Molecular Geometry & Bonding
what the shape decides
Theme

The thread: 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.

419 essays carry this theme. The first 24 are shown with their figures; the rest are listed without, so that the page does not grow without limit.

Chains of 2, 4, 8, 16, 40: the levels crowd, the edges do not move. Every level of a chain of 2, 4, 8, 16, 40 sites, drawn at its computed energy on one axis. Adding sites adds levels inside a fixed interval rather than widening it, which is what makes the large system a band rather than a wider molecule. When the molecule does not stop

A solid is a molecule that did not stop

Diagonalise a chain of two atoms, then four, then forty. Nothing new happens at any point, and by forty the levels are a band. The passage from molecule to solid is not a change of subject; it is the same matrix at a different size, and every step of it can be watched.

 methane — Td. 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. What is taught wrongly

Hybridisation does not explain

Methane's photoelectron spectrum has two bands, not one. Four equivalent sp³ bonding orbitals cannot produce that, and the resolution is that hybridisation was never a claim about what a measurement would find.

 H₂O: 3 distinct modes. The displacement of every atom in 3 normal modes of H₂O, drawn from the eigenvectors of the mass-weighted Hessian. Under each is the internal coordinate with the largest share of the motion and how large that share is; where no coordinate holds nine tenths, the mode is not a motion of one bond or one angle and is labelled so. What a spectrum settles

Normal modes are not bond stretches

Water has two stretching frequencies and two O–H bonds, and it is almost irresistible to pair them off. Computed, each mode is exactly half in one bond and half in the other, and neither frequency belongs to a bond at all.

 1s with 1s at 2.8 bohr. The two orbitals in the plane containing both nuclei, with the regions where their product is positive and negative shown faintly. The overlap integral is the signed volume of that product, and where symmetry makes the two regions mirror images it comes out exactly zero. Contours drawn: 1s at 50% of its density, |ψ| = 1.48e-1; 1s at 50% of its density, |ψ| = 1.48e-1. Bonding models

Overlap decides

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 energy. It is one integral, and almost everything about bonding follows from it.

 ammonia — C3v. 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. What symmetry decides

Point groups from coordinates

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 the search either finds an operation or it does not.

 A d shell in an octahedral field. The five d energies in octahedral coordination, computed and drawn in units of the octahedral splitting. The dashed line is the barycentre, which the five levels sum to whatever the arrangement. What the shape is for

The splitting is a symmetry statement

Put six ligands round a metal and the five d orbitals stop being degenerate. Which of them stay together, and how many sets there are, follows from the point group alone — before any account of what the ligands are made of, and before any number is computed.

 4 sites, minimised. The arrangement of 4 points on a sphere that minimises their mutual repulsion. The angles printed were measured off the result rather than quoted, and the shape was not assumed. Where the atoms go

VSEPR, computed

The tetrahedral angle is not 109.5 degrees because a textbook says so. It is arccos(−1/3), and it falls out of minimising the repulsion of four points on a sphere without ever being written down.

 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. Beyond the octet

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.

 1s with 2px at 2.8 bohr. The two orbitals in the plane containing both nuclei, with the regions where their product is positive and negative shown faintly. The overlap integral is the signed volume of that product, and where symmetry makes the two regions mirror images it comes out exactly zero. Contours drawn: 1s at 50% of its density, |ψ| = 1.48e-1; 2px at 50% of its density, |ψ| = 3.16e-2. Bonding models

Exactly zero

Where symmetry forbids an interaction the overlap is not small. It is zero — and computing it and finding arithmetic noise is a different kind of statement from computing it and finding a small number.

 Five sites are not five of a kind. The minimised arrangement of five points, with the two axial sites marked apart from the three equatorial ones. Their neighbour angles differ, so the two kinds of position are genuinely different places — which the shape's name does not convey. Where the atoms go

Five sites are not alike

Every other common arrangement has one or two distinct angles. Five has three, because two of its positions are on an axis and three are round an equator — and a molecule built that way does something about it.

 Choosing a contour for 3s. The fraction of the density enclosed by a contour, against the contour's level. Picking a level is picking a point on this curve, and the usual practice of picking one that looks right is picking a point without knowing which. Contours drawn: 3s at 50% of its density, |ψ| = 6.24e-3; 3s at 90% of its density, |ψ| = 2.64e-3; 3s at 99% of its density, |ψ| = 7.15e-4. Orbitals

Say what it encloses

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 different sizes with the same caption, and both be printed in good faith.

 H₂O and its D isotopologue. Every frequency of H₂O joined to the frequency the same force field gives when every H is replaced by D, with the ratio on each join. The force constants were not refitted and could not be: they do not depend on mass. The product of all the ratios is fixed by the masses and the moments of inertia alone, and is checked against that identity while this figure is drawn. What a spectrum settles

The isotope shift is arithmetic

Replace hydrogen with deuterium and every frequency drops. The usual rule says by a factor of the square root of two — and of water's three modes, not one of them does that. What is exact is a different identity, and the force constants cancel out of it.

 ⟨x²⟩ is the average number of neighbours. For each structure, the mean coordination counted off the edge list beside the mean of x² measured off the eigenvalues. They are equal by an identity about graphs, not by a limit — the full band width beside them obeys no such rule. When the molecule does not stop

The width of a band is a count of neighbours

The mean of the squared level energies equals the average coordination — exactly, for any structure, with no limit taken and no periodicity assumed. It is the one statement in this field that is arithmetic rather than physics, and the usual textbook formula for band width is a special case of something weaker.

 The same d shell in four fields. The five d energies in octahedral, tetrahedral, cubic, square planar coordinations, computed and drawn in units of the octahedral splitting. The dashed line is the barycentre, which the five levels sum to whatever the arrangement. What the shape is for

Two models, one ratio

A tetrahedron splits a d shell by four ninths of what an octahedron does. Two models that share nothing but the ligand directions — an integral over a point-charge potential and a rotated diagonal matrix — both produce that number to eight decimal places, and neither was told it.

 The density of states of a chain of 2000. The 2000 levels of a linear chain, binned into 34 intervals across the band, with the closed-form density drawn through them. The density piles up at both edges because that is where the level spacing turns over, and nothing periodic was assumed to get it. When the molecule does not stop

A density of states is not a spectrum

A molecule's spectrum is a list of positions and a solid's is a shape, and the shape is not the density of states. Between the two sits everything the count leaves out — which transitions are allowed, how strongly, and from where to where.

 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. Beyond the octet

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.

 ¹¹BF₃: what each mode is made of. ¹¹BF₃. Each row is one distinct frequency and each column one internal coordinate; the bar is the share of the motion in that coordinate. A mode whose largest share reaches nine tenths is a motion of one bond or one angle and is named for it. 1 of 4 here are. What a spectrum settles

Group frequencies, and where they stop

A carbonyl band sits near 1,700 wavenumbers in every ketone anybody looks at, and that regularity is real. Computed for a set of small molecules, four of twenty-one distinct frequencies belong to a single internal coordinate — and the four are exactly the coordinates symmetry leaves alone.

 sp3 hybrids. The directions the hybrids point, with the angle between them computed from the coefficients rather than quoted. The set is an orthogonal transformation of the atomic orbitals, so it describes the same space in different coordinates. Bonding models

Hybrids are a basis

An sp³ hybrid set is an orthogonal matrix applied to the atomic orbitals. Rotating a basis changes no observable, so asking whether the electrons are really in hybrids is asking which coordinate system nature prefers.

 The radial function of 3s. The radial part of the wavefunction, which changes sign at each node, and the radial distribution, which is the probability of finding the electron in a shell at that radius. The second vanishes at the nucleus and the first does not. Orbitals

Nodes

An orbital with quantum numbers n and l has exactly n−l−1 radial nodes and l angular ones. That is a count, it is exact, and it is the fastest way to catch a drawing that is wrong.

 What the group settles. For each molecule, the point group found from its coordinates and the two properties that follow from the group alone. Neither column required knowing anything about the bonds. What is taught wrongly

The dipole is not a sum of bonds

Adding bond dipoles as vectors gets the easy cases right and rests on a quantity with several incompatible definitions. The symmetry argument is exact, needs no electronegativities, and says when the answer must be zero.

 Where the d–d band falls, and where the eye is. Nine ligands' measured splittings as wavelengths, on a logarithmic scale, with the visible range shaded. Three fall inside it; the halides sit in the near infrared and carbon monoxide in the ultraviolet. The splitting decides where the band is and does not decide what is seen. What the shape is for

Where a d–d band falls

A splitting is an energy and an energy is a wavelength, so the ligand field fixes where a complex absorbs. Only three of nine common ligands put that band inside the visible range at all — and the most intensely coloured transition-metal compound in the cupboard has no d electrons to excite.

 water — C2v. 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. Where the atoms go

Why water is bent

The standard answer is lone pair repulsion, it predicts the right direction, and it cannot predict the magnitude. A better rule can, and the heavier hydrides show where both accounts run out.

 Chains of 4, 16, 64, 160: the levels crowd, the edges do not move. Every level of a chain of 4, 16, 64, 160 sites, drawn at its computed energy on one axis. Adding sites adds levels inside a fixed interval rather than widening it, which is what makes the large system a band rather than a wider molecule. When the molecule does not stop

A band with no structure in it

Everything in this field is computed from a finite matrix with no periodicity assumed, which is a real method and a real limitation. It produces a band and cannot produce a band structure — and the difference between those two words is worth an essay, because it is the boundary of what a finite matrix can honestly say.

 Eight points: the cube loses. The cube and the minimised arrangement of eight points on a sphere, with the repulsion energy of each computed. The minimum is a square antiprism — the cube twisted by forty-five degrees on one face — and the margin is about one part in three hundred. Where the atoms go

The shapes above six coordination

Eight points on a sphere do not arrange themselves in a cube. They twist one face by forty-five degrees, and above six the arrangements stop being the ones anybody would name and start being the ones a minimisation finds.

 What symmetry decides

Character tables and reduction

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 of orbitals into a statement about how many energies there can be.

 What the shape is for

Eighteen is a 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 ligand orbitals in the complex's own point group, match them against the metal's by symmetry, and the number that comes out is the count of orbitals lying below a gap — which is eighteen for an octahedron, sixteen for a square plane, and eighteen again for a tetrahedron for a different reason.

 What is taught wrongly

Electronegativity is not one quantity

Four scales, four definitions, four sets of units, and a defence — "they correlate well" — that answers a question nobody asked. Two scales can correlate at 0.99 and still put hydrogen on the wrong side of carbon.

 What a spectrum settles

How many frequencies, not how many modes

Benzene has thirty vibrations. It has twenty distinct frequencies, and of those, eleven can be seen — four in the infrared and seven in the Raman, with no band in common. The other nine are invisible to both experiments, exactly.

 Orbitals

Where the electron is

The wavefunction is largest at the nucleus, the electron is most likely to be found a bohr out, and the ninety-per-cent contour is at 2.66. Three numbers, all correct, all answering different questions.

 Beyond the octet

Where two-centre bonding stops

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, and the arithmetic that describes them is the arithmetic already used for benzene.

 What the shape is for

Back-bonding is two interactions

A carbon monoxide molecule bound to a metal donates from an orbital that is slightly antibonding and accepts into one that is strongly antibonding, so the two halves of the bonding move its stretching frequency in opposite directions. Five isoelectronic complexes differing only in charge settle which wins — and one of them stretches above free CO.

 Where the atoms go

Bent's rule, against the substituent series

More electronegative substituents get more p character, so the angle between them shrinks. The rule predicts a trend, the trend is observed, and the quantity it depends on turns out not to be one quantity.

 What is taught wrongly

Delocalisation is stabilising, and other things that are false in general

Spreading electrons over more centres is supposed to lower their energy. Cyclobutadiene is delocalised in exactly the same sense as benzene, gains exactly nothing by it, and the arithmetic says so before any experiment does.

 Bonding models

Hückel theory and what it gets right

A conjugated system's orbital energies are the eigenvalues of a matrix of ones and zeroes. Nothing about carbon enters, no geometry enters, and the results that survive are exactly the ones that depend on neither.

 What symmetry decides

Selection rules are one theorem

An integral over all space vanishes unless the integrand is totally symmetric. Every selection rule in spectroscopy is that sentence with a different integrand — and the rule of mutual exclusion falls out rather than being remembered.

 Beyond the octet

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.

 Orbitals

The radial distribution across the periodic table

A 4s orbital is bigger than a 3d by every measure of size except the one that decides which fills first. Penetration is a feature of a small inner peak, and the periodic table's shape depends on it.

 What a spectrum settles

Two structures, two spectra

A linear XY₂ gives two infrared bands, one Raman band and no band in common. A bent XY₂ gives three of each and three in common. Counting settles the shape, without a force constant, an assignment or a single measured frequency.

 When the molecule does not stop

Where a molecule stops being one

There is no size at which a molecule becomes a solid, and the useful question is a different one — how large must it be before a given property has stopped changing? The answers differ by a factor of several hundred between one property and the next, and every one of them is a measurement.

 What is taught wrongly

A double bond is not two single bonds

Carbon's single bond is 348 kilojoules a mole and its double is 614, which is not twice anything. The two halves are different integrals over different orbitals with different distance dependence, and computing them shows why no arithmetic could have made them add.

 Bonding models

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.

 What symmetry decides

Site symmetry, and what it constrains

A molecule's group is not the only group in the problem. Each atom sits at a position with a symmetry of its own, and what that local group permits decides how many kinds of environment a structure really has.

 What the shape is for

The trans influence is an overlap argument

Two ligands on opposite sides of a metal both bond through the same metal orbital, and there is only one of it. Strengthen one and the bond order to the other falls — computed exactly on three levels, and measured as a bond length that grows by a tenth of an ångström.

 Beyond the octet

Three-centre bonding, computed

Three orbitals in a line and four electrons: a bonding level, a level with exactly zero amplitude on the central atom, and an empty antibonding one. The middle atom never exceeds an octet, and the ligands carry the charge — which is why every molecule that needs this arrangement has electronegative ligands.

 Where the atoms go

What a lone pair is worth

A lone pair repels more than a bonding pair, says VSEPR, without saying how much more. Put a number on it and fit that number to water, and the same number is wrong for hydrogen sulfide by a factor of two — which means it was never a property of a lone pair.

 When the molecule does not stop

What a metal actually is

Not shiny, not a good conductor, not an element on the left of the table. A metal is a system with excitations of arbitrarily small energy, and that definition can be checked on a sequence of finite rings without drawing a band diagram or mentioning conduction at all.

 What a spectrum settles

What an absence proves

A band that symmetry forbids is not weak. Its intensity is zero, exactly, by a theorem — while a band that is merely too faint to see is absent for reasons no theorem covers. The two look identical in a spectrum and support completely different conclusions.

 When the molecule does not stop

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.

 What a spectrum settles

A spectrum counts environments, not atoms

Phosphorus pentafluoride has five fluorines in two inequivalent sets, so its magnetic resonance spectrum should show two signals. It shows one — and the reason is not a symmetry the molecule has but a motion faster than the measurement.

 Bonding models

Bond order from the eigenvectors

Having diagonalised the matrix, the coefficients are already there. Two sums over them give every bond's order and every atom's charge, and naphthalene's three kinds of bond come out in the order the measurements find them.

 What the shape is for

Copper is never quite octahedral

A d⁹ ion in an octahedral field has three electrons in a doubly degenerate pair, which cannot be shared evenly. The energy it gains by distorting is linear in the distortion and the elastic cost is quadratic, so no stiffness holds the symmetric structure — and the four configurations with an even occupation gain exactly nothing.

 What symmetry decides

Degeneracy is a group theorem

How many orbitals can share an energy is decided before any energy is computed. The dimensions of a group's irreducible representations are the only degeneracies it permits, and a molecule with no representation larger than one cannot have a degenerate level at all.

 What is taught wrongly

The aufbau order is not a property of the atom

Iron's 3d orbital is more than four times smaller than its 4s and, by every one-electron estimate available, far lower in energy. The 4s fills first anyway, and it empties first too — which is not a paradox but a sign that the filling order was never a list of orbital energies.

 Beyond the octet

The vibration that lowers the symmetry

A molecule in a degenerate electronic state distorts until the degeneracy is gone. Which distortion it needs is a direct product; whether it wins is a race between a π energy falling linearly and a σ frame resisting quadratically, and both powers are measured here.

 Orbitals

What an electron actually feels

A 3d orbital is less than half the size of the 4s beside it and fills second anyway. The charge an electron feels is not the nuclear charge, the correction is a fit rather than a derivation, and the two facts together explain the shape of the periodic table.

 Where the atoms go

Which angles are symmetry and which are the model

VSEPR says electron pairs repel and never says by what law. For four, five and six domains it makes no difference whatever — change the exponent by a factor of twelve and not one angle moves. For seven it decides the answer.

 What the shape is for

A moment counts electrons, not orbitals

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 first-row ions come back within a hundredth for five of them — and the five that miss all miss the same way, which is what a missing term looks like.

 What symmetry decides

Descent in symmetry

Lower a molecule's symmetry and its labels stop being available. Which of them survive, which split, and into what, is decided by restricting characters to the operations that are left — arithmetic, not a table to be looked up.

 Orbitals

How big is an orbital

Four measures of size, all computed from the same radial function, all correct, and spanning a factor of two and a half for a 1s orbital. The one that governs chemistry is a fifth, and it is not a measure of size at all.

 Beyond the octet

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.

 Where the atoms go

The angle does not fix the hybridisation

Two equivalent hybrids are orthogonal only at one relation between their s character and the angle between them. Run it backwards on measured bond angles and water's bonds come out sp³·⁹⁹, hydrogen sulfide's sp²⁷, and phosphine's lone pair takes eighty-four per cent of the one s orbital.

 What is taught wrongly

The force field is not in the spectrum

Water has three vibrational frequencies and its force field has four constants, so the spectrum cannot determine the field. A whole family of quite different fields reproduces all three frequencies exactly, and only a second isotopologue can tell them apart.

 When the molecule does not stop

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.

 Bonding models

The localisation transformation, demonstrated

Four equivalent bonds or one a₁ and three t₂ — two descriptions of methane's bonding electrons that disagree about everything except the electron density, which they agree about to the last bit a double can hold.

 What a spectrum settles

The rotational spectrum is a moment of inertia

Every line in a microwave spectrum sits at a multiple of one number, and that number is a conversion constant divided by a sum of mass times distance squared. No bonding argument appears anywhere in it.

 What a spectrum settles

A bond length out of a spectrum

A linear triatomic has two bond lengths and one moment of inertia, so one measurement cannot determine it. Substituting an isotope gives a second measurement on the same structure, and two equations in two unknowns have a solution — which comes out differently depending on which isotope is used.

 What symmetry decides

An infinite group, worked in a finite one

A linear molecule has infinitely many symmetry operations, and every formula in character theory divides by the number of them. The standard device is to work in a finite subgroup — and it is worth computing what that trade costs rather than putting it in a footnote.

 When the molecule does not stop

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.

 Where the atoms go

One coordinate, three point groups

Hydrogen peroxide has four atoms and one soft internal coordinate. Turning it from nought to a hundred and eighty degrees takes the molecule through C2v, C2 and C2h — so it is chiral at every angle but two, and forbidden a dipole at exactly one of them.

 Orbitals

One level is not one comparison

A plate of orbitals drawn at a single contour value looks like a comparison and is not one. At the level that encloses ninety per cent of a 1s, a 2s encloses four per cent, a 3s under one, and a 3d has no surface at all — its wavefunction never reaches that value anywhere in space.

 Bonding models

The antibonding level goes up more

The two-level diagram every course draws is symmetric, and the symmetry is an artefact of setting the overlap to zero. Keep it, and the upper level rises further than the lower one falls — which is why helium has no molecule and why closed shells push each other apart.

 What is taught wrongly

The frequency is not the bond strength

Sulfur dioxide's S–O force constant is larger than water's O–H constant, and its stretching bands sit at a third of the frequency. A vibrational frequency carries a mass as well as a force, and the two cannot be separated by looking at a spectrum.

 What the shape is for

The pairing energy decides the moment

Whether the sixth d electron pairs up in the lower set or goes alone into the upper one is a competition between the splitting and the cost of pairing. Run the filling rules over the whole shell and exactly four configurations have a choice — and every one of them changes state at Δ = P exactly.

 Beyond the octet

What one pair can hold together

Put a single electron pair into a ring of any size and it supplies a total bond order of exactly two and a π energy of exactly 4β — three atoms, eight atoms or six hundred. Spreading a pair over more centres divides the bonding among them; it neither creates nor destroys any.

 When the molecule does not stop

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.

 Bonding models

Hückel with a heteroatom

A hydrocarbon's Hückel matrix contains no fitted number at all. Put one nitrogen in the ring and two arrive at once, the levels stop being symmetric about α, and the delocalisation energy stops being quotable — all from changing three entries in a matrix.

 What symmetry decides

Six bonds and four orbitals

The six fluorine σ functions of sulfur hexafluoride span a₁g ⊕ eg ⊕ t₁u. Sulfur's 3s and 3p supply a₁g and t₁u and nothing else, so four bonding orbitals hold twelve electrons across six bonds — a bond order of two thirds, computed from characters with no energy anywhere in it.

 Beyond the octet

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.

 Where the atoms go

What a dipole cannot tell apart

A dipole moment is three numbers extracted from a whole charge distribution, and enormously many distributions give the same three. The moment above it is not even a property of the molecule unless the one below it vanishes — which is why a quadrupole is quoted with an origin and a dipole is not.

 What the shape is for

What couples two spins

Two magnetic ions a few ångströms apart interact far too strongly to be doing it magnetically — the dipole–dipole energy is about 0.06 wavenumbers and the measured couplings run to hundreds. What couples them is hopping, which the Pauli principle allows for antiparallel spins and forbids for parallel ones, and the exact answer is −4t²/U.

 Orbitals

What the screening model cannot see

Slater's rules put the 2s and the 2p in one group, so they give both orbitals exactly the same effective nuclear charge at every element from boron to neon. The two are separated by several electronvolts in all six, and the model has no term that could produce it.

 What a spectrum settles

Why a d–d band is weak

In a centrosymmetric complex the transition between the two halves of a split d shell is forbidden — exactly, by parity, with no small quantity anywhere. What makes it visible at all is that the molecule is never quite centrosymmetric, and the computation says which vibrations do the work.

 Beyond the octet

A cage needs one pair more than it has corners

Every closed borane holds n+1 skeletal electron pairs for n vertices, and the extra one is a theorem about connected graphs rather than an observation about boron. A cage's radial orbitals have exactly one nodeless combination, always, whatever its shape.

 Orbitals

A filled shell has no shape

Sum the angular densities of a complete p shell and the answer is 3/4π in every direction, to sixteen decimal places. A filled d shell gives 5/4π. The lobes are in the decomposition and not in the density, and nothing that measures a closed-shell atom can see them.

 What a spectrum settles

A spectrum that changes when only a mass does

Methane has four distinct vibrational frequencies and two infrared bands. Replace two of its hydrogens with deuterium and it has nine and eight — with every force constant identical, every nucleus where it was, and the potential energy surface unchanged.

 What symmetry decides

Every group a molecule can fall to

A distortion takes a molecule's symmetry away, and what is left is not a free choice — it has to be a group. Closing subsets of methane's twenty-four operations returns thirty of them, which is exactly the number the symmetric group on four letters has.

 Where the atoms go

Hybrids that were never orthogonal

Two sp³ hybrids are orthogonal at 109.47° and nowhere else. Water's are drawn at 104.5° and overlap by 0.062; cyclopropane's are drawn at 60° and overlap by 0.625, which is not a small correction to a basis but a description that has stopped being one.

 What the shape is for

Sixteen is also a count

A transition metal brings nine valence orbitals, and nine filled orbitals is eighteen electrons. A square plane leaves more of those nine unmatched than an octahedron does and still holds fewer electrons, because one of the leftovers is out of reach.

 When the molecule does not stop

The end is the hardest place to bind

A site in the middle of a chain traps a state for any energy difference however small. The site at the end demands a whole β before it traps anything — measured at 1.025, 1.013 and 1.006 on chains of forty, eighty and a hundred and sixty, converging on exactly one. The intuition runs the other way and is wrong.

 What is taught wrongly

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.

 What the shape is for

Overlap is not interaction

Two orbitals interact by the square of their coupling divided by the distance between them in energy. A coupling half again as large, with a gap four times worse, buys a third less stabilisation — so the pair that overlaps best is often not the pair that bonds best.

 Where the atoms go

The angle a ring cannot have

A closed ring of equal bonds has to turn through a full circle, so its bond angles cannot average more than 180°(n−2)/n. Three, four and five atoms are below the tetrahedral angle at every geometry whatever; six is above it, and reaches it only by leaving the plane.

 Orbitals

The atom does not bring its own orbital

Build a one-electron diatomic from two hydrogen 1s functions and it comes out 25 per cent too long and 37 per cent too weakly bound. Let the molecule choose how large those functions are and the bond length is right to three figures, at an exponent of 1.238 — the orbital contracts by a quarter when the bond forms.

 Beyond the octet

The hole that is not repulsion

Two electrons in a bond keep out of each other's way, and the obvious reason is that they repel. Setting the repulsion to zero and computing the spin correlation exactly gives −0.125 rather than nothing, and the number is reproduced to nine decimal places by a determinant with no repulsion in it at all.

 What a spectrum settles

The moment that is the sum of the other two

Three numbers computed from the coordinates and the masses decide whether a molecule is flat. For water, sulfur dioxide, formaldehyde, benzene and every other planar structure here the largest moment of inertia is the sum of the other two exactly; for ammonia it misses by 0.76 and for methane by 3.18.

 Bonding models

The smallest many-electron calculation

A Hückel energy comes from a model with one electron in it. Add a single term — a cost for two electrons on the same site — and the problem stops being a matrix of size n and becomes a matrix over configurations. Four sites give thirty-six of them, which is small enough to solve exactly, and the answers correct two things the one-electron model got wrong.

 When the molecule does not stop

What holds a solid together

Four kinds of interaction between two units of matter, each computed from a stated model and put on one logarithmic scale. The ordering is not the one usually taught — an ion pair at contact beats a shared pair, which is a fact about the comparison rather than about the numbers.

 What symmetry decides

Why a character table stops where it stops

A character table is square, and both of its dimensions are forced. The number of representations equals the number of classes, the squares of their dimensions sum to the order of the group, and between them there is no room for another row.

 What the shape is for

An orbital carries no angular momentum

The d orbitals every chemist draws carry exactly no orbital angular momentum, and the proof is one line about a matrix being antisymmetric. A set of three of them carries a whole unit, which is why the spin-only formula works for most ions and fails for cobalt by nearly a Bohr magneton.

 What a spectrum settles

Fewer bands than electrons

Methane has eight valence electrons and two photoelectron bands. Ammonia has eight and three; water has eight and four. The count is not of electrons, not of bonds and not of orbitals — it is of the symmetry species the occupied orbitals fall into, and it falls as the symmetry rises.

 Beyond the octet

Hypervalency is about the ligands

The three-centre four-electron bond is offered as the reason sulfur hexafluoride needs no d orbitals. Read it forwards instead of backwards and it is a requirement rather than a permission — the arrangement puts half an electron onto each ligand before any electronegativity difference is applied, which is why the hypervalent compounds are fluorides and SH₆ is not a compound.

 When the molecule does not stop

The lattice sum that depends on the order of adding

An ionic solid's binding is the sum of every pair of charges in it, and the series does not converge absolutely — rearranged, it gives a different answer. That is a genuine mathematical difficulty rather than a technicality, and it is the clearest example of something a real-space, neighbour-by-neighbour method cannot compute at all.

 Orbitals

The isovalue nobody chose

Every program that draws an orbital asks for a number, and almost every user accepts the default. At 0.02 atomic units that default encloses 96 per cent of hydrogen's 1s, 52 per cent of its 2s and four tenths of one per cent of its 4s — three pictures drawn to one rule, meaning three completely different things.

 Where the atoms go

The ring that cannot hold still

Cyclohexane's chair is rigid and its boat is not, and that is a statement about the rank of a matrix rather than about strain. Hold every bond length and every bond angle fixed and count what is left: the chair has nothing, and the boat sits on a continuous loop of shapes with the same bonds and the same angles.

 What is taught wrongly

The spectrochemical series is not electrostatics

Ligands can be put in order by how hard they split a d shell, and the order is highly reproducible. It is not the order of charge — three of twenty mixed-charge pairs come out the way a point-charge model predicts, which is worse than tossing a coin — and the two ligands at the strong end are electrically neutral.

 What symmetry decides

The tolerance is a decision

A measured structure is never exactly symmetric, so assigning it a point group means deciding how much error to forgive. Sweep that decision from a thousandth of an ångström to a third of one and benzene, bent by a hundredth, is assigned five different groups — and at one step the group named is not even a supergroup of the one named before it.

 Bonding models

Where molecular orbital theory dissociates

The molecular orbital description of a two-electron bond puts both electrons on the same atom half the time — at every bond length, including infinite. The exact answer falls from a half to 0.0039 as the atoms separate, and the point where the two standard models are equally wrong is exactly U = 4t.

 Orbitals

A Gaussian is the wrong shape

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 too fast far away, and no number of them fixes either — while three of them already reproduce hydrogen's 1s to better than 99.9 per cent by overlap, and that is why the method works.

 What the shape is for

The double hump and what removes it

The hydration enthalpies of the first transition series do not lie on a line — they rise, dip at manganese, rise and dip again at zinc. Subtract the ligand field stabilisation computed from the same model that describes their spectra and what is left is a line, with the one fitted parameter landing inside the range a spectrum measures.

 Beyond the octet

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.

 When the molecule does not stop

The insulator band theory cannot see

A half-filled ring of four sites has a degenerate shell and no gap at all in the one-electron picture, which is the definition of a metal at that size. Its exact charge gap is zero when the electrons do not repel and grows without limit when they do — so a material can have a half-filled band and not conduct, and here is the number.

 Where the atoms go

The lone pair is not the missing term

Ammonia's dipole is 1.47 debye and nitrogen trifluoride's is 0.235, although the N–F bonds are far more polar than the N–H ones. The bond sums explain the reversal exactly and point in opposite directions — and the lone pair that is supposed to make up the difference has to be worth 0.58 debye in one molecule and at least 1.48 in the other.

 What symmetry decides

The projector is unique, the basis is not

A reduction says how many times each representation appears. It cannot say which combinations of orbitals they are — that takes a projection operator, and for a degenerate representation the operator returns a different pair of orbitals depending on which function it is handed first. Both pairs are correct, they span the same space, and every energy computed from either is identical.

 Bonding models

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.

 What a spectrum settles

The six motions that are not modes

Three N minus six is quoted in every textbook and the six are almost never computed. Diagonalising a mass-weighted Hessian gives six eigenvalues at arithmetic noise, and projecting their vectors onto the three translations and three rotations written down from the geometry alone accounts for every one of them.

 What is taught wrongly

VSEPR does not reach a transition metal

Four ligands minimising their repulsion give a tetrahedron, whatever the metal. Half the four-coordinate complexes of the platinum group are square planar, which has larger repulsion, and the term that overrules it is largest at d⁸ and exactly zero at d⁰ and d¹⁰ — which is where the repulsion rule works again.

 Orbitals

A contraction is a decision made once

Every published basis set freezes its primitive functions into fixed combinations, on an isolated atom, before any molecule is in sight. Freeing one coefficient recovers three quarters of what that costs — and freeing it at the other end of the basis recovers one per cent.

 What is taught wrongly

A ranking is not a difference

Four electronegativity scales agree about order to a Spearman coefficient of 0.99 and disagree about size by a factor of sixty-two. Put each on its own range and the O–H bond is 39 per cent of the way across on three scales and 4.5 on the fourth.

 What a spectrum settles

Mutual exclusion does not prove a centre

A centrosymmetric molecule shows no band in both its infrared and its Raman spectrum. The rule is a theorem and its converse is read off as though it were part of it — but ferrocene in the gas phase has no centre of inversion and no coincidence either, and the reason is that a fivefold axis separates the coordinates from their products where a threefold or fourfold axis cannot.

 What the shape is for

The bonds are what is left over

Every metal wants eighteen electrons and a metal–metal bond gives one to each of its partners, so the number of bonds in a cluster is what is left over after the counting. It works for every carbonyl cluster up to five metals and fails at six by exactly one bond — where the other counting rule, the one boranes are analysed with, is right.

 What symmetry decides

The group of a molecule that will not hold still

A point group is a group of rotations of space, and it presupposes a structure for them to act on. Ethane has no one structure — its methyl groups turn billions of times a second — and the group that describes its spectrum has thirty-six elements where the point group has twelve, out of two thousand eight hundred and eighty conceivable.

 Where the atoms go

The sites are not the same size

Every arrangement in this collection puts its sites on one sphere, which is an assumption about bond lengths made silently. Give the repulsion model a bond length and it predicts that the long bond goes axial — the opposite of the rule the model is always cited for.

 When the molecule does not stop

Two defects, and the level between them

One impurity pulls a level out of a band at exactly −√(h²+4). Two of them pull out a pair, split by an amount that falls by a factor of 0.41421 for every site of separation — which is √2 − 1, predicted from the isolated level's energy alone and measured to six figures.

 Beyond the octet

Two kinds of correlation, and only one is small

Correlation energy is defined as a subtraction, and the definition hides that the thing subtracted is not one thing. In the two-site model the local power of the correlation energy in the interaction falls from two to one — and at every interaction strength it equals, exactly, the occupation of the bonding natural orbital.

 Bonding models

Two pictures, one plane

Molecular orbital theory and valence bond theory are taught as rival descriptions of a two-electron bond. In a model small enough to solve exactly they are two vectors in a two-dimensional space, the exact answer lies in the plane they span at every repulsion, and it is neither of them at any repulsion but two.

 What is taught wrongly

A better energy is not a better answer

The variational principle makes the energy a one-way test: lower is closer. It also makes the energy the least sensitive thing a wavefunction gets wrong — second order in the error where every other property is first — so the two diverge without limit as a calculation improves.

 Bonding models

A method that is not additive

Put a molecule next to a copy of itself, far enough away that they do not interact, and the exact energy doubles exactly. A truncated calculation does not — because the truncation forbids both halves being excited at once, which is something the pair can do and neither half can.

 Orbitals

A slice is not the surface

The page is flat, so every printed orbital is a slice through a contour rather than the contour itself — and a curve that leaves a tenth of the density outside it in space leaves about a thirtieth outside it on the page. Both claims are true of the same picture and only one of them is ever stated.

 Beyond the octet

Four centres, and the pair that will not localise

The occupied orbitals of a molecule can be mixed freely without changing anything observable, and the freedom is usually spent on making them as local as possible. For the methyllithium tetramer the answer is four centres — one carbon and the three lithiums of the face it caps — and no mixing of the four pairs reduces it.

 When the molecule does not stop

Half filled is as bonded as it gets

Sweep a band from empty to full and the binding it supplies rises to a maximum at half filling and returns to exactly zero when every level is occupied. A completely filled band holds a solid together no more than a filled shell holds two helium atoms together, and for the same reason.

 What symmetry decides

One table, three groups

A mirror plane, a centre of inversion and a twofold axis have the same character table — two rows, entries 1, 1 and 1, −1 — and one of the three describes a chiral molecule. A character table is a property of an abstract group; a point group is that plus an action on space, and the action is what a molecule has.

 Where the atoms go

The double bond a ring cannot hold

A trans double bond needs a ring of nine carbons to sit flat, and the eight-ring will hold one twisted by 40.7 degrees — against 136 degrees measured in trans-cyclooctene. The same eight-ring threshold, applied by counting ring sizes, sorts nine bridgehead alkenes correctly with no bridgehead anywhere in the argument.

 What the shape is for

The g-value is the orbital coming back

A ligand field quenches the orbital angular momentum of a d electron, and spin-orbit coupling gives some of it back — upward for a shell more than half full and downward for one less than half full. Which way a resonance line moves counts the electrons, and the size of the move comes from two matrix elements and one optical splitting.

 What a spectrum settles

The rotor that stretches

A rigid rotor's lines are evenly spaced, and a real molecule's are not — it pulls itself apart as it spins. How much is not a fitting parameter: it follows from the stretching frequency by one relation, and the prediction agrees with the measured constant to a few per cent across four molecules spanning three orders of magnitude.

 Bonding models

Closer is not more overlap

Two 1s orbitals overlap more the closer they are, and every curve drawn from that pair says the same thing. Put a radial node into one of them and the rule fails: a 1s with a 2s overlaps by 0.016 at contact, by 0.282 near four bohr, and less again beyond — and two 2p orbitals head-on change sign at 5.06 bohr and are more strongly coupled at eight bohr than at four.

 Beyond the octet

Hypervalency does not stop at three centres

The three-centre four-electron bond is written up everywhere as an arrangement peculiar to hypervalent molecules. It is the first member of a family — five centres and six electrons, seven and eight — and the family predicts alternating bond strengths that the polyiodide crystal structures have.

 What is taught wrongly

Koopmans' theorem is exact for nothing

Reading an ionisation energy off an orbital energy neglects two things that pull in opposite directions, and the cancellation between them is quoted as the reason it works. Compute all three energies in a model where the exact answer is available and the cancellation is real, partial, and gone by the time the repulsion is twice the hopping.

 When the molecule does not stop

The bond that weakens as neighbours multiply

Wrapped structures with two, four and six neighbours per site give a mean of x² of exactly 2, 4 and 6. Their binding per site goes 1.272, 1.611, 1.979 — and their binding per bond falls from 0.636 to 0.330, which is why a metal atom with twelve neighbours has weak bonds and a great many of them.

 What the shape is for

The electrons repel less inside the complex

Two measured bands determine two parameters in closed form, and one of them is the repulsion between the d electrons — which comes out below the free ion's value for every complex, by between two and forty-seven per cent. The ligands that split most are not the ligands that reduce the repulsion most, so a complex is characterised by two numbers rather than one.

 What symmetry decides

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.

 What a spectrum settles

When a mode becomes a bond stretch

Water's two stretching modes are each exactly half in one O–H bond and half in the other, which is why neither of them belongs to a bond. Change one hydrogen to deuterium and the same force field at the same geometry gives two modes that are 99.5 and 99.7 per cent in a single bond each. Nothing about the bonding changed; a mass did.

 Where the atoms go

Zero dipole is not no interaction

Benzene's dipole moment is exactly zero at every origin, and its quadrupole moment is large enough to decide a crystal structure. Two benzenes face to face repel by nine kilojoules a mole; edge to face they attract, and the sign changes on the way between.

 Orbitals

The orbital in momentum space

Every orbital has a second picture as complete as the first and almost never drawn. Nothing is added by taking it — it is the same function in the other variable — but the uncertainty product falls out of it, and the functions quantum chemistry is built from turn out to be the only ones that attain the bound.

 Orbitals

A function that is already there

Adding a function to a basis set can only lower the energy, so a bigger basis is a better one. What that leaves out is that the same act makes the functions less independent: an optimised basis's smallest overlap eigenvalue halves with every function added, and a function placed on top of one already present buys less than a millionth of what a well-placed one buys while driving that eigenvalue to 4×10⁻¹⁰ — past which the calculation is refused outright.

 What the shape is for

A moment between two integers

A magnetic moment is celebrated as one of the few chemical measurements that returns an integer: count the unpaired electrons, feed the count into a formula, and nine first-row ions come out right. That works when one state lies far below the others. Sit a complex at its own crossover and the same measurement returns 0.30 at 80 K and 3.61 at 400 K — a quantity that counts nothing and is a temperature in disguise.

 What is taught wrongly

A surface is not a count of broken bonds

Cut a crystal and every atom in the new face has lost one of its six neighbours. The standard estimate follows immediately: a surface costs one sixth of the cohesive energy per atom exposed. Computed, it costs a little over half that — the atom keeps 91.2 per cent of its binding while keeping only 83.3 per cent of its bonds, because the bonds that survive get stronger when their competitors are removed.

 Beyond the octet

A third kind of correlation

The two-site model gave an exact identity — the power of the correlation energy in the repulsion equals the occupation of the bonding natural orbital — and asked whether anything like it survives with more orbitals. It does not, and the way it fails is better than the identity was: a ring of four gives a power of one where every closed-shell system gives two, at repulsions fifty times weaker than the hopping, because its reference was never a single state.

 What a spectrum settles

The constant a spectrum cannot see

A symmetric top has two rotational constants and its microwave spectrum reports one of them. Not badly, not with difficulty: ammonia's A of 6.3406 wavenumbers appears in none of its lines at any J and any K, because the term it belongs to cancels exactly out of every transition. The molecule turns about that axis, the energy is real, and the measurement is blind to it.

 What symmetry decides

The degeneracy no group predicts

How many orbitals can share an energy is decided by a group before any energy is computed, and the rotation group of a central potential permits one, three, five and seven. Hydrogen's second shell has four orbitals at one energy and its third has nine. The extra degeneracy is not an accident of the arithmetic — it is the signature of a symmetry that has not been named, and it survives only for a potential that goes exactly as one over r.

 Bonding models

Two bent bonds, or a σ and a π

A carbon–carbon double bond is drawn two ways and the two look like rival claims about what is there. They are one occupied space in two bases, related by a rotation of exactly forty-five degrees: the density is identical to the last bit a double holds, the bent pair are sp⁵ hybrids at 50.77° to the axis, and their charge sits 0.235 Å off the plane of the molecule where neither canonical orbital's does.

 When the molecule does not stop

Where the states pile up

Two bands of the same width can be entirely different objects. Scale a chain, a square net and a cubic structure to one width and what is left varies by a factor of four — a chain puts more than half its levels in the outer thirds of its band and a cubic structure puts more than half in the middle third — and that difference alone decides how strongly each of them binds.

 Where the atoms go

The atoms are not at the points

Every structure in this collection is a set of points, and every bond angle it argues about is a property of that set. Computed from the fitted force fields it already has, water's hydrogens are 0.094 Å from where they are drawn and its bond angle has a spread of 8.9 degrees — larger than the difference between 104.5 and the tetrahedral value that half the essays here are about. This is at no temperature at all.

 Bonding models

A difference does not make a transfer

Every electronegativity scale reports one thing: how far apart two atoms are in their appetite for electrons. Put that difference into a model with repulsion in it and the charge it actually moves is not determined at all — the same difference of one moves 0.447 of an electron with no repulsion and 0.0019 with a strong one, a factor of two hundred and forty-two, and nothing on any scale distinguishes the two cases.

 Where the atoms go

A dipole is not what an infrared spectrum sees

Carbon dioxide has no dipole moment at all and three of its four modes are infrared active. Methane has none and six of nine; boron trifluoride none and five of six. Water, which has the largest dipole of the five, has three modes and three bands — and its dipole predicted neither number.

 Beyond the octet

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.

 What a spectrum settles

How much of a band is a bond stretch

Boron trifluoride's 1454 cm⁻¹ band is 36.7 per cent B–F stretch, or 49.9, or 97.9, depending on which of four standard ways of sharing a mode out among internal coordinates is used. All four are defensible, all four sum to one, and the spread between them is sixty-one percentage points on one band of one molecule.

 What the shape is for

The count that is not always eighteen

The eighteen-electron rule is a shell closure, and an octahedral level diagram has two of them — one at twelve electrons and one at eighteen. Which is deeper is decided by the sign of one parameter: with a π acceptor the gap above eighteen is 3.720 and above twelve 2.280, and with a π donor the two swap over exactly.

 When the molecule does not stop

The distortion the filling chooses

A half-filled chain of equal bonds is unstable and alternates — long, short, long, short. That is the case everyone is shown, and it is one case. Fill the chain a third of the way instead and the alternation is worthless: what wins is a pattern that repeats every three bonds, and the period is one over the filling at every filling tried.

 Orbitals

The measurement a basis was not fitted to

Six Gaussians reproduce hydrogen's energy to eleven parts in a hundred thousand and its Compton profile to three parts in a thousand — twenty-five times worse, on a quantity an X-ray scattering experiment measures directly. The gap between the two errors widens as the basis is improved, because the energy is the one property a variational fit is best at.

 What symmetry decides

The symmetry that is not a rotation

Hydrogen's n = 2 shell holds four states at one energy and its rotation group accounts for at most three. The operator that accounts for the fourth is built here out of computed integrals: three matrices whose commutators close into the rotations, whose product with the angular momentum vanishes, and whose Casimir comes out at exactly n² − 1.

 Orbitals

A bond is not two atoms overlapping

The surface enclosing ninety per cent of a σ orbital's density is one closed surface with both nuclei inside it, at a level of 0.0359. The two atomic surfaces usually drawn instead sit at 0.0394, are a different shape, and enclose 91.70 per cent of the same orbital — and whether the molecular one is one object or two is decided by the fraction the caption claims, anywhere between 2.1 and 5.2 ångström.

 What the shape is for

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.

 Beyond the octet

A mean field cannot get out of the way

The energy a mean field misses grows without limit as the repulsion rises — 27.82 at U = 32 for four electrons on four sites, against an exact energy of −0.2946, so the error is ninety-four times the answer. The rate it grows at is not an energy at all: it is n²/4N, a count of the coincidences a spread-out density cannot avoid.

 When the molecule does not stop

A vacancy is not an impurity

An impurity is a site whose energy has been changed, and everything about the level it produces depends on by how much. A vacancy is a site that is not there, and the levels it leaves sit at exactly zero for a reason that cannot be tuned, weakened or moved — the count of them is a difference between two numbers of atoms, and two vacancies do not split however far apart they are put.

 Bonding models

A weight that depends on how it is weighed

The ionic character of a two-electron bond is quoted as a percentage. For one wavefunction at hydrogen's bond length, three conventions in the literature give 18.73, 34.74 and 5.88 per cent — a factor of six — and on a wavefunction with no ionic structure in it at all, one of them still reports a quarter.

 What symmetry decides

How much symmetry is left

A point group is a verdict and every real structure fails it. Two distortions of benzene that are two per cent apart in how far they sit from D6h need tolerances 1.88 times apart before the search will call either of them D6h — and one tolerance, applied to six molecules, admits amounts of asymmetry differing by a factor of fifty-six.

 What is taught wrongly

The bond length that depends on the isotope

Hydrogen chloride and deuterium chloride have the same potential energy curve, and their measured equilibrium lengths agree to three hundredths of a milliångström. Their average bond lengths differ by 4.34 mÅ — a hundred times more — because a lighter atom explores more of a well that is not symmetric.

 Where the atoms go

The strain that is not in the angles

Cyclopentane's flat bond angles are 108°, a degree and a half from tetrahedral, and its angle strain computed from a standard bending constant is 0.4 kJ mol⁻¹. Its measured strain is twenty-six. The missing sixty kilojoules are torsional — one ethane barrier for every bond in the ring, which no account built on bond angles mentions.

 What a spectrum settles

The top that reports all three

A symmetric top hides one of its two rotational constants in every line of its spectrum. Break the symmetry and the hiding stops: for water, twenty-three of the twenty-five levels up to J = 4 move when A is changed, and the two that do not are the ground state and the one at B + C. There is no formula for any of them.

 What symmetry decides

How far, and along which coordinate

A continuous symmetry measure returns one number: how far a structure is from a shape. Projecting the same displacement onto the twelve symmetry species of benzene's group turns it into a list that sums to 1.000000000000, says which coordinates the structure left along, and shows that the totally symmetric part — 3.4 per cent of this one — moves every atom by 0.0200 ångström and lowers the symmetry measure by nothing at all.

 When the molecule does not stop

One defect is a level, many are a band

A single deepened site in a chain pulls one state out of the band to −√(h² + 4), exactly, and holds it on 1.42 sites. Put in more and the levels spread: at one site in ten they span 1.45 in the same units and have closed to within 0.69 of the host band, and above one site in eight the count of levels stops matching the count of defects, because two defects on neighbouring sites push one of their pair back into the band.

 Where the atoms go

The atoms that meet across a ring

Twelve closed conformers of a ten-membered ring at one bond angle, so every one has identical angle strain by construction. Two of them differ by 0.026 kilojoules a mole in torsional energy and by 0.80 ångström in how close two atoms on opposite sides of the ring come — 2.331 against 3.131, where two carbons are in contact at about 3.4. An account built from angles and torsions calls those two structures the same.

 What the shape is for

The moment a fit invents

One coupled pair of spins, its susceptibility computed exactly, fitted to a Curie–Weiss law over four temperature ranges. The moments reported are 2.471, 2.535, 2.566 and 3.590 Bohr magnetons, and the Weiss temperatures −51, −78, −78 and −292 K — from one sample, measured perfectly, with three of the four fits agreeing with their own data to better than a part in three hundred.

 Orbitals

The property that gets worse

One Gaussian fitted to hydrogen gets the mean radius exactly right — 1.500000, against an exact 1.5. Two Gaussians get it wrong by 1.4 per cent, which is three hundred and thirty-two thousand times further out, while the energy improves fivefold. The variational principle bounds one number and says nothing whatever about any other, and the sequence of errors in everything else need not even be monotone.

 Bonding models

The quantity no scale prints

Every electronegativity table is half of a calculation. The other half is the hardness — half the difference of the same two measurements the Mulliken scale is half the sum of — and it varies by a factor of 3.8 across eighteen elements. Put both halves in and B–F, with an electronegativity difference of 6.12 electronvolts, moves less charge than lithium iodide, whose difference is 3.75.

 Beyond the octet

The reference decides the correlation

The correlation energy of one exact state, measured against two references that are both called Hartree–Fock, is −27.82 and −0.107 at the same repulsion — a factor of 259, on a system whose exact energy is a single smooth curve. Below the instability at U = 2 the two agree to the last bit; above it one grows without limit while the other falls, and the reference that reports almost no correlation has ⟨S²⟩ = 1.99 where a singlet is zero.

 What is taught wrongly

The third way to be an insulator

A ring of two hundred sites with a half-filled band has its levels crowding together as 1/n, which is the usual electronic-structure criterion for a metal, and it goes on holding at every disorder tested. Meanwhile the states at the middle of the band go from occupying 127 sites to occupying 10 — and at that disorder the number stops depending on how large the ring is at all.

 What a spectrum settles

The width of a band is a bond length

Nitrogen's three photoelectron bands are one sharp line, a progression of five, and a line with a shoulder. Computed from the measured bond lengths of the three states of the ion, the intensities come out at 0.917, 0.263 and 0.880 in the first line of each — because removing a weakly bonding electron lengthens the bond by 18.7 thousandths of an ångström, a strongly bonding one by 77.2, and an antibonding one shortens it by 23.7.

 Beyond the octet

Four is all that s and p can match

Reduce the ligand σ set of ten molecules in each one's own point group and ask how many of its components transform as one of the central atom's four valence orbitals. The answer is never more than four — not by arrangement, in every geometry from linear to octahedral — and what is left over is n + L − 4, with exactly twice that many electrons in excess of an octet.

 Bonding models

One spectrum, a line of models

Hückel theory has three parameters and benzene's photoelectron spectrum supplies two numbers, so the fit has a curve of solutions rather than a point. Along it the resonance integral runs from −3.05 to −7.66 electronvolts, the empty level moves by eight, the terminal-to-central bond order ratio in butadiene goes from 2.000 to 2.671 — and the delocalisation energy is 6.100 electronvolts in every member.

 Where the atoms go

The mode that moves least radiates most

Boron trifluoride's strongest infrared band is the one in which the fluorines barely move: ninety per cent of the motion belongs to the boron, which is a fifth of the molecule's mass. The mode that moves the most mass is nine and a half times weaker. Across five molecules the rank correlation between band strength and how far the atoms actually go runs from +1 to −0.66, and every normal mode carries exactly the same weighted motion by construction.

 What the shape is for

The same count, two oxidation states

Count a complex by the neutral method and by the ionic one and the total is the same integer every time — eighteen for ferrocene, sixteen for tetrachloroplatinate, twenty for hexaaquanickel. The oxidation state and the d count are not: hexaaquairon is d⁶ on one convention and d⁸ on the other, and the two predict spin-only moments of 4.90 and 2.83 against a measured 5.40.

 Orbitals

The tenth that is not drawn

A ninety per cent contour of a hydrogen 1s orbital is a sphere of radius 2.661 bohr, and two of them stop touching at 5.322 bohr — where the overlap between the two orbitals is still 0.0768 and rising in importance. At a three-ångström contact, 36.9 per cent of the overlap integral lies outside both drawn surfaces, and holding nine tenths of it inside the picture would take a contour enclosing 97.28 per cent.

 What a spectrum settles

Three numbers is not a structure

Formaldehyde's rotational spectrum gives three constants, of which a planar molecule's are only two independent numbers, and its structure has three parameters. Seven structures are computed here that reproduce A and B to the last digit the solver carries: the C=O length runs from 1.000 to 1.300 ångström, the C–H length from 1.557 down to 0.952, and the HCH angle from 74.6 degrees to 164.5.

 What symmetry decides

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.

 When the molecule does not stop

Two structures with the same neighbours

Five structures in which every atom has exactly four neighbours. Their second moments are 4.000 to nine decimal places, because that identity is the coordination and nothing else. Their bindings per site run from 1.2756 to 1.6363 — a spread of twenty-two per cent — and two of them that agree on the second, third and fourth moments together still differ in the third decimal place.

 What is taught wrongly

Two wrong numbers and a right difference

A mean field gets the total energy of a four-site system wrong by 12.11 and of two two-site systems wrong by 12.49, and 96.9 per cent of that error cancels out of the difference between them. The residue is 0.38 — and the reaction energy it is a residue of is 0.09, so the cancellation improves and the answer gets worse at the same time. Change the pair being compared to a singlet and a triplet and nothing cancels at all: the sign goes.

 When the molecule does not stop

A band becomes a bell curve

The normalised fourth moment of a hypercubic structure's density of states is 3 − 3/2d exactly, from one dimension to six, to eight decimal places — because the spectrum is a sum of d independent one-dimensional ones and the central limit theorem is what it is obeying. Reach further in one dimension instead and the shape overshoots a Gaussian rather than approaching it: a chain touching its third neighbour has a cubic structure's coordination and a fourth moment of 3.389 against 2.500.

 What a spectrum settles

A band is a filter on the modes

A photoelectron band's vibrational structure reports the frequencies of a few of the ion's vibrations and is silent about the rest, and which few is decided by the point group before any geometry is known. Under a change of shape that lengthens every bond alike, methane's totally symmetric stretch gets a Huang–Rhys factor of 0.905 and its other eight modes get between 10⁻²⁸ and 10⁻³⁵.

 What symmetry decides

More coordinates than motions

Methane has ten internal coordinates and nine ways to vibrate, boron trifluoride seven and six, formaldehyde seven and six. The excess is not bookkeeping: it is a combination of coordinates that describes no displacement of any atom, and adding twenty-five units of force constant along it moves every frequency by five parts in a hundred million.

 Beyond the octet

One scale, from two centres to a cage

How many centres a pair of electrons holds together is two different numbers, and they separate exactly where the bonding is most deficient: the methyllithium tetramer's pairs sit on four atoms each and have a participation number of 2.54. Run the same measurement up the scale and the twelve-vertex borane refuses it — its localisation has at least ten maxima differing by six parts in a thousand, so the number of centres is not an output for it at all.

 Orbitals

The radius that was tabulated

A van der Waals radius is a fitted number that every structural argument in chemistry uses. Computing it instead — a repulsion taken from computed overlap integrals, an attraction taken from two measured scalars, and no length anywhere — puts helium's contact at 3.140 ångström against a tabulated 2.80, neon's at 3.226 against 3.08 and argon's at 3.996 against 3.76. And the surface two atoms actually stop at encloses 99.4 per cent of the density, not ninety.

 What is taught wrongly

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.

 What the shape is for

The splitting against something structural

The angular overlap model says a ligand field splitting is proportional to the square of one overlap integral and to no other power. Computing that integral from Slater functions at the measured bond lengths, for five chromium complexes whose splittings run from 13,600 to 26,700 wavenumbers, the ratio varies by thirty-one per cent with nothing fitted. And a power law on the donor's effective charge, at an exponent nobody predicted, does slightly better.

 Where the atoms go

Three bent bonds, and the same hybrid

A triple bond localises into three bent bonds at 101.537 degrees to one another, each an sp⁵ hybrid with exactly one sixth s character. A double bond's two bent components are sp⁵ hybrids at 101.537 degrees. The two are the same hybrid, built out of different frameworks and in different numbers, and the reason is that a third of a half is a half of a third.

 Bonding models

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.

 Beyond the octet

How many descriptions a cage has

A localisation is a maximisation, and running it once reports the maximum it reached rather than the maximum there is. Run to exhaustion on a twelve-vertex borane it finds ten answers and then three batches of twelve starts in a row that find nothing new — and none of the ten is another one seen from a different side, because a symmetry of the cage cannot change a multiset of participation numbers and all ten multisets differ.

 What a spectrum settles

The coordinate an isotope reports

Kraitchman's equations return an atom's position from the change in the moments when that atom alone is made heavier, and for a rigid structure they are an identity — formaldehyde's four atoms come back to a part in ten million. What they return is the square of each coordinate, so both hydrogens at b = ±0.9348 come back at +0.9348; every out-of-plane coordinate comes back imaginary at a moment error of one part in a hundred thousand; and the famous error cancellation, measured at a factor of thirteen, still leaves the answer two and a half times worse than a direct fit.

 Bonding models

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.

 What is taught wrongly

The metal a thermometer cannot find

A metal is a system with excitations of arbitrarily small energy, which is a claim about a sequence of finite systems rather than about any one of them. Put a temperature on it and the claim needs a second limit, and the two do not commute: a uniform ring of forty-two at kT = 0.002 carries exactly as much as an alternating one, and at kT = 0.1 a ring of three hundred and twenty-two carries only four times as much.

 What the shape is for

The model is what is fitted

Fit a pair of coupled spins with the two-spin expression and it hands back the coupling exactly, from any temperature range. Fit a chain of eight with the same expression and it hands back −66.7 where the sample has −50, with a residual of 0.998 and a g factor of 1.973 — three numbers of which only the last says anything is wrong, and it is the one nobody looks at.

 Orbitals

The surface a table draws

Three noble gases stop at a surface enclosing between 99.38 and 99.75 per cent of their density — a near-constant, and an argument that a contour is a real boundary. Charge the atoms and it collapses. Across ten electrons the tabulated radius encloses anything from 94.4 to 99.98 per cent, it peaks at the neutral rather than trending through it, and radii built at a fixed enclosure do not add up to a single measured separation.

 When the molecule does not stop

Two bands, and the shape of each

A band's shape has a closed form when the structure it belongs to factorises. Put a second band beside it and the form survives exactly while the two do not talk, and fails as the square of the coupling once they do. Make one direction weaker instead and the form survives everywhere — 3 − 3(2 + λ⁴)/(2(2 + λ²)²), agreeing with eigenvalues to nine decimal places — so how anisotropic a structure has to be before its band is two-dimensional is a number: λ = 0.46518.

 Where the atoms go

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.

 What symmetry decides

The one intensity symmetry does fix

Symmetry says which bands exist and declines to say how strong they are. It makes exactly one exception, and it is a ratio: every Raman band that is not totally symmetric is depolarised by exactly three quarters, whatever the molecule and whatever the model — and methane's symmetric stretch is polarised by exactly zero, because its group is cubic.

 When the molecule does not stop

A mixture is not the average of its ends

Half of a structure's sites raised and half lowered, and the line between the two pure ends is arithmetic — no diagonalisation needed. Every mixture lies below it, by 0.42987 per site for the ordered arrangement and 0.23690 for the segregated one at the same composition, so composition fixes neither the binding nor the gap. And the departure is not quadratic in the contrast: it grows as its 1.79 power in a chain and its 1.28 power on a square net.

 What a spectrum settles

More bands than there are orbitals

A photoelectron spectrum is read as a list of orbital energies, one band per occupied orbital. Computed exactly for a six-orbital ring, it has three bands with no repulsion and a hundred with eight — and by then fifty-three per cent of the intensity is in lines that no orbital corresponds to. The total intensity is three at every repulsion, exactly, because that is a sum rule and not a fit.

 Orbitals

The basis the other atom lent

Two atoms in a molecule are described in each other's functions and the separated atoms are not, so the molecule is treated better than the pieces and the binding comes out too large. That is the basis set superposition error, it is removed by a standard correction, and for H₂⁺ in four Gaussians a centre it is six tenths of a microhartree against an incompleteness error of twelve millihartree — a factor of eighteen thousand the other way.

 What symmetry decides

The coordinate it was already soft along

A distorted structure can be resolved into the symmetry species of its ideal group, and that resolution cannot say which coordinate a molecule fell down by itself and which one something outside pushed it along. The force field answers that, and the two halves agree on the case where they must — a unit distortion along a normal mode costs exactly that mode's frequency, to a part in a million, computed from masses and force constants by one side and from coordinates and characters by the other.

 What is taught wrongly

The correction that was computed somewhere else

Every composite method rests on one assumption: that an expensive correction computed on a small case can be added to a cheap calculation on a large one. Tested on four sites where both answers are exact, it removes 99.74 per cent of the cheap method's error near the reference and −1450 per cent of it further away — at which point the recipe is fifteen times as wrong as the calculation it was improving.

 Beyond the octet

The count is the population

The census counted how many ligand combinations have no partner on the central atom and called the count n + L − 4. A σ-only model built from each molecule's own coordinates says what that count is worth: with no electronegativity difference anywhere, the mean charge on a ligand is minus the orphan count divided by the ligand count, exactly, in all ten cases. And the prediction the census made — that the charge grows with the orphan count — is refused by the divisor.

 What the shape is for

The distortion the ends decide

A chain of an even number of sites has an odd number of bonds, so its two dimerisations are different molecules rather than one molecule translated. Held at the same distortion they differ by 1.08715 in units of the hopping, whatever the length — a fixed amount of energy living at the two ends, with the per-site difference falling as one over the length at a fitted exponent of −0.99986. And below a hundred and twenty-eight sites the second dimerisation does not exist at all.

 Where the atoms go

The explanation with the wrong sign

Two methyl groups on one carbon make a ring easier to close, by up to eleven thousand-fold, and the textbook reason is that they compress the ring's internal angle. Computed from a standard bending term, that compression helps a three-ring and a four-ring and hinders every ring from five up — predicting a slowing of 0.754-fold for the five-ring measured to speed up 250-fold. The account with the right sign is about rotations rather than angles, and it has a ceiling of 36.5 that the measurement is already above.

 Bonding models

The value that only exists in the bond

A difference in electronegativity moves charge, and moving charge closes the difference — until every atom in the molecule has the same chemical potential. Solving that gives one number per molecule and a charge per atom, and carbon's runs from +0.0692 in methane to +0.3074 in tetrafluoromethane. Hydrogen's changes sign between the first and the second, so the same C–H bond is polarised one way in one compound and the other way in the next.

 Where the atoms go

A ceiling that rises where the measurements fall

The rotamer account of the gem-dimethyl effect has a largest possible acceleration, which looks like a limitation. It is a prediction: the ceiling is a closed form in the number of rotations a closure freezes, it rises steeply with ring size, and the measurements fall — so the account is refuted for the five-membered ring and more than sufficient for the six.

 Orbitals

A correction computed at one length

The counterpoise correction is expensive, so it is evaluated once at a reference geometry and subtracted across a whole potential surface. A constant does not move a minimum — so a frozen correction returns the uncorrected bond length exactly, at every reference geometry and in every basis, and everything the correction does to a structure is the part that has just been thrown away.

 What a spectrum settles

A hundred lines and no way to sort them

A spectrum with a hundred lines has six fundamentals in it somewhere. Sorting by height works until the tallest satellite is as tall as the shortest band, and sorting by position works until satellites start arriving between the bands — and on a ring of six both stop working at the same repulsion.

 What is taught wrongly

A mean that is low rather than right

Sanderson's rule takes the geometric mean of the atoms' electronegativities, and it works better than the plain average. Computed against the exact equalised answer for twelve molecules it is better by erring downwards — and two of the twelve have exact answers above the plain average, where no geometric mean can go at any parameter.

 What the shape is for

An integer nobody measured

The oxidation state of chromium in the hexacarbonyl is zero. Its charge, computed from the same wavefunction, is anywhere between −3.04 and −0.98 depending on how the shared electrons are divided — and the integer sits outside that whole range. The electron count, meanwhile, is eighteen at every point on it.

 Bonding models

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.

 Beyond the octet

The answer a search is most likely to give

If a cage has ten equally good localised descriptions, why does the literature agree about its picture? The hoped-for answer was that one basin is very large. Counting four hundred and eighty starting points says it is not: two independent searches agree one time in ten, and the description they most often return is not the best one.

 What symmetry decides

The current does not divide

A fused ring system's response was computed from the areas of its rings, and the obvious next question was whether the current divides between them the way it divides between two resistors. Giving each ring its own flux and taking the second derivatives says no: the response is a matrix, its off-diagonal entries are nearly half its diagonal ones, and anthracene's middle ring carries 1.18 times what its outer rings do.

 When the molecule does not stop

Two bands, if the chain is short enough

Take a chain, raise half its sites and lower the other half, and ask when the band comes apart into two. The ordered arrangement splits at once, the segregated one at a contrast equal to the band width, and the random one splits earlier than either — and then closes again as the chain is made longer, because a long chain contains a long run of like atoms and a long run is a narrow sub-band.

 What a spectrum settles

A ratio that squares what it measures

A depolarised Raman band sits at exactly three quarters because symmetry says its mean polarisability derivative is zero. Distort the molecule and it comes off — by 4.5 × 10⁻⁴ for a hundredth of an ångström and 0.042 for a tenth, going as the square of the distortion, which makes a ratio measured to three decimals a length known to one and a half.

 What symmetry decides

How nearly a broken symmetry survives

The hydrogen shell's extra symmetry is what makes its Stark effect linear, and a real atom does not have it. Screening splits the shell, and the field needed to overcome the splitting and restore the linear behaviour is a curve — from forty thousand volts a centimetre at a quantum defect of 0.0004 to thirty million at a defect of 0.21.

 When the molecule does not stop

The arrangement a count cannot pick

Three arrangements of one composition came out ordered by their count of unlike bonds, which looked like a rule. Enumerating every arrangement instead of three shows it is not one: it holds at every composition on a square net at a small contrast, fails at four of them at a large contrast, and fails on a triangular net at any contrast at all.

 Bonding models

The hybrids that point outside the bonds

Coulson's relation says two equivalent hybrids sharing an s orbital are orthogonal only between ninety and a hundred and eighty degrees. Cyclopropane's carbons make sixty, so its ring hybrids cannot point at the atoms they bond to — and the same relation says by how much they miss: 22.75 degrees each, falling to 0.03 in cyclohexane.

 What the shape is for

The integral that cannot count electrons

Adding the π channel to a ligand-field splitting means one more overlap integral over the same two orbitals at the same distance. It removes a fifth of the error. Telling the model which ligand is a π acceptor — one word per ligand, quoted rather than computed — removes forty-five per cent, because an overlap cannot know whether the orbital it reaches is full or empty.

 Orbitals

The surface a neighbour moves

An ion in a crystal sits in the field of the ion next to it, and that field moves its contour. The displacement has a closed form, it is checked against the polarisability it implies, and it turns out to be almost perfectly anti-correlated with the discrepancy it was proposed to explain — the pairs that need the most correction get the least.

 Where the atoms go

The table that could not have mattered

Charges taken from one of four electronegativity tables invite a worry, because the tables disagree with each other. They do disagree — hydrogen cyanide's dipole runs over a factor of eleven between them — and for the molecules in question the worry could not have applied, because a molecule of two elements has charges that are one number times a fixed pattern and a rank correlation does not notice a rescaling.

 What is taught wrongly

The warning a cheap calculation gives

A correlation correction computed on one system and carried to another works until it does not, and nothing in the scheme says in advance which. The mean field's own symmetry breaking says: it collapses at a definite field, and the transfer fails where it goes. Across thirty-two systems the two rank together at 0.90.

 Beyond the octet

Where the electrons are, without subtracting anything

A correlation energy is an exact energy less a mean-field one, so the answer depends on which mean field was picked — by a factor of 259. The pair distribution says the same thing with no subtraction in it, and two systems matched to the same correlation energy turn out to have electrons in visibly different places.

 Orbitals

A control that outranked the mechanism

Ruling polarisation out left one candidate, and the closed-shell overlap ranks at 0.857 against the additivity shortfall — which looked like the answer until the control was read. The cation's formal charge, which cannot be a mechanism, ranks at 0.9524. Eight pairs split four and four by charge cannot separate anything, and within a charge group two candidates both rank perfectly.

 Bonding models

An anomaly that is not the first of a series

Naphthalene's shared bond behaves differently when the geometry is allowed to answer back, and the obvious reading is that two rings feeding one bond is what does it. Run it up the acenes and the effect falls at every step and changes sign: +0.02036 for two rings, +0.00678 for three, −0.00029 for four, −0.00603 for five. Two rings feeding a bond is not the beginning of anything.

 Where the atoms go

An estimate that can be wrong by two

The ceiling on the gem-dimethyl effect is exponential in the number of rotations a closure freezes, and that number was taken as n − 2 without counting — which left the refutation at five rings probable rather than airtight. It is airtight. The ceiling does not reach the measured 250 until five rotors, on a ring that has three, and no hindering of the tether can raise it.

 What the shape is for

How many parameters a curve is worth

A susceptibility curve routinely carries four fitted parameters and the question of whether it can support them is never asked. It has an arithmetic answer: the four directions the fit sees span a factor of five hundred, so one per cent data fix the first two to under three per cent and the last to thirty-seven — and forty points reaching two kelvin are worth more than sixteen thousand starting at twenty.

 What symmetry decides

Ten directions no frequency can see

A redundant force field has one obvious flat direction. There are ten: methane has fifty-five independent force constants and a ten-dimensional subspace of them that no spectrum can touch. The literature's repair is to project — and the metric a vibrational analysis is naturally written in cannot define the projection at all, because a redundancy is a null vector of it.

 What is taught wrongly

The carriers a distortion was hiding

A half-filled ring of 4m carries exactly one pair of thermal carriers at every temperature, which is true only of a ring held rigid. Allowed to move, it does not carry them: it alternates, opens a gap of 0.4927, and carries none at all until a temperature that undoes the distortion.

 When the molecule does not stop

The chain distorts hardest where it stops

Holding the alternation uniform is what made the end energy a clean constant, and it is the one assumption the end-energy calculation had to make. Letting every bond find its own value shows the distortion is largest at the end and decays inwards over a measurable length — one and a half bonds in a strongly dimerised chain, five in a weak one.

 What a spectrum settles

The correction that was invented

A standard error model puts the zero-point error in a rotational constant at a few tenths of a per cent, shared between the three moments by invented weights, with the parent and its deuterated form differing by five. Computed from a force field it is 1.88 per cent, one of the three shares is negative, and the mismatch is 35 — so the cancellation the substitution method rests on is worth a factor of 2.5 and not thirteen.

 Beyond the octet

Two models that disagree about the shape

The identity says how much charge a hypervalent molecule's ligands must share and nothing about how. Working out which arrangement shares it most evenly puts the σ model and the repulsion model on one axis for the first time: the even sharer is the pentagonal plane, which is the arrangement the repulsion likes least — and along the interchange chemistry actually uses, one model sees 0.15 per cent of a change and the other sees 54.

 Orbitals

A correction that is two functions

A symmetric pair's counterpoise correction splits exactly in half, at every separation, to the last digit — which is why one frozen number describes it. Give the two atoms different charges and the split runs from 0.03 per cent to 67, changing places at 6.29 bohr: the correction a single number was standing in for is two functions of different shapes.

 What symmetry decides

A label that prices nothing

Pricing a distortion by its symmetry species works when the species appears once. It appears more than once for every one of seventeen molecules — 71.4 per cent of their vibrations on average belong to a species that is not unique — and for nine of the seventeen nothing forces it: their groups have room to spare and they repeat anyway.

 Beyond the octet

Half of it is given back at one bond

The correlation hole of a ring removes 1.2653 pairs from zero separation and puts 1.1126 of them one site away. With an on-site repulsion that second number costs nothing, because the interaction is zero there — but with anything that reaches a neighbour it costs 44 per cent of what the hole saved, and with a Coulomb tail 46.9.

 What a spectrum settles

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.

 What the shape is for

The count that cannot be broken by strength

Back-donation puts electrons into orbitals that are not the metal's, and the eighteen-electron rule counts the metal's nine. Turning the π channel up as far as it will go never breaks it: the counted orbital's metal share falls from 100 per cent to 54.67 and approaches a half from above without reaching it. What does flip it is not strength but order.

 Where the atoms go

The curve between two rows

A rotation a ring closure has to freeze was treated as free or as absent, and the two answers sat in adjacent rows of a table. A real hindered rotor is neither. The factor one rotor contributes runs from 3.316 to one along a curve nobody had drawn, and where it matters is between one and eight kilojoules a mole — which is a torsional barrier rather than a conformational preference.

 What is taught wrongly

The exponent was the window's

A fit over the last decade before a distortion vanishes gives an exponent of 0.44, and running it on larger rings should say whether the number belongs to the transition or to a forty-site ring. It belongs to neither. The local slope runs to 0.5020 as the transition is approached, and 0.44 is what a fit over that particular decade returns — on every ring size and every stiffness, because the whole curve is one curve.

 When the molecule does not stop

Two ways of being second order

A band's shape and a band's gap are both second order in the coupling that mixes two bands, which sounds like a reason to measure only one of them. They are second order in different ways — one goes as the square of the ratio and the other as the square over the separation — and that single difference of one power is what turns a curve of possible answers into a point.

 Bonding models

Two systems a model cannot tell apart

Two molecules that share a Hückel eigenvalue but were measured to differ bound a whole family of models at 0.456 eV. That is a reusable instrument, and chemistry has plenty of predictors of exactly the same shape. Turned on themselves: VSEPR cannot account for 92 per cent of the variation in the four angles it predicts, and no fitting is involved anywhere.

 When the molecule does not stop

A particle in a box the alloy made

A random alloy's gap is set by the longest run of like sites. What sits at the edge of that gap turns out to be the simplest state in quantum mechanics: a particle in a box of L sites, occupying 2(L+1)/3 of them, to within three per cent and with nothing fitted — except for the two states in forty that found a second run to share.

 What symmetry decides

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.

 Orbitals

A size a confound cannot supply

A rank correlation of 0.857 was beaten by a control that cannot be a mechanism, so a size is the next thing to ask for: does a closed-shell repulsion of the computed magnitude displace two ions by the tenths of an ångström the additive radii are wrong by. It does not. The balance of a Madelung attraction against six computed repulsions predicts six separations to 0.242 ångström where adding two tabulated radii predicts them to 0.183, and the displacement it produces ranks at 0.14 against the shortfall it was proposed to explain.

 Where the atoms go

Four tables and one molecule to disagree about

A molecule of two elements is provably safe from the choice of electronegativity table, and a series down a group should land where the tables do disagree. It does. All four agree that hydrogen iodide is the exception — and they disagree about what its dipole is by 1.245 debye, which is two and a half times the 0.448 that was measured.

 Bonding models

The angle that does not have to be searched for

A carbonyl's two bent components have no symmetry making them equivalent, so the mixing that best localises them looks like something to search for and their s characters look like two different numbers. Neither happens. The localisation functional is a quadratic with no linear term, so its maximum is at forty-five degrees or at the ends — bent bonds or canonical ones, decided by one inequality, with nothing in between.

 What a spectrum settles

The forty-five that are fixed

Methane's fifty-five-dimensional space of force constants has ten directions no frequency can see, and the useful thing to report is the forty-five that are fixed. It is a table: every stretching constant is fixed on its own, no bending constant is, and projecting a fitted field onto the determined subspace takes four fits that span 0.88 mdyn per ångström down to four that span 0.0027.

 What the shape is for

The product a curve measures

A susceptibility curve's fourth parameter is undetermined, and the question is which combination the free direction actually is. It is a product of powers, because the Jacobian is logarithmic — and at the usual window it is the temperature-independent term divided by the 0.40 power of the monomer fraction, fixed to forty per cent, while the product one place up is fixed to 6.7. A paper could print that instead of four numbers.

 What is taught wrongly

Where a closed form stops being one

What happens when the quadratic energy is not enough? A cubic makes the equalisation condition a quadratic with two roots, and something has to choose between them. The choice is easy and the finding is somewhere else — a cubic fitted through the dication gives lithium an electronegativity of −7.907 eV, a capacity of 0.073 of an electron, and a molecule of two alkali metals no solution at all.

 Beyond the octet

The square that wastes an orbital

The orphan count that prices hypervalency was treated as a property of a molecule's composition — ligands plus lone pairs minus four. Run on twenty-six arrangements of the same ten molecules it is right for twenty-three and wrong for three, and all three are flat. A planar arrangement gives a main-group centre three usable orbitals rather than four, so the count is a property of the shape.

 When the molecule does not stop

A decay that keeps slowing down

A healing length fitted over the first twelve bonds of each relaxed chain goes as the gap to the power −0.60, against an argument that says −1. A fitted exponent can belong to its window, and this one does — but not because the window is too short for the chain. The profile is not an exponential at all, so there is no single length for an exponent to be of.

 What a spectrum settles

An expression for what was a warning

The zero-point correction to a moment of inertia comes out at 1.88 per cent where a few tenths had been assumed, and heavy molecules are safer. Written out, the correction is a mean curvature times the sum of reciprocal wavenumbers over the moment — one line, evaluated from things a spectroscopist has before starting. One coefficient serves four molecules whose corrections span a factor of eleven.

 What is taught wrongly

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.

 Where the atoms go

A verdict inside its own error bar

The tightest comparison in the rotamer argument is a computed 10.99 against a measured 10 — a nine per cent margin, offered as a verdict. The computed side is built on one quoted energy known to ±0.4 kJ/mol, and that alone puts a band of twenty-three per cent on it. The verdict turns on four tenths of one standard deviation of a number nobody had put an error bar on.

 Orbitals

The half that cannot be computed

How many points does each half of a counterpoise correction need to interpolate? The natural expectation is two different numbers. The answer is that the question is not yet askable: the lighter centre's half is a difference of two energies agreeing to six figures, its second differences are seven per cent of its own value, and no interpolation of it means anything. The third thing worth checking — the symmetric-pair check — works perfectly.

 Bonding models

The pair that is not a tie

An exact tie bounds every model of a form with no fitting, and it is available only for predictors that read integers. A near tie bounds the model's derivative instead, and a pair the predictor orders the wrong way round refuses monotonicity outright. Run on four standard predictors, the new instrument ranks them in a different order from the old one — and its worst case is the control the old one could not see.

 Beyond the octet

Where the count stops being an effort

A cage's localised descriptions were counted by a search that stopped when it stopped finding new ones, which makes the count a property of the stopping rule. Run to four thousand starts the count is fourteen and the last new description appears at start 124 — after which three thousand eight hundred and seventy-six starts add nothing. The four rarest are found six times in a thousand, which is what says nothing rarer is hiding.

 What the shape is for

The orbital a ligand cannot reach

The sixteen-electron count of a square plane is a statement about an energy rather than about symmetry matching, so it was the count that ought to be sensitive to a π channel where the eighteen-electron one is not. It is not sensitive either — and for a sharper reason. The orbital that sets its gap is d(z²), and a square-planar ligand set contains nothing of that symmetry, so the gap is exactly 2eσ until the π strength reaches a quarter of the σ one.

 What symmetry decides

A formula that predicts minus eleven vibrations

A molecule's count of totally symmetric vibrations is often explained as one per orbit of internal coordinates, less one per redundancy, with methane as the worked example. Computed for fifteen molecules it is right for seven and wrong for eight — and for sulfur hexafluoride, benzene and both ferrocenes it returns a negative number. Two independent corrections turn it into an identity.

 What is taught wrongly

A capacity that is largest where there is none

A cubic through four charge states implies a largest amount of electron an atom can accept, and it is natural to ask whether the idea survives its own model. It does not. The three atoms whose anions are not bound — beryllium, magnesium and nitrogen — have the largest capacities in the set, two of them infinite; the three smallest capacities all belong to atoms whose anions are bound. And among the fourteen where the test cannot bite, the ordering is sensible.

 Bonding models

A bond with nothing in the middle

Two head-on 2p functions have an overlap that changes sign at 5.03 bohr, and the picture of what that means is worth drawing. Below that separation the combination the model fills has a nodal plane through the midpoint of the bond, so two electrons in it put exactly nothing between the nuclei; above it the same model fills the other one. Which picture a bonding orbital has is decided by a separation.

 Where the atoms go

A dispute is a small difference

Four electronegativity tables disagree about the direction of a quarter of a set of hydrogen bonds, and the interhalogens look like the natural test: five compounds with no hydrogen in them, to say whether the disagreement is about one awkward element or about the whole idea. All four tables agree about every interhalogen. They agree because the differences are large, and that is not a fact about halogens.

 What a spectrum settles

One number was one direction

The departure of a depolarisation ratio from three quarters goes as the square of a distortion, and the exponent was first measured along one bond stretch. Computed along every non-symmetric coordinate the exponent is always two and the coefficient is not: it spans a factor of thirty inside methane. And the ratio is not preferentially sensitive to the coordinates a molecule is soft along — in two molecules of three the stiffest coordinate is the most sensitive.

 Orbitals

The assembly that counts one share twice

A counterpoise correction is divided unequally between its two centres, and the first place that matters is a three-fragment system, where the pairwise corrections are added up and the assembly must double-count one share and undercount another. It does: the heavy centre is over-corrected by seventeen per cent and the light ones under-corrected by two and a half, and the two do not cancel.

 What the shape is for

The gap that would have to be smaller

An angular overlap parameter is an overlap squared over an energy denominator, and the usual fit folds the denominator away. Put the measured ionisation energies back in and the denominator alone over-predicts the trend down the halide group at every metal level a donor permits — the smallest it can give is 1.67 against a fitted 1.43. So the overlap has to shrink down the group, which is the opposite of the usual expectation.

 Beyond the octet

The give-back that turned into a saving

Take a wavefunction optimised for an on-site repulsion, weight its correlation hole with an interaction that reaches one neighbour, and the enhancement at one bond gives back forty-four per cent of the on-site saving. Putting that neighbour term into the Hamiltonian and solving exactly does not shrink the give-back. It reverses its sign.

 When the molecule does not stop

The length at which levels become a band

Two runs of low sites share a state when they are close enough, and the splitting falls exponentially over two sites. A longer chain brings some pair closer while spreading its levels thinner, so the two quantities run against each other and cross — at about a thousand sites for runs of four and eighty thousand for runs of eight. Chains of four hundred sites are far below that, which is why every gap state on them is a box.

 What symmetry decides

Two events where there was one

A broken symmetry returns at a field where the coupling matches the gap it has to overcome, and in a shell with two levels that field is a single number. The next shell up has three, two gaps and two dipoles — so there are two crossovers, a factor of 3.66 apart, both of them below the single one of the shell below, and the exponent takes more than a decade of field to travel between them.

 What a spectrum settles

A contrast with a closed form

Below half filling the satellite test flattens instead of failing, and the value it flattens at could be above or below the factor of two the test needs. It is — on all eight systems, by between 1.25 and 3.7 times. And on a ring the limit is (1 + 2cos(π/n))², to six figures, on every ring tried.

 What is taught wrongly

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.

 What symmetry decides

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.

 Where the atoms go

One number decides which way it breaks

The six-membered verdict gives way at 312.1 K, and that is a statement about the ceiling rather than a prediction about the ratio — because a real rate ratio has a temperature dependence the model has no term for. Putting that term in moves the crossing, and at 5.308 kJ/mol it removes it entirely.

 When the molecule does not stop

The constant that belonged to one net

Divide one second-order departure by another and the coupling cancels, leaving one constant times the gap — which reads as arithmetic. It was arithmetic about a square net. On a triangular one — the same graph for both bands, nothing else changed — the quotient drifts by a hundred and twenty-eight per cent.

 What the shape is for

The gap that only a tetrahedron closes

The sixteen-electron gap is 2eσ exactly, and a square-planar π set cannot touch it because d(z²) has no partner there. Fold the ligands out of the plane and the gap survives almost intact for fifteen degrees, closes to nothing only at the tetrahedron — and loses its exactness at the very first degree.

 Bonding models

The node that decided a picture

A hydrogenic calculation cannot say whether formaldehyde's localised description is two bent components or σ and π, because the integral that decides it came out at 0.00718 where it should be the largest in the molecule. Replacing the hydrogenic radial functions with nodeless Slater ones moves that overlap by a factor of ninety-seven, leaves the π overlap identical to the last bit, and flips the answer: the description is bent.

 Orbitals

The residue that is two numbers

Is a shortfall between a sum of radii and a measured separation a property of the pairs, or of the compromise a universal table makes? The eight separations are two complete two-by-two blocks, so what no assignment of radii can reproduce is not a residual at all — it is an alternating sum, computed by subtraction, and it comes to three hundredths of an ångström and six.

 Beyond the octet

Three shapes from one search

A cage's fourteen localised descriptions fell into two groups with a factor of five between them and nothing in the gap, which made counting the big basins look like a stopping rule. Two more cases from the same family give a smooth tail over sixty-three descriptions and a single basin holding ninety-eight per cent. One search, one family, three shapes.

 What the shape is for

A denominator that fails both ways

The energy gap an e_π folds away over-predicts the halide trend at every metal level a donor allows, and the acceptors look out of reach because a π* is not an atomic level. It is measurable — a slow electron is captured by it — and on that side the same denominator under-predicts. No metal level fixes either, and the two want it moved in opposite directions.

 Where the atoms go

A gap that was a choice of bonds

One number separates every disputed bond from every agreed one with nothing in between — on thirty-one bonds, which is enough to see a gap and not enough to know it is real. It is not real. The line is in the same place on all one hundred and fifty-three pairs the four tables cover, and twenty-seven agreed pairs now sit below the largest disputed one.

 Beyond the octet

Expensive is not the same as unadopted

A counting formula right for twenty-three arrangements and wrong for three invites a reading: the failures are the arrangements nothing adopts, so the formula is reliable because chemistry stays away from where it breaks. Put the repulsion energy on the same axis and the reading fails — the most expensive arrangement in the census is one the formula gets right.

 What is taught wrongly

The amplitude the collapse left behind

Five Peierls curves became one curve when each was divided by the alternation its ring settles at cold, so that amplitude is the whole of what distinguished them — and it was five golden-section searches over diagonalisations with no formula anywhere. It has one, exactly, as a sum of square roots; and writing it down says that one of the five rings was never measuring a long chain.

 What a spectrum settles

The axis that goes the other way

One dimensionless coefficient turned a zero-point correction into an expression a spectroscopist could evaluate, and the three principal axes were averaged over to get it. Split by axis it gets worse, not better — the coefficients span 2.3 where the molecule-averaged ones span 1.36 — and water's smallest moment does not grow at all. It shrinks.

 Orbitals

The correction that gets harder to assemble

Summed across a trimer, pairwise counterpoise corrections come to fifteen per cent more than the trimer's own. Three Gaussians a centre is a small basis, so the natural question was whether the fraction shrinks with a better one or stays put. It does neither. The correction falls by three orders of magnitude and the fraction grows fivefold.

 Bonding models

The floor was in the bookkeeping

A heteroatom in one ring of a fused chain is a perturbation with a place, and the relaxation carries it along the molecule. Measured directly, the response stops falling after five rings and sits at four hundredths of a millionth for ever. That floor is not the molecule. It is the relaxation measuring each system's couplings from its own mean, and removing it recovers nine orders of magnitude.

 When the molecule does not stop

Twelve basins where there were two

Sixteen sites can be searched exhaustively and thirty-six cannot, so the only instrument available past about twenty sites is a local search — and how much weaker it is has never been measured against the answer on a net where both can be run. On sixteen it is barely weaker at all. On thirty-six it fails nine starts in ten, and the arrangement it is beaten by has fewer unlike bonds than the one it starts from.

 What symmetry decides

Three events, and a ratio of two dipoles

The m = 0 half of a shell has two crossovers because it has three levels and two coupled pairs. The other half has two levels and one, so the whole shell has three distinct fields rather than four — and two of the three share a gap exactly, which makes the ratio between them a ratio of two dipoles, 2/√3.

 When the molecule does not stop

A band that is a hundred and seventy decades of nothing

The chain length at which a defect's levels become a band can be located by asking when the coupling between the closest pair exceeds the level spacing. A band also has a width, and a width is set by the typical coupling rather than the closest one. For runs of eight at half concentration those two numbers differ by a hundred and seventy-three orders of magnitude.

 Bonding models

An interior maximum a third orbital allows

Two orbitals related by a mirror plane give a Boys functional with no linear term, so it has no interior maximum and no angle is ever searched for. Add the oxygen lone pair and the search over a three-dimensional rotation group finds one — eight and a half degrees of lone pair mixed into two bent bonds, worth a further two and a half per cent that no pair of orbitals could reach.

 Beyond the octet

Fifty descriptions of one molecule

A search whose largest basin takes ninety-eight per cent of its starts will report one description however long it is run, and nobody runs four thousand starts when the first fifty agree. Whether the rare ones are worse descriptions or merely rarer is one number per description, already computed and never looked at. On two of three cages they are not worse — they are the same answer, to parts per million.

 Orbitals

One basis size where it is worth doing

The pairwise assembly's error grows with the basis — 5.25 per cent at one Gaussian a centre, 37.98 at six — while the correction itself falls by a factor of three thousand. Which of the two should a practitioner care about? Drawing a line at a kilocalorie a mole answers it: there is exactly one basis size at which the correction is worth computing and its pairwise assembly is accurate enough to use.

 What is taught wrongly

The half of the square a ring of four cannot show

There is a warning that says in advance whether a transferred correction will hold: the mean field's own symmetry breaking, which collapses at a definite site-energy modulation and takes the transfer with it. The other axis of the same square has a threshold too — on the other side — and the ring of four it was all measured on is the one system with no threshold to find.

 What the shape is for

The ligand the rule was waiting for

A sixteen-electron complex is called reactive because it can add a ligand, and the gap that makes it sixteen points straight at where the ligand arrives. Bringing one in closes the gap exactly linearly — and leaves its exactness completely untouched, which is the opposite of what bending the same complex does.

 What a spectrum settles

Two moments about two different lines

Asked axis by axis, the substitution method's near-cancellation gives numbers as large as 163 per cent. The arithmetic is the smaller half of the answer. A principal axis is an eigenvector of a tensor built from the masses, so one deuterium turns water's frame by 21.12° — and the two moments being compared are not moments about the same line.

 What symmetry decides

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.

 What is taught wrongly

A bond order between atoms that do not interact

A diatomic held where its overlap changes sign has no interaction between its two orbitals at all — which is what the sign change of its overlap means. Put a third orbital beside it and the pair is still split, one line sits exactly at the free-atom energy at every third-orbital energy, and the bond order between the two runs to −0.9999. Three measures of the same bond disagree completely.

 Beyond the octet

A sign change is not always a zero

The solved give-back changes sign somewhere between a neighbour repulsion of two and one of four, and a bisection looks like the way to find the value where the structure beyond contact contributes exactly nothing. It changes sign twice. One crossing is that value; at the other the quantity the fraction is a fraction of has vanished instead, and a bisection reports the two in identical words.

 What symmetry decides

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.

 When the molecule does not stop

Seven points that looked like a switch

A constant belongs to a square net and not to a triangular one, which points at the coupling graph. Seven wrapped nets say which property of it: the third moment, and neither the dimension nor the coordination. They also say it is a switch — and made continuous, it is a crossover that every one of the seven sits twenty-five times past.

 What the shape is for

The distortion that opens the gap

Two distortions close the sixteen-electron gap — one by bending all four ligands, one by adding a fifth. Folding two of the four makes it larger, by five per cent, before it makes it smaller. And it costs the exactness at the first degree, while the gap is still growing, so the size of a gap and whether it is exact are not one measurement.

 Bonding models

The reach is the molecule's

Every measurement of the relaxation's reach so far puts the heteroatom on the end ring, because that gives the longest run to measure a decay over — and the response of a molecule to a perturbation at its end is not the response to one in its middle. Moved inward, the decay is the same in both directions and the same as the end's. The amplitude is not: it halves, and splits across two rings.

 Orbitals

The residue is below its own noise

Two interaction terms, both negative, leave the sign a coin toss — and a larger block would settle it. There is a larger block: a model that needs no measured separations supplies twenty-one. It cannot settle anything, because a fourfold alternating difference of distances known to a quarter of an ångström cannot resolve three hundredths of one.

 What a spectrum settles

The term a harmonic field cannot produce

The usual zero-point correction to a moment of inertia comes from a harmonic force field, which contains the mean square displacement and nothing else. A rotational constant does not average that. It averages one over r squared, whose leading correction is the mean displacement — zero in any symmetric well — and which enters with the opposite sign and about twice the size.

 Beyond the octet

A count that changes at one point

Where does the orphan count step along a distortion, relative to where the energy's minimum sits? A rule that depends on an exact symmetry may have no answer for a real molecule. On the path from a tetrahedron to a square plane the count is the same at every angle up to 89.99° and changes only at 90° exactly — which is the energy's maximum, not its minimum, and a single geometry out of a continuum.

 What is taught wrongly

A filled shell is not an empty statement

A bond order of −0.954 between two orbitals with no overlap and no resonance integral invites the prediction that at six electrons — every level occupied, the sum over a complete set — it would be exactly zero. It is exactly one seventh, and the reason is that a complete set in a non-orthogonal basis sums to the inverse of the overlap matrix, which has entries where the overlap has none.

 What a spectrum settles

A ratio of exactly one is a tie

Does the intensity contrast fall below two at half filling? It does — it falls to exactly one. But one is the floor of a ratio between two ranked quantities, and it is reached here because the cut between fundamental and satellite lands between two lines of identical weight. The guard installed to catch that case tests the wrong degeneracy, and the guard installed to license the extrapolation cannot tell an exact answer from a divergent one.

 What symmetry decides

Four angles the shell chooses

A field along one axis gives a shell of nine functions three fields with an event, and tilting the field should separate the coincident ones and raise the count towards the number of coupled pairs. It does — from three to six. But not monotonically: at four angles two events collide again, and every one of those angles is the arctangent of a ratio of the shell's own angular integrals.

 Bonding models

The error bar that would be needed

Two things a pair of measurements can say without any model both treat their numbers as exact. The quoted measurements carry no uncertainties, and inventing some would be worse than having none — so the question is asked the other way round. Not what the errors are, but how large they would have to be. The three refutations that seemed most worrying turn out to be the sturdiest of the lot.

 What the shape is for

The overlap the model is not proportional to

Without a computed π overlap, the natural argument reasons about one instead: the denominator over-predicts the halide trend, so the overlap must shrink down the group to cancel part of it. Computed, it grows — 3.5 times from fluoride to chloride. And the fitted parameter changes sign across the series, which no ratio of squared overlaps can do.

 When the molecule does not stop

The triangles that were never in the bands

A switch in how a gap scales is usually attributed to a band's third moment, and the attribution cannot be tested while the coupling runs along one of the bands. Separated, the bands turn out to decide nothing. Two triangular bands coupled along a matching — which cannot close a triangle across the gap — behave exactly like square ones.

 When the molecule does not stop

A count rather than an average

Two couplings quoted from one distribution sit a hundred and seventy decades apart. Neither is a summary of it. The quantity that decides how much of a spectrum near a box level is resonant pairs is a count of pairs above a threshold — and it has a closed form, which is a density times a reach times the logarithm of ten.

 Beyond the octet

Counting was right except where it mattered

A cage whose localisation gives dozens of descriptions that are all the same answer raises a worry: if degeneracy is common across the family, counting descriptions is the wrong measure of ambiguity. Across forty-eight cage-and-filling pairs it is the right measure on eleven of the thirteen that have anything to count — and it fails on the one leaned on hardest.

 What symmetry decides

None of the six was a crossing

Counting events in a tilted field finds the count rising from three to six, dropping again at four angles that are exact arctangents of the shell's own integrals. Every one of those statements is true of the two-state estimates. Diagonalising the five-level problem exactly finds one avoided crossing at every tilt, in a field that moves by a third across ninety degrees, and no feature whatever at any of the four angles.

 What a spectrum settles

The distortion the ratio cannot see

A depolarisation ratio can be made into a structural probe: break a molecule's symmetry and a band fixed at three quarters comes off it by an amount that depends on the distortion. The question is whether two distortions the ratio can see separately might cancel into one it cannot. They do — and the cancellation is exact, at any size, because the blind direction is the one that keeps the symmetry.

 Bonding models

The five figures were an identity

Two lone pairs satisfy the two-orbital mixing criterion by a margin of 1.732128, against √3 = 1.732051 — agreement to five figures on a number assembled from three integrals over a numerical grid. It is exact. The criterion is the reciprocal of the p amplitude of the outward hybrid, and everything else in the molecule cancels out of it.

 Beyond the octet

The group nobody wrote a table for

The orphan count is predicted to be constant along the Bailar twist, because the twist keeps D3 the whole way. The premise is exactly right — at every angle strictly between the prism and the octahedron the symmetry finder assembles six operations that hold to four parts in 10¹⁶. The conclusion cannot be tested here, because a count is a reduction, and the calculation's nineteen character tables did not include D3.

 What is taught wrongly

A symmetry holds or it does not

One level of a three-orbital trio sits at the free-atom energy exactly, at every third-orbital energy, because the antisymmetric combination of the pair has nothing of its own symmetry to mix with. Detuning one of the two by a twentieth of an electron volt moves it by half of that — first order, immediately, with no protected regime at all.

 What the shape is for

The sign a frustrated ring changes

There is a sharper question than whether a low-temperature feature buys back a fourth parameter: does it change which combination is free? It does, and by a sign. Every odd ring of spins leaves free the monomer fraction times a negative power of the temperature-independent term, and every even ring a positive power, with no case in between.

 What the shape is for

A contraction that cannot reach three of them

The angular overlap model's own derivation gives a π/σ ratio that disagrees with the fitted parameters by up to sixfold, and the metal's contraction is the obvious candidate to account for it. The whole range Slater's rules allow moves the ratio by a factor of two. Two ligands need charges far outside it, one needs a charge past where the overlap rule can be trusted at all, and two are unreachable at any charge because a quotient of squared overlaps cannot be zero or negative.

 When the molecule does not stop

The composition that is hard is not the full one

A landscape of arrangements measured at one composition on each of two nets raises the question of where the hardest one sits — and the natural guess is the half-filled one, where there is most to arrange. It is the easiest. On thirty-six sites at a contrast of one, half filling has one local optimum and nine billion arrangements, and a quarter filling has six optima and a hundredth as many.

 Beyond the octet

The count the table was hiding

The interior of the Bailar twist looks uncountable — exactly D3 at every angle, and D3 is a point group whose character table is rarely written out. Writing it takes nine lines and settles the prediction made for the path: the orphan count is two at every geometry on the path that can be answered, in three different groups, out of three different decompositions.

 What a spectrum settles

The number the tie got right

On a ring of six the contrast at half filling comes out exactly one, and the one is an artefact — the rank cut had landed between two lines of identical weight, so the ratio was a quantity divided by itself. The chain of six has no such pair anywhere, at any repulsion, at any filling. Its contrast at half filling converges to one anyway.

 What is taught wrongly

The second number is the error, rearranged

A cheap diagnostic for a composite method leaves a scatter it cannot explain, and the number that ought to close it is the change in the correlation energy, already computed at every point, so the test is arithmetic rather than a calculation. It is arithmetic, and the arithmetic is the answer. The composite's error is that change with a sign on it.

 Bonding models

The other end of the bracket is not a number

Every claim can be priced by the measurement error it would take to overturn it, assuming the two errors independent, and the correlated extreme ought to turn each price into a bracket. At perfect correlation between equally precise measurements the difference is exact and the price is infinite — and every price is multiplied by the same factor, so the ordering of prices cannot change.

 What symmetry decides

The variation was the basis

Solved in the five functions a field in one plane couples, the tilted Stark problem has one avoided crossing at every tilt and a crossing field that moves by a third across ninety degrees. Solved in the whole nine-function shell the field does not move at all — the same number at every direction, to eleven decimal places — and the thirty-four per cent was the truncation.

 What is taught wrongly

A correlation is not an account

A quantity called the chemical capacity is ordered against the thing it was supposed to predict and correlates instead with the second ionisation energy, which invites asking how much of it that accounts for — on the reasoning that a quantity which is ninety per cent of one input has a simpler name than the one it carries. It is three per cent of it. Nitrogen and potassium sit two electronvolts apart in the second ionisation energy and differ two hundredfold in capacity.

 What a spectrum settles

One number was a direction too

The depolarisation probe is exactly blind to the totally symmetric direction, which suggests it reads the distortion's component outside that species. What is left is a plane, and over a circle in it the reading varies by a factor of 1.98 — smallest for one bond against another, largest for two bonds against one, repeating every sixty degrees.

 Where the atoms go

The count that was never written down

The rotamer ceiling for a ring closure needs a count of frozen rotations, and the counts in use disagree in direction: one gives the six-membered ring more frozen rotations than the five, another fewer. Every measured verdict survives all three defensible conventions unchanged — and the margins move by a factor of eleven, which is why a verdict decided by ten per cent was worth being nervous about.

 When the molecule does not stop

The exponent was the floor

Fitting the local decay rate against the reciprocal distance reads a power off the slope. It runs from 0.41 to 0.66 across ten stiffnesses and appears to settle near two thirds. It is not settling. The tail is dropping below the arithmetic's own floor sooner at every step, so each case's power is taken over a shorter piece of the curve than the last.

 What symmetry decides

Consistently wrong is not a limit

Does the two-state picture of a tilted Stark shell become right as the quantum defect closes the l degeneracy? Swept over two decades it does not move: one avoided crossing against eight estimates at every defect. The reason is that every dimensionless quantity settles — the crossing sits at 0.04000 of the zero-field gap and the nearest estimate at 0.9067 of the crossing, and neither is heading anywhere.

 What the shape is for

It was the count, not the frustration

The free combination's exponent comes out negative on every odd ring and positive on every even ring and on an open chain, and the sign was put down to frustration. The control was a chain of eight. A chain of five is not frustrated in any sense — a chain is bipartite and every bond can be satisfied — and its exponent is −0.792.

 Bonding models

A bend is not an end

Moving a heteroatom from the end of a straight chain of twelve rings to its middle leaves the decay unchanged and halves the amplitude, which raises the question of whether a bend in the chain is a boundary of the same kind. It is not a boundary of the same kind: the kink — two adjacent angular fusions — leaves the fitted decay length within two per cent and costs the heteroatom's own ring a fifth only on the kink's first angular ring. It does multiply down the response on the rings beyond it, which a decay length does not see.

 What is taught wrongly

Five failures in five different places

Seven quantities a mean field produces for nothing have been tried as diagnostics, and none of them is usable. The question left is whether that is one finding or seven — whether the same awkward corner of the square breaks every candidate, or each is broken somewhere else. Each is broken somewhere else. Five candidates, five failing pairs, ten systems, and not one of them appearing twice.

 When the molecule does not stop

Five rings that were five different sizes

Five warmed rings have scaled alternation curves that lie on one another to 3.41 per cent, and the departure from the bulk amplitude turns out to be a function of the ring measured in its own alternations. The five cases span a factor of seven in that quantity. Choosing sizes that make them comparable halves the residual — and runs into a floor the lattice itself imposes.

 What the shape is for

The direction the gap cannot see

A two-ligand fold opens the sixteen-electron gap and a four-ligand bend closes it, so a distortion mixing the two must pass through a direction the gap does not move along. It does, and the direction is the antisymmetric fold — one pair of trans ligands up, the other down. Its blindness is exact, because exchanging the two pairs is a symmetry of the arrangement.

 What a spectrum settles

The sum was flat all along

The depolarisation reading varies by a factor of two round a circle in the non-symmetric plane, and the anisotropy may belong to taking a maximum over four bands rather than to the physics. Summing the four instead gives a reading constant to four parts in ten thousand, against a maximum that varies by ninety-eight per cent.

 What a spectrum settles

The suspect that did not fit

The sum over depolarised bands is flat to four parts in ten thousand rather than exactly, and the quartic term is the obvious suspect. Two tests say otherwise. The residual scales as amplitude to the 1.248, which is neither candidate — and reversing a distortion changes the sum by as much as the residual is, which only an odd power can do. The leading term is the cubic, and the suspect was wrong by one order.

 What symmetry decides

Five coordinates for six vibrations

A torsion is reversed by every improper operation that carries it onto itself and by no proper one, so adding torsions tests the orbit rule on a second kind of signed coordinate — and the rule holds on every molecule. It also moves hydrogen peroxide's count of totally symmetric vibrations from three to four, which a property of a molecule cannot do. Its five coordinates never spanned its six vibrations, and three of fifteen coordinate sets had been counting vibrations they did not describe.

 When the molecule does not stop

The rule of thumb was on the flat part

Fitting the Peierls tail discards the first few bonds of every profile, on a rule of thumb — three coherence lengths. Does that unexamined choice hide a second exponent? It does not. From six bonds outward the fitted power moves by half a per cent to nine; below six it moves seven times as much, and starting at two would have halved the very trend the fit reports.

 When the molecule does not stop

The window that was not a plateau

The near and far ends of this fit both sit on plateaus, and it is tempting to expect the same of every window. The third choice — that the local decay rate is read from the first quarter of the chain — is not a plateau. It never bound the soft half of the series and it was setting the answer for the stiff half, where opening it moves an exponent by a seventh, always downward. And a profile allowed to end on its own always runs 13.26 of its own decay lengths, which is the ruler that shows three cases are still cut at the midpoint.

 What is taught wrongly

The worst of the six was the one we asked about

The capacity's correlation with the second ionisation energy explains 2.9 per cent, and the natural next step is the same pair test against every other candidate input. Every input fails it — but the second ionisation energy fails it by a factor of twenty-four more than the best, and the test itself had to be repaired first, because the version the question implied reports the capacity failing to be a function of itself.

 What the shape is for

The frustrated cluster with an even count

A parity account of the exponent's sign replaced a frustration account and left the two still confounded: every case tested had frustration and odd parity aligned. A tetrahedron of four spins is frustrated and even. Its exponent is +0.2084, and so are those of three more clusters the two accounts disagree about — but the mechanism proposed with the parity rule is refuted along with the account it replaced.

 What symmetry decides

The crossing nothing couples

A tilted Stark shell's one avoided crossing settled at 0.04000 of the zero-field gap, and the natural guess was a ratio of angular integrals. With the quantum defect taken out the limit is 0.0399865, not four hundredths — and the two levels at that minimum belong to different symmetries about the field, which no element of the field connects. It was never an avoided crossing. The one minimum between levels that do interact sits at half the field, behind a level of the other kind.

 What a spectrum settles

The cubic a Morse curve guesses

The two terms in a vibrationally averaged rotational constant were computed on Morse curves built from measured constants, and the vibration–rotation constant αₑ is the measurement that tests them. In all four molecules the Morse curve's αₑ is short, by four to fifteen per cent. The averaging is not the error — it matches the closed form to four parts in ten thousand. The curve's cubic is, and the measurement asks for more of exactly the term a harmonic field cannot produce.

 What the shape is for

A blindness that is inherited

There is a direction the sixteen-electron gap does not move along, and it is tempting to call it the one a complex is softest along without paying for it. That second half is a claim about an energy the gap model has no term for. Put the ligand repulsion on the same family and the answer splits: where a symmetry fixes the blind direction the repulsion is blind to it too, exactly, and slightly downhill — and where no symmetry fixes it, the two are nearly perpendicular.

 When the molecule does not stop

Three points, and they all go down

Matching five rings at one value of n·δ∞ tightens the temperature collapse from 3.41 per cent to 1.62, and what is left might be the even-site rounding rather than anything physical. At three targets the residual falls monotonically — and at the smallest one it is a third of what the rounding leaves, which the rounding cannot explain.

 Where the atoms go

Two numbers caught what one could not

No single threshold on the electronegativity difference separates the bond pairs whose polarity the four tables dispute from those they agree on — the best misses six of a hundred and fifty-three. Adding how much the tables disagree halves that to three and catches every disputed pair. And three is what four coin-flips produce: the rule flags nineteen pairs, an eighth of which should look agreed for no reason at all, which is 2.375 against the three observed.

 What is taught wrongly

No panel of this kind can find an exception

Four electronegativity tables leave three exceptions to a rule for which bond polarities they dispute, and four independent coins would produce two and a half. How many tables before one exception would mean something? Ten, if tables were coins. Near the boundary they are not: they correlate at 0.32, two of the four are nearly one table, and the requirement becomes seventy-three — with a plausible range running past any panel that could exist.

 What symmetry decides

Two levels cannot make a minimum

The one minimum between coupled levels in a Stark shell sits at 0.0196 of the s–p gap, and the nearest two-state estimate at 0.0192 — two per cent away, which reads as the estimates having been aimed at the right feature all along. They were not. Two coupled levels only ever separate, so no estimate can be where its own pair is closest. The minimum belongs to a third level, exists only while the d level sits within a quarter of the s–p gap, and meets the estimate by crossing it.

 Where the atoms go

A barrier is not what a splitting measures

Ammonia's inversion barrier is quoted everywhere as 2020 wavenumbers. Put that number into the simplest double well its own measured geometry allows and the ground-state splitting comes out at 1.3508 against a measured 0.7935, and the excited one at 68.37 against 35.81. Both are too large because a splitting is an area under a barrier and a height is only one of its two dimensions.

 Orbitals

The line was holding the answer up

There is exactly one basis size where the three-body correction is worth computing and the pairwise assembly reproduces it, and the accuracy line is the choice that whole picture is most sensitive to. Sweeping the line over two decades gives four different answers, long stretches with no answer at all, and a published window whose lower edge sits seven tenths of a per cent below the standard kilocalorie a mole.

 When the molecule does not stop

The other window was a plateau too

Sweeping where the fit begins finds a plateau. The far end is the other window and nobody had swept it: inside each profile's own reach the exponent moves by at most 5.3 per cent, and past that reach every larger window returns exactly the same fit — because there are no more points to add. What the reach is depends on the stiffness, and for half the series it is an arbitrary rule rather than the physics.

 What the shape is for

The sign rule holds between two poles

The sign of a susceptibility fit's exponent was put down to the parity of the spin count, then to the spin of the ground state, and every cluster tried had the two aligned. A star of four spins is even with a ground spin of one, and its exponent is −6.84. But swept across the coupling, every cluster's sign changes, through poles the ground spin places, and the spin rule is right on all sixteen clusters only between 20.1 and 47.0 cm⁻¹.

 What is taught wrongly

Six of fifteen change verdict

The chemical capacity is finite or infinite according to the sign of a coefficient that is one sixth of a second difference of three measurements, and the pair test asked how much of its reported ordering is the quantity and how much is the fit's resolution. Adding a fifth point changes the verdict on six of fifteen atoms — and the five it unbounds are the five it had largest.

 What a spectrum settles

The second molecule with a blind spot

Methane's fifty-five force constants have ten directions no spectrum can touch, and the forty-five that are fixed were worth a table. Boron trifluoride is the only other molecule here with a redundant coordinate set, and it is not the same case: eight kinds of constant rather than five, a quarter of its field invisible rather than a fifth, and an out-of-plane coordinate the redundancy cannot reach.

 What symmetry decides

The quarter, generalised

A shell's one coupled minimum exists because its d level sits within a quarter of the s–p gap, and the quarter was found by bisecting a numerical search on one shell. It is exactly 2(n² − 4)/(5(n² − 1)) for every shell — zero at the second, a quarter at the third, two fifths in the limit — while the offset it is compared against is one fifth whatever the shell.

 Where the atoms go

The exponent that runs both ways

How hard does a splitting depend on a barrier? Locally, as the power −3.5628 — and the local slope runs from −2.53 to −6.15 across the same sweep, so there is no power law. What is exact is stranger: rescaling the equation forces the mass exponent to be one below the barrier's and the geometry exponent to be twice the mass's, so the model's three sensitivities are one number and the arithmetic reproduces both identities to six decimals.

 Bonding models

The ranking moved and the headline did not

One correlation applied across a collection multiplies every price by the same number, so it slides the fragility ordering rigidly and cannot scramble it. Give each predictor its own correlation and the ordering does move — but not at the top. The two most fragile claims come from the same predictor, share a factor, and are locked in order at every correlation whatever; the easiest swap anywhere below them needs only ρ = 0.15.

 What the shape is for

The correction that moves three of them backwards

Every ligand radial function in the angular overlap sweeps was a neutral atom's, while three of the five donors carry a formal charge. Giving each one the charge it actually has moves three of the five computed ratios — and moves all three away from the fitted parameter, none towards it. The whole window each donor's own oxidation states allow sits above the value it was meant to reach.

 When the molecule does not stop

A net with no two-colouring

Half filling is the easiest composition to search and the reason given was the two-colouring: on a bipartite net it is the unique arrangement with every bond unlike, so the optimum has nothing competing with it. A triangular net has no two-colouring, its half-filled composition is reached by every one of two hundred starts, and its best arrangement is two bonds short of what counting allows.

 What a spectrum settles

Adding data made it worse

A projection leaves a residual spread across four starting points, and an earlier essay attributed it to the eigenvalue problem rather than to the coordinate set — an attribution rather than a measurement. Refitting with two isotopologues was the proposed test. Run, it produces a runaway to minus ninety-seven thousand that one isotopologue alone does not, and the reason is that the direction it walks along is flat at every mass.

 What symmetry decides

No length separates them

A bond list here is one distance cutoff with a clause about hydrogen, added when peroxide came back with five bonds instead of three. The clause repaired one molecule. Across the twenty-three molecules drawn here, the longest bond is 2.32 ångström and the shortest pair that is not a bond is 1.51 — so no cutoff can work at all, and five molecules currently come back with no bonds.

 Where the atoms go

It was never the mass

Phosphine does not invert, and the reason given is that phosphorus is heavier than nitrogen. Three things about phosphine differ from ammonia. Substituting each into ammonia's own well one at a time, the reduced mass costs a factor of 1.61, the pyramid height costs 9,600 and the barrier 94,000 — and the mass is the smallest of the three by four orders of magnitude.

 Orbitals

The error was the row, not the charge

An ionic model checked against six measured separations has a wildly uneven error — under one per cent on two pairs, thirteen to eighteen on three others — and the pattern is not obviously size or charge. It is the row of the periodic table. Counting how many of a pair's two ions have a third-row valence shell separates the errors completely, with an eleven-point gap; counting the charge separates nothing.

 What the shape is for

The channel that points at the metal

Two ligands in the spectrochemical series carry a fitted π parameter no quotient of squared overlaps can produce, because it is negative. Giving the derivation the second interaction it lacks makes both of them negative at every metal level — and the reason is not the energy denominators, which favour the donor channel in all three cases. It is where each orbital keeps its amplitude.

 What is taught wrongly

A size the fit was not made from

Every input tested against the chemical capacity so far has been inside the cubic that produced it, so the search was constrained to fail. A size is not: a hydrogenic orbital at the effective charge Slater's rules give an atom's valence shell has an exact radius, on one scale for every atom, and the cubic knows three energies and no length at all. It fails the test by sixteen times the floor.

 What a spectrum settles

The residual was a loop

Projecting away boron trifluoride's flat direction left four fits disagreeing by a tenth in the B–F stretch constant, and the disagreement was put down to the search or to the arithmetic. It is neither. Two frequencies cannot fix three constants in one symmetry block, so the exact fits form a closed curve along which the stretch constant runs from 3.9 to 12.1, and the four fits are four points on a short arc of it.

 What symmetry decides

Four alkalis the model cannot hold

A shell's s and p levels have a coupled minimum in an electric field when its d level sits within a threshold fraction of the s–p gap above p, and a screening model put every shell's d level at one fifth — inside the threshold from the third shell on, with more room the larger the defect. Computed from measured quantum defects, lithium keeps the minimum at every shell. Sodium, potassium, rubidium and caesium lose it at every shell, and the screening model has no member that resembles any of them.

 When the molecule does not stop

The gap follows the winner late

On a triangular net at half filling, the arrangement that binds best was expected to be the one that opens the widest gap at the Fermi level. Across twenty cases it is, eighteen times. The two exceptions are not noise: the frustrated net's winner changes at a contrast of 3.790, from an arrangement with thirty unlike bonds to one with thirty-two, and the gaps of the two do not cross until 4.849. For a whole unit of contrast the better binder has the narrower gap.

 Where the atoms go

The mass nobody chose

Every one-dimensional treatment of ammonia's inversion needs a mass, and the measurement does not supply one. Three constructions are defensible and they give 1.35075, 0.94420 and 0.0000055 wavenumbers. The honest one is not a constant at all, and it moves the answer thirty per cent towards the measurement — which means the usual choice is the wrong one.

 Bonding models

The criterion that has never said no

The two-orbital mixing criterion is a tautology on a carbonyl's lone pairs, and it is tempting to contrast that with the double bond, where the criterion seems to discriminate. Both halves are wrong. The double bond's margin is wider, ethene's is infinite because symmetry puts both centroids at the same point, and the criterion has never returned a negative on any case it is usually put to — though the same three orbitals supply two pairs it refuses outright.

 Orbitals

The overshoot was one arrangement

A fourth fragment leaves four three-body terms out of a pairwise counterpoise assembly as well as the four-body one, so the window in which the assembly is usable was expected to narrow. Asked of three arrangements that each gain one centre, it widens twice and narrows once, the uniform chain loses its answer at a kilocalorie a mole altogether, and the overshoot every earlier calculation reported turns out to belong to the arrangement with the heavy centre inside.

 What is taught wrongly

Four quantities go and one question stays

The exact theory says an atom's energy against electron count is straight segments between integers, and that model gets the alkali metals right for free — which is exactly where the fitted curves give lithium a negative electronegativity. What it costs is the hardness, the capacity, the electronegativity and equalisation, and what it leaves is a question whose answer is an integer.

 What a spectrum settles

Two coefficients are not a potential

A Morse curve built from measured constants gets the vibration–rotation constant αₑ short by four to fifteen per cent, and the measured αₑ and anharmonicity imply the cubic and quartic a real potential should have. Built with exactly those two coefficients and solved, the potential overshoots instead. The reason is that the Morse curve's own series, cut after its quartic, moves αₑ by a sixth to a half of the error being repaired — so a quartic cannot tell whether the shortfall was the cubic.

 What symmetry decides

The census a bond rule was hiding

Every internal coordinate, redundancy and totally symmetric count here is built on a bond list, and the bond list came from a length cutoff that gave five molecules no bonds. Rebuilt on the radius rule, the census reaches nineteen molecules instead of fifteen, the orbit identity holds on every newcomer, ferrocene's coordinates finally span all its vibrations — and a different gap appears: no bond rule can give a square-planar centre its two out-of-plane vibrations.

 What the shape is for

Purity renames the poles

A susceptibility fit's sign poles were located at one monomer impurity, two per cent, and a real sample's impurity is rarely known. Swept from half a per cent to sixteen, the couplings where the fit's nuisance parameters separate barely move for ground spins of one and above and move a tenth for doublets. But the same separation stops being a pole and becomes a zero of the exponent at a few per cent, and above six to eight per cent every singlet acquires separations of its own.

 When the molecule does not stop

The share that was read as a line

The share of defect runs sitting in a resonant pair was read as a straight line in the logarithm of the chain length, rising by a density times a reach per decade and never saturating. The exact rise is that amount times the share of runs not yet resonant. At the concentration first studied the difference is small over nine decades and large beyond them; swept to the concentration with the most runs, the rise falls to a third, and half of all runs are resonant ten orders of magnitude later than the straight line says.

 Where the atoms go

The lever that was supposed to be smaller

Sweeping the rotor conventions leaves every verdict unmoved, and suggests that the other quoted input will be a smaller lever. It is a larger one. Sweeping butane's gauche energy over its reported range moves the ceiling by 8.7-fold, flips the six-membered verdict at 3.630 kilojoules a mole — inside the quoted error bar — and takes the five-membered refutation with it at 4.422.

 Beyond the octet

A second criterion left a gap too

The localisation spreads are bimodal — an empty factor of two hundred around the degeneracy threshold — and the gap might belong to the criterion rather than to the cages. Boys localisation leaves a gap of seven decades on the same forty-eight pairs, so it belongs to the cages. But the two criteria disagree about six of them, in both directions, and the family's most ambiguous cage under one is exactly degenerate under the other.

 Orbitals

A repair that costs more than the whole

Four fragments are the first system in which a counterpoise correction can be assembled from something between pairs and the whole. Adding the three-body increments leaves what is still missing below a kilocalorie a mole on every cell, widens the usable range for every arrangement and turns a single usable basis size into two — and under a cubic model of cost it is dearer than the full calculation it stands in for until the cluster has eleven fragments. Keeping only the consecutive triples, which pays from five, works for two arrangements and does worse than pairs for the third.

 Bonding models

A level no symmetry was protecting

A three-orbital trio keeps one level at the free-atom energy exactly, and the reason given was that its two outer orbitals are equivalent. Make them inequivalent by changing one overlap rather than one energy and the level does not move at all — not to first order, not to any order, at any energy of the third orbital. It was never the symmetry. It is allyl's non-bonding orbital, held by a count.

 What a spectrum settles

Two integers made one exponent

A sum over boron trifluoride's depolarised bands is nearly isotropic, and its residual scaled as the amplitude to the 1.248 — no integer, and a two-term fit left a pattern it could not remove. Averaging each distortion with its reverse splits the residual exactly into an even half that scales as the square, to 2.005, and an odd half that scales as the first power until a fifth-order term turns it over. And the sum was a stand-in: what an unresolved pair of bands would actually show is twenty times less flat.

 What the shape is for

Five more clusters break the band

Seven spin clusters put their upper separation coupling times the ground spin plus a half inside a band from 115 to 129 cm⁻¹, which looked like a law of where a susceptibility fit's nuisance parameters decouple. Five clusters built to test it — two larger stars and three larger bipartite graphs — land inside it once. Stars drift above and bipartite graphs below, and neither the shape of the curve nor the first excitation places the separation instead.

 What is taught wrongly

The rule is not the lever

One exception among nineteen flagged bond pairs would need seventy-three electronegativity tables to mean anything, and a rule that flagged fewer pairs looked like the way to need fewer. Searched over sixty-one thousand boundaries, nineteen is already the fewest that misses no disputed pair; each missed pair buys a table or five; and a rule flagging a single pair would still need fourteen. The requirement lives in the correlation between the tables, which moves it nine times as far as any rule can.

 Where the atoms go

Two sweeps and one lever

The rotamer ceiling depends on the gauche energy and on the temperature only through their ratio, so a sweep of either traces the same curve. Measured in that one variable, the temperature sweep lies entirely inside the gauche sweep — it covered no ground the other does not, and the whole reported spread in one energy is worth a temperature swing from 196 to 353 kelvin.

 Beyond the octet

The cage is on both sides

Two localisation criteria classify six cage-and-filling pairs differently, and a symmetry explanation for the six is the natural first guess. The icosahedron's graph has a hundred and twenty automorphisms, the most in the family, and supplies three of the six disagreements and nine of the agreements. What the six do have in common is sharper than a symmetry: in every one, one criterion's spread is not small but zero.

 Orbitals

The sum of the exponents, not the softer ion

The row of the periodic table sorted an ionic model's errors into three groups, and a count that takes three values can say nothing inside a group. Made continuous, the variable the proposed mechanism names — how diffuse the softer ion is — carries no information: an oxide's 2p and a chloride's 3p have the same exponent to a hundredth. The sum of the two exponents carries nearly all of it, orders the middle group, and, asked about that group without having seen it, predicts its spread at twice the size.

 Bonding models

Three chains and ninety orderings ruled out

Two claims priced against the same predictor share its error correlation, so no correlation structure can reorder them. That makes the collection's fragility ranking three chains rather than one list — and thirty of the hundred and twenty orderings of five claims are reachable, with the other ninety forbidden before a single measured difference is looked at.

 What is taught wrongly

The limit of one is a parity

A half-filled chain of six has no degenerate removal lines, and its intensity contrast still goes to one — after dipping near U = 32 and rising again to U = 128. Followed line by line, every removal line is smooth and monotone; the dip is an exact tie between two lines at U = 28.916, and the rise ends where two satellites change places. At large repulsion the lines pair up at ±E with equal weights, so the contrast goes to one exactly when the cut falls inside a pair. On a chain of four it does not, and the limit is 1.3125.

 Beyond the octet

The leftover changes sides

A main-group centre brings four valence orbitals against six ligand combinations, so two are orphaned. A transition metal brings nine, so the arithmetic inverts and three metal orbitals are left instead — three at every geometry of the Bailar twist, out of decompositions that share no species. Run past the whole arrangement census, exactly one arrangement orphans anything at a metal, and it needs an f orbital to fix.

 Orbitals

One contraction for two conditions

The ionic model's errors run with the row of the periodic table, and the test proposed for that — stiffen the repulsion and watch for second-row pairs moving out and third-row pairs moving in — produces exactly that pattern from a repulsion that knows nothing about shells. The test that can fail contracts the third-row shells alone and asks one factor to bring two different groups of pairs onto the second-row ones. The two factors needed are 1.35 and 1.38.

 Where the atoms go

An ordering worth half a per cent

A mass that varies along a coordinate has no unique quantum kinetic energy, and the choice among the Hermitian orderings in use was the one thing left that could undo a forty-three per cent correction to ammonia's splitting. It cannot. The five orderings anybody uses span 0.47 per cent between them, a ninety-second of the correction, and the family only reaches the measurement at exponents three times larger than any of them.

 Bonding models

A denominator needs three currencies

Every price in this argument has been a price on a measured rise. A slope floor is a rise over a run, the run is computed rather than measured, and pricing it turns out to need three currencies rather than one — because one predictor's run is a graph eigenvalue, one is a convention, and one is a difference of square roots of integers that nothing defensible moves.

 Beyond the octet

The gap found on purpose

A twist between two coordination arrangements passes through a point group with no character table among the twenty in use, and that was found by accident. Running every path between the arrangements — eighteen of them, two hundred and seventy geometries — finds exactly one more gap, at a group of order ten. It also finds something the search was not looking for: a group without a table had been reported as an infinite group.

 Where the atoms go

Deuterium cannot tell the masses apart

A reduced mass built by holding ammonia's bonds rigid predicts a deuterated molecule differently from any constant mass, and ND₃'s splitting is measured. Run as a test, it cannot choose. Every well and every mass needs a barrier for ND₃ several per cent lower than for NH₃, every prediction of the isotope ratio from a well fitted to NH₃ overshoots by eighteen to twenty-nine per cent, and the two masses differ by less than either misses — in opposite directions in the two wells.

 Orbitals

Three contractions for one shell

One contraction of the third-row p shells removed the row pattern from an ionic model's errors, with two conditions met by factors 2.2 per cent apart, and left the order of three pairs untouched. Taken ion by ion, chloride needs 1.302, potassium 1.387 and calcium 1.441 — the pairs' own order — and potassium chloride, fitted on nothing, lands among the second-row pairs. The single factor's two conditions agreed because both were averages of these three.

 Bonding models

A floor on models written in one scale

A tie asks whether two predicted values are equal and a discordance asks whether two differences have the same sign. Both are questions about order, and a strictly increasing change of scale preserves order — so both are exactly invariant under thirteen reparameterisations of all three predictors. The slope floor asks for a ratio of differences, and it moves by factors of nineteen, ten and eleven thousand.

 Beyond the octet

Six disagreements and three calculations

Two localisation criteria classified six of the family's forty-eight cage-and-filling pairs differently, and three of the six were consecutive fillings of one icosahedron with spreads identical to three figures. They are not three coincidences and not one degeneracy being filled: they are one calculation, because the survey's forty-eight rows are thirty-three distinct questions.

 Orbitals

The nodes in the other variable

An orbital has n − l − 1 radial nodes, and it has exactly that many in momentum too — the two radial functions are polynomials of the same degree. Nothing pairs one node with another: they are zeros of two different classical families. What is exact is the product over all of them, which is a ratio of factorials and does not depend on the nuclear charge at all.

 Where the atoms go

The two that are not on the line

Ammonia and its fully deuterated twin are two points, and two points cannot show a curve. Putting the partly deuterated molecules between them needs a term neither symmetric one has — three ligands of unequal mass move their own centre of mass sideways as the apex descends — and it moves the prediction by half a per cent, which is what a whole change of mass construction was worth.

 Bonding models

The spin the count does not hold

Three orbitals in a row keep one level at the free-atom energy however the overlap of one end is changed, and at three electrons that level holds the radical's unpaired electron. Its energy does not move. Its spin does: from half on each end to 0.236 and 0.764 as one overlap goes from 0.25 to 0.45, exactly the squared ratio of the two overlaps, at every energy of the middle orbital. A coupling between the ends moves the level and cannot move the spin. The energy and the spin are answering to different things.

 Beyond the octet

The basis a diagonaliser happened to return

Six of the cage family's thirty-three inputs have their occupied set cut through the middle of a degenerate shell, and the members of a degenerate shell are interchangeable. Re-orienting the shell changes nothing about the cage, the filling or the criterion — and it moves the best localisation functional by up to twenty-one per cent, moves the count of descriptions from ten to fourteen, and flips the degeneracy label on three of the six.

 Orbitals

Oblate in the picture nobody draws

Every drawing of a σ bond shows a density stretched along the bond, and the position-space calculation agrees: the second moment along the axis is twice the one across it. In momentum the same orbital is flattened in the same direction, because the interference between the two centres cuts the distribution off at π over the bond length — and that cut-off is a zero a measurement could find.

 Where the atoms go

The ordering a manifold picks

A position-dependent mass leaves the kinetic energy with no unique quantum form, and an earlier sweep of the five orderings in use found half a per cent between them. A one-dimensional reduction is a one-dimensional manifold, a manifold has a distinguished Laplacian, and carrying it to the flat measure lands on exactly one of those five — not the one with no extra potential, and not the one anybody reaches for.

 Bonding models

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.

 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.

 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.

 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.

 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

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.

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