Density Functional Theory (DFT) - Basics

The information of a quantum system can be obtained from the state function (or wave function), which is described by the Schrödinger equation as $\hat{H} \Psi=E \Psi$. In this equation, $\hat{H}$ is the Hamiltonian operator and $\Psi$ is a set of solutions of the Hamiltonian. Each of these solution $\psi_{n}$ has an associated eigenvalue $E_{n}$ that satisfies the eigenvalue equation. For a system consisting of N electrons, the time independent Schrödinger equation is a function of 3N coordinates. To obtain an approximation solution of Schrödinger equation, define the electron density as $n(\mathbf{r})=2\sum_{i}\psi_{i}^{*}(\mathbf{r})\psi_{i}(\mathbf{r})$, where $n(\mathbf{r})$ is a function of only 3 coordinates and contains a great amount of the information that is actually physical observable. The density functional theory (DFT) rests on the Hohenberg and Kohn theorem and Kohn-Sham equation.

The Hohenberg and Kohn theorem states:

  1. The ground-state energy from Schrödinger's equation is a unique functional of the electron density.
  2. The electron density that minimizes the energy of the overall functional is the true electron density corresponding to the full solution of the Schrödinger equation.
The Kohn-Sham equation is written as $\left[ -\frac{h^{2}}{2m}\nabla^{2}+V(\mathbf{r})+V_{H}(\mathbf{r})+V_{XC}(\mathbf{r}) \right]\psi_{i}(\mathbf{r})=\varepsilon_{i}\psi_{i}(\mathbf{r})$. Here, $V_{H}$ is the Hartree potential, and $V_{XC}$ is the exchange-correlation potential.

The approximation of exchange-correlation potential includes local density approximation (LDA), generalized gradient approximation (GGA), etc. Two of the most widely used GGA functionals are the Perdew-Wang functional (PW91) and the Perdew-Burke-Ernzerhof functional (PBE).

Application of density functional theory (Read more...)