Density Functional Theory - A Practical Introduction

Section 1 What is DFT

1. Schrödinger equation:

   \[\hat{H}\Psi=E\Psi\]

   where $\hat{H}$ is Hamiltonian operator, $\Psi$ is a set of solutions, $E$ is the ground-state energy.

   The more complete description:

   \[[\underbrace{-\frac{\hbar^2}{2m}\sum^N_{i=1}\nabla^2_i}_\text{Kinetic energy} +\underbrace{\sum^N_{i=1}V(\mathbf{r}_i)}_\text{Potential}+\underbrace{\sum^N_{i=1}\sum_{j<i}U(\mathbf{r}_i,\mathbf{r}_j)}_\text{interaction of electrons}]\Psi=E\Psi\]

2. The density of electrons at a particular position in space:

   \[n(\mathbf{r})=2\sum^{N/2}_i\psi^*_i(\mathbf{r})\psi_i(\mathbf{r})\]

3. “Functional”: A functional takes a function and defines a single number form the function (e.g. $F[f]=\int^1_1f(x)dx$).

4. Simplify by the Hohenberg-Kohn therorem:

\[E[\{\psi_i\}]=E_{known}[\{\psi_i\}]+E_{xc}[\{\psi_i\}]\] \[\begin{align*} E\_{known}[\{\psi_i\}]&=-\frac{\hbar^2}{2m}\sum_i\int\psi^*_i\nabla^2\psi_i d^3r+\int V(\mathbf{r})n(\mathbf{r})d^3r\\ &+\frac{e^2}{2}\int\int\frac{n(\mathbf{r})n(\mathbf{r}')}{\lvert r-r' \rvert}d^3rd^3r'+E_{ion} \end{align*}\]

1. The Kohn - Sham equation:

\[[-\frac{\hbar^2}{2m}\nabla^2+V(\mathbf{r})+V_H(\mathbf{r})+V_{XC}(\mathbf{r})]\psi_i(\mathbf{r})=\varepsilon_i\psi_i(\mathbf{r})\]

• The interaction between an electron and the collection of nuclei: $V(\mathbf{r})$
• Hartree potential (Coulomb repulsion between electron and electron density):

\[V_H(\mathbf{r})=e^2\int \frac{n(r')}{\lvert r-r' \rvert}d^3r'\]

• Functional derivative of the exchange-correlation energy: \(V_{XC}(\mathbf{r})=\frac{\delta E_{XC}(\mathbf{r})}{\delta n(\mathbf{r})}\)

1. Iterative algorithm:
   1. Define an initial, trial electron density, $n(\mathbf{r})$
   2. Solve Kohn-Sham equation to find the single-particle wave function, $\psi(\mathbf{r})$
   3. Calculate the electron density, $n_{KS}(\mathbf{r})=2\sum_i\psi^*_i(\mathbf{r})\psi_i(\mathbf{r})$
   4. Compare the electron density, updated trial electron density or start step 2.