# Determination of +/- reflections in sigma molecular term symbols

This page, which depicts the molecular term symbols for the $\ce{O2}$ and $\ce{O2-}$ (Edit: Mistaken on $\ce{O2-}$) molecules, perhaps best summarizes the full scope of my questions. In general, I don't understand how a specific electron configuration relates to either +,- or both reflections.

1. Why does the linked page switch reflections for different spin multiplicities, ie. why does $^3\Sigma^-$ have a minus reflection but $^1\Sigma^+$ a positive reflection?

2. For $\ce{O2-}$, the left configuration leads to $^3\Sigma^- + ^1\Sigma^+$. Why are two sigma terms needed to describe this configuration?

3. As a more general question, how accurate is it to use the molecular orbital representation for deriving term symbols? Most sources I have seen describe electron configurations as three degenerate p-orbitals or 5 degenerate d-orbitals and never invoke MO diagrams. However, it seems necessary for deriving both parity and reflections. Is the MO approach always correct? Can the term symbols be derived without using MO diagrams, specifically for the parity and reflections?

Thank you for any advice.

## 1 Answer

The $\pm$ superscripts refer to symmetry operations defined in the $D_{\infty h}$ point group. The operation on the molecular orbital is reflection in a plane containing the nuclei. If the orbital in unchanged then the sign is $+$, if changed then $-$.

The electron configuration for ground state oxygen is $(1\sigma _g)^2(1\sigma _u)^2(2\sigma _g)^2(2 \sigma _u)^2(3\sigma _g)^2(1\pi _u)^2(1\pi _g)^2(1\pi _g)^1(1\pi _g)^1$

However, the only electrons we need to consider are the two in the $\pi_g$ orbitals. Adding up the individual orbital and then spin angular momenta separately (equivalent to Russell-Saunders coupling in atoms) then combining both gives the term symbols $^3\Sigma, ^1\Delta ,^1\Sigma$. The calculation is done in a similar way to that for atoms and is shown below in the notes.

The subscript g is found because there is an even number of $\pi_g$ electrons. The superscript $\pm$ is because any state ( singlet or triplet) must be asymmetrical overall to exchange and as the triplet has a symmetrical spin wavefunction the spatial part is asymmetrical, and vice-versa for the singlet, the superscript for the triplet is $-$. Thus the superscript gives the term symbol $^3\Sigma_g^-$. Hund’s rule dictate that this is the lowest energy level, followed by $^1\Delta$ at $7918 \pu{cm^{-1}}$ and $^1\Sigma_g^+$ at $13195.22 \pu{cm^{-1}}$.

The $\ce{O_2^-}$ has one unpaired electron in a $\pi$ orbital (as does $\ce{O_2^+}$) so the term symbol is $^2\Pi_g$.

As to your third question the overall symmetry should not change when you go from atomic to MO's.

Notes

The individual $m_l$ and $m_s$ are listed below. The values restricted by the Pauli principle are removed. The $m_l$ values are $\pm 1$ and the $m_s \pm 1/2$. The the table the spins are labelled as $\pm$ only. The total angular momentum is given as $\Omega = \Lambda + \Sigma$ by analogy with $J=L+S$ in atoms.

$$\begin{matrix} m_{l1} & m_{l2} & m_{s1} & m_{s2} & \Lambda & \Sigma \\ 1 & 1 & + & + &\text{etc. Pauli forbid'n}\\ \\ 1 & 1 & + & - & 2 & 0 & ^1\Delta\\ 1 & 1 & - & + & 2 & 0 & .. \\ -1 & -1 & + & - & -2 & 0 & ..\\ -1 & -1 & - & + & -2 & 0 & ..\\ \\ 1 & -1 & + & + & 0 & 1 & ^3\Sigma\\ -1 & 1 & + & + & 0 & 1 & ..\\ 1 & -1 & - & - & 0 & -1 & ..\\ -1 & 1 & - & - & 0 & -1 & ..\\ \\ 1 & -1 & + & - & 0 & 0 & ^1\Sigma\\ 1 & -1 & - & + & 0 & 0 & ..\\ \end{matrix}$$