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.

Why use DFT?




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.

Getting to know the terminology




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.

Coulomb potential:

Remember, this is just the classical potential that arises from any charged object.

Jellium

Hartree Potential:

The interaction between the electrons in the system is approximated by the Coulomb potential arising from a system of fixed electrons. Alternatively we say that each individual electron moves independently of each other, only feeling the average electrostatic field due to all the other electrons plus the field due to the atoms. In other words, this is the potential due to the electron density distribution and ionic lattice but neglects exchange and correlation effects.

Pauli Exclusion Principle:

This is the rule that disallows two identical particles to lie in the same quantum state.

Exchange interaction:

This is due to the Pauli Exchange Principle. In this case it states that if two electrons have parallel spins then they will not be allowed to sit at the same place at the same time. This phenomenon gives rise to an effective repulsion between electrons with parallel spins. Not only, then do we have two electrons interacting via their electronic charge, but also by their spins.

Correlation interaction:

The correlation interaction is also a result of the Pauli Exchange interaction. In this case there is a correlated motion between electrons of anti-parallel spins which arises because of their mutual coulombic repulsion. See alsoJellium

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

Hartree-Fock:

This method employs the Hartree potential but also forces the exchange interactions by forcing the antisymmetricity of the electronic wavefunction. This acts to lower the total binding energy of atoms by ensuring that parallel spin electrons stay apart. The downside to the theory is that it neglects correlations in the motion between two electrons with anti parallel spins.

Density Functional Theory




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.