Density functional theory for beginners

The purpose of this site is to explain in a non-mathematical way what density functional theory is and what it is used for. I hope to familiarise the reader with the syntax of solid state theory and give a basic understanding of the related topics. The content is suitable for anyone with a physics background and should serve as a useful preamble to a more mathematical approach.

The quantum mechanical wavefunction contains,in principle, all the information about a given system. For the case of a simple 2-D square potential or even a hydrogen atom we can solve the Schrödinger equation exactly in order to get the wavefunction of the system. We can then determine the allowed energy states of the syatem. Unfortunately it is impossible to solve the Schrödinger equation for a N-body system. Evidently, we must involve some approximations to render the problem soluble albeit tricky. Here we have our simplest definition of DFT: A method of obtaining an approximate solution to the Shrodinger equation of a many-body system.

DFT computational codes are used in practise to investigate the structural, magnatic and electronic properties of molecules, materials and defects.

I would like to remind the reader of some of the ideas used in this field and introduce them to some more. Take, as an example of a many-body problem, the case of a regular crystal. The electrons are not only affected by the nuclei in their lattice sites, but also by the other electrons.

Now that you are familiar with the terms involved, here is an example of a widely used method for solving the many-body problem.

Firstly we need to reduce as far as possible the number of degrees of freedom of the system. Our most basic approximation does just this. It is called the **Born-Oppenheimer approximation**.

A functional is a function of a function. In DFT the functional is the electron density which is a function of space and time. The electron density is used in DFT as the fundamental property unlike Hartree-Fock theory which deals directly with the many-body wavefunction. Using the electron density significantly speeds up the calculation. Whereas the many-body electronic wavefunction is a function of 3N variables (the coordinates of all N atoms in the system) the electron density is only a function of *x, y, z* -only three variables. Of course simply doing any old calculation fast is not good enough - we also need to be sure that we can derive something significant from it.
It was Hohenburg and Kohn who stated a theorem which tells us that the electron density is very useful. The Hohenburg-Kohn theorem asserts that the density of any system determines all
**ground-state** properties of the system. In this case the total ground state energy of a many-electron system is a functional of the density. So, if we know the electron density functional, we know the total energy of our system.

By focusing on the electron density it is possible to derive an effective one-electron-type Schrödinger equation.

We can now write the total energy of our system in terms of which are all functionals of the charge density. These terms are:

- Ion-electron potential energy
- Ion-ion potential energy
- electron-electron energy
- kinetic energy
- exchange-correlation energy

The two difficult terms to calculate here are the **kinetic energy** and the **exchange correlation energy**.