Molecular Geometry & Bonding
what the shape decides
The collection

Every essay — page 8

Page 8 of 42, 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.

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 One steps, the other slides — and the step is at the far end. The repulsion energy and the orphan count along the path from a tetrahedron to a square plane. The energy rises smoothly and monotonically, lowest at the tetrahedron and highest at the plane. The count is zero everywhere — including at 89.99° — and becomes one only at 90° exactly. The step is not near the energy's minimum; it is at its maximum, and it is at a single point.

A count that changes at one point

Where does the orphan count step along a distortion, relative to where the energy's minimum sits? A rule that depends on an exact symmetry may have no answer for a real molecule. On the path from a tetrahedron to a square plane the count is the same at every angle up to 89.99° and changes only at 90° exactly — which is the energy's maximum, not its minimum, and a single geometry out of a continuum.

6 figures How many descriptions, against how much they differ. Every cage-and-filling pair, by the number of distinct descriptions its localisation finds and by how far apart they are in the functional. If the count measured ambiguity the points would rise from left to right. The case with the most descriptions — 44 of them — has a spread of two parts in a hundred thousand, and sits at the bottom right.

Counting was right except where it mattered

A cage whose localisation gives dozens of descriptions that are all the same answer raises a worry: if degeneracy is common across the family, counting descriptions is the wrong measure of ambiguity. Across forty-eight cage-and-filling pairs it is the right measure on eleven of the thirteen that have anything to count — and it fails on the one leaned on hardest.

7 figures The energy runs the whole way; the count exists at the two ends. The Coulomb repulsion of six ligands along the Bailar twist, from the trigonal prism at 0° to the octahedron at 60°, with the geometries that have an orphan count marked underneath. The energy is smooth and monotone, lowest at the octahedron. The count is defined at the two ends and at the handful of angles the symmetry finder rounds into them, and nowhere else — not because the geometry is unsymmetrical, but because its group is not tabulated.

The group nobody wrote a table for

The orphan count is predicted to be constant along the Bailar twist, because the twist keeps D3 the whole way. The premise is exactly right — at every angle strictly between the prism and the octahedron the symmetry finder assembles six operations that hold to four parts in 10¹⁶. The conclusion cannot be tested here, because a count is a reduction, and the calculation's nineteen character tables did not include D3.

6 figures The same number, in three different groups, all the way along. The orphan count along the Bailar twist once the D3 character table is written. It is 2 at every one of the 15 geometries the symmetry finder answers for, across D3h at the prism, D3 through the whole interior and Oh at the octahedron. The lower row shows what the same sweep gave before the table existed: two ends and nothing between. This constancy was predicted from the premise that the twist keeps D3 throughout, and the premise was right.

The count the table was hiding

The interior of the Bailar twist looks uncountable — exactly D3 at every angle, and D3 is a point group whose character table is rarely written out. Writing it takes nine lines and settles the prediction made for the path: the orphan count is two at every geometry on the path that can be answered, in three different groups, out of three different decompositions.

6 figures Both criteria leave a gap, and Boys leaves a chasm. Every non-zero relative spread under each criterion, on one logarithmic axis, with the largest empty stretch shaded. Pipek–Mezey's runs a factor of 206; Boys's runs 2.8e+7 — seven decades, from numerical zero to a real spread with nothing between. So the bimodality belongs to the cages rather than to the functional, and under the second criterion the threshold matters even less.

A second criterion left a gap too

The localisation spreads are bimodal — an empty factor of two hundred around the degeneracy threshold — and the gap might belong to the criterion rather than to the cages. Boys localisation leaves a gap of seven decades on the same forty-eight pairs, so it belongs to the cages. But the two criteria disagree about six of them, in both directions, and the family's most ambiguous cage under one is exactly degenerate under the other.

6 figures