Ladder

Delocalisation — the ladder

4 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. HHCCHCCHCCHHD6hprincipal axis C67 mirror planeshas an inversion centrecannot be polarcannot be chiralgroup recovered from the coordinates12 atoms

    Delocalisation

    Benzene does not alternate between two structures. It has one structure, and the two Kekulé forms are basis functions in a description of it — which is a different and much less exciting claim than the one usually made.

    rung 1 · beyond
  2. αα + 2.000βα + 1.000β×2α − 1.000β×2α − 2.000βethenebenzene · 6 π electronstotal π energy 6α + 8.0000βagainst 3 isolated double bonds: 6βdelocalisation +2.0000βeigenvalues of the adjacency matrixHückel, no repulsion

    Hückel theory and what it gets right

    A conjugated system's orbital energies are the eigenvalues of a matrix of ones and zeroes. Nothing about carbon enters, no geometry enters, and the results that survive are exactly the ones that depend on neither.

    rung 2 · bonding
  3. 0.5550.7250.6030.7250.5550.5550.7250.6030.7250.5550.518naphthaleneorders spread by 0.20610 π electronscharge density 1.000 on every atomfrom the eigenvectors, at no extra costHückel, no repulsion

    Bond order from the eigenvectors

    Having diagonalised the matrix, the coefficients are already there. Two sums over them give every bond's order and every atom's charge, and naphthalene's three kinds of bond come out in the order the measurements find them.

    rung 3 · bonding
  4. αα + 1.802βα + 1.247βα + 0.445βα − 0.445βα − 1.247βα − 1.802βethenehexatriene · 6 π electronstotal π energy 6α + 6.9879βagainst 3 isolated double bonds: 6βdelocalisation +0.9879βeigenvalues of the adjacency matrixHückel, no repulsion

    Conjugation, and its limits

    Every extra double bond in a chain lowers the gap between the highest occupied and lowest empty orbital, which is why long conjugated molecules are coloured. The trend has a limit, and the limit is not where the arithmetic says it should be.

    rung 4 · beyond

All ladders