Electrons and Nuclei
Molecules and solids consist of atoms that bind to each other due to the interaction of the positively charged nuclei with the electrons. In this interaction for instance covalent bonds are formed among pairs of atoms. The mutual interaction of nuclei and electrons is based on the Coulomb law, however, as a large number of particles are involved, the resulting bonding can be complex. Generally, one distinguishes different types of bonding
- covalent bonding
- metallic bonding
- ionic bonding
- van der Waals bonding
Any description of a molecule or solid from first principles starts with the full Hamiltonian $\hat{H}$ of the $N_\mathrm{k}$ nuclei and $N_\mathrm{e}$ electrons. The electrons have coordinates $\mathbf{r}_j$, with $j=1,\ldots,N_\mathrm{e}$ and mass $m_\mathrm{e}$. The nuclei posses the mass $M_{J}$, charge $Z_J\,e$, and coordinates $\mathbf{R}_J$, where $e$ is the elementary charge and $J=1,\ldots,N_\mathrm{k}$. The Hamiltonian $\hat{H}$ consists of the kinetic energy operators $\hat{T}_\mathrm{e}$ and $\hat{T}_\mathrm{k}$ of electrons and nuclei and the operators for the Coulomb interaction between the charged particles
\begin{equation}
\hat{H}= \hat{T}_\mathrm{e}+\hat{T}_\mathrm{k}+\hat{V}_{\mathrm{ee}}+
\hat{V}_{\mathrm{ek}}+\hat{V}_{\mathrm{kk}}\quad.
\end{equation}
The kinetic energy operators $\hat{T}_\mathrm{e}$ and $\hat{T}_\mathrm{k}$ are
\begin{equation}
\hat{T}_\mathrm{e}=-\sum_j\frac{\hbar^2}{2 m_\mathrm{e}}\frac{\partial^2}{\partial \mathbf{r}_j^2}
\qquad\mathrm{and}\qquad
\hat{T}_\mathrm{k}=-\sum_j\frac{\hbar^2}{2 M_J}\frac{\partial^2}{\partial \mathbf{R}_J^2}\quad,
\end{equation}
where $\frac{\partial}{\partial \mathbf{r}_j}$ and $\frac{\partial^2}{\partial \mathbf{r}_j^2}$ represents the gradient and Laplacian with respect to the coordinate $\mathbf{r}_j$. The mutual electron-electron and nuclei-nuclei interaction $\hat{V}_{\mathrm{ee}}$ and $\hat{V}_{\mathrm{kk}}$, and the interaction of the electrons with the nuclei read
\begin{align}
\hat{V}_\mathrm{ee}=\frac{1}{2}\sum_{i\neq j} \frac{e^2}{4\pi\varepsilon_0}
\frac{1}{|\mathbf{r}_i-\mathbf{r}_j|}\qquad&\qquad\qquad
\hat{V}_\mathrm{k\,k}=\frac{1}{2}\sum_{I\neq J} \frac{e^2}{4\pi\varepsilon_0}
\frac{Z_I Z_J}{|\mathbf{R}_I-\mathbf{R}_J|}\\
\hat{V}_\mathrm{ek}&=-\sum_{i J} \frac{e^2}{4\pi\varepsilon_0}
\frac{Z_J}{|\mathbf{r}_i-\mathbf{R}_J|}
\end{align}
The former two interactions are purely repulsive (the factor $\frac{1}{2}$ above accounts for double counting in the sums). Only the interaction between nuclei and electrons is attractive and thus is the source of bonding of molecules and solids.
In the absence of external potentials, the ground state properties and excited states are described by the Schrödinger equation
\begin{equation}
\hat{H}\,\Psi\left(\{\mathbf{r}_j\},\{\mathbf{R}_J\}\right)
=E\,\Psi\left(\{\mathbf{r}_j\},\{\mathbf{R}_J\}\right)\quad.
\end{equation}
The wave function $\Psi\left(\{\mathbf{r}_j\},\{\mathbf{R}_J\}\right)$ of the interacting electrons and nuclei is a function of $3\,(N_\mathrm{e}+N_\mathrm{k})$ degrees of freedom. This complexity of the Schrödinger does not allow for closed solutions. Physical approximations must be made in order to separate the problem into tractable parts. Such an approximation is the Born-Oppenheimer approximation or more correctly the adiabatic approximation. It allows to separates the nuclear motion from the electron-problem.
Before this, a note on units is due. It is common practice to drop the SI-units and to refer to atomic units, which greatly simplifies the equations. This means to set
\begin{equation}
\hbar \rightarrow 1\qquad\frac{e^{2}}{4 \pi\,\varepsilon_{0}} 1
\quad\mathrm{and}\quad m_{\mathrm{e}}\rightarrow 1\quad.
\end{equation}
Thereby the unit of length becomes Bohr=$\frac{4 \pi\,\varepsilon_{0}\,\hbar^{2}}{m_{\mathrm{e}}\,e^{2}}=0.529\,10^{-10}$m and the unit of energy is Hartree (Ha)=$\frac{m_{\mathrm{e}}\,e^{4}}{16 \pi^{2}\,\varepsilon_{0}\,\hbar^{2}}$ $=27.2116$eV.