Units
Since these notes are concerned about molecular structure, the most natural choice for a system of units is the Hartree atomic units system. In this system, the following units are informally set to \(1\): \[\begin{equation} \require{si} \text{m}_{\text{e}} % = \hbar % = \text{e} % = \frac{1}{4 \pi \epsilon_0} % = 1 % \thinspace , \end{equation}\] in which _{} is the mass of the electron, \(\hbar\) is the reduced Plank constant, is the magnitude of the charge of the electron and \(\epsilon_0\) is the permittivity of the vacuum.
The fine-structure constant \[\begin{equation} \label{eq:fine-structure_constant} \alpha = % \frac{ % \text{m}_{\text{e}}^2 }{ (4 \pi \epsilon_0) \hbar c } % \approx % \frac{1}{137} \end{equation}\] has the same numerical value in every system of units, so from equation \(\eqref{eq:fine-structure_constant}\) we can derive that the speed of light in atomic units is approximately \(137\) because \[\begin{equation} \alpha c = 1 % \thinspace . \end{equation}\]
The atomic unit for energy is called the hartree: \[\begin{equation} \text{Ha} % = \text{m}_{\text{e}} \alpha^2 c^2 % = 1 \end{equation}\] and the atomic unit for length is the bohr (radius) \[\begin{equation} \text{a}_0 = % \frac{ \hbar } { \text{m}_{\text{e}} \alpha c } % = 1 % \thinspace . \end{equation}\]
At first sight, we could go on and define a.u.-analogues of all SI units that we know, but once we reach the domain of electromagnetism, it seems that everything starts becoming ambiguous. This is because the form of the Maxwell equations (by which I mean where the \(c\), \(4 \pi\), , are placed) depends on the very definition of the units that are used for the electric and magnetic fields.
In order to make everything as transparant as possible, we will follow the convention that is used by Helgaker (T. Helgaker et al. 2012), who uses SI-based atomic units also in the domain of electromagnetic applications.