For heteronuclear diatomic molecules, the u/g symbol does not correspond to any exact symmetry of the electronic molecular Hamiltonian.
In linear molecules, however, the lack of spherical symmetry destroys the relationship
Since these operators commute with each other and with the Hamiltonian on the limit of negligible spin-orbit coupling, their eigenvalues may be used to describe a molecule state through the quantum numbers S, MS, ML and MJ.
The cylindrical symmetry of a linear molecule ensures that positive and negative values of a given
for an electron in a molecular orbital will be degenerate in the absence of spin-orbit coupling.
Now, the total z-projection of L can be defined as As states with positive and negative values of ML are degenerate, we define and a capital Greek letter is used to refer to each value: Λ = 0, 1, 2, 3... are coded as Σ, Π, Δ, Φ... respectively (analogous to S, P, D, F for atomic states).
The molecular term symbol is then defined as and the number of electron degenerate states (under the absence of spin-orbit coupling) corresponding to this term symbol is given by: Spin–orbit coupling lifts the degeneracy of the electronic states.
This is because the z-component of spin interacts with the z-component of the orbital angular momentum, generating a total electronic angular momentum along the molecule axis Jz.
Choosing the pair of values with ML positive will give a Ω = 3/2 for that state.
With this, a level is given by Note that Ω can have negative values and subscripts r and i represent regular (normal) and inverted multiplets, respectively.
Approximating the spin–orbit Hamiltonian to first order perturbation theory, the energy level is given by where A is the spin–orbit constant.
Including higher-order effects can lead to a spin-orbital levels or energy that do not even follow the increasing value of Ω.
This degeneracy is lifted when spin–orbit interaction is treated to higher order in perturbation theory, but still states with same |MS| are degenerate in a non-rotating molecule.
Taking the molecular center of mass as origin of coordinates, consider the change of all electrons' position from (xi, yi, zi) to (−xi, −yi, −zi).
If the resulting wave function is unchanged, it is said to be gerade (German for even) or have even parity; if the wave function changes sign then it is said to be ungerade (odd) or have odd parity.
For a molecule with a center of inversion, all orbitals will be symmetric or antisymmetric.
An alternative method for determining the symmetry of an MO is to rotate the orbital about the axis joining the two nuclei and then rotate the orbital about a line perpendicular to the axis.
Witmer proposed rules to determine the possible term symbols for diatomic molecular states formed by the combination of a pair of atomic states with given atomic term symbols.
Electronic states are also often identified by an empirical single-letter label.
In polyatomic molecules (but not in diatomic) it is customary to add a tilde (e.g.