What does this notation mean: $\mathrm{O}_{2} \ \ a\,{}^{1} Δ_{g} ← X\,{}^{3}Σ^{-}_{g} $? The notation (which I found in the abstract of this paper) is 
$$\mathrm{O}_{2} \ \ a\,{}^{1} Δ_{g} ← X\,{}^{3}Σ^{-}_{g}. $$
Any help with this?
I understand it's talking about quantum states of the molecule but I can't find a concise source that explains all this notation. The paper it's from is talking about oxygen absorption bands.
 A: The notation is molecular term symbols and, the wikipedia article does seem to provide what you are asking for.  The larger subject is molecular spectroscopy.
Lets look at the term on the right.  The capitalized sigma refers to the angular momentum of this orbital along the molecular z axis, in this case it is zero.  A capital pi would mean it is 1.  Where atomic orbitals are labeled s, p, d etc., these are labeled Sigma, Pi, Delta etc.   The "3" preceding it should be shifted up (as it is in the article), it indicates that this is a triplet.   The g (for gerade, even) means it has inversion symmetry, the alternative is u (ungerade or odd) for antisymmetry under inversion.
You might find an introduction to the topic in the textbook for an undergraduate level course in physical chemistry.
A: The objects to the left and the right of the arrow are molecular term symbols, as pointed out by the initial answer (thin as it is on actual content). Molecular term symbols specify the parity and angular momentum characteristics of the state, in the format
$$
{}^{2S+1}\Lambda^p_{(g|u)}
$$
where


*

*$\Lambda$ is an uppercase Greek letter from the sequence $\Sigma$, $\Pi$, $\Delta$, $\Phi$, $\ldots$, in strict analogy to the atomic $S$, $P$, $D$, $F$, $\ldots$, which specifies that the state is an eigenstate of the projection of the total angular momentum on the molecular (usually internuclear) axis with corresponding value $L_z=0,1,2,3,\ldots$. 
The change to Greek letters indicates the passage from total angular momentum to the indication of a single (conserved) component, and the use of upper-case vs lower-case letters to indicate total vs single-electron angular momenta is identical to the usage in atomic term symbols.

*The spin multiplicity $2S+1$ is indicated as a pre-superscript exactly as in atomic term symbols.

*The $g$ or $u$ subscript indicates the parity of the state (under a full space inversion, $\mathbf r\mapsto -\mathbf r$, though it's really a proxy for the behaviour under reflections in a plane orthogonal to the internuclear axis), and it stands for the German 'gerade' and 'ungerade'. Gerade states have definite parity $+1$ and ungerade states have definite parity of $-1$. (Mind the pronunciation, by the way, with a hard g and short vowels.)

*The $p$ superscript indicates the parity behaviour under a reflection in a plane that contains the internuclear axis. This is normally omitted if $L_z>0$, since the term symbol will indicate an orbital doublet with one $+$ state and one $-$ state (i.e. if the internuclear axis is the $z$ axis and you set the reflection plane as the $xz$ plane, then the $\Pi_x$ state is even and the $\Pi_y$ state is odd). For single electrons in $\sigma$ states this parity must always be even, but in multi-electron systems it is possible to combine higher-$\ell_z$ states down to $L_z=0$ such that they're odd under this parity, in which case it's indicated as $\Sigma^-$. Sometimes the $+$ superscript is also omitted in $\Sigma^+$, so $\Sigma$ always means $\Sigma^+$.

*The letter preceding the term symbol is a label used to separate the different states with identical angular-momentum behaviour, in order of ascending energy. The precise usage is captured well by the Wikipedia page:

The ground state is labelled X, excited states of the same multiplicity (i.e., having the same spin quantum number) are labelled in ascending order of energy with capital letters A, B, C...; excited states having different multiplicity than the ground state are labelled with lower-case letters a, b, c...

If you want further sources for this, then any textbook on molecular physics will have a section devoted to the notation.
