# 9.15: Molecular Term Symbols Designate Symmetry

- Page ID
- 210867

The quantum numbers for diatomic molecules are similar from the atomic quantum numbers. Be cautious, because the rules for finding the possible combinations are different. The total orbital angular momentum quantum number For the molecular case, this number is called \(Λ\) instead of \(L\). It follows the same naming convention as \(L\), except that instead of using capital English letters, it uses capital Greek letters:

- \(Λ = 0 \rightarrow Σ \)
- \(Λ = 1 \rightarrow Π\)
- \(Λ = 2 \rightarrow Δ \)
- \(Λ = 3 \rightarrow Φ \)

Unlike \(L\), there is not a general formula for finding the possible combinations of \(Λ\). You have to examine the individual microstates. This is easier than it sounds.

**The total magnetic quantum number**\(M_L\): \(M_L\) works like \(M_l\) with atoms, except that there is no formula for finding the combinations.**The total spin magnetic quantum number**\(M_S\): \(M_S\) works exactly like \(M_s\). Electrons can either point with or against the z ‐ axis, and being in a molecular orbital versus an atomic orbital doesn’t change this. \(M_S\) can range from \(m_{s1} + m_{s2}\) to \(m_{s1} ‐ m_{s2}\).

## Two new components: parity and reflection

Molecular orbitals are more complex than atomic ones and require more modifiers to completely define. **Parity **(sometimes called “inversion”) tells you if the orbital is *symmetric *or *anti‐symmetric* when an inversion operation is performed. The symmetry notation **u** and **g** are sometimes used when describing molecular orbitals. This refers to the operation of inversion, which requires starting at an arbitrary point in the orbital, traveling straight through the center, and then continuing outwards an equal distance from the center. The orbital is designated **g **(for gerade, even) if the phase is the same, and **u** (for ungerade, uneven) if the phase changes sign.

To determine whether or not a given state is \(g\) or \(u\), find the parity of each individual open‐shell electron and uses these simple (Laporte rules):

- \(g + g \rightarrow g \)
- \(g + u \rightarrow u \)
- \(u + u \rightarrow g \)

Reflection determines if a given orbital is symmetric or anti‐symmetric upon reflection through a plane that contains both nuclei. The choice of symmetry planes is arbitrary. As long as you pick a plane and stick with it, you will always get the right answer. When an orbital is symmetric, it is labeled +. When an orbital is anti ‐ symmetric, it is labeled ‐ . To find the overall reflection of a state, use these rules:

- (+)(+) \rightarrow +
- (+)(‐) \rightarrow ‐
- (‐)(‐) \rightarrow +

**Reflection only applies to Σ states!** For Λ > 0, there are no reflection labels! If you experiment with the rules, you will quickly realize why this is the case.

## Contributors

Mattanjah de Vries (Chemistry, University of California, Santa Barbara)