Configurations, aufbau and Hund's rules

In hydrogen, 2s and 2p are degenerate; nothing in a pure 1/r potential distinguishes them. The self-consistent central field of a many-electron atom does. A low-l orbital keeps more amplitude close to the nucleus, so it penetrates inside the screening cloud of the inner electrons, feels a larger effective charge, and is pulled down relative to the high-l orbitals of the same n. Penetration therefore fights the cost of raising n, and it can win: in potassium the 4s orbital lies about 2.7 eV below 3d, so the nineteenth electron goes into 4s and the configuration is [Ar] 4s¹, not [Ar] 3d¹. The aufbau order is nothing more than a record of which side of that competition each subshell falls on, atom by atom — which is exactly why it has exceptions and a derived law would not.

To reproduce the sequence, group the subshells by the value of n + l and, inside a group, take the smaller n first. n + l = 1 gives 1s; 2 gives 2s; 3 gives 2p then 3s; 4 gives 3p then 4s; 5 gives 3d, 4p, 5s; 6 gives 4d, 5p, 6s; 7 gives 4f, 5d, 6p, 7s. Fill each to its capacity of 2(2l + 1) and keep a running total, and the periodic table appears: 2, 4, 10, 12, 18, 20, 30, 36, and onward. The closures that make noble gases fall at 2, 10, 18, 36, 54 and 86: helium closes 1s, and every later one closes an np subshell, which from argon on is not a complete shell. Argon is inert at Z = 18 with 3d entirely empty, because the next orbital available, 4s, sits across a large gap.

The rule is often mis-taught as parallel spins repelling magnetically. Estimate that energy: two electron magnetic moments an ångström apart interact with roughly μ₀μ_B²/4πr³ ≈ 5 × 10⁻⁵ eV. The splitting the rule actually describes is four orders of magnitude larger — carbon's ³P and ¹D terms are 1.26 eV apart. The mechanism is exchange. A two-electron state must be antisymmetric overall, so a symmetric spin function (parallel spins, S = 1) forces an antisymmetric spatial function, and that function vanishes whenever the two electrons sit at the same point. Parallel spins therefore keep the electrons further apart on average and their Coulomb repulsion is smaller. The magnetism of an open-shell atom is a consequence of the alignment, not its cause.

The rules are lexicographic: maximise S first, then maximise L among only those states that already have that S. Carbon's 2p² makes the point. Put both electrons spin-up, giving S = 1; being parallel they must take different mₗ values from (+1, 0, −1), and the largest sum available is +1 + 0 = 1, so L = 1 and the term is ³P. You could instead put both electrons in mₗ = +1 with opposite spins and reach L = 2 — but that costs S, giving ¹D, which sits 1.26 eV higher, with ¹S higher still, 2.68 eV above ³P. The bookkeeping is worth automating: draw the mₗ boxes, fill one electron per box spin-up from the top mₗ downward, then start again pairing from the top.

Inside a term, spin-orbit coupling splits the levels as E_J = (A/2)[J(J+1) − L(L+1) − S(S+1)], so adjacent levels are separated by A⋅J with J the larger of the two — the Landé interval rule. Everything in Hund's third rule sits in the sign of A. Below half filling A is positive and J = |L − S| lies lowest: carbon's ³P₀ is the ground level, with ³P₁ at 16.4 cm⁻¹ and ³P₂ at 43.4 cm⁻¹ above it, near the 1 : 2 spacing the interval rule predicts. Above half filling the subshell is better counted in holes, A changes sign, and the multiplet inverts: oxygen's 2p⁴ carries the same ³P term, but now ³P₂ is the ground level, with ³P₁ 158 cm⁻¹ and ³P₀ 227 cm⁻¹ above it. Exactly half filled, L = 0 and only one level exists.

Chromium is [Ar] 3d⁵ 4s¹ and copper [Ar] 3d¹⁰ 4s¹, not the 3d⁴ 4s² and 3d⁹ 4s² the rule predicts; niobium, ruthenium, rhodium, palladium and platinum break it as well, so the exceptions are not a short list of curiosities. The reason is that the 3d and 4s one-electron energies differ by only a few tenths of an electronvolt in these atoms, so the total energy decides instead: the 3d orbitals are compact, so extra d occupancy costs repulsion, while an extra parallel d electron gains exchange, lifting the count of parallel d pairs from C(4,2) = 6 to C(5,2) = 10 at chromium. Hund's rules are more robust but still bounded: they fix only the lowest term of the ground configuration and say nothing dependable about excited terms, and the third rule holds only while the residual repulsion outweighs spin-orbit coupling, so it degrades down the table as spin-orbit grows and jj coupling takes over.