Computational Chemistry
Table of Contents
1. Born-Oppenheimer Approximation
- BO Approximation
The atomic nuclei is approximated to be fixed in space.
2. Density Functional Theory
The electron density \(\rho(\mathbf{r})\) is calulated as a proxy for the entire wave function. \[ \rho(\mathbf{r}) = N \int \dd[3]{\mathbf{r}_2} \cdots \int\dd[3]{\mathbf{r}_N} |\Psi(\mathbf{r}, \mathbf{r}_2,\dots, \mathbf{r}_N) |^2 \] where \(N\) is the total number of electrons.
2.1. Honherberg-Kohn Theorems
First Theorem Electron density \(\rho\) uniquely determines \(\hat{V}_{\rm ext}\).
Second Theorem The minimum energy for electron density of given number of electrons is the ground state energy: \[ \min_\rho E[\rho] = E_0. \]
The minimum energy is obtained using the variational method.
2.2. Density Function
The density function is approximated to be constant when many electrons are distributed uniformly.
In particles in a box problem with mols of electrons, \[ E \to \frac{\pi \hbar^2}{6m_{\rm e}L^2} N^3 \] and assuming uniformity \[ \frac{E}{N} = \frac{\pi\hbar^2}{6m_{\rm e}} \rho^2. \]
3. Programs
3.1. Molecule Skeleton Builder
3.1.1. Avogadro
- free and open source
3.1.2. Chemcraft
- proprietary
- many academical use
3.2. Calculation
3.2.1. VASP
- Vienna Ab inito Simulation Package
- proprietary, the baseline
3.2.2. Gaussian
- proprietary
3.2.3. ORCA
- free for academic use