Molecular Geometry & Bonding
what the shape decides
The collection

Every essay — page 9

Page 9 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.

Six pairs off the diagonal, and every one of them on an axis. Each cage-and-filling pair's Pipek–Mezey spread against its Boys spread, both logarithmic, with the two thresholds drawn. Agreements sit in the two opposite corners: spreads that are zero in both, or large in both. The six disagreements do not sit between them — they sit on the axes, with one coordinate at the floor. A criterion-dependent cage is not one the two criteria half-agree about; it is one where the difference between its descriptions is invisible to one of them entirely.

The cage is on both sides

Two localisation criteria classify six cage-and-filling pairs differently, and a symmetry explanation for the six is the natural first guess. The icosahedron's graph has a hundred and twenty automorphisms, the most in the family, and supplies three of the six disagreements and nine of the agreements. What the six do have in common is sharper than a symmetry: in every one, one criterion's spread is not small but zero.

5 figures Two centres, one path, and the leftover changes sides. The count of leftover orbitals along the Bailar twist, for a main-group centre and for a transition metal. At four valence orbitals against six ligand combinations, two combinations are left with no partner and the count is a count of orphans; at nine against six, every combination finds one and three metal orbitals are left instead. Both are constant along the whole path, in three different point groups, out of decompositions that share no species — which is the replacement rule holding in a case where the arithmetic runs the other way.

The leftover changes sides

A main-group centre brings four valence orbitals against six ligand combinations, so two are orphaned. A transition metal brings nine, so the arithmetic inverts and three metal orbitals are left instead — three at every geometry of the Bailar twist, out of decompositions that share no species. Run past the whole arrangement census, exactly one arrangement orphans anything at a metal, and it needs an f orbital to fix.

5 figures Eighteen paths, two hundred and seventy geometries, one gap. Every one-parameter path between two arrangements of the same ligand count, each sampled at 15 geometries, with each geometry coloured by whether its group is named and tabulated, refused by the finder's tolerance, or a finite group with no table. The one gap is a group of order 10 at the pentagonal-pyramidal end of two paths — C5v, which no table here reaches. The path with a linear end is excluded, since a continuous group is declined deliberately rather than missing.

The gap found on purpose

A twist between two coordination arrangements passes through a point group with no character table among the twenty in use, and that was found by accident. Running every path between the arrangements — eighteen of them, two hundred and seventy geometries — finds exactly one more gap, at a group of order ten. It also finds something the search was not looking for: a group without a table had been reported as an infinite group.

5 figures Forty-eight rows, thirty-three calculations. Every cage-and-filling pair the family survey covers, one cell per pair, with cells that hand the localisation the same set of orbitals joined. A localisation here selects its occupied orbitals by asking which have any occupation at all, and Hund's rule puts one electron into each member of a degenerate shell before pairing any of them — so adding two electrons to a half-filled shell pairs a spin and changes nothing the search can see. Fifteen of the forty-eight rows repeat an input already in the table.

Six disagreements and three calculations

Two localisation criteria classified six of the family's forty-eight cage-and-filling pairs differently, and three of the six were consecutive fillings of one icosahedron with spreads identical to three figures. They are not three coincidences and not one degeneracy being filled: they are one calculation, because the survey's forty-eight rows are thirty-three distinct questions.

6 figures The boundary falls inside a shell, and the shell has no inside. The 12-vertex cage at 10 electrons: its Hückel levels, with each degenerate shell drawn as its members and the occupied ones filled. The occupied set takes 2 of the 5 members of one shell — and the members of a degenerate shell are not distinguishable. Whichever combinations the eigenvalue routine happened to return are the ones that get occupied, so the density being localised is a choice made by a diagonaliser rather than a property of the cage.

The basis a diagonaliser happened to return

Six of the cage family's thirty-three inputs have their occupied set cut through the middle of a degenerate shell, and the members of a degenerate shell are interchangeable. Re-orienting the shell changes nothing about the cage, the filling or the criterion — and it moves the best localisation functional by up to twenty-one per cent, moves the count of descriptions from ten to fourteen, and flips the degeneracy label on three of the six.

7 figures Angular rings leave the heteroatom's own ring alone and scale the rest. Each ring's response as a fraction of the straight chain's, with the heteroatom on ring 0 and the first one, two, four or ten interior rings angularly fused, on a logarithmic axis. The heteroatom's ring stays within one and a half per cent in every case. The rings beyond the angular fusions fall, to under half by the third ring once two are angular, and to between an eighth and a third along the fully angular chain, alternating from ring to ring.

An angular ring rescales what lies beyond it

A heteroatom's influence along a chain of fused rings decays with a length near 0.7 rings, and one bend was found to change almost nothing. Counted properly, that bend was two angular rings, and counting angular rings one at a time shows what they do: the heteroatom's own ring never moves by more than one and a half per cent, the rings beyond angular fusions fall to between an eighth and a half of their response — or rise by up to two fifths — and the decay length changes by a tenth.

6 figures Folding the ring off its plane never gives the orbital back. For rings of four to eight ligands round a centre with no lone pair, the orphan count minus the formula n + L − 4, at polar angles from 60° to 120°. The upper dot in each row is the bare ring and the lower the same ring with a ligand on the axis. The bare ring is one over the formula at every angle, including 90°, where the ring is flat and its group is D₄ₕ, D₅ₕ or D₆ₕ; there is no table here for D₇ₕ or D₈ₕ. The capped ring agrees with the formula at every angle, 90° included, where the ring is exactly flat and the apex alone keeps its group C₄ᵥ to C₈ᵥ.

Folding the ring does not give the orbital back

Three arrangements break the orphan count n + L − 4, all flat, and the reason given was flatness: the p orbital perpendicular to the ring has no ligand combination of its species. Fold the ring into an umbrella at any angle and that orbital becomes totally symmetric — and the count stays wrong by exactly one, for rings of four to eight. The ring offers one symmetric combination to a centre with two symmetric orbitals. Writing C₅ᵥ to see it also counts the pentagonal pyramid at last, and the formula holds there.

6 figures The rank and the character table agree on every geometry both can count. Every geometry counted both ways — 451 of them, from the census of arrangements, every folded, capped and flat ring, and every point on every path between arrangements where the point group has a table — sorted by how many central orbitals the table matches and how many the rank of the coupling rows reaches. Every geometry is on the diagonal. The last column is the 69 geometries no table could count, which the rank counts anyway.

The count needed no table

Every orphan count behind the three-centre account of hypervalency has been a reduction in a character table, and the tables ran out three times. The count never needed them. In a σ-only model the number of central orbitals the ligands can reach is the rank of one small matrix, and that rank is one more than the dimension of the flat space the ligand directions span — four unless the ligands lie on one circle of the sphere. It agrees with the tables on all 451 geometries both can count, counts the 69 they could not, and finds a case they get wrong: ligands on a circle with no symmetry, where the table matches four orbitals and only three can be reached.

6 figures The rule says in advance which cut shells cannot move the answer, and it is right twenty times. Every input with a cut degenerate shell — the six of the cage family and four cut Platonic cages the family never contained — with the shell's polynomial degree, the cage's spherical-design strength, and how far re-orienting the shell moves the best localisation functional under each criterion. A shaded cell is one the rule predicts cannot move: a degree-one shell on a cage whose design strength reaches four for Pipek–Mezey or three for Boys. Every shaded cell is exactly one and every unshaded cell moved.

The cage that averages like a sphere

Re-orienting a half-filled degenerate shell moved five of the cage family's six affected inputs and left one alone: the icosahedron at four electrons, which gave the same answer to every digit while an octahedron with the same shape of cut moved by fifteen per cent. The icosahedral group cannot explain that; it is too small. What explains it is a property the icosahedron shares with the best quadrature grids on a sphere. Its twelve vertices average every polynomial up to the fifth degree exactly as the sphere does, and a localisation functional on a shell of degree one is a polynomial of degree four. The rule that follows predicts all twenty cases tried, including four cages the family never contained.

6 figures The energy sees the frame, not the missing orbital. Every census arrangement of four, five and six ligands, in the order its σ-only bond energy puts it at weak coupling, most stable first. The bars are the fourth-order cost — the ligands' net direction and their excess over a tight frame, weighted as the series weights them — and they rise down each group without exception. The right-hand column is the Gram matrix's smallest eigenvalue, the measure of how nearly an orbital is unreached: it does not fall down each group, and the square with an eigenvalue of zero sits above the seesaw.

What a missing orbital costs

A flat ring of ligands leaves one central orbital unreached, and a continuous measure of how nearly an orbital is unreached — the smallest eigenvalue of a Gram matrix — goes to zero there. Whether that makes a weak bond is a question a σ-only bond energy can answer exactly. It answers no, twice. At second order the energy is the same for every arrangement of the same ligands. At every order it sees the ligand directions only through their sum and their second moment, so an octahedron and a suitably tilted prism have one energy at every coupling — and a square with an orbital missing is more stable than a seesaw that reaches all four, until the coupling is strong.

7 figures Five ligands: which shape the σ energy wants, by one number. The fourth-order cost of the trigonal bipyramid (flat at one sixth) and of the best square pyramid, against q — the ratio of the net-direction coefficient to the frame coefficient, which with equal overlaps is r(1 + r) for p at r times s's height. The two cross at q = 0.862. The bipyramid is a minimum only above q = 2/3 and the pyramid only below 0.955, so between them both are minima. The model's own heights give q = 3.75, and any main-group centre, with p above s, has q above 2.

Five ligands, and no ideal to aim at

A σ-only bond energy prefers, at the order where geometry first enters, an arrangement that points nowhere and treats every direction alike — a spherical 2-design. Four ligands have one and so does every count from six. Five does not, and the energy has to trade one virtue against the other. One number decides the trade, and across it the trigonal bipyramid and the square pyramid exchange the roles of minimum and transition state on Berry's pseudorotation path.

7 figures A ligand repulsion makes a main-group square pyramid only if it is steeper than the fifteenth power. At a σ parameter q = 2, where the bipyramid is the only σ minimum, the contact weight at which the square pyramid takes over, as a share of the bipyramid's σ cost, against the power s of a 1/rˢ contact between the ligands. For every power below 15.05 — Coulomb, 1/r⁶, 1/r¹² — no weight does it. Above it every power does, at 1123 at s = 16, 233 at s = 20, 101 at s = 30.

Only a steep contact makes five square

A σ-only bond energy gives a main-group centre with five ligands one minimum, the trigonal bipyramid, and yet pentaphenylantimony is a square pyramid. The obvious repair is a repulsion between the ligands. A repulsion does not help by being there: five points repelling as any power of their distance below about the fifteenth prefer the bipyramid themselves, so Coulomb and every soft law only widen the bipyramid's region, at any weight. Above the fifteenth power they prefer the square pyramid, and then every main-group centre switches at a weight that rises with its σ parameter. An exponential contact between ligand atoms 2.2 Å out does it only below a range of 0.215 Å — which is the fifteenth power again, measured at the ligands' separation.

7 figures