Say what it encloses.
An orbital picture is not a picture of a cloud. It is a contour — the surface on which the wavefunction takes one chosen value — and the value was chosen by whoever drew it. "The ninety per cent surface" is the stock phrase and very often not what has been plotted, because nobody did the integral. These are essays about molecular shape and bonding with the integrals done: every orbital here is drawn at a level found by integrating the density, and says what fraction it encloses.
430 essays · 9 fields · 31 series · 366 named objects · every figure computed from its model
The parts of the subject
9 fields — what each one is for
Orbitals
A one-electron wavefunction, drawn as a contour surface at a level somebody chose. Nodes, signs, and what the picture is a picture of.
Where the atoms go
VSEPR as a repulsion minimisation rather than a table, the angles that fall out of it, and the case where five sites are not all alike.
Bonding models
Valence bond, molecular orbital, and hybrids — three descriptions of one thing, related by transformations that change no observable.
What symmetry decides
A molecule's point group follows from its coordinates, and it settles whether the molecule can be polar or chiral with no reference to bonding.
Beyond the octet
Hypervalency without d orbitals, delocalisation, aromaticity, and the structures the first-year rules quietly cannot describe.
What a spectrum settles
A spectrum is a list of positions, and how many there can be is decided by the shape before any of them is measured. Counting them, and reading a structure back out.
When the molecule does not stop
A chain of two hundred atoms is a molecule and behaves like a solid. Bands, gaps, metals, defects and surfaces, every one of them out of a finite matrix — and a clear account of what that route cannot reach.
What the shape is for
Coordination compounds, where the shape decides a colour, a magnetic moment and a bond length — and where the one-electron picture used everywhere else in the collection stops being enough.
What is taught wrongly
The explanations that are confident, memorable and false — stated fairly and then tested against a calculation rather than an opinion.
How far each idea goes
31 series — every one of them
Representation
A molecule's point group is a list of matrices, not a label. Generating them, sorting them into classes, and dividing by the group order turns any set...
Contour
An orbital picture is a contour at a level somebody chose, and almost no source says which. Two textbooks can draw the same orbital at visibly...
Basis
Sixty years of molecular calculation are built on functions that get the two ends of an orbital wrong. A Gaussian has no cusp at the nucleus and dies...
Hypervalency
Sulfur hexafluoride is not d²sp³ hybridised. The d orbitals are far too high in energy to contribute meaningfully, the bonding is three-centre...
Magnetism
A magnetic moment is one of the few chemical measurements that returns an integer. Feed the count of unpaired electrons into √(n(n+2)) and nine...
Multicentre
A bond between two atoms is a special case, not the general one. Rings, clusters and metals are held together by orbitals spread over many centres,...
Point group
A molecule's symmetry is not a label to be looked up. It is decidable from the atom positions by searching for the operations that permute them, and...
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...
Threads running through
themes, not chapters
Say what it encloses
An orbital picture is a contour at a level somebody chose, and almost no source says which. Every one here is drawn at a level found by integrating the density, and states the fraction it encloses.
Exactly zero
Where symmetry forbids an interaction the overlap is not small, it is zero. Computing it and finding arithmetic noise is a different kind of statement from computing it and finding a small number.
A basis is not a thing
Hybrid orbitals, localised bonds and canonical molecular orbitals are related by transformations that leave every observable unchanged. Arguing about which is real is arguing about a coordinate system.
Symmetry settles it
Whether a molecule can have a dipole, or be chiral, or show a particular spectral line, follows from its point group alone — with no reference to what the bonds are made of.
One electron only
Every orbital drawn here is a hydrogen-like solution. A many-electron atom has no exact orbitals at all, and keeping that in view is the difference between a model and a picture of reality.
Taught confidently, and wrong
This subject has an unusual number of explanations that are memorable, widespread and false. Each is stated fairly here and then tested against a computation rather than an opinion.
Measured from somewhere
A great many numbers in this subject are differences, fits or projections: a resonance energy against a reference state, a mode's percentage composition in a set of coordinates, a structure's weight under a convention, a moment from a fitted law. Each is quoted as though it were measured, and each changes when the choice behind it changes.
What the measurement does not fix
A spectrum with two lines in it constrains two numbers, and a model with three parameters fitted to it has a curve of answers rather than an answer. These essays are about measurements that leave a family of possibilities standing — and about what it takes to tell the members of the family apart.
Counted, not quoted
Node counts, bond angles, overlap integrals and point groups are all computed from the geometry and the wavefunctions themselves. None of them is a number recalled from a table.
Just written
the most recent — what arrived when
Orbitals
Length did not rescue the consecutive triples
On four fragments, keeping only the consecutive triples of a counterpoise assembly worked for two arrangements and did worse than pairs for the uniform chain, and a short chain was the obvious excuse. Carried to eight fragments the excuse fails: the uniform chain's consecutive assembly settles at 48 per cent of the line against 68 for pairs. And the heavy-outside chain turns the lesson over — from five fragments every triple together covers less than the consecutive ones alone.
Orbitals
The zero is a parity, not a bond
A hydrogen-like σ bond has a momentum profile along its axis that vanishes at π/R, and it is natural to read that zero as a bond's signature. Built from 2p functions pointing along the axis, the σ bond carries a sine instead and sits at 83 per cent of its peak there. Which factor an orbital carries is decided by whether its inversion parity matches its atom's, and bonding has nothing to do with it.
Beyond the octet
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.
Orbitals
The zero belongs to one determinant
A bonding orbital's momentum profile along the bond is exactly zero at π/R, and that zero reads a bond length with nothing fitted. It is a property of putting every electron into that one orbital. Any antibonding occupation fills it in linearly and drags the minimum outward, a tenth of an electron erases it, and the valence-bond wavefunction built from the same two functions never has one at any separation.
Where the atoms go
The long bond goes to the crowded site
Given one bond longer than the others, the repulsion model puts it axial in a trigonal bipyramid — against the rule it is usually cited for. The reason is a crowding count, and at seven sites the count reverses: the pentagonal bipyramid's crowded site is equatorial, so a long bond goes equatorial and a short one axial. PF₅'s long bonds are axial and IF₇'s are equatorial. And at seven the site a bond avoids is not even a minimum.
Beyond the octet
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.