Molecular Term Symbol - Λ Quantum Number

Λ Quantum Number

For atoms, we use S, L, J and M_J to characterize a given state. In linear molecules, however, the lack of spherical symmetry destroys the relationship, so L ceases to be a good quantum number. A new set of operators have to be used instead:, where the z-axis is defined along the internuclear axis of the molecule. 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, M_S, M_L and M_J.

The cylindrical symmetry of a linear molecule ensures that positive and negative values of a given m_l for an electron in a molecular orbital will be degenerate in the absence of spin-orbit coupling. Different molecular orbitals are classified with a new quantum number, λ, defined as

λ = |m_l|

Following the spectroscopic notation pattern, molecular orbitals are designated by a smallcase Greek letter: for λ = 0, 1, 2, 3,... orbitals are called σ, π, δ, φ... respectively.

Now, the total z-projection of L can be defined as

As states with positive and negative values of M_L are degenerate, we define

Λ = |M_L|,

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. The molecular term symbol is then defined as

2S+1Λ

and the number of electron degenerate states (under the absence of spin-orbit coupling) corresponding to this term symbol is given by:

• (2S+1)×2 if Λ is not 0
• (2S+1) if Λ is 0.