Half filling forbids the anomaly: an exchange ledger for the transition-metal ground-state configurations

Abstract

The anomalous ground-state configurations of the transition metals are conventionally attributed to a special stability of half-filled subshells. Scerri has shown that explanation to be neither necessary nor sufficient. We show that counting the same-spin pairs available for exchange derives both of his results rather than restating them. Promotion s²d^k → s¹d^(k+1) opens two exchange channels, not one: a d–d channel of gain G(k) and an s–d channel of gain S(k), the second arising because promotion also discards an s electron that was itself exchanging. Both vanish identically at half filling, and so do the direct F² and F⁴ terms, at that occupancy and at no other. Half filling is therefore the unique position at which the excitation energy contains no angular two-electron contribution whatever, and the half-filled element's regularity reduces to the sign of a single quantity that no exchange bookkeeping can reach. That quantity is measured directly at manganese (+2.114 eV), technetium (+0.319 eV) and rhenium (+1.457 eV). The half-filled shell does not cause the anomalies; it is the one occupancy at which one is forbidden, and the textbook rule looked right because the anomalies sit immediately beside half filling, where the exchange gain is maximal. For the energies we replace the usual fitted model with a decomposition in which the exchange payoff is computed and only the promotion cost is fitted — every angular coefficient from Gaunt algebra, every radial integral from each atom's own orbitals, and two free numbers per series. Against term energies, which is what such a model predicts, this halves the error of a three-parameter fit in all three series and reproduces the sign of all twenty-two elements measured. The promotion cost so obtained is flat across 3d and falls monotonically through zero in 4d and 5d, which is why the three series look nothing alike: where the cost is constant the payoff decides and only the maximal gain clears it, giving chromium and copper; where the cost goes negative promotion is downhill regardless. Nickel and tungsten, long treated as separate unexplained cases, are shown to be one case: both prefer promotion at the level of terms and both are regular because spin–orbit coupling reverses it. The construction is not specific to the d block. Rebuilt at ℓ = 1 it reproduces the same parameter-free result at p³, with the s–p exchange coefficient 1/3 against 1/5 for s–d and nothing carried over. And across all five blocks computed — 2p, 3p, 3d, 4d, 5d — the promotion cost is depressed at half filling relative to its own two neighbours, deepest there in every one, at a position specified in advance. That depression has the same cause as the vanishing of the angular terms: half filling is spherical, so the Fermi hole is maximal, so it is the occupancy at which Hartree–Fock is most accurate and the residual cost is least. One geometric fact produces both.