Spectroscopy Symbols Explained

In spectroscopy and related fields, a notation for states is used that holds a lot of symmetry information to support employing selection rules. The only problem I have is that I use this to rarely to remember all the meanings. Moreover, this notation is exceptionally hard to resolve using search engines: “digit in front of Mulliken Symmetry labels” contains not many helpful resources. Here, I document the notation in all variants I know of.

If you have seen any variants or want to correct details, please comment.

In the following, “symmetric” means that the orbital wave function does not change the phase upon the symmetry operation. So it is not the orbital shape alone to care about, but also the phase of which. Hence, “anti-symmetric” means that the shape stays the same under the symmetry operation but the phase sign changes.

Single electron orbital

(2t’¹*_2g↑)¹

Not all elements are required. From left to right:

2: running number

If you have a list of orbitals that have the same symmetry but different energy, you want to assign different symbols to them. This counter runs from the single electron orbital with the lowest energy and starts with 1. The counter does not distinguish between occupied and unoccupied orbitals. So if you have two orbitals t_2g (occupied) and one orbital t*_2g (unoccupied), then they can be further distinguished by labelling them 1t_2g, 2t_2g, and 3t*_2g.

t: degeneracy

Also referred to as Mulliken label. Gives the degeneracy of the orbital. a is degeneracy of one (non-degenerate), e and t (sometimes f is used here) are for twofold and threefold degenerate, respectively. Higher values are rarely used: 4=g, 5=h. Rarely, b is used for the non-degenerate case, as well. The difference is that a is symmetric with respect to rotation of the principle between the nuclei while b is anti-symmetric. While b is only used for highlighting this property, it is not safe to assume that all a orbitals are symmetric, as this letter is used for labelling the degeneracy alone in most cases.

‘ (prime): counter or symmetry

In most cases, this is only a way to distinguish orbitals of the same symmetry. For multi-electron orbitals, this has a symmetry meaning that is sometimes transferred onto single electron orbitals, as well, namely if the orbital is symmetric under a reflection in a plane. In this case, double prime ” is used to denote an anti-symmetry under reflection in a plane.

¹: running number

Another counter. Rarely used. If it is a counter, it is always a superscript. Do not confuse with the subscript at this place.

* (asterisk): bonding character

A star here means this orbital is anti-bonding. Omitting this star means that this orbital is bonding only if the same source used the star in some other place. Otherwise, the star just has been omitted.

2: symmetry

If this subscript is 1, this means that the orbital is symmetric with respect to the C_n axis of symmetry. If it is 2, this means that the orbital is anti-symmetric.

g: symmetry

Stems from the German words gerade and ungerade. If there is a g here, this means that the orbital is symmetric under inversion. Less surprising, u means that the orbital is anti-symmetric in this case.

↑: spin state

If ↑, it is to emphasise that this is spin-up electron. ↓ means spin-down state. If omitted, no information on the spin state is given.

1: number of electrons

If the parentheses are present, the superscript following immediately after the closing one is the number of electrons in this orbital.

Multi-electron state

⁵E_g(2)

Not all elements are required. From left to right:

5: multiplicity

Spin multiplicity or 2S+1. Identical to the number of unpaired electrons plus one.

E: degeneracy

Also referred to as Mulliken label. Gives the degeneracy of the state (not the electron orbitals). A is degeneracy of one (non-degenerate), E and T (sometimes F is used here) are for twofold and threefold degenerate, respectively. Higher values are rarely used: 4=G, 5=H. Rarely, B is used for the non-degenerate case, as well. The difference is that A is symmetric with respect to rotation of the principle between the nuclei while B is anti-symmetric. While B is only used for highlighting this property, it is not safe to assume that all A orbitals are symmetric, as this letter is used for labelling the degeneracy alone in most cases.

(2): degeneracy

Redundant information to the E in this case.

Molecular Orbitals

2π*⁺_2p(2p)

The subscript and the parentheses are mutually exclusive.

2: counter

Counts the molecular orbitals of same symmetry from low energy to top. E.g. the order could be 1σ_g, 1σ_u, 2σ_g.

π: bond type

May be σ (angular momentum 0), π (angular momentum 1) or δ (angular momentum 2). Sigma bonds have the interatomic axis as symmetry axis of the orbital. For pi bonds, there is no electron density along the interatomic axis.

* (asterisk): bonding character

A star here means this molecular orbital is anti-bonding. No star here means that this molecular orbital is bonding.

⁺ (plus): symmetry

If present, a plus means that the molecular orbital is symmetric upon reflection in a plane containing the interatomic axis. If a minus is given, the molecular orbital is anti-symmetric under this operation. If no sign is given, no statement on the symmetry operation is made.

2p / (2p): original atomic orbitals

The type of the atomic orbitals that form this molecular orbital.