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.

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+

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 overview.
 

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:
 

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