Many-electron atoms and the exclusion principle

For N electrons the Hamiltonian is a sum of hydrogen-like terms, pᵢ²/2mₑ − Ze²/(4πε₀rᵢ) for each electron, plus e²/(4πε₀rᵢⱼ) for every pair. The pair term depends on two electrons' coordinates at once, so the equation no longer separates. Helium shows how big it is. Dropping it puts both electrons in hydrogen-like 1s states with Z = 2, at −2 × 4 × 13.6 eV = −108.8 eV; the measured ground-state energy, minus the sum of both ionisation energies, is −79.0 eV. Averaging the repulsion over that unperturbed state gives (5/4)Z × 13.6 eV = 34.0 eV, and a first-order energy of −74.8 eV. Better, let each electron see a trial charge Z_eff and minimise the energy: the optimum is Z_eff = Z − 5/16 = 1.69, giving −77.5 eV, within 2% of the measurement. That optimum is the lesson. Each electron behaves as if the other cancels 0.31 of a proton: screening, measured by an energy.

Electrons are identical, so swapping two labels cannot change any probability: |ψ(x₂, x₁)|² = |ψ(x₁, x₂)|², where each x includes spin. For electrons, as for every half-integer-spin particle, nature takes the minus sign: ψ(x₂, x₁) = −ψ(x₁, x₂). Build ψ from one-electron states a and b as [a(x₁)b(x₂) − b(x₁)a(x₂)]/√2 and set a = b: ψ vanishes. That is Pauli's rule, no two electrons sharing all of n, l, mₗ and mₛ, and it is a consequence of symmetry, not an extra force. The same sign has a second effect. Two electrons with parallel spins have a symmetric spin part, so their spatial part must be antisymmetric, and it vanishes wherever r₁ = r₂: they keep apart and repel less. Helium's 1s2s configuration measures it. The triplet ³S sits 19.82 eV above the ground state and the singlet ¹S at 20.62 eV, a 0.80 eV splitting produced by the Coulomb force alone. This exchange energy is the origin of Hund's first rule.

The central-field approximation replaces the pair terms by what one electron feels on average from all the others, smeared into a spherically symmetric cloud. Because the cloud is spherical, Gauss's law on a sphere of radius r says only the charge inside r screens the nucleus: the electron feels a force −Z_eff(r)e²/(4πε₀r²) with Z_eff(r) = Z − N(r), falling from Z at the nucleus to 1 outside a neutral atom, and its potential energy U(r) is the integral of that force from r out to infinity. The cloud depends on the orbitals and the orbitals on the cloud, so the field is found self-consistently, iterating until the two agree: the Hartree method. Because U is central, the angular equation is hydrogen's, and Yₗₘ, l and mₗ carry over unchanged. Because U is not 1/r, energy now depends on l. Near the nucleus Rₙₗ grows as rˡ, so an s orbital has density at the nucleus while a d orbital is held out; the s electron penetrates the core and sees more of Z. Sodium's valence levels show it: 3s at −5.14 eV, 3p at −3.04 eV, 3d at −1.52 eV, against −1.51 eV for hydrogen's n = 3. The 3d electron stays outside the core and sees one net charge, as hydrogen's electron does.

Each orbital (n, l, mₗ) takes two electrons of opposite mₛ, so a subshell holds 2(2l + 1). Feeding electrons into the lowest free levels gives an order summarised by the n + l rule: increasing n + l, and for a tie the smaller n first, which reads 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p. It is penetration in bookkeeping form: 4s (n + l = 4) comes before 3d (n + l = 5) because its inner lobes reach through the argon core. The biggest gaps follow each np⁶, so the noble gases close at Z = 2, 10, 18, 36, 54 and 86, not at the 2, 10, 28 and 60 that whole shells of 2n² would give. An open subshell still holds choices. Carbon's two 2p electrons have C(6, 2) = 15 arrangements, all degenerate in the central field. The leftover repulsion splits them into ³P (9 states), ¹D (5) and ¹S (1), and Hund's rules name the lowest: largest S, then largest L, then J = |L − S| for a subshell less than half full, which gives ³P₀.

Across period 2 the first ionisation energy climbs from 5.39 eV (Li) to 21.56 eV (Ne). Each added proton is screened only partly by the added electron, which sits in the same shell, about as far out as the one being removed, so Z_eff = n√(E_I/13.6 eV) rises from 1.26 to 2.52. At sodium the next electron starts n = 3 outside a full n = 2 core, and E_I collapses to 5.14 eV. Two dips inside the period are the model's fingerprints. Boron (8.30 eV) falls below beryllium (9.32 eV) because its electron is in 2p, which penetrates less than 2s. Oxygen (13.62 eV) falls below nitrogen (14.53 eV) because nitrogen's three 2p electrons sit in separate orbitals with parallel spins, the ⁴S₃/₂ term, and oxygen's fourth must pair with one of them, adding repulsion and gaining no exchange. The same two dips recur at aluminium and sulfur in period 3. Down a group E_I changes slowly, Li 5.39, Na 5.14, K 4.34 eV, because the growth of n is nearly offset by the growth of Z_eff: 1.26, 1.84, 2.26.

The n + l rule predicts chromium [Ar]3d⁴4s² and copper [Ar]3d⁹4s²; spectroscopy finds [Ar]3d⁵4s¹ and [Ar]3d¹⁰4s¹. The cause is what the averaging discarded. The central field ranks configurations by one-electron energies in a single averaged potential, but the ground configuration is the one of lowest total energy, which includes the repulsion and exchange among the valence electrons themselves. Across the 3d series the 4s and 3d orbitals lie close in energy, so that neglected part decides. Moving one 4s electron into 3d gives chromium six parallel spins (S = 3, the ⁷S₃ ground term) instead of four, and gives copper a closed 3d¹⁰ shell. The rejected configuration has not vanished: copper's 3d⁹4s² is a real excited level, ²D₅/₂, 1.39 eV above the ground state. The field also changes with the electron count, so iron, which fills 4s before 3d, loses 4s first: Fe²⁺ is [Ar]3d⁶. About twenty elements break the rule, among them Nb, Mo, Pd, Ag, Pt and Au; palladium, [Kr]4d¹⁰, leaves 5s empty.