Molecular Geometry & Bonding
what the shape decides
The collection

Every essay — page 7

Page 7 of 36, continuing through the fields in the same order.

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

Beyond the octet

Hypervalency without d orbitals, delocalisation, aromaticity, and the structures the first-year rules quietly cannot describe.

Where the orphan pair actually sits. A σ-only Hückel model of each molecule, built from its own coordinates: the central atom's four valence orbitals, one σ orbital on each ligand, and the coupling between them the direction cosine of that ligand. Lone pairs need no special handling — they come out of the diagonalisation as the central orbitals no ligand combination transforms like. The charge is measured rather than assigned, and PF₅ comes out with two kinds of fluorine at 3 at -0.13 and 2 at -0.3, the more charged pair being the axial one that carries the orphan.

The count is the population

The census counted how many ligand combinations have no partner on the central atom and called the count n + L − 4. A σ-only model built from each molecule's own coordinates says what that count is worth: with no electronegativity difference anywhere, the mean charge on a ligand is minus the orphan count divided by the ligand count, exactly, in all ten cases. And the prediction the census made — that the charge grows with the orphan count — is refused by the divisor.

5 figures Four bonds, or one a₁ and three t₂. The same four occupied orbitals written in two bases. On the left each orbital sits on one bond; on the right one is shared over all four hydrogens and three follow the Cartesian directions. The transformation between them is orthogonal, so the density is unchanged.

The answer a search is most likely to give

If a cage has ten equally good localised descriptions, why does the literature agree about its picture? The hoped-for answer was that one basin is very large. Counting four hundred and eighty starting points says it is not: two independent searches agree one time in ten, and the description they most often return is not the best one.

8 figures Where the electrons are, without subtracting anything. The opposite-spin pair distribution of a half-filled ring of 6 at six repulsions, by separation, each divided by what uncorrelated electrons of the same density would give. At no repulsion it is one everywhere; at a repulsion of 16 the chance of finding two electrons on one site is 0.0430 of that, and what is missing has turned up next door. Nothing here is a difference between two calculations.

Where the electrons are, without subtracting anything

A correlation energy is an exact energy less a mean-field one, so the answer depends on which mean field was picked — by a factor of 259. The pair distribution says the same thing with no subtraction in it, and two systems matched to the same correlation energy turn out to have electrons in visibly different places.

8 figures The even sharer is the one the repulsion likes least. trigonal bipyramid: 2 kinds of ligand, spread 0.1137, repulsion 6.4747; square pyramid: 2 kinds of ligand, spread 0.1658, repulsion 6.4844; pentagonal planar: 1 kind of ligand, spread 0.0000, repulsion 6.8819. The planar arrangement gives all five ligands exactly the same charge and costs 6.3 per cent more in repulsion than the bipyramid, which is the arrangement chemistry actually adopts — so the two models disagree about which arrangement is preferred, and about how much charge is moved.

Two models that disagree about the shape

The identity says how much charge a hypervalent molecule's ligands must share and nothing about how. Working out which arrangement shares it most evenly puts the σ model and the repulsion model on one axis for the first time: the even sharer is the pentagonal plane, which is the arrangement the repulsion likes least — and along the interchange chemistry actually uses, one model sees 0.15 per cent of a change and the other sees 54.

5 figures The same hole, priced three ways. The correlation hole of a ring of 6 at a repulsion of 8, weighted by three interactions. With an on-site interaction the answer is 100 per cent at separation zero — as an identity, since the interaction is zero everywhere else. With one that reaches a neighbour, the enhancement at separation one costs rather than pays, and gives back 44.0 per cent of the on-site saving; with a Coulomb tail, 46.9. Everything beyond one neighbour is worth under a twentieth of the on-site term.

Half of it is given back at one bond

The correlation hole of a ring removes 1.2653 pairs from zero separation and puts 1.1126 of them one site away. With an on-site repulsion that second number costs nothing, because the interaction is zero there — but with anything that reaches a neighbour it costs 44 per cent of what the hole saved, and with a Coulomb tail 46.9.

6 figures Ten molecules, twenty-six arrangements, three disagreements. Every arrangement of every one of the ten molecules, with the point group recovered from the arrangement's own coordinates and the ligand σ set reduced in it. 27 of them can be worked in a tabulated group; the formula n + L − 4 is right for 23 and wrong for 4. Every failure is a planar arrangement of four or more ligands, and every one of them is a molecule that does not adopt that arrangement.

The square that wastes an orbital

The orphan count that prices hypervalency was treated as a property of a molecule's composition — ligands plus lone pairs minus four. Run on twenty-six arrangements of the same ten molecules it is right for twenty-three and wrong for three, and all three are flat. A planar arrangement gives a main-group centre three usable orbitals rather than four, so the count is a property of the shape.

7 figures The count is a curve, and the curve flattens. How many distinct localised descriptions of a twelve-vertex cage had been found after each batch, out of 4,000 random starts. The last new one appears at start 124; the remaining 3,876 add nothing. The dashed curve is what the basin sizes measured here predict — the chance of having hit each description at least once — and it is a closed form rather than a fit to the points.

Where the count stops being an effort

A cage's localised descriptions were counted by a search that stopped when it stopped finding new ones, which makes the count a property of the stopping rule. Run to four thousand starts the count is fourteen and the last new description appears at start 124 — after which three thousand eight hundred and seventy-six starts add nothing. The four rarest are found six times in a thousand, which is what says nothing rarer is hiding.

6 figures A ring of 6 as the neighbour repulsion is turned up. At an on-site repulsion of 8, three quantities against the nearest-neighbour repulsion: the alternating structure factor, the double occupancy, and the nearest-neighbour opposite-spin pair distribution. The rise is steepest at V = 4.5, which is 0.563 times the on-site repulsion. Far past it the ring is charge ordered — nearly every electron paired on alternate sites, which is what a double occupancy approaching a half means.

The give-back that turned into a saving

Take a wavefunction optimised for an on-site repulsion, weight its correlation hole with an interaction that reaches one neighbour, and the enhancement at one bond gives back forty-four per cent of the on-site saving. Putting that neighbour term into the Hamiltonian and solving exactly does not shrink the give-back. It reverses its sign.

6 figures Three basin populations from one search. The share of 4000 random starts landing on each localised description, in rank order, on a logarithmic axis, for three cages and fillings. a twelve-vertex cage at twelve electrons gives 14 descriptions and two groups with a gap; a nine-vertex cage at eight electrons gives 63 descriptions and a smooth tail; a twelve-vertex cage at twenty electrons gives 15 descriptions and one basin and dust. Only the first is the twelve-vertex shape, and the rule it suggests is a rule about that shape.

Three shapes from one search

A cage's fourteen localised descriptions fell into two groups with a factor of five between them and nothing in the gap, which made counting the big basins look like a stopping rule. Two more cases from the same family give a smooth tail over sixty-three descriptions and a single basin holding ninety-eight per cent. One search, one family, three shapes.

6 figures The repulsion against the count, and they do not sort together. Every arrangement of the census by how many ligand σ combinations are left without a central partner and by how far its ligand repulsion sits above the best arrangement of that many points. The three that break the formula are marked. They are not the expensive ones: they sit at 4.2, 6.3, 9.8 per cent while the arrangements the formula gets right run to 26.5.

Expensive is not the same as unadopted

A counting formula right for twenty-three arrangements and wrong for three invites a reading: the failures are the arrangements nothing adopts, so the formula is reliable because chemistry stays away from where it breaks. Put the repulsion energy on the same axis and the reading fails — the most expensive arrangement in the census is one the formula gets right.

6 figures Three cages, three answers, and none of them reassuring. For each cage: how many descriptions the search finds, what share the commonest takes, how far the whole set spreads in the functional, whether the commonest is the best, and the verdict. A search that always agrees with itself is agreeing about a choice that does not matter; a search whose descriptions genuinely differ does not return the best one.

Fifty descriptions of one molecule

A search whose largest basin takes ninety-eight per cent of its starts will report one description however long it is run, and nobody runs four thousand starts when the first fifty agree. Whether the rare ones are worse descriptions or merely rarer is one number per description, already computed and never looked at. On two of three cages they are not worse — they are the same answer, to parts per million.

6 figures One sign change is a root and the other is not. The solved give-back against the neighbour repulsion on a ring of 6 at U = 8. It crosses zero at V = 3.895, where the price of everything beyond contact really is nothing, and changes sign again at V = 4.159, where the quantity it is a fraction of has vanished instead. The curve is broken at the second because it is an asymptote and not a crossing; four of the 12 points fall outside the band drawn here.

A sign change is not always a zero

The solved give-back changes sign somewhere between a neighbour repulsion of two and one of four, and a bisection looks like the way to find the value where the structure beyond contact contributes exactly nothing. It changes sign twice. One crossing is that value; at the other the quantity the fraction is a fraction of has vanished instead, and a bisection reports the two in identical words.

6 figures