Ladder

Hybrids — the ladder

3 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. sp3109.471°between every pair4 hybridsworst off-diagonal 0e+0an orthogonal transformation of the atomic orbitalsone electron

    Hybrids are a basis

    An sp³ hybrid set is an orthogonal matrix applied to the atomic orbitals. Rotating a basis changes no observable, so asking whether the electrons are really in hybrids is asking which coordinate system nature prefers.

    rung 1 · bonding
  2. HHCHHTdprincipal axis C36 mirror planesno inversion centrecannot be polarcannot be chiralgroup recovered from the coordinates5 atoms

    Hybridisation does not explain

    Methane's photoelectron spectrum has two bands, not one. Four equivalent sp³ bonding orbitals cannot produce that, and the resolution is that hybridisation was never a claim about what a measurement would find.

    rung 2 · wrong
  3. localised — four equivalent bondscanonical — a₁ and t₂H1H2H3H4a₁t₂ (x)t₂ (y)t₂ (z)C 2s-0.219-0.219-0.219-0.2190.438000C 2px0.299-0.299-0.2990.29900.59800C 2py0.299-0.2990.299-0.299000.5980C 2pz0.2990.299-0.299-0.2990000.598H1 1s0.558-0.040-0.040-0.040-0.2190.2990.2990.299H2 1s-0.0400.558-0.040-0.040-0.219-0.299-0.2990.299H3 1s-0.040-0.0400.558-0.040-0.219-0.2990.299-0.299H4 1s-0.040-0.040-0.0400.558-0.2190.299-0.299-0.299the transformation between the two is orthogonal, and the density matrices differ by 1.1e-16both descriptions hold 8.000000 electronsa mixing that is not orthogonal changes the density by 0.214 — the test can failthe same eight electrons, in two coordinate systemsorthogonal, so identical

    The localisation transformation, demonstrated

    Four equivalent bonds or one a₁ and three t₂ — two descriptions of methane's bonding electrons that disagree about everything except the electron density, which they agree about to the last bit a double can hold.

    rung 3 · bonding

All ladders