Connection between irreducible representations and electronic states in diatomic molecules

Question

I am trying to understand the connection between irreducible representations and molecular electronic states. To explain the problem: I found some potential energy curves (PECs) for $\ce{N2+}$ in the form of irreducible representations such as $\mathrm{A_g}$, $\mathrm{A_u}$, $\mathrm{B_{2g}}$, ...

However, in the literature, I normally find the PECs in the form of $\Sigma$, $\Pi$, $\Delta$... states. How do we connect these two pictures? I am not an expert in quantum chemistry calculations. A simple explanation will be highly helpful.

Answer 1

In the linear $C_{\infty \mathrm{v}}$ and $D_{\infty \mathrm{h}}$ point groups there are two notations for the irreducible representations which are equivalent, in that

$$\begin{align} \mathrm{A_1} &\equiv \Sigma^+ \\ \mathrm{A_2} &\equiv \Sigma^- \\ \mathrm{E_1} &\equiv \Pi \\ \mathrm{E_2} &\equiv \Delta \\ \mathrm{E_3} &\equiv \Phi \\ &\,\,\vdots \end{align}$$

For $C_{\infty \mathrm{v}}$, these are the actual irreps; for $D_{\infty \mathrm{h}}$ you simply need to tack on a g or u label to denote the symmetry with respect to inversion. See e.g. the $D_{\infty \mathrm{h}}$ character table found at http://symmetry.jacobs-university.de/cgi-bin/group.cgi?group=1001&option=4 which contains both notations.