The electrons revolving on an orbital generate an angular moment.

M_{L} is the quantic number associated to the projection of L on the internuclear axis.

The projection is degenerated because it can either be in the positive values of the z axis or in the negative ones. The projection of L can thus give M_{L} or –M_{L}. We can define a new quantic number L=∣M_{L}∣.

The fact that there are two projections explains the dimension 2 or the orbitals π, δ, … of linear molecules. The length of the projection of L grow if L gets larger but the degeneration is always 2. The only exception is Σ for which M_{L}=0. In the atoms we had the possibility to choose the orientation but we cannot do that with molecules.

As a result, the operator σ_{v} doesn’t commute with L_{z}:

except for L=0 that gives the states Σ^{+} and Σ^{ –}. This distinction + or – is not present for the states Π, Δ, …

The fact that σ_{v} doesn’t commute with L_{z} induces the degeneration of the states.

The inversion operator Î still commutes with Ĥ, L_{z} and σ_{v} in the case of centrosymmetric molecules.

As a resume, the operator σ_{v} is separated from the other operators of the CSCO of linear molecules because it does not commute with L_{z} anymore.

**Application to H _{2}**

Let’s take a look at the possible electronic configurations of the molecule H_{2}.

The unexcited state of H_{2} has two equivalent electrons on the ground orbital 1σ_{g}:

This state is binding: between the nuclei, the probability of presence of electrons is positive. In antibonding states, there are some points between the nuclei where the probability to find electrons is zero.

The first excited state is the configuration 1σ_{g}1σ_{u}. In this configuration the electrons are not equivalents and the degree of degeneration is 4 (2×2):

To determine the states, we take the emergent (M_{L}=0,M_{S}=1). It corresponds to the triplet ^{3}Σ_{ u}^{ +} (the pairs (0,1), (0,0) and (0,-1)). The second emergent (0,0) corresponds to the singlet state ^{1}Σ_{ u}^{ +}(the pair (0,0)).

To determine the energy of the states, we proceed as for the atoms with the determinants of Slater

In the case of an emergent such as (0,1) we have a proper function and thus the energy can be determined. From this function we find the other functions with operators of rise/descent S_{+} and S_{–} and with the principle of orthogonality we find the energy of the singlet state.

The dashed curves are correspond to unstable states because there is no minimum of energy: the atoms are get more stable as they get away one from each other.

**Application to O _{2}**

The same method can be applied to O_{2}. Its fundamental configuration is

As usual, we only consider the highest occupied molecular orbitals (HOMO). Those are the 2 1π_{g} orbitals with 2 electrons to place, represented above by the circles: they can be on the same orbital or separated. There is thus a degeneration of 6:

The first emergent is (2,0). That corresponds to the group 1Δ_{g}. Remember that the degeneration for linear molecules is 2 except for M_{L}=0. In the atomic case, we had a degeneration of 2L+1 (i.e. M_{L}, M_{L-1}, …, 0, …, –M_{L-1}, -M_{L}) but here we just have M_{L} and –M_{L}. The next emergent is (0,1), a triplet ^{3}Σ_{g}. Finally there is a singlet ^{1}Σ_{g}. To know if it is Σ^{+} or Σ^{–}, we have to apply the operator σ_{v}. In this case we have ^{3}Σ_{g}^{–} and ^{1}Σ_{g}^{+}.

Ozone is obtained by the excitation of one molecule of O_{2} at its fundamental state into two oxygen atoms in the ^{3}P state. In this state, they react with another molecule of O_{2} and a catalyst to produce the ozone.