?
ab initio methods, while retaining much of their accuracy. Nowadays, traditional wavefunc-
To give an idea of what DFT is all about, and why it is so useful, we start with a very simple
tion methods, either variational or perturbative, can be applied to find highly accurate results
example, the hydrogen molecule, H 2 . Throughout this book, we make the Born-Oppenheimer
on smaller systems, providing benchmarks for developing density functionals, which can then
be applied to much larger systems approximation, in which we treat the heavy nuclei as fixed points, and we want only to solve
the ground-state quantum mechanical problem for the electrons.
But DFT is not just another way of solving the Schr¨odinger equation. Nor is it simply a
In regular quantum mechanics, we must solve the interacting Schr¨odinger equation:
method of parametrizing empirical results. Density functional theory is a completely different,
 
formally rigorous, way of approaching any interacting problem, by mapping it exactly to a  1 - 2 1 - 
much easier-to-solve non-interacting problem. Its methodology is applied in a large variety of  − 2 i=1,2 ∇ + |r 1 − r 2 | + i=1,2 v ext (r i ) Ψ(r 1 , r 2 ) = EΨ(r 1 , r 2 ), (1.1)
i

fields to many different problems, with the ground-state electronic structure problem simply
being the most common. where the index i runs over the two electrons, and the external potential, the potential
experienced by the electrons due to the nuclei, is
The aim of this book is to provide a relatively gentle, but nonetheless rigorous, introduction
to this subject. The technical level is no higher than any graduate quantum course, or many v ext (r) = −Z/r − Z/|r − Rˆz|, (1.2)
advanced undergraduate courses, but leaps at the conceptual level are required. These are
almost as large as those in going from classical to quantum mechanics. In some sense, they where Z = 1 is the charge on each nucleus, ˆz is a unit vector along the bond axis, and R is
are more difficult, as these leaps are usually made when we are more advanced (i.e., set) in a chosen internuclear separation. Except where noted, we use atomic units throughout this
our thinking. text, so that 2
Students from all areas of modern computational science (chemistry, physics, materials e = ¯h = m = 1, (1.3)
science, biochemistry, geophysics, etc.) are invited to work through the material. The only where e is the electronic charge, ¯h is Planck’s constant, and m is the electronic mass. As
1 c !2000 by Kieron Burke. All rights reserved. a consequence, all energies are in Hartrees (1 H = 27.2114 eV= 627.5 kcal/mol) and all
15