Term Symbols and Selection Rules

for Electronic Transitions

The symbols for the total electronic state of a molecule are composed of a superscript 1, 2, 3, ... that represents the spin multiplicitiy, a Greek capital letter Σ Π Δ (standing for the projection of the total angular momentum of electrons with respect to the bond's axis) and, for homonuclear molecules, g or u as a subscript. The first index indicates the spin multiplicity 2S+1 where S is the sum of the electrons' spins. For an MO with only one electron, the total spin S = ½ and therefore the multiplicity is 1. The symbols Σ Π Δ denote an angular momentum. Note that the analogy between quantum number l of atoms and Λ is limited to the nature of this quantity; both of them are angular momentums but Λ is just the projection of this momentum on the axis connecting the nuclei. For Λ = 0, we have an state Σ, for Λ = 1, we have state Π and so on. For molecular orbitals with only one electron, this is the projection of λ of the single electron, i.e Σ (Λ = 0) for &sigma (λ = 0) and Π (Λ =1) for π (λ = 1) etc. For molecules with several electrons, the question is whether these electrons are equivalent and the Pauli principle rules their allocation, or, whether they are not equivalent and no further restrictions apply. From full molecular orbitals, there are no contributions to the term scheme presented, therefore, they can be ignored when deriving the term symbol.

Table 1 lists, according to Pauli's principle, the possible molecule terms for given electronic configurations. For example, it is clear that a σ MO occupied by two electrons results to the term 1Σ as the angular momentum of both electron is zero and therefore they must differ with respect to their spin. This leads to a total spin of zero, a spin multiplicity of 1 and a molecule in singlet state.

In contrast, for molecules with one unpaired electron to which a spin of + ½ or - ½ is attributed, we get a duplet state. With two unpaired electrons, we obtain a triplet state in a situation of parallel spin of the electrons (S = 1) and a singulet state where with electrons of

antiparallel spin (S = 0). Decisive with respect to the question whether a singlet or a triplet state is lower in energy, is, provided there are no other differences in electronic configuration, Hund's rule

For molecules with no equivalent orbitals in their electronic configuration, Pauli's principle is without relevance. For example, electrons within two different π molecular orbitals contribute with λ ± 1 to Λ, therefore the values of +1-1 = -1+1 = 0 ≡ Σ and +1+1 = +2 resp. -1-1 = -2 ≡ Δ can result. As there are no restrictions for the combination of the two electrons' spins, total spin S = ½ + ½ = 1 (triplet state) and S = ½ − ½ = 0 (singlet state) are possible. Table 2 offers an

Tab. 1: Terms for equivalent electrons

electronic configuration the molecule's term

σ1 2Σ

σ2 1Σ+

π1 2Π

π2 1Σ+, 3Σ-, 1Δ

π3 2Π

π4 1Σ+

δ2 1Σ+, 3Σ-, 1Γ

δ3 2Δ

δ4 1Σ+

overview.  

Tab. 2: Terms for electrons not equivalent

electronic configuration term symbol for the molecule

σ 2Σ+

π 2Πr

σσ 1Σ+, 3Σ+

σπ 1Π, 3Πr

σδ 1Δ, 3Δr

ππ 1Σ+, 3Σ+, 1Σ-, 3Σ-, 1Δ 3Δr

πδ 1Π, 3Π, 1Φ, 3Φr

δδ 1Σ+, 3Σ+, 1Σ-, 3Σ-, 1Γ, 3Γr

σσσ 2Σ+(2), 4Σ+

σσπ 2Π(2), 4Πr

σσδ 2Δ(2), 4Δr

σππ 2Σ+(2), 4Σ+, 2Σ-(2), 4Σ-, 2Δ(2), 4Δr

σπδ 2Π(2), 2Π, 2Φ(2), 4Φr

πππ 2Π(6), 4Π(3), 2Φ(2), 4Φr

ππδ 2Σ+(2), 4Σ+, 2Σ-(2), 4Σ-, 2Δ(4), 4Δ(2), 2Γ(2), 4Γr

The numbers in brackets indicate how many of these states exist. The subscript r denotes regular (normal) multiplets

Subscript g and u classify the total wave function with respect to an inversion using the middle between both nuclei as centre. Of course, this is only relevant for homonuclear molecules as for heteronuclear diatomic species, no centre of symmetry exists. For u-Term molecules, such an inversion is equivalent to the wave function's multiplication with -1. In contrast, g-term molecules remain unaffected. With the exception of molecules with an odd number of occupied molecular orbitals of u-type, the total wave function is always of g-type.

The superscript ± with Σ states indicates the wave function's change when reflected on a plane containing both nuclei.  

A quick overview on how linear molecules are classified by term symbols is given here. The selection rules (one photon processes) for electronic transitions of these molecules are:  

Allowed transition Examples

Δ Λ = 0, ±1 Σ <=> Σ, Σ <=> Π, Δ <=> Π

Δ S = 0 singlet <=> singlet, triplet <=> triplet

+ <=> + − <=> −

Σ+ <=> Σ+ Σ− <=> Σ−

g <=> u Σ+g <=> Σ+

u, Δg <=> Πu