Screening and shell structure

In any central field the Hamiltonian commutes with L² and L_z, so eigenstates carry the labels l and mₗ and every level keeps its 2l+1-fold m-degeneracy. That follows from rotational invariance alone, and screening cannot touch it. Hydrogen has more than that. For an exact 1/r potential the Runge–Lenz vector M = (p × L − L × p)/2mₑ − e²r̂/4πε₀ is also conserved, and on the E < 0 subspace the rescaled M and L close into so(4); its multiplets are labelled by n = nᵣ + l + 1 alone, which is why 3s, 3p and 3d coincide in hydrogen. Screening makes Z_eff depend on r, the potential is no longer 1/r, [H, M] ≠ 0, and the extra symmetry is gone. What remains is so(3), so E = Eₙₗ. Nothing about the split is small: in sodium the 3s and 3d terms lie 5.139 eV and 1.522 eV below the ionisation limit, a gap of 3.62 eV — more than five times hydrogen's whole n = 3 to n = 4 spacing of 0.661 eV.

Near the origin the centrifugal term dominates the radial equation, and the regular solution behaves as uₙₗ(r) ∝ r^(l+1); the second Frobenius root r^(−l) is rejected by the boundary condition u(0) = 0. The weight a state carries inside a core of radius r_c is therefore ∫₀^(r_c) r^(2l+2) dr = r_c^(2l+3)/(2l+3) — two extra powers of r_c for every unit of l. With r_c ≈ 0.5 a₀ in sodium that is a factor of four lost per unit of l, and it is precisely the amplitude in that region which samples an unscreened charge climbing towards +11 instead of +1. Read the same fact through the barrier: ħ²l(l+1)/2mₑᵣ² at r = 0.5 a₀ is 12 hartree, 327 eV, for l = 2 and 4 hartree for l = 1, against nothing at all for l = 0. Be careful with the argument, though. Hydrogen carries the identical barrier and shows no l-splitting whatever. What creates a defect is the barrier combined with a Z_eff(r) that varies across the very region the barrier excludes.

The quantum defect is measured, not postulated. Fit sodium's series to Eₙₗ = −13.606 eV/(n − δₗ)²: the sharp series (ns → 3p) returns δₛ ≈ 1.37, the principal series (np → 3s) gives δₚ = 0.883 at n = 3, 0.866 at n = 4 and 0.861 at n = 5, and the diffuse series (nd → 3p) gives δ_d = 0.010. Two things deserve attention. First, δₗ barely moves with n — it falls 2.5% from 3p to 5p — because it describes the core the electron crosses, not the orbit; quantum-defect theory sharpens this by identifying πδₗ with the threshold scattering phase shift of the core, and what residual drift is left is core polarisation, the valence electron's own field distorting the core it flies past, which a Rydberg–Ritz expansion δ(n) = δ₀ + δ₁/(n − δ₀)² absorbs. Second, δ_d ≈ 0.01 says a sodium 3d electron is hydrogenic to better than 1%, a claim about geometry rather than about nuclear size. For δₗ from a model rather than a fit, put Z_eff(r) on a radial grid of a few thousand points, assemble the tridiagonal radial Hamiltonian in numpy, and hand it to scipy.linalg.eigh_tridiagonal; the eigenvalues invert straight back to δₗ.

A subshell (n, l) holds 2(2l+1) electrons: 2l+1 values of mₗ times two spin projections, with antisymmetry forbidding any repeat. Group the subshells by n + l, breaking ties by increasing n, and the Madelung order appears: 1s | 2s | 2p 3s | 3p 4s | 3d 4p 5s | 4d 5p 6s | 4f 5d 6p 7s. The group with n + l = k contains l = 0 up to m − 1, where m = ⌈k/2⌉, because l ≤ n − 1 forces n ≥ l + 1, and its capacity is Σ 2(2l+1) = 2m². A period opens at an ns and closes at the np of the group above, and since an ns and an (n+1)s both carry 2, the length of period p is the capacity of group k = p + 1: 2, 8, 8, 18, 18, 32, 32 for the seven periods that exist. The repeats are the pairing k = 2j − 1 and k = 2j sharing m = j; only the leading 2 stands alone, because the group that would pair with it is the 1s that opens the table rather than closing a period. Notice what 2m² is not: it is a group capacity, not the 2n² of a hydrogenic shell. Both are the same sum Σ2(2l+1), but hydrogen's runs over l = 0…n−1 at fixed n while this one runs over l = 0…m−1 at fixed n + l — which is why period lengths repeat and hydrogenic shell capacities never do.

About twenty of the first 103 elements disobey the n + l rule, and every one of them sits in the d or f block: chromium is [Ar]3d⁵4s¹ rather than 3d⁴4s², copper is [Ar]3d¹⁰4s¹, niobium is [Kr]4d⁴5s¹, and palladium drops the 5s subshell altogether at [Kr]4d¹⁰. No s- or p-block element breaks it, and that pattern is the clue: the exceptions appear only where two configurations are already nearly degenerate in total energy — in nickel the lowest 3d⁹4s¹ level lies just 205 cm⁻¹, 0.025 eV, above the 3d⁸4s² ground level. Two effects the rule never mentions then decide the winner. One is exchange: each pair of parallel-spin d electrons gains an exchange integral K, and 3d⁵4s¹ has ten such pairs against six for 3d⁴4s². The other is the compactness of 3d, where ⟨1/r₁₂⟩ between two d electrons is far larger than between two diffuse 4s electrons, so double occupancy of 3d is expensive. Both are two-electron quantities, invisible to any ordering of one-electron levels — which is the honest description of the Madelung rule: an empirical regularity of measured ground states with no derivation from the Schrödinger equation behind it.

What the shell model buys is a reason for the columns. Lithium, sodium, potassium, rubidium and caesium share an ns¹ valence configuration outside a closed core, and their first ionisation energies — 5.39, 5.14, 4.34, 4.18 and 3.89 eV — span just 1.5 eV while Z runs from 3 to 55. The nuclear charge changes by a factor of eighteen and the chemistry does not, because the valence electron still sees a screened charge near +1 and a fixed l. Halogens repeat ns²np⁵, noble gases close at np⁶, and the table becomes a statement about Z_eff and l rather than about Z. Two limits bite. Screening says nothing about correlation, and a single determinant misses real energy — 1.14 eV in helium alone, where Hartree–Fock gives −2.8617 Ha against −2.9037 Ha exact — though cancellation between atom and ion leaves Koopmans ionisation energies only a few tenths of an electron-volt out. And in heavy atoms the 1s speed is of order Zαc — 0.58c in gold — so relativistic contraction pulls 6s down; gold's 5d → 6s gap falls near 2.4 eV, absorbing blue at 517 nm and leaving the metal yellow, while mercury's contracted, closed 6s² leaves it liquid at 234 K. Neither follows from Schrödinger; both need Dirac.