 ## Density Functional Theory

Density Functional Theory (DFT) is a method to describe the dispersion relation and density of states of electrons in crystals. It is popular because it provides relatively good accuracy for a modest computational effort.

The starting point for the calculations is the electronic Hamiltonian for electrons in a solid.

\begin{equation} \label{eq:helec} H_{\text{elec}}= -\sum\limits_i \frac{\hbar^2}{2m_e}\nabla^2_i -\sum\limits_{a,i} \frac{Z_ae^2}{4\pi\epsilon_0 |\vec{r}_i-\vec{r}_a|}+\sum\limits_{i< j} \frac{e^2}{4\pi\epsilon_0 |\vec{r}_i-\vec{r}_j|} +\sum\limits_{a< b} \frac{Z_aZ_be^2}{4\pi\epsilon_0 |\vec{r}_a-\vec{r}_b|}. \end{equation}

The first sum describes the kinetic energy of the electrons. The electrons are labeled with the subscript $i$. The second sum describes the attractive Coulomb interaction between the positively charged nuclei and the negatively charge electrons. The atoms are labeled with the subscript $a$. $Z_a$ is the atomic number (the number of protons) of nucleus $a$. The third sum describes the repulsive electron-electron interactions. Notice the plus sign before the sum for repulsive interactions. The fourth sum describes the repulsive nuclei-nuclei interactions. This Hamiltonian exploits the Born-Oppenheimer approximation where the atomic nuclei are assumed to be fixed.

The ground state wavefunction $\Psi(\vec{r}_1,\,\vec{r}_2,\cdots ,\, \vec{r}_N)$ depends on the coordinates of all of the electrons and minimizes the energy,

$$E=\frac{\langle \Psi |H_{elec}|\Psi\rangle}{\langle \Psi |\Psi\rangle}.$$

In principle, this wavefunction can be found by starting with an initial guess for $\Psi$ and then adjusting the wavefunction in such a way that the energy always decreases. For many electrons, this approach is impractical because the space of variables is so large. Hohenberg and Kohn proved that the energy of this system of electrons can be expressed as a functional of the electron density $E=E[n(\vec{r})]$. Here the electron density $n(\vec{r})$ is a function of position. A functional, such as $E[n(\vec{r})]$, assigns a scalar value to every function it is given. Hohenberg and Kohn also showed that for a constant number of electrons $N$, the electron density that minimizes the energy is electron density of the ground state.

Density functional theory introduces a system of fictional non-interacting quantum particles moving in a potential $V_{scf}(\vec{r})$. The potential is constructed with some clever approximations so that the density of the quantum particles is the same as the density of electrons in the ground state of the electronic Hamiltonian. The ground state is calculated with a self-consistent loop. The self-consistent-field potential $V_{scf}(\vec{r})$ is calcualted using the ion positions and the electron density. Using this potential, the electron density is recalculated. This process is repeated until self-consistency is achieved and the electron density does not change anymore.