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.
498 essays · 9 fields · 31 series · 369 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...
Hypervalency
Sulfur hexafluoride is not d²sp³ hybridised. The d orbitals are far too high in energy to contribute meaningfully, the bonding is three-centre...
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...
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...
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...
Orbital
Not a region the electron occupies, not a path it follows, and for any atom but hydrogen not an exact anything. An orbital is a one-electron...
Overlap
Two orbitals interact in proportion to how much they overlap, and the sign of the overlap decides which of the two combinations is the lower in...
Electron count
The eighteen-electron rule is usually justified by adding up an s, three p and five d orbitals. That is a restatement rather than a reason. Reduce the...
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
Where the atoms go
The lone pair has no bond to stretch
Put every one of xenon hexafluoride's bonds on a spring and let the repulsion set the six lengths, and which three come out long is decided by something the model has never had to say: how far from the xenon the lone pair sits once the bonds have moved. Three equally defensible answers give three different verdicts — and none of them produces a split within a factor of twenty of the measured one without stretching every bond beyond recognition.
Bonding models
A verdict that flips and changes nothing
Run the two-orbital mixing criterion along H₂C=CH⁻, H₂C=NH, H₂C=O and H₂C=F⁺ and one pair finally crosses its boundary — the π and the lone pair, a third of the way from oxygen to fluorine. It is the first real test the criterion has had, and the localised orbitals on the far side of it are the ones on the near side: two bent bonds and a lone pair that stays itself.
When the molecule does not stop
The chain that pays for nothing
Held to a fixed length, a Peierls chain pays for its ends' extra alternation with a compression of every bond. Let the length go and nothing is paid: the middle is the free infinite chain to twelve figures, the ends adjust locally — and every tail exponent comes out where the fixed chain put it, to under a thousandth. The constraint was never in the exponents. Nor is the drift in them a missing 1/b term: fitting one makes the exponent wander three times further.