Molecular Geometry & Bonding
what the shape decides
Concept

Aromaticity — where it appears

The extra stability a cyclic conjugated system has over an open one of the same size, which for a monocycle needs 4n+2 electrons. The size of it depends entirely on what reference state the comparison is made against, and the reference is a choice.

Named by 14 essays across 3 fields — each of them below, with the objects they name alongside it.

benzene — D6h. The molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.

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.

beyond · Delocalisation Which rings close a shell. Each ring is filled with its own number of pi electrons and asked whether the highest occupied shell came out full. Of the rings drawn here, C6 close — at 6 electrons — which is Hückel's 4n+2, produced here rather than recalled.

Aromaticity as a computed shell closure

Hückel's 4n+2 rule is not a rule. It is what comes out when every ring size is filled and asked whether its highest occupied shell came out full — and the calculation that answers does not know the phrase.

bonding · Aromaticity pyrrole against cyclopentadienyl anion. The Hückel levels of pyrrole beside those of cyclopentadienyl anion, which is the same graph with one diagonal entry and the bonds touching it changed. The parent's levels are symmetric about α because its matrix has nothing on the diagonal; the substituted system's are not, and the asymmetry is the size of the fitted parameter rather than a result.

Six electrons in a ring that is not all carbon

Pyrrole and the cyclopentadienyl anion hold the same six π electrons in the same five-membered ring. The anion spreads them perfectly evenly; pyrrole's nitrogen keeps 1.720 of them and furan's oxygen keeps 1.791 — the more electronegative atom donates less, and the bond orders to it fall with it.

beyond · Delocalisation Levels of a ring closed with a half turn in it. The orbital energies of a ring with one resonance integral reversed in sign, which is what half a turn in the ribbon of p orbitals does to it. The levels come in degenerate pairs from the bottom up rather than singly, so the count that closes a shell is 4n rather than 4n + 2. Degenerate levels share a line. The electrons are placed by aufbau with Hund's rule, so a half-filled degenerate shell shows as two unpaired spins.

The ring with a twist in it

Reverse the sign of one resonance integral in a ring and its levels stop being one nodeless orbital above degenerate pairs and become degenerate pairs all the way up. The count that closes a shell changes from 4n+2 to 4n, and every aromatic ring size and antiaromatic ring size exchange places.

beyond · Aromaticity benzene: what alternation costs and gains. The π energy of benzene as its bonds are alternated by β(1 ± δ), the elastic cost of alternating them at a stated stiffness, and the sum. The π curve falls quadratically in δ, which is measured here and is the whole difference between a molecule that distorts and one that does not. The stiffness is a stated parameter, so where the minimum falls is not a claim; which power of δ each curve goes as is.

The hexagon is the frame's doing

Benzene's delocalisation energy is quoted as 2β and read as the reason its bonds are equal. Let the bonds alternate and the π energy goes down, not up — at every ring size, for 4n+2 as much as for 4n. The π electrons are not what keeps the hexagon regular; the σ frame is, and the margin between them is uncomfortably thin.

beyond · Delocalisation Hückel levels of cyclopropenyl cation. The orbital energies of the pi system, computed as the eigenvalues of the molecule's adjacency matrix. Degenerate levels share a line. The electrons are placed by aufbau with Hund's rule, so a half-filled degenerate shell shows as two unpaired spins.

The same ring, three charges

Three carbons in a ring are aromatic with two π electrons, a doublet with three, and a triplet with a delocalisation energy of exactly zero with four. Nothing about the molecule changed but the count, and adding electrons to a π system can make its π binding energy fall.

bonding · Aromaticity The characters of C6, and the levels they are. The 6 representations of the ring's rotation group, drawn as points on the unit circle at 2πk/6. Each level is twice the horizontal coordinate: a representation and its complex conjugate have the same real part, so they are degenerate, and the levels pair up automatically. Only k = 0 — and k = n/2 when n is even — lands on the real axis, so one level is unpaired at the bottom and the shell closes at 4n + 2. Nothing here has been diagonalised.

The ring's levels are its group's characters

The Hückel energies of a cyclic system are twice the real part of the characters of its own rotation group, so the level pattern — one level, then pairs, then one more if the ring is even — is a group theorem rather than a calculation. Hückel's 4n+2 rule is a statement about the representations of a cyclic group and about nothing else.

symmetry · Aromaticity The two figures, per electron. For each ring, the delocalisation energy against isolated double bonds and against the same ring with one bond deleted, divided by the number of pi electrons. The upper bar of each pair is the usual reference; the lower one is the cycle's own contribution, and the two are nowhere proportional.

A stabilisation is measured from somewhere

Benzene's delocalisation energy is 2β against three isolated double bonds and 1.0121β against the same six carbons with one bond deleted. Cyclobutadiene's is exactly zero against the first reference and −0.4721β against the second, so one of the two references records the most famously destabilised ring in the subject as neutral.

beyond · Delocalisation Two kinds of answer to a flux, and only one of them is a curve. The π binding of three rings against the magnetic flux through them, in units of beta and of the flux quantum, measured from each ring's own value at zero flux. Beta is negative, so a binding that FALLS is an energy that rises: benzene's does, which is what a diamagnetic ring current is. Cyclobutadiene's rises in both directions from a corner — its energy has no second derivative at zero field at all, and the two one-sided slopes differ by 12.57.

Two rules that share no arithmetic

Hückel's rule is a statement about a gap. Put a magnetic flux through the same rings instead and ask which of them push the field out, and the answer is the same set — eighty cases, no exceptions — although the second calculation counts nothing and has no shells in it. And the rings the rule excludes turn out to have no magnetic susceptibility at all: their energy has a corner at zero field.

symmetry · Aromaticity A length that keeps growing, and one that stops. The fitted decay length of the ring-current response, against the number of rings. The bare acene's runs 1.397, 1.669, 1.896, 2.104, 2.302 — up by a factor of 1.65 and still climbing — while its own gap falls from 0.590 to 0.1102. With a gap held open the same measurement gives 0.678, 0.642, 0.636, 0.636, 0.639, which has stopped moving by the third molecule. There is a magnetic reach, and an acene is too nearly gapless to have one.

A reach that has no length

A ring current's response to a neighbouring ring falls with distance, which invites asking for the length. Every acene computed gives a longer one — 1.397, 1.669, 1.896, 2.104, 2.302 rings — because the gap that would set the length is closing at the same time. Give the same molecule a gap that stays open and the number settles at 0.636 by the third one and does not move.

symmetry · Aromaticity 6 rings fused two ways. Two catacondensed chains of 6 hexagons: the linear one, where every fusion continues the line, and the angular one, where the fusions alternate. They have the same formula and the same number of bonds and they are not the same graph. Ring centres are numbered; in the linear molecule two rings k steps apart have centres 1.7321k units apart and in the angular one they do not, which is the whole reason this pair can be asked the question.

Neither of the two separations

A linear acene cannot pose the question, because the number of fusions between two rings and the distance between their centres are the same variable there. Bending the molecule pulls them apart — and the response follows neither. Two pairs of rings the same distance apart differ by two thirds, and the larger one is at the greater distance.

symmetry · Aromaticity The end pair against the deepest pair, at each separation. For each separation, the response of the pair that touches an end divided by the response of the pair at the same separation sitting deepest in the molecule. A straight chain is below one at every separation and a zigzag is above one at every separation, so the end effect has opposite signs on the two shapes. The third chain — two straight arms meeting at one angular ring — is above two at three separations and below one at the fourth, which is a third behaviour and not an intermediate one.

An end effect with two signs

Neither of two separations accounts for the scatter in a fused ring system's response, and the natural guess is the end: pairs with more molecule outboard should behave differently from pairs at an edge. They do. In a straight chain an end pair responds a third less than an interior one, in a zigzag a quarter more, and in a chain of two straight arms meeting at one angular ring the anomaly is in the middle.

symmetry · Aromaticity One turned fusion, moved along the chain. Chains of 9 rings differing in one integer: which fusion's direction is turned. Turned at the first or last fusion, that leaves one angular ring beside an end; anywhere between, it leaves two adjacent angular rings whose turns cancel. The widest ratio between two pairs at the same separation, against which fusion is turned. A straight chain gives 1.520 and every bent one gives more — from 3.052 to 4.600. The two ends of the curve are the one-ring members; every interior point is a two-ring step.

One integer, and everything it changes

Eight molecules with the same rings, the same carbons and the same graph distance between every pair, differing in which fusion's direction is turned — one angular ring when the turned fusion is at an end, two adjacent ones anywhere else. The scatter within a separation class runs from three to four and a half times, against a straight chain's one and a half — and the two ends of one molecule disagree by up to a factor of four.

symmetry · Aromaticity The one-bend family was one angular ring at its ends and two everywhere else. Filled circles: the worst spread within a separation class for the eight nine-ring chains built by bending one fusion, placed at the angular rings each actually has. The chains bent at the first and last fusions have one angular ring, next to an end; the six bent in between have two adjacent angular rings whose turns cancel. The lower line is a single angular ring at each interior position, running 3.05, 3.66, 3.46, 3.03, 3.46, 3.66, 3.05; the upper line is two adjacent ones, running 4.30, 4.60, 4.06, 4.06, 4.60, 4.30. Every published point lies on one line or the other.

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.

bonding · Aromaticity

Named alongside it

The objects these essays reach for when they reach for this one.

Hückel theoryDelocalisationDegeneracyAntiaromaticityConventionHückel's 4n+2 ruleModel limitReference stateConjugationLeast-squaresRing currentShell closure

All concepts