# Letter codes in molecular term symbols

I'm trying to find an explanation for the letter codes (X, A, B, C, etc) when you get a term symbol like this

$$X\,{}^3{\Sigma}^-_g$$

Can anyone point me to some literature that explains these letter codes?

• en.wikipedia.org/wiki/… In this case it is referring to the ground state (I guess probably $\ce{O2}$). – orthocresol Sep 17 '16 at 11:40
• have a look also in the tables at the end of any of the Herzberg books 'Molecular Spectra & Molecular Structure' – porphyrin Sep 17 '16 at 12:49

Quoting from Hollas, Modern Spectroscopy, 4th ed. (p 236):

There is a convention, which is commonly but not always used, for labelling electronic states. The ground state is labelled $X$ and higher states of the same multiplicity are labelled $A, B, C\ldots$ in order of increasing energy. States with multiplicity different from that of the ground state are labelled $a, b, c \ldots$ in order of increasing energy.

As such, for the oxygen molecule, where the lowest three states are ${}^3{\Sigma}_g^-$, ${}^1{\Delta}$ and ${}^1{\Sigma}^+_g$ (in increasing order of energy), the triplet ground state would be labelled $X$ and the next two states $a$ and $b$ respectively.

A word of caution: it seems that this convention is not always obeyed. On the next page, Hollas writes (for $\ce{I2}$, where the ground state is a singlet)

The transition $B\,{}^3{\Sigma}_{0_u^+} - X\,{}^1{\Sigma}_g^+$ involves the $\Omega = 0$ component of the $B$ state [...] The labelling of the $B\,{}^3{\Sigma}_{0_u^+}$ state follows general usage rather than convention, which would label it $b\,{}^3{\Sigma}_{0_u^+}$.

Wikipedia also adds, although without a citation:

In polyatomic molecules (but not in diatomics) it is customary to add a tilde (e.g. $\tilde{X}$, $\tilde{a}$) to these empirical labels to prevent possible confusion with symmetry labels based on group representations.

• I don't think you explained what ${}^3{\Sigma}^{-}_g$ means. – DHMO Sep 17 '16 at 12:24
• That explanation is here. – pentavalentcarbon Sep 17 '16 at 14:11
• The $\ce{^3\Sigma ^-_g}$ is a term symbol for the not-fully filled orbitals of a homo-nuclear diatomic molecule and comes from the $\ce{D_{\infty h}}$ point group. The 3 indicates a triplet spin state, $\ce{\Sigma}$ zero orbital angular momentum, g (gerade) means orbitals have a centre of inversion, otherwise u (ungerade). The superscript - indicates -1 to reflection in plane containing nuclei, i.e. no plane of reflection. A superscript + would indicate a +1 reflection, i.e. has plane. The $\ce{O2}$ ground state is $\ce{^3\Sigma ^-_g}$ . – porphyrin Sep 17 '16 at 16:00
• Thanks for mentioning your source. I totally missed the footnote on that page. It's been a while since I last had to use a lot of quantum mechanics thanks for the quick refresher. – Arno van der Weijden Sep 17 '16 at 17:25
• The question didn't ask for an explanation of the term symbol anyway, but I appreciate the links/explanations in the comments. :) Just want to point out that the superscript minus indicates that the wavefunction is antisymmetric with respect to reflection in a plane containing the nuclei, not that such a plane does not exist. – orthocresol Sep 18 '16 at 18:14