# Molecular Term Symbols

Molecular term symbols specify molecular electronic energy levels. Term symbols for diatomic molecules are based on irreducible representations in linear symmetry groups, derived from spectroscopic notations. They usually consist of four parts: spin multiplicity, azimuthal angular momentum, total angular momentum and symmetry. All molecular term symbols discussed here are based on Russel-Saunders coupling.

### Introduction

Molecular term symbols mark different electronic energy levels of a diatomic molecule. These symbols are similar to atomic term symbols, since both follow the Russell-Saunders coupling scheme. Molecular term symbols employ symmetry labels from group theory. The possibility of an electronic transition can be deducted from molecular term symbols following selection rules. For multi-atomic molecules, symmetry labels play most of term symbols' roles.

For homonuclear diatomics, the term symbol has the following form:

\[ ^{2S+1}\Lambda_{\Omega,(g/u)}^{(+/-)} \]

whereas Λ is the projection of the orbital angular momentum along the internuclear axis: Ω is the projection of the total angular momentum along the internuclear axis; *g*/*u* is the parity; and +/− is the reflection symmetry along an arbitrary plane containing the internuclear axis. Λ may be one of the greek letters in the sequence: Σ Π Δ Φ... when Λ = 0, 1, 2, 3..., respectively. For heteronuclear diatomics, the term symbol does not include the *g*/*u* part, for there is not inversion center in the molecule.

### Determining term symbols of diatomics

Let's start with CO again. As we have seen before, the molecule has a close-shell configuration. Its ground state is a totally symmetric singlet, ^{1}Σ^{+}, since the only possible values of (*S*, Λ) are (0, 0). If one of the HOMO electrons on the 5σ^{+} orbital has jumped to the LUMO, this molecule will be in an excited state as follows.

Suppose a CO molecule is in the excited state shown above. In order to know the term symbol of this state, a direct product of the labels is required for the two MO's with unpaired electrons. The multiplication is such as \(\Pi \times \Sigma^+ = \Pi\). According to Pauli's exclusion rule, these two unpaired electrons can never share the same set of quantum numbers, therefore the spin degeneracy *S* can reach its maximum 3. The resulting term symbols are ^{1}Π and ^{3}Π.

Now if we look at O_{2}, it does not have a close-shell configuration at its ground state. There are two unpaired electrons each occupying one of the two degenerate 2π orbitals, which can be seen in the diagram below.

The term symbol for oxygen molecule at its ground state is therefore derived such as Π x Π = Σ^{+} + Σ^{-} + [Δ], as the symbol in brackets does not allow the oxygen atoms to commute.

### Transition between electronic states of diatomics

We'll focus on selection rules. Like atomic electronic states, different selection rules apply when differently incurred transitions occur. Usually for electric dipole field induced transitions, the selection rules are the same as for atoms.

- Δ
*Λ*= 0, ±1 except*Λ*= 0 ‡*Λ*' = 0 - Δ
*S*= 0 - Δ
*Ω*= 0; ±1 except*Ω*= 0 ‡*Ω*' = 0

### References

- Harris, Daniel C. (republished in 1989).
*Symmetry and Spectroscopy: An Introduction to Vibrational and Electronic Spectroscopy.*Dover Publications. pp. 421-478. ISBN 0-486-66144-X - D. J. Willock (2009).
*Molecular Symmetry*. John Wiley & Sons Ltd. ISBN 0-470-85348-4

### Contributors

- Anda Zheng (UCD) and Sudarson S Sinha (TAU)