Term Symbols and Russell–Saunders Coupling, Step by Step

What a term symbol encodes

S is the total spin quantum number, so 2S+1 is the spin multiplicity. L is the total orbital angular momentum, written as a letter: L = 0 is S, 1 is P, 2 is D, 3 is F, 4 is G. J is the total angular momentum, obtained by combining L and S.

The symbol therefore states, compactly, how many unpaired electrons a state has and how their orbital and spin momenta are arranged. That is what spectroscopic selection rules operate on, which is why the notation exists.

Finding the ground-state term — the five steps

1. Write the electron configuration and identify the partly filled shell. Filled shells contribute nothing, since all their momenta cancel.
2. Place electrons in the orbitals of that shell to maximise total spin, filling singly with parallel spins before pairing.
3. Given that spin arrangement, place them to maximise the sum of m_l. That maximum sum is L.
4. Compute S as half the number of unpaired electrons, and write the multiplicity 2S+1.
5. Determine J: it is |L − S| if the shell is less than half filled, and L + S if it is more than half filled. For an exactly half-filled shell L is zero, so J equals S.

Worked examples

d¹

One electron, so S = 1/2 and multiplicity is 2. It occupies the orbital with m_l = +2, so L = 2, which is D. The shell is less than half filled, so J = |2 − 1/2| = 3/2. The ground term is ²D_3/2.

d⁵ high spin

Five unpaired electrons give S = 5/2 and multiplicity 6. Each of the five d orbitals holds one electron, so the m_l values +2, +1, 0, −1, −2 sum to zero, giving L = 0, which is S. With L = 0, J = S = 5/2. The term is ⁶S_5/2.

This is the term symbol for a half-filled d shell, and its L of zero is why such ions show no orbital contribution to their magnetic moment — a connection frequently asked.

d⁹

One vacancy behaves like one electron with reversed sign, so d⁹ gives the same L and S as d¹: S = 1/2, L = 2. But the shell is more than half filled, so J = L + S = 5/2, giving ²D_5/2. The hole formalism makes several configurations quick once recognised.

Why it matters beyond the notation

Term symbols are the labels used in Orgel and Tanabe–Sugano diagrams, which predict the electronic spectra of coordination complexes. Selection rules are stated in terms of changes in S, L and J: a transition is spin-allowed only if S does not change, which is why spin-forbidden bands are so weak. Without term symbols those rules cannot be stated at all.

Frequently asked questions

Why do filled shells contribute nothing?

Because for every electron with a given m_l and spin there is another with the opposite values, so both the orbital and spin momenta cancel exactly. Only partly filled shells need considering.

When does Russell–Saunders coupling break down?

For heavy elements. It assumes spin–orbit coupling is weak compared with the electrostatic interactions between electrons, which fails for heavy atoms where jj coupling becomes the better description.

How do I find excited terms, not just the ground one?

By enumerating all allowed microstates for the configuration and grouping them. It is laborious for configurations with several electrons, which is why standard tables of terms exist and are usually supplied.

What is the connection to magnetic moment?

The full expression for magnetic moment includes both spin and orbital contributions. The spin-only formula works when the orbital contribution is quenched, which happens in many first-row complexes but not in lanthanides — where term symbols are essential for predicting moments correctly.