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.
444 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...
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...
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...
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,...
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
A floor no charge transfer explains
A σ bond between two identical atoms has a momentum profile that is exactly zero at π/R, because the two atomic contributions are identical and cancel. Two different atoms cannot cancel there at any coefficients, so the depth of the minimum becomes a measure of polarity — and it is one, monotone in the charge imbalance in the direction chemistry moves charge. What it is not is a measure with a zero: it has a floor set by how far the two radial functions are from proportional, and B–N's floor is deeper than C–O's although C–O is the more polar bond.
Where the atoms go
The steps were a square root of a mass
Replacing ammonia's hydrogens one at a time does not lower its splitting in equal steps, and the obvious readings are the potential's shape or the tunnelling arithmetic. Both lines — the ground doublet and the first excited one, which nothing in this argument has ever been fitted to — come out straight in the square root of the reduced mass and bent against the count, by a factor of eight, so the unequal steps are the shape of a square root drawn against the wrong variable. And the excited line is the one test the missing correction cannot pass by construction: it improves it threefold and over-shoots it.
Orbitals
The pair that cancels only at zero overlap
A directional Compton profile along a bond is exactly zero at π/R for every cosine combination, which makes the depth there a count of parity mismatches. A gas measurement averages over orientations, and the question was whether the count survives. It does not, for a reason nothing in the directional picture shows: the bonding and antibonding combinations of one atomic function are normalised by 2 + 2S and 2 − 2S, so a filled pair leaves a residue proportional to its overlap — and in nitrogen that residue is the largest single term, from a pair whose imbalance is zero.
Where the atoms go
The band that cannot supply what it changes
A missing transverse zero-point term was sized at five per cent of the difference between two molecules' frozen frequencies, as though the five frozen modes were one thing. They are three bands carrying 26, 50 and 24 per cent of that difference, so the requirement is not one fraction but three, spanning a factor of two — and the band a flattening pyramid changes most is the one that would have to move furthest. What survives the choice is the barrier: every version implies the same bare electronic value to one and a third per cent, because every band of a pyramidal hydride has the same isotope frequency ratio.
Where the atoms go
The same fraction at six times the barrier
Ammonia's two fitted barriers leave a gap that an isotope-dependent term would have to fill, and the term was sized at five per cent of the transverse zero-point difference. Phosphine can be asked the same question and cannot answer it the same way: its splitting is twenty-four orders of magnitude below anything the solver can represent, and the solver does not fail cleanly — it reports values that rise as the barrier rises. Calibrated where a splitting is still visible, the action carries the question across, and the same fraction moves phosphine's isotope ratio by sixteen per cent against ammonia's twenty-one.
Bonding models
A cancellation between two separations
The give-back's numerator has one term in it, so its root is the repulsion at which the pair count at one bond returns to its free value. Let the interaction reach two bonds and the numerator has two terms with opposite signs, and its root becomes a cancellation between an enhancement at one separation and a depletion at the next — at a repulsion where neither is at its free value. Which side of the old zero the new root falls on reverses between a reach of a third and a half, because whether two bonds are enhanced or depleted is the interaction's own doing.