LS coupling and term symbols

The self-consistent central field has already swallowed the spherical average of the electron-electron repulsion. Two things are left over: the non-spherical remainder of that repulsion, and the spin-orbit interaction, a sum of terms in l⋅s for each electron. Both act on a configuration whose states are still degenerate — 2p² carries fifteen of them before either switches on — and degenerate perturbation theory does not let you choose an order. The dominant term must be diagonalised first, because only the basis it picks out makes the other one a genuine small correction. In carbon the ordering is not close: repulsion spreads 2p² over 21648 cm⁻¹ while spin-orbit moves things by tens of cm⁻¹, a ratio near 500. In lead the same two are within a couple of per cent of each other, and the question has no clean answer at all.

For non-equivalent electrons every combination occurs. The excited configuration 2p3p holds 6 × 6 = 36 states and gives six terms — ¹S, ³S, ¹P, ³P, ¹D, ³D — carrying 1 + 3 + 3 + 9 + 5 + 15 = 36 states between them. Equivalent electrons are different. Two 2p electrons share six spin-orbitals, so there are only C(6,2) = 15 antisymmetric states, and the Pauli principle keeps just the terms with L + S even: ¹S, ³P and ¹D, carrying 1 + 9 + 5 = 15. That is not a convention to memorise; it is the same Slater determinant that gives exclusion, doing more work than the slogan about two electrons in one state. Always count states before and after: if the term list does not reproduce the configuration's state count, a term has been invented or lost.

Within a term L and S are fixed, so squaring J = L + S gives L⋅S = ½[J(J+1) − L(L+1) − S(S+1)]ℏ² and the spin-orbit energy collapses to one constant: E = (A/2)[J(J+1) − L(L+1) − S(S+1)]. Take ³P, with L = S = 1. Then E(J=0) = −2A, E(J=1) = −A and E(J=2) = +A, so the gaps are A and 2A — the Landé rule, E(J) − E(J−1) = A J, with each gap proportional to the upper J. Carbon's first gap is 16.40 cm⁻¹, so the rule places ³P₂ at 16.40 + 32.80 = 49.20 cm⁻¹ above ³P₀. Measurement says 43.40, so the rule overshoots by 5.80 cm⁻¹, 13% of the measured spread. That failure is informative, and it is not term mixing: spin-orbit does couple ³P₂ to ¹D₂ and ³P₀ to ¹S₀, but with ζ about 30 cm⁻¹ against gaps of 10⁴ cm⁻¹ those second-order shifts, of order ζ²/ΔE, stay below 0.1 cm⁻¹. The culprit is the magnetic spin-spin interaction between the two electrons, which inside a term goes as (L⋅S)² as well as L⋅S, so it bends the ladder even where LS coupling is excellent.

A is positive for a subshell less than half filled and negative past half filling, which is Hund's third rule stated in one symbol. Carbon's 2p² puts two electrons in six places, so its ³P is normal and ³P₀ is the ground level. Oxygen's 2p⁴ puts in four, so the same ³P is inverted: ³P₂ sits at 0, ³P₁ at 158.27 cm⁻¹ and ³P₀ at 226.98 cm⁻¹. The interval rule still applies, now read downward — the gaps 158.27 and 68.71 cm⁻¹ stand in the ratio 2.30 against the predicted 2. What never moves is the degeneracy-weighted mean. For ³P, 1(−2A) + 3(−A) + 5(+A) = 0, so the split levels straddle the unsplit term exactly. That sum is the fastest check on any level scheme you write down.

The spin-orbit parameter ζ of a valence electron is picked up mostly inside the core, where the electron sees nearly the full nuclear charge, so it grows roughly as Z² down a group while the electrostatic parameters barely change, and the ratio of spin-orbit to repulsion climbs steeply. Compare the group-14 np² sequence. Carbon's whole ³P spread is 43.40 cm⁻¹ against a ³P-to-¹D separation of 10149.23, a ratio of 0.0043. Lead's lowest triplet spreads over 10650.33 cm⁻¹ against a 10807.47 cm⁻¹ gap to the next J = 2 level: a ratio of 0.985. Lead's first two gaps, 7819.26 and 2831.07 cm⁻¹, stand in the ratio 0.36 where the interval rule demands 2, so the LS labels have stopped describing anything. In the jj limit each electron's own l and s couple first into j, and the j add to J. What survives the change of scheme is J and parity; L and S were only ever names for a choice of basis.

A measured group of close levels can be decoded without calculating anything about the atom. Count the levels: that number is min(2S+1, 2L+1). Then compare successive gaps, because the interval rule makes them proportional to the J values of the upper members, and those climb in steps of one. A five-level group whose gaps stand as 2 : 3 : 4 : 5 therefore has J = 1, 2, 3, 4, 5, so |L − S| = 1 and L + S = 5. That pins the pair 3 and 2 but not which is which: ⁵F, with L = 3 and S = 2, and ⁷D, with L = 2 and S = 3, give identical ladders, and a Landé g factor or the configuration has to decide. The state count, 3 + 5 + 7 + 9 + 11 = 35 = 7 × 5, checks either. And if the ratios come nowhere near consecutive integers, like lead's 0.36 where 2 was due, that is not a bad measurement; it is the spectrum telling you this atom is not LS coupled.