The periodic table from shell structure

An orbital is fixed by n, l and mₗ, and an electron in it by one further label, mₛ = ±½. For a given l there are 2l + 1 values of mₗ, so a subshell holds Nₗ = 2(2l + 1) electrons: 2 for s, 6 for p, 10 for d, 14 for f. Nothing in the Schrödinger equation forbids a third 2s electron — antisymmetry does. Write the state as a Slater determinant of one-electron spin-orbitals and it vanishes identically the moment two columns repeat, so each set of labels is used once and once only. Sum over l = 0 to n − 1 and a principal shell holds 2n²: 2, 8, 18, 32. Those are the only capacities on offer, which is why the periodic table has exactly four block widths and not, say, a block of seven. Below Z = 118 only l = 0, 1, 2 and 3 are ever occupied in a ground state, so s, p, d and f exhaust the vocabulary the table is written in.

A row opens at ns and closes at np, and what fills between them is set by increasing n + l, lower n first on a tie. Period 4 therefore reads 4s (n + l = 4), then 3d (5), then 4p (also 5, but 4 > 3, so 3d goes first): 2 + 10 + 6 = 18 elements, potassium to krypton. Period 6 reads 6s (6), 4f (7), 5d (7), 6p (7): 2 + 14 + 10 + 6 = 32, caesium to radon. Now look at what those lists have in common. Each is exactly one subshell of every l from 0 up to some lₘₐₓ, drawn from whichever n the ordering happens to supply. So the length is Lₖ = Σ 2(2l + 1) over l = 0 to lₘₐₓ, which is 2(lₘₐₓ + 1)², with lₘₐₓ = ⌊k/2⌋. Feeding k = 1 to 7 gives 2, 8, 8, 18, 18, 32, 32. The repetition is not a coincidence to be memorised: lₘₐₓ advances only on every second period, because each l first appears at n = l + 1, on the diagonal n + l = 2l + 1, so successive new l values sit two diagonals apart while each period advances by one. The formula also predicts forward — period 8 would have lₘₐₓ = 4, an 18-wide g block, and 2 × 5² = 50 elements.

In hydrogen 3s, 3p and 3d are degenerate and the n + l rule would be meaningless. In a screened field they are not. A low-l radial function has inner lobes that reach inside the core, where the electron sees close to the bare nuclear charge; a high-l orbital is held out by the centrifugal term ℏ²l(l + 1)/2mr², sits outside the screening cloud, and sees something near Z minus the core count. At potassium the 4s orbital, despite its larger principal number, is bound more tightly than 3d, and the nineteenth electron goes into 4s. That single competition is the whole reason the d block is inset by one period and the f block by two. It is also a competition barely won: across the 3d series the 4s and 3d levels sit within about an electronvolt of each other, which is why the order already fails at chromium ([Ar]3d⁵4s¹) and copper ([Ar]3d¹⁰4s¹), and why once d electrons are present the 3d level drops below 4s, so transition-metal cations lose their 4s electrons first — Fe²⁺ is [Ar]3d⁶, not [Ar]3d⁴4s².

Two terms compete in every trend. Across a period Z rises by one per step, but the new electron joins a shell already occupied, where Slater's rules charge it only 0.35 of a unit of screening against its neighbours. So Z_eff climbs by roughly 0.65 per element — 1.30 at lithium to 5.85 at neon — while n stays at 2. The orbital contracts and binds harder: covalent radii fall from 128 pm at lithium to 57 pm at fluorine, and the first ionisation energy rises from 5.39 eV to 21.56 eV, a factor 4.0. Down a group the other term wins. A new shell adds a radial node and puts the valence electron outside a much larger core, and the extra protons are screened almost completely — 0.85 per n − 1 electron, 1.00 per deeper one. Lithium, sodium and potassium have radii 128, 166 and 203 pm and ionisation energies 5.39, 5.14 and 4.34 eV. Keep the model honest about magnitudes, though: Z_eff rises by a factor 4.5 across period 2 while the measured radius falls by only about 2.2, so r ≈ n*²a₀/Z_eff gets the direction and the mechanism right and the size wrong.

The rise across a row is not monotonic, and the two kinds of stumble have different sizes and different causes. The first is a change of subshell. Beryllium is 2s² and boron is 2s²2p¹, so boron's outermost electron leaves a 2p level that penetrates the core less than 2s and is screened by the filled 2s² pair: the first ionisation energy falls from 9.32 eV to 8.30 eV. Magnesium to aluminium repeats it, 7.65 eV down to 5.99 eV. The second is the first forced spin pairing. Nitrogen is 2p³ with one electron in each of the three 2p orbitals and their spins parallel; oxygen is 2p⁴ and must put a second electron into one of them, and that pairing repulsion is refunded when an electron leaves: 14.53 eV falls to 13.62 eV. Phosphorus to sulfur repeats it, 10.49 eV to 10.36 eV. Compare the scales in period 3: 1.66 eV for the subshell step against 0.13 eV for the pairing step, a ratio of about 13. That both breaks recur in period after period is the evidence that block structure is a real feature of the level scheme.

All of this rests on orbital energies, and Koopmans' theorem tempts you to read them straight off as ionisation energies, IEᵢ ≈ −εᵢ. The identification assumes the other N − 1 orbitals stay exactly where they were. They do not: they contract around the hole, lowering the ion's energy and therefore the true ionisation energy, so −ε is an overestimate, and the deeper the hole the larger the relaxation. For neon, −ε(2p) = 23.13 eV against a measured 21.565 eV, an error of 1.57 eV; −ε(1s) = 891.8 eV against 870.2 eV, an error of 21.6 eV. Correlation pushes the other way, since the neutral correlates more strongly than the cation, and the partial cancellation is why valence errors stay at a few per cent. Relativity moves the levels again in heavy atoms: gold's 6s contracts by about 18%, and francium's first ionisation energy, 4.07 eV, is higher than caesium's 3.89 eV, reversing the group-1 trend the geometry alone would predict. The lanthanide contraction leaves hafnium the same size as zirconium, 32 protons later. The table's combinatorics are exact; its energetics are not.