Molecular Geometry & Bonding
what the shape decides
Concept

HOMO–LUMO gap — where it appears

The distance from the highest occupied level to the lowest empty one. For a molecule it is a finite number that falls as the system grows, and calling it a band gap requires that limit to be non-zero.

Named by 17 essays across 7 fields — each of them below, with the objects they name alongside it.

Hückel levels of hexatriene. 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.

Conjugation, and its limits

Every extra double bond in a chain lowers the gap between the highest occupied and lowest empty orbital, which is why long conjugated molecules are coloured. The trend has a limit, and the limit is not where the arithmetic says it should be.

beyond · Delocalisation Rings of 6, 10, 20, 60 and the band at 2000. The Hückel levels of rings of 6, 10, 20, 60 atoms, all of them inside the same interval from −2 to +2, beside the density of states of a ring of 2000. The histogram is the computed levels; the line through it is the closed-form density, which diverges at both band edges.

The band limit

Every level of a ring of n atoms lies between −2β and +2β, however large n gets. The levels do not spread out as the molecule grows; they crowd into a fixed interval — and that crowding, computed, is a band with its density of states diverging at both edges.

beyond · Multicentre A chain of 100 is more stable alternating. The electronic energy gained by alternating the bonds of a half-filled chain, the elastic cost of doing it, and their sum, all per site and all against the alternation. The sum has its best value away from zero, so the evenly spaced chain is not the stable structure.

A chain cannot stay even

Diagonalise a half-filled chain, alternate its bonds slightly, and diagonalise again. The electrons gain more than the springs lose, and they do so for every spring constant whatever — because the gain is steeper than a parabola near the origin and a logarithm beats any constant.

solids · Peierls distortion The gap against size: uniform against δ = 0.15. The HOMO–LUMO gap of a half-filled chain plotted against the number of sites, on log axes, for a uniform chain and for one whose bonds alternate. The uniform sequence falls without limit; the alternating one settles at four times the alternation.

The gap is not the band width

Two numbers describe a band and they answer different questions. The width is set by how many neighbours an atom has; the gap is set by how unequal they are. A structure can have a wide band and no gap, a narrow band and a large one, and changing one leaves the other alone.

solids · Peierls distortion 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.

Counting electrons in an extended structure

The octet rule, Hückel's 4n + 2 and the 8 − N rule that predicts the structures of the main-group elements are one rule counted three ways. Each says the same thing — close the shell — and each stops being reliable at exactly the point where closing it becomes impossible.

solids · Metal The state on a defect at site 31 of 60. The amplitude of one eigenvector at each site of a 60-site chain whose site 31 has its energy raised by 1.6β. The two colours are the two signs. A state belonging to the whole chain would reach across it; this one does not.

A defect is a level in the gap

Change one site's energy in a chain of a hundred and sixty and a level leaves the band, carrying a state that lives on a handful of atoms. In one dimension it happens for any change however small — the threshold measured on chains of forty, eighty and a hundred and sixty halves with every doubling, so there is no threshold at all.

solids · Defect 60 electrons in 60 levels. The density of states of a ring of 60, drawn with the energy up the page, and the 60 electrons filled in from the bottom. Where the filling stops is what decides whether the system has cheap excitations.

A half-filled band is not always a metal

Every band picture rests on an approximation that a whole class of materials refuses — each electron moving in an average field, never seeing another one individually. Where the repulsion is strong enough, a half-filled band describes an insulator, and no amount of care with the band fixes it.

wrong · Metal The gap against size: uniform against δ = 0.15. The HOMO–LUMO gap of a half-filled chain plotted against the number of sites, on log axes, for a uniform chain and for one whose bonds alternate. The uniform sequence falls without limit; the alternating one settles at four times the alternation.

A band gap is not a bond energy

Silicon's gap is 1.1 electronvolts and its Si–Si bond is 2.3. Both are quoted in the same units, both describe the same material, and neither is convertible into the other — one is the cost of promoting an electron and the other is the cost of taking two atoms apart.

wrong · Bands in a solid The total energy against the distortion, at several gaps. The elastic cost plus the second-order lowering, for five gaps between the ground state and the excited state it mixes with. The critical gap is 1.00: above it the symmetric structure is the minimum, below it the minimum has moved off zero, and nothing about the molecule is degenerate in either case.

A distortion needs two states

A degenerate electronic state cannot survive — that is the Jahn–Teller theorem, and it can be computed. A closed shell can fail to survive too, and the condition is a number: the symmetric structure holds only while the nearest excited state of the right symmetry lies above 2λ²/k, and one of ten symmetry species in an octahedron is the right one.

applied · Peierls distortion The bonding follows the overlap over, and turns where it turns. The stabilisation of three pairs of orbitals against how far apart they are, each scaled to its own largest, with the coupling taken from the computed overlap. Two orbitals with no radial node are stabilised less at every separation further out; the two with one turn over — and the maximum of the bonding is at exactly the separation of maximum overlap, to five decimal places, because a coupling proportional to the overlap makes the stabilisation a strictly increasing function of it.

The same overlap, a different bond

A 1s and a 2s change their overlap by two thirds between two and seven bohr, and change what they are bonded by by 0.154 per cent. The overlap turns over and the bonding turns over with it, at exactly the same separation to five decimal places — the arithmetic refused the expectation that the secular denominators would move it — and what separates one pair from another is not where the maximum is but how little of it there is.

wrong · Overlap Which numbers survive a change of frame, and which are coordinates. Every quantity this field quotes, against the four things that can be changed without changing the molecule: the zero of energy, the unit, the reference state a stabilisation is measured from, and what a "per" quantity is divided by. A filled mark is a quantity that moves. three of the 9 survive all four, and every one of them is a property of the eigenvectors rather than of the energies. The first two changes are exact symmetries of the model, so a quantity that moves under either is a coordinate and not a quantity at all.

Which numbers carry a frame

A Hückel calculation is written in two numbers nobody computes and quoted against reference states nobody measures, so every quantity it prints is a quantity in a frame. Nine of them, tested against four changes of frame: three survive all four, and the one that survives both exact symmetries and looks safest — a dimensionless ratio between two molecules — is the most fragile of all, because against one reference its denominator is exactly zero.

bonding · Delocalisation One pi energy, two delocalisation energies. For each of six rings: the computed pi energy, the energy of the same atoms with one bond deleted, and the delocalisation energy that follows from each of the two reference states. Every entry is an eigenvalue sum; the two right-hand columns differ only in what was subtracted from the second column.

The frame that was allowed to relax

Four changes of frame were tested on nine quantities and none of them could move a bond order, because a bond order is a property of the eigenvectors and every change left the eigenvectors alone. Letting the geometry answer back does move them — butadiene's central bond falls from 0.4472 to 0.3676 — and it moves naphthalene's the other way, because its weakest bond is the one two rings share and relaxation strengthens it.

bonding · Delocalisation An effect that cannot distort a molecule, deciding how far it distorts. The energy along one distortion coordinate, three times. With only the first-order term the minimum is at 0.6; with only the second-order term there is no minimum away from zero at all, because the gap of 1.5 is above the critical 1 for a closed shell on its own. With both, the molecule distorts to 1.06 — well past the first-order answer — and gains 0.09 more than the two separate stabilisations add up to.

Two distortions in one coordinate

A second-order Jahn–Teller effect that cannot distort a molecule by itself — its gap is half again above the critical value — nearly doubles the distortion when a first-order effect is already acting. The molecule goes to 1.075 instead of 0.600 and gains twice the energy, and it does it while the gap the second-order term divides by is opening rather than closing.

shape · Peierls distortion Two molecules, one eigenvalue, and 1.27 eV between them. Six alternant hydrocarbons placed by the Hückel eigenvalue of their highest occupied level — computed by diagonalising each molecule's own adjacency matrix — against the measured first π ionisation energy. Ethene and benzene share an eigenvalue of exactly 1 and their measurements differ by 1.27 eV; butadiene and naphthalene share 0.618 and differ by 0.94. A model that reads only the eigenvalue is a function of it, so it must give each pair one answer, and the two vertical pairs are the whole of its error.

A parameter that never finds a value

Show a two-parameter model six measurements instead of two and it stops being underdetermined and starts being wrong. Adding the third parameter improves the fit by one part in eighty, moves the resonance integral by a factor of four, and never finds a best value at all — because nine tenths of the error is a term the model does not have.

bonding · Models The boundary belongs to the gap, not to the repulsion. The repulsion at which a satellite stops being tellable from a fundamental by intensity, against the system's own one-electron gap. Four systems: ring of 6, gap 2.000, boundary 8; chain of 4, gap 1.236, boundary 4; chain of 6, gap 0.890, boundary 2; ring of 4, gap 0.000, boundary 0.25. The three with a gap order exactly with it, and the ring of four — whose half-filled ground state is degenerate and whose gap is zero — has no boundary at all: its contrast is one at every repulsion, so its satellites are never distinguishable and there is nothing for a boundary to separate.

The boundary belongs to the gap

A satellite stops being tellable from a fundamental somewhere, and it can be located on one ring at one filling. Four systems put it at repulsions of 2, 4 and 8 — ordering exactly with each one's own one-electron gap and not with its band width — and the fourth, whose gap is zero, has no boundary at all: its satellites are indistinguishable at every repulsion including none.

spectra · Photoelectron The pair's regime is not a property of the pair. How much the pair's bonding responds to its own overlap — the ratio of what it is bonded by at an overlap of 0.4 to what it is bonded by at 0.1 — with and without a third orbital coupled to both. Alone it is 11.83, which is the regime in which bonding tracks overlap. With a third orbital present it falls to 1.46, 0.92, 0.74 — and two of those are below one, meaning a fourfold increase in the overlap between the two atoms buys them less bonding rather than more. The coupling comes from the overlap by the Wolfsberg–Helmholz rule with K = 1.75, which is fitted rather than derived. Every stabilisation here inherits that; the shape of the curve against separation does not, because K is a constant.

A regime that belongs to the neighbours

Two orbitals at the same energy are bonded in proportion to their overlap and two far apart are barely bonded at all — two regimes, and the natural question is whether the regime is a property of the pair. It is not. Put a third orbital beside them and the pair's response to its own overlap falls from twelvefold to less than one: more overlap buys less bonding.

wrong · Overlap How much stronger a fundamental is than a satellite, against the repulsion. The weakest fundamental divided by the strongest satellite, for a six-site ring and chain at every filling from a third to a half, against the on-site repulsion. Below the line at two the two kinds of line cannot be told apart by their height. The half-filled systems cross it and the third-filled ones do not — not at any repulsion up to sixty-four times the hopping, where the third-filled ring is still at 8.3.

A satellite that never loses its place

The repulsion at which a satellite stops being tellable from a fundamental orders exactly with the one-electron gap across four systems. Changing the gap by the filling instead is the sharper test, and the ordering does not survive it: a six-site chain has a larger gap at half filling and a smaller boundary. Below half filling there is no boundary at all, at any repulsion up to sixty-four times the hopping.

wrong · Photoelectron

Named alongside it

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

Model limitBands in a solidEigenvalueDegeneracyConjugationConventionHückel theoryBond alternationDelocalisationIonisation energyMetalPeierls distortion

All concepts