Density Functional Theory

Introduction

Density Functional Theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (principally the ground state) of many-body systems, in particular atoms, molecules, and the condensed phases. The use of DFT in the calculation of electronic structure is among the most popular and versatile methods available in condensed-matter physics, computational physics, and computational chemistry.

A computer generated image of a quantum mechanical model showing the electronic structure of a molecule.

Theoretical Background

DFT is based on the principle that the ground state properties of a many-electron system can be determined by an energy functional of the electron density. This principle is known as the Hohenberg-Kohn theorem, after Pierre Hohenberg and Walter Kohn. The original Hohenberg-Kohn theorems, however, did not provide a practical way to calculate the ground state density. This was achieved by the development of the Kohn-Sham equations, a set of differential equations which are derived from the variational principle.

The Kohn-Sham equations have been very useful for calculations in solid-state physics and quantum chemistry. However, the exact functional for exchange and correlation is not known. Therefore, approximations must be made. The most widely used approximations are the local density approximation (LDA) and the generalized gradient approximation (GGA).

Local Density Approximation

The local density approximation (LDA) is the simplest approximation and it is based on the uniform electron gas model. In this model, the exchange-correlation energy is assumed to be a function of the local (or average) electron density at each point in space. The LDA has been remarkably successful in predicting the properties of many systems, especially solids.

Generalized Gradient Approximation

The generalized gradient approximation (GGA) is a further refinement of the LDA. In the GGA, the exchange-correlation energy is assumed to be a function of not only the electron density, but also its gradient. The GGA often gives better results than the LDA, especially for molecules.

Applications of DFT

DFT has been widely used in many areas of physics, chemistry and materials science. It has been used to study the electronic structure of atoms, molecules and solids, the optical properties of solids, the electronic transport properties of devices, and the dynamics of chemical reactions, among other things.

Limitations of DFT

Despite its success, DFT is not without its limitations. The most notable of these is the self-interaction error, which arises because the electron density functional includes the Coulomb interaction of each electron with itself. Other limitations include the lack of exact exchange, the difficulty in treating van der Waals forces, and the inability to accurately describe systems with strong electron correlation.