Every essay — page 14
Orbitals Where the atoms go Bonding models What symmetry decides Beyond the octet What a spectrum settles When the molecule does not stop What the shape is for What is taught wrongly Series Named objects Orbitals Refutations Search
Bonding models
Valence bond, molecular orbital, and hybrids — three descriptions of one thing, related by transformations that change no observable.
The spin the count does not hold
Three orbitals in a row keep one level at the free-atom energy however the overlap of one end is changed, and at three electrons that level holds the radical's unpaired electron. Its energy does not move. Its spin does: from half on each end to 0.236 and 0.764 as one overlap goes from 0.25 to 0.45, exactly the squared ratio of the two overlaps, at every energy of the middle orbital. A coupling between the ends moves the level and cannot move the spin. The energy and the spin are answering to different things.
The scatter counts angular rings
Eight nine-ring chains built by turning one fusion looked like one bend moved along a molecule. They were two families: one angular ring when the turned fusion is at an end, two adjacent angular rings everywhere else. Built from stated angular rings instead, twenty-seven chains show the ring-current scatter ignores which way a ring turns, does not grow with how many turn, and is mostly explained by one count per pair — each angular ring between two rings multiplies their response by 0.60, each one under them by 0.65.
A cancellation between two separations
The give-back's numerator has one term in it, so its root is the repulsion at which the pair count at one bond returns to its free value. Let the interaction reach two bonds and the numerator has two terms with opposite signs, and its root becomes a cancellation between an enhancement at one separation and a depletion at the next — at a repulsion where neither is at its free value. Which side of the old zero the new root falls on reverses between a reach of a third and a half, because whether two bonds are enhanced or depleted is the interaction's own doing.
Two changes that do not add
A three-orbital radical keeps its unpaired electron's level fixed when one end's overlap changes, and keeps its spin fixed when the two ends are coupled — two changes, two quantities, and each change touches only one of them. Put both on at once, as in a bent radical with one end nearer the middle, and the separation dissolves. The level feels the overlap asymmetry, the spin feels the coupling, the electron reaches the middle orbital, the middle orbital's energy starts to decide the spin, and the spin becomes a matter of how overlap is partitioned — except at one special point, where every site energy is equal and a different reason from the one given keeps the partitions together.
Spin below zero
A three-orbital radical read from its singly occupied orbital has no spin on its middle orbital unless its ends are both unequal and coupled — and never negative spin anywhere. Solved exactly with an on-site repulsion, the middle orbital goes negative at once, reaches the allyl radical's measured −0.15 to −0.18 at a repulsion of five and a half to seven electronvolts, and responds to the end–end coupling at first order where the one-determinant leak responds at second. The two have opposite signs, so in a bent, unequal radical the middle orbital's spin falls and then rises, and the unequal ends that let the leak in drain most of the polarisation first.
Six electronvolts was a difference
A three-orbital radical solved with an on-site repulsion reproduced the allyl radical's negative central spin at a repulsion of about six electronvolts — six tenths of what a carbon atom's own charges cost. Add the repulsion between neighbouring sites and the repulsion allyl needs rises almost one for one with it. With the neighbour values π-electron theory uses, it lands within a quarter of an electronvolt of Pariser's 11.13 for carbon. And the band McConnell's constant left can be closed without that constant at all.
The agreement was hydrogen's energy
A three-orbital radical solved exactly with on-site and neighbour repulsion reproduced the allyl radical's central spin at an on-site repulsion of 10.89 eV, a quarter of an electronvolt from Pariser's 11.13 for carbon. Its hopping, 2.91 eV, was never allyl's: it came from a round overlap of 0.25 and hydrogen's energy of −13.6 eV. Computed from carbon, the overlap barely changes — carbon's own 2pπ overlap at 1.39 Å is 0.248 — but the energy does, and with carbon's −11.4 eV the hopping is 2.34 eV, Pariser and Parr's own β. Then the repulsion allyl needs is 9.49 eV with one standard form of the neighbour terms and 12.68 with the other. Pariser's value sits between them, not on either.
A root that moves when nothing else does
Give the ring of six an interaction that falls off as one over the distance and the give-back's numerator has three terms. It still has one root, and at that root every separation's pair change has the opposite sign from the two-bond case. But the root has a weakness the earlier essays never tested: lower the interaction by the same amount at every separation and not one state changes, while the root moves — by as much as the reach of the interaction did, and out of existence altogether if the zero is put at one bond.
The angular factor belongs to the chain
Each angular ring between two rings of a fused chain multiplied their ring-current response by 0.60, on nine rings. Refitted on every chain of seven, nine and eleven rings, the same two counts explain the same 93.9 per cent of the variation at all three lengths — and the factor runs 0.656, 0.593, 0.542. The form transfers and the number does not, because it is the ratio of a zigzag's decay, which is the same at every length, to a straight chain's, which lengthens as its gap closes.
A verdict that flips and changes nothing
Run the two-orbital mixing criterion along H₂C=CH⁻, H₂C=NH, H₂C=O and H₂C=F⁺ and one pair finally crosses its boundary — the π and the lone pair, a third of the way from oxygen to fluorine. It is the first real test the criterion has had, and the localised orbitals on the far side of it are the ones on the near side: two bent bonds and a lone pair that stays itself.
Orbitals
A one-electron wavefunction, drawn as a contour surface at a level somebody chose. Nodes, signs, and what the picture is a picture of.
What an orbital is
Not a region the electron occupies, not a path it follows, and for any atom but hydrogen not an exact anything. An orbital is a one-electron wavefunction, and almost every difficulty in this subject comes from forgetting that.
Say what it encloses
An orbital picture is a contour at a level somebody chose, and almost no source says which. Two textbooks can draw the same orbital at visibly different sizes with the same caption, and both be printed in good faith.