Atomic Term Symbols and LS Coupling Scheme

Core Idea

Term symbols ²ˢ⁺¹Lⱼ encode the electronic configuration of an atom: 2S+1 is the multiplicity (spin degeneracy), L is the total orbital angular momentum (S, P, D, F, ...), and J is the total angular momentum. In LS coupling, individual electron spins and orbital momenta couple to form L and S, which then couple via spin-orbit interaction to form J. Term symbols predict level structure and allowed transitions.

How It's Best Learned

Derive term symbols for simple atoms (He, Li, C) using Russell-Saunders rules. Predict J-levels and their relative energies. Use term symbols to apply selection rules (ΔL = 0,±1, ΔS = 0, ΔJ = 0,±1) for allowed transitions.

Common Misconceptions

Term symbols apply to the whole atom, not individual electrons (electron labels are not meaningful in the quantum case). The ordering of J-levels (normal vs inverted multiplicity) requires knowledge of whether shells are less than or more than half full.

Explainer

You already know that each electron in an atom is assigned four quantum numbers: the principal quantum number n, the orbital angular momentum quantum number l (0, 1, 2, 3 → s, p, d, f), the magnetic quantum number m_l (ranging from −l to +l), and the spin quantum number m_s (±1/2). The Pauli exclusion principle tells you no two electrons in the same atom can share all four. LS coupling (also called Russell-Saunders coupling) goes one step further: it combines all the electrons' individual angular momenta into collective quantum numbers that characterize the atom as a whole.

In LS coupling, the individual orbital momenta l⃗ᵢ couple together to give a total orbital angular momentum L⃗ = Σ l⃗ᵢ, with quantum number L = 0, 1, 2, 3, ... (labeled S, P, D, F, ... following the same letter convention as single-electron states). Simultaneously, the individual spins s⃗ᵢ couple to give total spin S⃗ = Σ s⃗ᵢ, with quantum number S = 0, 1/2, 1, 3/2, ... The multiplicity 2S+1 counts the number of distinct m_S values and gives the number of levels the spin degeneracy splits into. Finally, L⃗ and S⃗ couple via spin-orbit interaction to give the total angular momentum J⃗, with J ranging from |L − S| to L + S in integer steps. The term symbol ²ˢ⁺¹Lⱼ compactly encodes all three: multiplicity, total orbital angular momentum, and total angular momentum.

To build intuition, consider carbon (1s² 2s² 2p²). The closed 1s² and 2s² subshells contribute L = 0 and S = 0, so only the two 2p electrons matter. Each has l = 1, so L can be 0, 1, or 2. Each has s = 1/2, so S can be 0 or 1. But not all combinations are allowed — the Pauli exclusion principle restricts which (m_l, m_s) pairs can both be occupied. Applying the rules carefully yields the terms ³P (the ground term), ¹D, and ¹S in order of energy. The ³P term (S = 1, L = 1) further splits into ³P₀, ³P₁, and ³P₂ as J takes values 0, 1, 2. For shells less than half-full, the lowest J level lies lowest in energy (normal multiplicity); for shells more than half-full, the highest J is lowest.

Term symbols are indispensable for spectroscopy because selection rules for dipole-allowed transitions are stated in terms of them: ΔL = 0 or ±1, ΔS = 0, ΔJ = 0 or ±1 (but J = 0 → J = 0 is forbidden). A transition from ³P₁ to ¹S₀ is forbidden by ΔS = 0; a transition from ³P₂ to ³D₃ is allowed. These rules explain why certain spectral lines appear in atomic spectra and others are absent — they are the quantum-mechanical statement that the atom and emitted photon must together conserve angular momentum. Mastering term symbols transforms spectral line tables from empirical catalogs into a deducible consequence of quantum mechanics.