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?
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asked Sep 17, 2016 at 11:18
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1$\begingroup$ en.wikipedia.org/wiki/… In this case it is referring to the ground state (I guess probably $\ce{O2}$). $\endgroup$– orthocresolCommented Sep 17, 2016 at 11:40
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1$\begingroup$ have a look also in the tables at the end of any of the Herzberg books 'Molecular Spectra & Molecular Structure' $\endgroup$– porphyrinCommented Sep 17, 2016 at 12:49
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1 Answer
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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.
answered Sep 17, 2016 at 12:00
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$\begingroup$ I don't think you explained what ${}^3{\Sigma}^{-}_g$ means. $\endgroup$– DHMOCommented Sep 17, 2016 at 12:24
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1$\begingroup$ 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 $\ce{^3\Sigma ^-_g}$ . $\endgroup$ Commented Sep 17, 2016 at 16:00
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$\begingroup$ 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. $\endgroup$ Commented Sep 17, 2016 at 17:25
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2$\begingroup$ 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. $\endgroup$ Commented Sep 18, 2016 at 18:14