True as far as it goes, and silent on what decides

“A vibrational calculation predicts the frequencies of a molecule.”

The force constants used here were fitted to the observed frequencies by least squares, so the positions are what was fitted to and are not a prediction of anything. What the calculation does predict is everything the fit did not see: the mode shapes, their symmetry species — which must and do agree with the group-theoretic count made with no force constant at all — and every isotopologue.

Tested in Normal modes are not bond stretches · the series on normal mode

“The d orbitals split because negatively charged ligands repel the electrons in the orbitals pointing at them.”

The repulsion story gets the pattern right and cannot be the reason, because the pattern follows from the point group alone. Reducing the five d functions in Oh gives Eg ⊕ T2g — two levels, of two and three — with no charge, no distance and no interaction strength anywhere in the calculation. The same reduction gives four levels in D4h, and the computed angular overlap eigenvalues reproduce the dimensions in all three geometries.

Tested in The splitting is a symmetry statement · the series on ligand field

“The tetrahedral angle is 109.5 degrees.”

It is arccos(−1/3) = 109.4712°, and it is never written down here: four points are pushed apart on a sphere until the repulsion stops falling, and the angle is measured off the answer. The minimiser reaches the published Thomson minima to about 2e-10.

Tested in VSEPR, computed · the series on VSEPR

“A published orbital picture shows the ninety per cent probability surface.”

The level is a free choice and almost no source states it. Solved for here, 1s at fifty, ninety and ninety-nine per cent comes out at |ψ| = 1.48e-1, 3.94e-2 and 8.44e-3 — radii differing by more than a factor of two under one caption.

Tested in Say what it encloses · the series on contour

“A molecule is chiral when it has a carbon bearing four different groups.”

Chirality is the absence of any improper operation whatever. Searching the coordinates for improper axes Sₙ and for mirrors catches the chiral allenes and biphenyls that have no stereocentre at all, and refuses the meso compounds that have two.

Tested in Chirality is a symmetry statement · the series on point group

“A functional group has a characteristic frequency because its bond has a characteristic force constant.”

The force constant is necessary and not sufficient. Computed compositions across six molecules show that a mode belongs to one coordinate only when no symmetry operation carries that coordinate into another: four of twenty-one distinct frequencies here are localised, and all four are unique coordinates. Every stretching mode of water, carbon dioxide and methane is spread evenly over symmetry-equivalent bonds whatever their force constants are.

Tested in Group frequencies, and where they stop · the series on normal mode

“A molecule's dipole moment is the vector sum of its bond dipoles.”

The sum leaves out the lone pairs and rests on a quantity with four incompatible definitions. Where the answer must be zero, the point group settles it exactly and needs no electronegativity at all.

Tested in The dipole is not a sum of bonds · the series on dipole

“A transition-metal complex is coloured because its d electrons absorb light, and the colour seen is the complement of the d–d band.”

Permanganate has no d electrons at all — manganese(VII) is d⁰ — and is one of the most intensely coloured compounds in ordinary use. Its absorption is a ligand-to-metal charge transfer, an entirely different transition, and it is some ten thousand times more intense than a d–d band. Of the nine ligands in the tabulated series, six put the d–d band outside the visible range: four in the near infrared and two in the ultraviolet.

Tested in Where a d–d band falls · the series on ligand field

“Water is bent because lone pairs repel more strongly than bonding pairs.”

The rule gives a direction and no magnitude. It cannot say why the angle is 104.5° in water and 92.1° in hydrogen sulfide, and the trend down the family is the one Bent's rule predicts from s character rather than the one repulsion predicts.

Tested in Why water is bent · the series on VSEPR

“A tight-binding calculation on a large enough cluster gives the same answers as a band-structure calculation.”

It gives the same density of states and the same total energy, both of which are sums over all states. It cannot give a dispersion relation, because that is a function of a wavevector and a finite cluster with no periodicity has no wavevector to be a function of. Direct and indirect gaps are the same number here and different physics in a laboratory.

Tested in A band with no structure in it · the series on Bands in a solid

“A transition metal wants eighteen electrons because it has nine valence orbitals — one s, three p and five d — and nine orbitals hold eighteen electrons.”

That arithmetic is the same for every geometry and every ligand, and the counts are not. The same nine metal orbitals give sixteen in a square-planar complex, because the reduction leaves the metal's perpendicular p orbital with no ligand partner and far too high to occupy. What the count measures is how many orbitals lie below the antibonding set, which is an output of matching the ligand combinations to the metal orbitals by symmetry species.

Tested in Eighteen is a count · the series on electron count

“A chemical bond is a pair of electrons shared between two atoms.”

The same eigenvalue arithmetic that describes benzene describes rings, clusters and the band limit. Two centres is the special case, and electron-deficient clusters have no two-centre description at all.

Tested in Where two-centre bonding stops · the series on multicentre

“A ligand binding to a metal donates electrons to it, so the ligand's own bonds are weakened.”

Carbon monoxide's donor orbital is slightly C–O antibonding, so emptying it strengthens the C–O bond. The hexacarbonyl dication stretches at 2204 cm⁻¹, above free CO's 2143, which no account in which binding can only weaken the ligand can produce. What lowers the frequency in ordinary carbonyls is the other half of the interaction — density donated back from the metal into the ligand's π* orbital.

Tested in Back-bonding is two interactions · the series on electron count

“Bond angles are set by the repulsion between electron domains alone.”

Bent's rule sends s character to the less electronegative substituent and predicts the direction of the angle change across the substituent series, which the repulsion account does not. The quantity it depends on then turns out not to be one quantity.

Tested in Bent's rule, against the substituent series · the series on VSEPR

“4s fills before 3d because the 4s orbital is lower in energy and smaller.”

The 4s function is larger than the 3d by every measure of size computed here. What decides the filling order is penetration — a small inner peak of the 4s radial distribution lying inside the 3d maximum — which is a feature of the shape rather than of the size.

Tested in The radial distribution across the periodic table · the series on orbital

“A cluster of a few hundred atoms is effectively bulk material.”

It depends entirely on the property. A chain's band edges are within a part in ten thousand of the limit by two hundred sites, while its energy per site is still about 0.7β/n away — three parts in a thousand at that size, and needing roughly seven hundred sites to reach a thousandth of a β. The same cluster is converged for one question and not for another.

Tested in Where a molecule stops being one · the series on Bands in a solid

“A lone pair repels more strongly than a bonding pair, which is why water's angle is less than tetrahedral.”

Written as a weight in the repulsion sum, the clause reproduces water's 104.5° at a weight of 1.24 — and the same clause needs 1.15 for ammonia, 2.26 for hydrogen sulfide and 2.87 for phosphine. A parameter that varies by two and a half across four molecules of two shapes is not a property of a lone pair.

Tested in What a lone pair is worth · the series on VSEPR

“A metal is a material that conducts electricity.”

Conduction is a consequence and a poor definition: a doped semiconductor conducts and is not a metal, a metal at absolute zero with no field applied conducts nothing, and the conductivity of a real metal is set by scattering rather than by the electronic structure. The property that survives all three cases is the availability of excitations of arbitrarily small energy, which is checkable on a level ladder.

Tested in What a metal actually is · the series on metal

“Metallic behaviour comes from a partly filled band.”

That is the mechanism in this model and not the definition, and the model is wrong for a class of real materials. A partly filled band gives arbitrarily cheap excitations in a one-electron picture; strong repulsion between electrons can remove them without changing the band at all, and the resulting insulator has a half-filled band throughout.

Tested in What a metal actually is · the series on metal

“Counting the bands in a spectrum counts the allowed transitions.”

It counts them from below. Every band seen is allowed; not every allowed band is seen, because an allowed transition with a small dipole derivative can be arbitrarily weak and symmetry says nothing about magnitudes. So an observed count is a lower bound on an allowed count, and structural arguments built on it have to run in the direction that bound supports.

Tested in What an absence proves · the series on spectrum

“Symmetry decides how many signals a spectrum shows.”

It decides how many the static structure can show. Whether the measurement sees that number depends on how its timescale compares with any exchange the molecule undergoes: the same molecule gives two environments to a vibrational spectrum, which is fast, and one to a magnetic resonance spectrum, which is slow.

Tested in A spectrum counts environments, not atoms · the series on spectrum

“Degeneracy comes out of the calculation.”

Which degeneracies are possible is fixed by the point group before any energy exists: the irreducible representations of Td have dimensions 1, 1, 2, 3, 3, so a tetrahedral molecule can have levels that are single, double or triple and nothing else. The calculation decides which occur, not which are allowed.

Tested in Degeneracy is a group theorem · the series on representation

“Benzene is regular because its π system prefers equal bonds.”

Its π energy also falls when the bonds alternate; what differs is the power. Benzene's closed shell makes the fall quadratic rather than linear, so a stiff enough σ frame can hold it regular, and which wins is a quantitative question rather than a symmetry one. The computed slope is zero for benzene and finite for cyclobutadiene, and that is the whole difference.

Tested in The vibration that lowers the symmetry · the series on aromaticity

“The nuclear charge an outer electron feels is the nuclear charge minus the number of electrons inside it.”

Counting inner electrons at one apiece gives carbon's 2p electron a charge of 2.00; Slater's rules give 3.25 and spectroscopy supports something near it. The difference is that electrons in the same shell screen partially and electrons in the shell below screen incompletely, and neither number can be counted — both are fitted.

Tested in What an electron actually feels · the series on orbital

“A distortion splits a degenerate level because the atoms have moved.”

Whether it splits is decided before anything moves, by whether the lower group has a representation of the same dimension. Oh's T2g must split in D4h because D4h has nothing three-dimensional; Td's E does not split in C3v because C3v has a two-dimensional representation to hold it. The displacement decides the size, not the fact.

Tested in Descent in symmetry · the series on point group

“An orbital has a size.”

Four measures computed from one radial function give 1.00, 1.50, 1.73 and 2.66 bohr for a 1s orbital — a spread of two and a half — and every one of them is the correct answer to a question somebody actually asks. A size quoted without saying which is a number without a question.

Tested in How big is an orbital · the series on contour

“A heteroatom that contributes a lone pair to an aromatic ring donates it, so the ring carries the charge and the heteroatom is left positive.”

Pyrrole's nitrogen retains 1.720 π electrons of the two it brings and furan's oxygen retains 1.791, so 0.280 and 0.209 are donated. The direction of the sentence is right and its magnitude is wrong by a factor of seven: the donation is between a fifth and a quarter of a lone pair, not a whole one, and the difference between the two heteroatoms is decided by the same matrix entry that makes oxygen more electronegative.

Tested in Six electrons in a ring that is not all carbon · the series on delocalisation

“The pairing theorem — levels symmetric about α — is a property of alternant hydrocarbons and follows from the ring being even-membered.”

It follows from the graph being bipartite, and it is destroyed by a diagonal entry regardless of the graph. Pyridine's carbon skeleton is bipartite and its levels are 2.107, 1.167, 1.000, −0.841, −1.000, −1.934 — visibly unpaired. The theorem is a statement about a matrix, not about a drawing of a molecule.

Tested in Six electrons in a ring that is not all carbon · the series on delocalisation

“Furan is aromatic in the same way pyrrole is, both being five-membered rings with six π electrons.”

Both have six π electrons in five levels with a closed shell, and the resemblance ends there. The π bond orders to the heteroatom are 0.440 in pyrrole and 0.385 in furan, and the oxygen retains 1.791 π electrons against nitrogen's 1.720. The ring with the more electronegative heteroatom is the one whose π system is least shared, which is the computed content of furan's well-known reluctance to behave aromatically.

Tested in Six electrons in a ring that is not all carbon · the series on delocalisation

“Water is sp³ hybridised, and its 104.5° angle is the tetrahedral 109.5° reduced by lone-pair repulsion.”

Inverting the orthogonality relation cos θ = −a/(1−a) on 104.5° gives an s fraction of 0.2002 per bond hybrid, which is sp³·⁹⁹ rather than sp³. The account is not wrong about the direction; it is wrong that the hybridisation is a fixed starting point which something then distorts. The s character is whatever the angle says it is, and the two lone pairs take the remaining 0.5995 of the one s orbital — 0.2998 each, which is sp²·³⁴.

Tested in The angle does not fix the hybridisation · the series on hybrids

“A force field that reproduces every observed frequency is the right force field.”

Every member of the computed family reproduces every observed frequency of H₂O. They differ by up to 1.1% on D₂O, which none of them was fitted to. Reproducing the data a model was fitted to is not evidence about the model, and the family exists precisely to make that measurable.

Tested in The force field is not in the spectrum · the series on normal mode

“The band gap is what an infrared or optical measurement returns.”

The measurement returns the lowest energy at which an allowed transition occurs, which equals the gap only when the states either side of it are connected by a non-zero transition moment. A forbidden edge transition puts the measured onset above the true gap, and the difference is why some materials have an optical gap larger than the gap deduced from conduction.

Tested in The gap is not the band width · the series on Peierls distortion

“Which kind of top a molecule is has to be worked out from its moments.”

The moments settle it, and so does the point group independently: an axis of order three or higher makes two perpendicular moments equal, because the inertia tensor is invariant under the group and a threefold rotation admits no other possibility. Both are computed here for every molecule in the collection and are required to agree.

Tested in The rotational spectrum is a moment of inertia · the series on rotation

“A rotational spectrum gives the structure of a molecule.”

It gives one moment of inertia per isotopologue, and a molecule with n independent geometric parameters needs n of them. A linear triatomic has two bond lengths and one moment; the classic inversion uses two isotopologues, and the three available pairs of carbonyl sulfide give C–O lengths of 1.16272, 1.15533 and 1.16246 ångström — a spread of 0.0074, which is far larger than the measurement error in any of them.

Tested in A bond length out of a spectrum · the series on rotation

“Linear molecules are a special case that group theory cannot handle.”

It handles them; the reduction formula does not, because it divides by the group order. Everything that does not need the order — which operations exist, what a basis's character is, which transitions vanish — works in the infinite group directly. Only the reduction has to be moved into a finite subgroup, and the count it produces is checked against 3N−5 there.

Tested in An infinite group, worked in a finite one · the series on point group

“Metals are elements that give up electrons easily.”

That describes what they do in a reaction, not why they are metals. The structural statement is that an element with too few valence electrons has no arrangement of bonds that closes a shell — sodium has one electron and would need seven bonds — so it is left with partly filled levels whatever structure it adopts. The metallic state is what remains when the counting cannot be satisfied.

Tested in Counting electrons in an extended structure · the series on metal

“Whether a molecule is polar is a property of the molecule, so it can be read off the formula and the bond polarities.”

Hydrogen peroxide has the same two O–H bonds at every dihedral angle and the same O–O bond, and it may be polar at every angle except 180°, where it may not. The forbidding operation is an inversion centre that exists only at that one value of one coordinate. Nothing in the formula or the bond list changes across the sweep, and the answer does.

Tested in One coordinate, three point groups · the series on dipole

“Helium forms no molecule because the bonding and antibonding contributions cancel.”

They do not cancel; they add to a positive number. Four electrons in two levels at S = 0.25 come out 1.067|β| above where they started, so He₂ is not merely unbound but actively repulsive — which is what a cancellation would not predict.

Tested in The antibonding level goes up more · the series on overlap

“Strong-field ligands give low-spin complexes and weak-field ligands give high-spin ones.”

Only for four of the eleven fillings. Running the two filling rules over d⁰ to d¹⁰ shows that d¹, d², d³, d⁸, d⁹ and d¹⁰ give the same unpaired count either way, so no ligand can change their spin state — and the four that can be changed, d⁴ to d⁷, switch at Δ = P exactly, which makes the pairing energy as much a part of the prediction as the ligand is.

Tested in The pairing energy decides the moment · the series on magnetism

“A three-centre two-electron bond is a weaker version of an ordinary bond.”

Its total bond order is 2, the same as a two-centre pair's, distributed as 0.7071 over each of two links rather than 1.0000 over one. The pair is not doing less work; it is doing the same work across more links, and the individual links are correspondingly less able to resist being broken one at a time.

Tested in What one pair can hold together · the series on multicentre

“Hückel theory gives the delocalisation energy of pyridine.”

A delocalisation energy is a difference from a stated reference, and the reference for pyridine has to include the nitrogen's own α shift — a fitted parameter. The difference would then be a difference between two calibrations. Benzene's exactly 2β survives because nothing in benzene's matrix was fitted.

Tested in Hückel with a heteroatom · the series on delocalisation

“Every bond in SF₆ is a normal two-electron bond, so the molecule has twelve bonding electrons in six bonds.”

Twelve electrons occupy four bonding orbitals and two non-bonding ones, so eight electrons are doing the bonding across six bonds — a σ bond order of two thirds. PF₅ gives four bonding orbitals for five bonds, four fifths; XeF₄ gives three for four, three quarters. The series runs the way the electron-rich picture predicts and the way an octet-expansion picture does not.

Tested in Six bonds and four orbitals · the series on hypervalency

“The 4n+2 rule is a property of a ring of that many electrons.”

It is a property of a ring whose orbitals are all in phase around the loop. Reverse one interaction — half a turn in the ribbon of p orbitals — and the eigenvalues become 2cos((2k+1)π/n), which are degenerate pairs from the bottom up rather than a single lowest level and then pairs. The closure count is then 4n, and the two rules are required to disagree at every ring size and every electron count they are both defined for.

Tested in The ring with a twist in it · the series on aromaticity

“Cyclobutadiene is antiaromatic.”

Flat, yes: its four electrons give a π energy of exactly 4.000β, which is the same as two isolated double bonds, so the delocalisation energy is exactly zero. Twisted, the same four electrons in the same four orbitals give 5.657β and a delocalisation energy of 1.657β — larger than benzene's per electron. Antiaromaticity is a statement about a ring with a particular topology and not about a count of atoms.

Tested in The ring with a twist in it · the series on aromaticity

“A molecule's quadrupole moment is a property of the molecule, like its dipole.”

Only when the dipole vanishes. Moving the origin by 1.6 bohr leaves carbon dioxide's and benzene's quadrupoles unchanged to every digit and changes water's from 0.182 to 0.291 in the same units. The rule is that the lowest non-vanishing moment is origin-free and every moment above it is not, so a polar molecule has no quadrupole moment until an origin is named.

Tested in What a dipole cannot tell apart · the series on dipole

“The number of photoelectron bands equals the number of occupied orbitals.”

It equals the number of distinct ion states the molecule's symmetry permits, which is smaller wherever a species is degenerate: methane has four occupied valence orbitals in two symmetry species, a₁ and t₂, and shows two bands in a 1:3 intensity ratio rather than four.

Tested in What a photoelectron spectrum measures · the series on photoelectron

“The 2s orbital lies below the 2p because it is less shielded, so it feels a larger effective nuclear charge.”

Slater's rules place the 2s and 2p in a single screening group, so the effective charge they give is identical: 2.60 at boron, 3.25 at carbon, 3.90 at nitrogen, 4.55 at oxygen, 5.20 at fluorine, 5.85 at neon. The same holds for 3s against 3p from silicon to argon. The sentence names a real effect and attributes it to a quantity the standard model of it computes as equal — the difference lives in the shape of the radial function, not in a screening constant.

Tested in What the screening model cannot see · the series on orbital

“Boron hydrides are electron deficient because there are not enough electrons for two-centre bonds.”

True as a count and it does not explain the regularity. B₆H₆²⁻ has fourteen skeletal electrons for twelve edges, so a two-centre picture is short by ten; but every closo ion from five vertices to twelve has exactly n+1 pairs and no other number, which is a pattern a shortage does not predict. The n+1 is a count of bonding combinations, and the +1 is forced by the cage being connected at all.

Tested in A cage needs one pair more than it has corners · the series on multicentre

“The arrangement that minimises repulsion on a sphere is the borane deltahedron.”

For five, six, seven, nine, ten, eleven and twelve vertices it is. For eight it is not: the repulsion minimum is a square antiprism, which has two square faces and is not a deltahedron at all, while B₈H₈²⁻ is a dodecahedron. The cage builder refuses eight rather than accepting a polyhedron whose shortest non-edge is exactly as long as its longest edge.

Tested in A cage needs one pair more than it has corners · the series on multicentre

“Since filled shells are spherical, orbital pictures are of no use for real atoms.”

A shell one function short is not spherical, and the departure is large. Remove the dz2 from a d shell and the angular density runs from 0 to 0.398 across directions — a spread as large as the whole constant. Every ligand-field argument starts from exactly that: an incompletely filled shell whose density has a shape, and whose shape is what the ligands push on.

Tested in A filled shell has no shape · the series on contour

“A molecule's point group is a property of where its atoms are.”

The nuclei of CH₄ and CH₃D are in identical positions and their groups are Td and C3v. An operation belongs to the group only if it permutes identical atoms among themselves, and two atoms of different mass are not identical for that purpose. The positions fix the candidate operations; the masses decide which of them survive.

Tested in A spectrum that changes when only a mass does · the series on spectrum

“A distortion can lower a molecule's symmetry to whatever is left after the atoms move.”

What is left has to be closed under multiplication, because the product of two operations a structure has is an operation it has. Methane's twenty-four operations have 2²⁴ subsets and exactly thirty of them are groups. Removing the three S₄ axes and keeping everything else, for instance, leaves a set whose products put them straight back.

Tested in Every group a molecule can fall to · the series on point group

“Water's oxygen is sp³ hybridised, with two bonds and two lone pairs in four equivalent orbitals.”

Two hybrids each holding a quarter s character are orthogonal at exactly 109.47°, and water's bonds are at 104.5°. Built at the measured angle with a quarter s character apiece they overlap by 0.062 — small, but not zero, and a set of four such orbitals is not a basis. Either the label or the angle has to give, and the arithmetic that relates them is one line.

Tested in Hybrids that were never orthogonal · the series on hybrids

“Square-planar complexes are sixteen-electron because they have four ligands instead of six.”

Tetrahedral complexes have four ligands and obey eighteen. The count is not the number of ligands but the number of metal orbitals that end up filled: four sigma combinations take four of the nine in both geometries, and the difference is that a square plane leaves the metal p perpendicular to the plane unmatched and far too high to occupy, while a tetrahedron leaves nothing out of reach.

Tested in Sixteen is also a count · the series on electron count

“A clean surface with no impurities has no states in the gap.”

It has none in this model unless the surface site's energy differs from the bulk by more than a β, which is a strong condition. Real surfaces usually do satisfy it, because a surface atom's missing neighbour changes its energy substantially and because surfaces reconstruct; the point is that the existence of surface states is a quantitative question with a computable threshold rather than something a missing bond guarantees.

Tested in The end is the hardest place to bind · the series on defect

“The photoelectron spectrum proves the lone pairs are inequivalent, so the hybrid description is wrong.”

The spectrum measures the states of the ion, and those have definite symmetry because the ion is a molecule with a point group. The neutral molecule's density is identical in either description — the localisation transformation changes no observable. What is refuted is the claim that the two lone pairs are separately observable, not the use of localised orbitals as a basis.

Tested in Water's lone pairs are not a pair · the series on photoelectron

“One of water's lone pairs is in a p orbital and the other in an sp³ hybrid, so they differ.”

That is a better description and it is still a description of a basis. What the group settles is that no two of water's four valence orbitals can be equivalent: C₂ᵥ has four one-dimensional representations and no degenerate one at all, so every occupied orbital has a distinct symmetry label and no pair of them can be exchanged by any operation.

Tested in Water's lone pairs are not a pair · the series on photoelectron

“A HOMO–LUMO gap and a band gap are the same quantity at different sizes.”

They are the same *definition* — the distance from the highest occupied level to the lowest empty one — and they behave completely differently as the system grows. A half-filled chain's HOMO–LUMO gap falls as 1/n toward zero, measured at 0.299, 0.153, 0.078 and 0.039 for chains of 20, 40, 80 and 160. A band gap is the limit that sequence approaches, and for a uniform chain that limit is nothing at all.

Tested in A band gap is not a bond energy · the series on Bands in a solid

“The orbitals that overlap best form the strongest bonds.”

Stabilisation goes as the square of the coupling divided by the energy gap between the two orbitals. Taking a coupling of 1.5 with a gap of 8 against a coupling of 1.0 with a gap of 2 gives 0.272 against 0.414: the larger coupling loses by a third, because the gap moved by a factor of four and the coupling only by a half.

Tested in Overlap is not interaction · the series on overlap

“Cyclopropane is strained because its 60° bond angles are far from the 109.5° a tetrahedral carbon prefers.”

True as far as it goes and it omits why nothing can be done about it. Three equal bonds meeting at equal angles are an equilateral triangle and nothing else — the closure search refuses every angle but 60°, with a residual of 0.05 two degrees away in either direction. The ceiling is not a preference the ring fails to reach; it is the only geometry the ring has.

Tested in The angle a ring cannot have · the series on strain

“A molecular orbital is built from the atomic orbitals of the atoms in the molecule.”

It is built from functions of the same shape, at a size the molecule chooses. Holding the exponent at hydrogen's own value gives a bond of 2.49 bohr against the exact 2.00; letting it vary gives 2.00, at an exponent of 1.238. The orbital that goes into the bond is a quarter smaller than the one the free atom has, and the difference is the whole error in the bond length.

Tested in The atom does not bring its own orbital · the series on orbital

“As the repulsion grows, the electrons localise and the double occupancy goes to zero.”

Both halves are right and they are not the same statement. Double occupancy falls from 0.250 to 0.002 by U = 32, and the neighbouring spin correlation deepens from −0.125 to −0.246 — but the local moment rises over the same range from 0.125 to 0.246, and it is the moment rather than the double occupancy that says how much of a spin each site is carrying. Three quantities from one wavefunction, moving at different rates.

Tested in The hole that is not repulsion · the series on correlation

“Whether a molecule is planar is a structural fact that has to be determined by fitting a structure to the data.”

Three moments of inertia settle it by an identity: for a rigid planar body Ic = Ia + Ib exactly, because every atom's squared distance from the perpendicular axis is the sum of its squared distances from the two in-plane ones. Computed here the defect is 0 to 10⁻¹⁴ for water, sulfur dioxide, formaldehyde, boron trifluoride, benzene, xenon tetrafluoride and ethene, and −0.76, −3.18 and −1.08 for ammonia, methane and hydrogen peroxide. There is nothing between the two groups.

Tested in The moment that is the sum of the other two · the series on rotation

“A molecule with two nearly equal moments of inertia is nearly a symmetric top, so its spectrum is nearly evenly spaced.”

Hydrogen peroxide's two larger moments differ by three per cent and its asymmetry parameter is −0.994, which is very near the prolate limit; water's differ by fifty-three per cent and κ is −0.432, near the middle. The parameter that decides the spectrum is built from all three moments as (2B − A − C)/(A − C), and two moments being close matters only relative to the spread of the three.

Tested in The moment that is the sum of the other two · the series on rotation

“A one-electron model gives the wrong energy because it neglects a small correction.”

What it neglects is not small and is not a correction to an energy. In an independent-electron ground state two electrons are found on the same site a quarter of the time at half filling, whatever the repulsion — that is what independence means. The exact ground state has 0.0023 at U = 32, so the electrons' positions have stopped being independent, and no adjustment to the orbital energies can produce that.

Tested in The smallest many-electron calculation · the series on correlation

“The spin-only formula fails for cobalt(II) because its field is weak.”

The group decides which ions are permitted a contribution at all, and it permits exactly the high-spin configurations d¹, d², d⁶ and d⁷ — those whose t₂g set is partly filled. Of nine measured moments, the two furthest above spin-only are Co²⁺ at +0.93 and Fe²⁺ at +0.50, and both are permitted; the largest departure among the forbidden ones is Ni²⁺ at +0.37. But permission is not prediction: Ti³⁺ is permitted and measures exactly spin-only.

Tested in An orbital carries no angular momentum · the series on magnetism

“Three-centre four-electron bonding explains how sulfur can have six bonds.”

It explains it at a price the account rarely states. The second occupied level of a three-centre system has exactly zero amplitude on the middle atom, by symmetry, so its two electrons sit entirely on the ends: each end carries −0.5 and the middle +1.0 even with all three atoms identical. Three such systems put +3 on the central atom of an SF₆, and the arrangement is available only where the ligands will take it.

Tested in Hypervalency is about the ligands · the series on hypervalency

“The counting rule — 2n − 3 freedoms against n + 3 conditions — gives the number of shapes a ring has.”

It gives the generic answer and is measured to be right for seven, eight and nine, where the families come out at dimension 1, 2 and 3. For six it predicts 0 and both answers are found: an isolated rigid chair with family 0 and a loop with family 1. A count of freedoms against conditions cannot see that a solution set has two pieces of different dimension.

Tested in The ring that cannot hold still · the series on strain

“A strong-field ligand is one that forms a strong bond to the metal.”

The splitting is the separation between two sets of metal orbitals, not the metal–ligand bond energy. Carbon monoxide splits a d shell harder than any common ligand and its contribution comes substantially from accepting electron density into an empty π* orbital, which lowers the lower set rather than raising the upper. Δ = 3eσ − 4eπ: a ligand can raise Δ by 4eπ without changing its σ bonding at all.

Tested in The spectrochemical series is not electrostatics · the series on ligand field

“A molecule has a point group, which can be read off its structure.”

Only an exact structure does. Benzene with its ring alternately in and out by 0.02 Å is assigned D3h at a tolerance of 0.04 Å and D6h at 0.08 Å; bent out of plane by the same amount it is C6v at 0.02 and D6h at 0.04; stretched along one axis it is D2h, then C6h, then D6h. All five verdicts are correct answers to the question actually asked, and the question contains a number nobody states.

Tested in The tolerance is a decision · the series on point group

“A big enough Gaussian basis reproduces the orbital.”

It reproduces the energy: one to six Gaussians give −0.42441, −0.48581, −0.49698, −0.49928, −0.49981 and −0.49995 hartree against an exact −0.5, converging from above as a variational method must. It does not reproduce the shape at either end. The slope at the nucleus is exactly zero for every finite sum of Gaussians and should be −1; at eight bohr the best six-function fit has 58 per cent of the amplitude it should, against 0.002 per cent for one function — better, and still the wrong functional form.

Tested in A Gaussian is the wrong shape · the series on basis

“The double hump in the hydration enthalpies shows that the ligand field stabilisation is large.”

It shows that the stabilisation varies with the d count, which is a different statement. The stabilisation itself is at its largest for d³ and d⁸ and exactly zero for d⁰, d⁵ high spin and d¹⁰ — so what the humps measure is the DIFFERENCE across the series, not the size. Removing it takes the fit of a straight line from R² = 0.732 to 0.965.

Tested in The double hump and what removes it · the series on ligand field

“A closed shell means a molecule is stable against distortion.”

It means the flat geometry is a stationary point, which is a weaker statement. The measured exponent of the gain is 1.996 for benzene and 1.989 for a ten-ring — second order, so there is no first-order push — against 1.000 for cyclobutadiene and 1.110 for a twelve-atom chain, which have no choice about distorting. A stationary point survives only if something else resists, and whether it does is a quantitative question.

Tested in The hexagon is the frame's doing · the series on delocalisation

“Band theory is a good approximation that gets the gap slightly wrong for strongly correlated materials.”

The two answers differ by the whole of the result rather than by a correction. Band theory returns zero for this system at every U, and the exact gap grows without bound, so no adjustment of the band-theory parameters produces the right answer — the quantity band theory computes is a difference of one-electron levels and the charge gap is a difference of three many-electron energies.

Tested in The insulator band theory cannot see · the series on metal

“Ammonia has a large dipole and nitrogen trifluoride a small one because the lone pair opposes the bonds in one and reinforces them in the other.”

The first half is right and computable: the bond sums are +0.888 D for ammonia and −1.713 D for the trifluoride, pointing in opposite directions along the threefold axis, because hydrogen is less electronegative than nitrogen and fluorine is more. The second half requires the lone pair to be worth 0.582 D in ammonia and at least 1.478 D in the trifluoride, and it is the same lone pair on the same atom.

Tested in The lone pair is not the missing term · the series on dipole

“Reducing a representation tells you what the symmetry-adapted orbitals are.”

It gives multiplicities and nothing else. Benzene's six π functions span B2g ⊕ E1g ⊕ A2u ⊕ E2u, which is four numbers; the orbitals themselves come from projection operators built from the same characters and the operation matrices, which the reduction throws away. For the four one-dimensional entries the projector fixes the combination completely; for the two-dimensional ones it does not.

Tested in The projector is unique, the basis is not · the series on representation

“Aromaticity is a property of a molecule: benzene is aromatic, cyclobutadiene is not.”

The cyclopropenyl skeleton gives a delocalisation energy of 2β with two π electrons, 1β with three and exactly 0 with four, and the same three carbons carry all three. The cyclopentadienyl skeleton gives 2.472β with six electrons and 1.236β with four, the second having two unpaired electrons. The skeleton does not decide it; the skeleton and the count together do.

Tested in The same ring, three charges · the series on aromaticity

“A molecule of N atoms has 3N−6 vibrations because three coordinates describe its position and three its orientation.”

That is the counting argument and it is right; it is also not a computation. Diagonalising the mass-weighted Hessian of water gives six eigenvalues at 10⁻⁷ of the largest and three that are not, and the six eigenvectors lie entirely inside the six-dimensional space spanned by three mass-weighted translations and three infinitesimal rotations written down from the geometry — with the largest leak of any vibration into that space being 10⁻⁹.

Tested in The six motions that are not modes · the series on normal mode

“A linear molecule has 3N−5 vibrations because it has only two rotational degrees of freedom.”

The reason is visible in the construction rather than needing to be quoted. The rotation vector about an axis is r cross the axis direction, and for atoms lying on that axis the cross product is identically zero at every atom. So the sixth rigid vector is not a small vector; it is the zero vector, and the rigid space has rank five. Carbon dioxide's rigid motions span five dimensions and it has four vibrations, not three.

Tested in The six motions that are not modes · the series on normal mode

“The electronegativity scales correlate well, so which one is used does not matter.”

The rank correlations run from 0.944 to 0.995, so the scales agree about order. Normalised to each scale's own range, the N–H bond's difference is 0.266, 0.015, 0.273 and 0.222 — a factor of eighteen between the largest and the smallest — and the P–H bond's is 0.003, 0.195, 0.044 and 0.014, a factor of sixty-two. Any use that needs a magnitude is using the part they disagree about.

Tested in A ranking is not a difference · the series on electronegativity

“Mutual exclusion is a statement about parity.”

It is where there is a parity to speak of. In each of the eight centrosymmetric groups here, every representation carrying a coordinate has character −d under inversion and every one carrying a quadratic has +d, which is the whole proof and is checked for all of them. In D5h there is no inversion class at all, so no representation is even or odd under anything, and the exclusion holds for a different reason entirely.

Tested in Mutual exclusion does not prove a centre · the series on spectrum

“Methane's symmetry group is the twenty-four permutations of its hydrogens.”

There are 24 permutations and each can be taken with or without the inversion of all coordinates, giving 48 conceivable operations. Exactly half are symmetry operations, and the half is selected by a rule nobody puts in: an even permutation comes with no inversion and an odd one comes with one. Closing the group on two generators reproduces all 24 and every one of them obeys it.

Tested in The group of a molecule that will not hold still · the series on point group

“A defect level is a property of the defect, so it sits at the same energy however many defects there are.”

The mean of the pair's two levels converges on the isolated value −2.828427β = −2√2 as they separate, and departs from it when they are close. At two sites apart the two levels are −2.500 and −3.035, neither of them the isolated value. A defect level is a property of a defect and its neighbourhood, and another defect is part of the neighbourhood.

Tested in Two defects, and the level between them · the series on defect

“Correlation energy measures how badly a single determinant describes a state.”

In the two-site model at strong repulsion the exact energy tends to zero and the single-determinant energy tends to +U/2, so the correlation energy grows without limit — while the natural occupations converge on 1 and 1, which is a state no single determinant has at all. The energy error and the description error do not track each other; the first is unbounded and the second saturates.

Tested in Two kinds of correlation, and only one is small · the series on correlation

“Molecular orbital theory and valence bond theory are alternative descriptions that give the same answer when carried far enough.”

They are two vectors in one plane and the exact state is a third. The exact ground state of a two-site model lies in the span of the two functions to twelve decimal places at every repulsion, and its overlap with each is less than one everywhere except at the ends: 1.000 with the molecular-orbital function at U = 0, and approaching the valence-bond function only as U goes to infinity. At U = 4t the two overlaps are equal, both 0.9239.

Tested in Two pictures, one plane · the series on models

“The variational principle says a lower energy is a better wavefunction.”

It says a lower energy is closer to the ground-state energy, which is a statement about one number. It gives no bound at all on any other property, and the second-order behaviour of the energy is exactly why: the same insensitivity that makes the energy easy to get nearly right makes it uninformative about the rest of the state. Two trial functions with the same energy can differ in every other expectation value.

Tested in A better energy is not a better answer · the series on approximation

“A ninety per cent orbital picture shows ninety per cent of the electron.”

The surface encloses 90.0 per cent of the density in space. The same contour drawn in the plane the picture is printed in encloses 96.9 per cent of the density in that plane, for a 1s, and between 95.4 and 96.9 per cent for every hydrogenic orbital tested. Both numbers are correct statements about the same curve; the caption names neither.

Tested in A slice is not the surface · the series on contour

“A four-centre bond is a pair divided into four.”

Fifty-nine per cent of each localised pair sits on its carbon and about twelve per cent on each of the three metals, with three per cent leaking to the fourth. The participation number is 2.54 rather than 4, so the orbital spans four atoms without being spread evenly over them, and an electron-deficient bond is mostly on the electronegative atom.

Tested in Four centres, and the pair that will not localise · the series on multicentre

“Trans-cyclooctene is strained because its ring is too small for a double bond.”

At ordinary bond lengths and angles the eight-ring will close on a trans double bond only if the bond is twisted to 139.3°, against the 180° a flat double bond wants. The measured torsion in trans-cyclooctene is 136°. That is a model with bond lengths and bond angles in it and no energy anywhere agreeing with a crystal structure to a few degrees.

Tested in The double bond a ring cannot hold · the series on strain

“The g-value is a property of the metal ion.”

Using the free-ion spin-orbit constant, the computed shifts exceed the measured ones by factors of 0.93 for copper(II) and 0.72 for vanadium(IV) along the axis. What is left over is the orbital reduction factor: the electron spends part of its time on the ligands, where the metal's own spin-orbit coupling does not reach. A covalency, measured with a magnet.

Tested in The g-value is the orbital coming back · the series on magnetism

“A rotational spectrum gives the bond length.”

It gives a bond length that depends on which lines were used. For carbon monoxide, the spacing between the first two lines corresponds to 1.130913 Å and the spacing at J = 40 to 1.149966 Å — seventeen parts per thousand apart, moving one way all the way up the spectrum. The lowest lines give the equilibrium value back to two parts in a hundred thousand, which is why the defect stays hidden.

Tested in The rotor that stretches · the series on rotation

“The three-centre four-electron bond is a special arrangement peculiar to hypervalent molecules.”

A chain of m centres holding m+1 electrons is the same situation for every odd m. At three the ends carry 1.5 electrons each and the two bonds have order 0.707; at five the ends carry 1.333, the outer bonds 0.789 and the inner ones 0.577; at seven and nine the pattern continues with the alternation growing. Nothing about the calculation is specific to three.

Tested in Hypervalency does not stop at three centres · the series on hypervalency

“Koopmans' theorem works because relaxation and correlation cancel.”

They do have opposite signs, which is checked at every repulsion here, and the cancellation is neither exact nor stable. At U = t the two are +0.010 and −0.057 and the residue is −0.047; at U = 4t they are +0.337 and −1.049 and the residue is −0.712; at U = 8t they are +1.764 and −4.055 and the residue is −2.291. The two errors grow at different rates and the leftover grows with them.

Tested in Koopmans' theorem is exact for nothing · the series on photoelectron

“The cohesive energy goes as the square root of the coordination number.”

It is a second-moment argument that assumes the band's shape is fixed and only its width scales. Measured, the exponent between two and four neighbours is 0.341 and between four and six is 0.507. The square root is right asymptotically and wrong for the chain, whose density of states piles up at both band edges and binds more at half filling than its moment alone would allow.

Tested in The bond that weakens as neighbours multiply · the series on cohesion

“A complex is characterised by its ligand-field splitting.”

Ordering seven chromium(III) complexes by Δ gives F⁻ < urea < H₂O < oxalate < NH₃ < en < CN⁻. Ordering the same seven by how much they reduce the electron-electron repulsion gives F⁻ > H₂O > NH₃ > urea > en > oxalate > CN⁻. The two orderings are different — urea and oxalate move several places — so the two parameters are not one quantity read twice.

Tested in The electrons repel less inside the complex · the series on ligand field

“A normal mode is never a bond stretch.”

In H₂O and D₂O the two stretching modes are 50.0 per cent in each O–H coordinate, so neither belongs to a bond. In HOD, with the same force field and the same geometry, one mode is 99.5 per cent in the O–H coordinate and the other 99.7 per cent in the O–D. The statement is true of symmetric molecules and false as a generality.

Tested in When a mode becomes a bond stretch · the series on normal mode

“Mixing between coordinates is decided by the force field.”

The force field is identical across all three isotopologues — the same four constants, fitted to H₂O and D₂O — and the geometry is identical. What changes is one nuclear mass, and it takes the mixing from exactly half and half to less than one part in two hundred.

Tested in When a mode becomes a bond stretch · the series on normal mode

“The quadrupole-quadrupole interaction falls off as the fifth power of the distance.”

Fitted on a log-log plot of the computed energy between 8 and 34 Å it comes out at 5.00 for benzene and 5.00 for nitrogen, whose moments differ by a factor of six — so the exponent is a property of the moments and not of their size. A pair of dipoles over the same range gives 3.00, and a charge set with no dipole and no quadrupole gives more than 6.5.

Tested in Zero dipole is not no interaction · the series on dipole

“A Gaussian is a poor approximation to an orbital in every respect.”

In one respect it is exactly optimal. The uncertainty product ⟨r²⟩⟨p²⟩ is bounded below by 9/4, and a normalised Gaussian attains 9/4 exactly at every exponent. Hydrogen's ground state sits at 3, a third again above the bound, and every excited orbital sits higher — 7.5 for a 2p, 10.5 for a 2s, 20 for a 3p. The functions every basis set is built from are the minimum-uncertainty functions and the function they are fitted to is not one.

Tested in The orbital in momentum space · the series on orbital

“A larger basis set is a better one.”

Optimised bases of one to six Gaussians reach −0.4244, −0.4858, −0.4970, −0.4993, −0.4998 and −0.49995 hartree, so each is better than the last. Their smallest overlap eigenvalues run 1.000, 0.445, 0.220, 0.114, 0.0615 and 0.0342 — halving with each function added. The two properties move together and only one of them is usually reported.

Tested in A function that is already there · the series on basis

“A magnetic moment counts unpaired electrons.”

For an iron(II) complex whose two spin states are 400 wavenumbers apart the computed moment is 0.300 at 80 K, 2.296 at 200 K and 3.609 at 400 K. The nearest integer count's value at 200 K is 2.83, which is more than half a Bohr magneton away, and no count reproduces the temperature dependence at all.

Tested in A moment between two integers · the series on magnetism

“The state lower in energy is the one observed.”

At 400 K the high-spin form is 54 per cent of the sample although it is the higher of the two in energy, because there are five of it and one of the low-spin form. At a temperature far above the gap the populations settle at five to one, which is the degeneracy alone and has nothing to do with the energies.

Tested in A moment between two integers · the series on magnetism

“Correlation energy is second order in the interaction for a weak interaction.”

For every closed-shell system tried the exponent tends to two as the repulsion vanishes. For a half-filled ring of four it is 1.0080 at a repulsion of one fiftieth of the hopping and never exceeds 1.36 at any repulsion measured. A degenerate reference is shifted at first order, and no amount of weakening the interaction recovers second-order behaviour.

Tested in A third kind of correlation · the series on correlation

“The invisible constant is recovered by looking harder.”

Within the rigid rotor it cannot be recovered at all: the K² term is identical in the two levels of every allowed transition and cancels identically. What brings K back is a centrifugal term, which exists only because the rigid model is wrong — so the constant is measurable only through the failure of the model that hides it.

Tested in The constant a spectrum cannot see · the series on rotation

“The degeneracies a system shows are the ones its symmetry group requires.”

The rotation group of a central potential permits degeneracies of 1, 3, 5 and 7 — the dimensions of its representations. Hydrogen's n = 2 shell holds four orbitals at one energy, computed here from ⟨1/r⟩ and the virial theorem to a spread of 1.2×10⁻⁷ hartree. Four is not one of the permitted dimensions, so the group is not the whole symmetry.

Tested in The degeneracy no group predicts · the series on representation

“The bent-bond picture is a rough sketch and the σ–π one is the calculation.”

The bent pair are sp⁵ hybrids — one part s to five parts p exactly — pointing 50.768° off the internuclear axis, 101.537° apart, with a centre of charge 0.235 Å off the plane of the molecule. Every one of those is a computed number, and the picture is as quantitative as the other.

Tested in Two bent bonds, or a σ and a π · the series on hybrids

“A band is described by its width.”

Three structures whose bands are scaled to the same width put 53.6, 23.5 and 12.3 per cent of their levels in the outer thirds of that band. At half filling they supply 0.318, 0.203 and 0.167 of the width in binding per site — a factor of 1.91 between the first and the last, with the width held identical by construction.

Tested in Where the states pile up · the series on Bands in a solid

“A molecule has a bond angle.”

Water's H–O–H coordinate has a root-mean-square spread of 8.9 degrees in the vibrational ground state, methane's H–C–H of 8.4, ammonia's H–N–H of 8.5. Every angle this collection quotes to a tenth of a degree is the centre of a distribution several degrees wide, and the width is computed from the same force field the frequencies come from.

Tested in The atoms are not at the points · the series on approximation

“The electronegativity difference between two atoms fixes the polarity of the bond between them.”

A difference of one moves 0.4472 of an electron at zero repulsion, 0.1783 at U = 2, 0.0637 at 4, 0.0128 at 8 and 0.0019 at 16 — a factor of 242 across the range, with the difference held constant. The scales report the difference and no scale reports the repulsion.

Tested in A difference does not make a transfer · the series on electronegativity

“Whether the bands overlap is a detail of the material.”

It is an inequality between four numbers: the two bands are clear of one another exactly when the separation of their centres exceeds twice the sum of the two couplings, and the gap is then that difference exactly. Nothing about it is approximate, and nothing about it involves the electron count.

Tested in A full band is not an insulator · the series on metal

“Benzene's delocalisation energy is 2β.”

It is 2β against three isolated double bonds. Against the same six carbons and the same six electrons with one bond deleted — hexatriene — it is 1.0121β, because the open chain is already 0.9879β below three ethenes. Both numbers are exact eigenvalue sums and they answer different questions; neither is the delocalisation energy.

Tested in A stabilisation is measured from somewhere · the series on delocalisation

“Cyclobutadiene is not stabilised by delocalisation, and benzene is.”

Against isolated double bonds cyclobutadiene comes out at exactly 0.0000β — neither stabilised nor destabilised, which is not what any chemist means by antiaromatic. Against the open chain it comes out at −0.4721β, destabilised. The word antiaromatic is a claim the first reference cannot express and the second can.

Tested in A stabilisation is measured from somewhere · the series on delocalisation

“Transition metal complexes want eighteen electrons.”

An octahedral level diagram closes twice, at twelve and at eighteen. With the angular overlap parameters that reproduce carbon monoxide's splitting the gap above eighteen is 3.720 eσ and above twelve 2.280; with chloride's the two are 2.360 and 3.640. The rule as usually stated is the first case with the second suppressed.

Tested in The count that is not always eighteen · the series on electron count

“The eighteen-electron rule fails for early transition metals because their d orbitals are too high.”

The parameter that decides it is the ligand's, not the metal's. Every π donor in the spectrochemical series puts the deeper closure at twelve and every π acceptor at eighteen, with the crossover at exactly eπ = 0 — and the ligands early metals are found with are halides and oxides, which are π donors.

Tested in The count that is not always eighteen · the series on electron count

“A one-dimensional chain of equal atoms distorts by alternating its bonds.”

At half filling it does: the alternating pattern is worth 2.479 against 0.407 for the best of the others, at equal elastic cost. At one-third filling the alternating pattern is worth 0.344 and a pattern of period three is worth 1.538. The alternation is not the distortion a chain chooses; it is the distortion a half-filled chain chooses.

Tested in The distortion the filling chooses · the series on Peierls distortion

“The Stark effect is quadratic in the field.”

It is quadratic for every atom whose shell is split and linear for one whose shell is not. The eigenvalues of z inside hydrogen's n = 2 shell are −3, 0, 0 and +3 in units of ea₀, so the shift is first order in the field. For a pair split by 2.1 eV the same calculation is quadratic until 2.7 × 10⁷ volts per centimetre.

Tested in The symmetry that is not a rotation · the series on representation

“An orbital has no dipole moment, because the density is symmetric about the nucleus.”

Each of hydrogen's n = 2 eigenstates in a field is an equal mixture of 2s and 2pz with a computed dipole of 3ea₀ — 7.6 debye, larger than any molecule in this collection. The symmetric density belongs to the states labelled by l, and those are not the states a field picks out.

Tested in The symmetry that is not a rotation · the series on representation

“A bonding orbital piles up density between the nuclei and an antibonding one removes it.”

At two bohr the bonding combination holds 0.5284 of its density between the nuclei and the antibonding one 0.2580 — a factor of 2.05, which is the statement. At eight bohr the two are 0.5043 and 0.4956, within two per cent, because at that separation there is nothing between the nuclei for either of them to differ about.

Tested in A bond is not two atoms overlapping · the series on contour

“The Jahn–Teller effect explains distortions and the second-order version is a refinement of it.”

They are different in kind. The first-order effect is linear in the distortion, so no stiffness whatever holds the symmetric structure — d⁹'s slope computes as linear and d⁶'s as exactly zero. The second-order effect is quadratic, so whether it wins is a comparison of two numbers and a stiff enough molecule keeps its symmetry.

Tested in A distortion needs two states · the series on Peierls distortion

“A mean field fails because the interaction is strong.”

It fails because it cannot move the density. Release its spin symmetry and the same iteration on the same system at U = 16t goes from an error of 12.11 to one of 0.21 — the interaction is exactly as strong and the failure has gone, at the price of a spin density the exact state does not have.

Tested in A mean field cannot get out of the way · the series on correlation

“A missing atom is a very strong impurity.”

An impurity's level leaves the band by an amount that depends on its strength — at h = 0.2, 0.5, 1 and 2 the level sits at 2.0100, 2.0616, 2.2361 and 2.8284. A vacancy's level sits at zero, and stays at zero under every change to the rest of the structure that keeps the colouring. The first is a number that can be tuned and the second is a theorem.

Tested in A vacancy is not an impurity · the series on defect

“Two defects interact more weakly the further apart they are.”

Two impurities split by a factor of 0.41421 for every site of separation. Two vacancies on the same sublattice give two levels at zero to within 3×10⁻¹⁷ whether they are two sites apart or four, and two on opposite sublattices give none at all at any separation. Separation is not what decides it.

Tested in A vacancy is not an impurity · the series on defect

“Choosing a convention is arbitrary, so the percentages are meaningless.”

At zero overlap all three conventions are identical and equal to the squared coefficient, to the last bit a double holds. At λ = 1 — the molecular orbital wavefunction — all three give exactly fifty per cent, because symmetry forces it. The disagreement is confined to the middle, and grows monotonically with the overlap.

Tested in A weight that depends on how it is weighed · the series on models

“A molecule either has a symmetry or it does not.”

No measured structure has any symmetry exactly. What a point-group search reports is whether the largest displacement an operation produces is below a number somebody chose — and at 0.06 Å that number admits structures whose distance from their own ideal group differs by a factor of 56 across six molecules.

Tested in How much symmetry is left · the series on point group

“Two isotopes of the same molecule have the same bond length.”

They have the same potential, so the same equilibrium length: the measured values for HCl and DCl agree to 0.03 mÅ. Their ground-state averages are 1.28985 and 1.28551, differing by 4.34 mÅ, because the heavier molecule has a lower zero-point energy and sits deeper in a well whose walls are not symmetric.

Tested in The bond length that depends on the isotope · the series on approximation

“Cyclohexane is unstrained because its angles can reach the tetrahedral value.”

A flat six-ring has angles of 120°, which is 10.5° from tetrahedral and 25 kJ mol⁻¹ of angle strain, and 72.6 of torsional strain on top of it. The chair reaches 111.5° and staggers every bond: 0.94 of angle strain and 1.41 of torsion, against a measured strain of nothing. Both terms had to be relieved and the angles alone did not do it.

Tested in The strain that is not in the angles · the series on strain

“Rotational energy levels are given by a formula.”

They are for a linear rotor and a symmetric top. For an asymmetric one the Hamiltonian connects K to K ± 2 and is diagonal in no basis chosen in advance: each J is a matrix of side 2J+1 whose eigenvalues are the levels. Water's five levels at J = 2 are 71.08, 80.03, 95.24, 133.61 and 134.85 cm⁻¹, and none of them is an expression in J and K.

Tested in The top that reports all three · the series on rotation

“An asymmetric top's spectrum reports all three of its constants, so nothing is hidden.”

Two levels still cannot see A: the ground state, which is zero whatever the constants are, and the one at B + C. Every other one of the twenty-five up to J = 4 moves when A is changed. So the count is two rather than the symmetric top's one per J, and it does not grow with J.

Tested in The top that reports all three · the series on rotation

“A molecule is either a symmetric top or it is not.”

Ray's parameter is continuous and runs from −1 to +1. Computed from this collection's own coordinates it is −0.4322 for water, −0.9414 for sulfur dioxide and −0.9618 for formaldehyde — all asymmetric, two of them so nearly prolate that a symmetric-top formula fits their low-J levels to within a few per cent.

Tested in The top that reports all three · the series on rotation

“A symmetry measure says how symmetric a structure is.”

It says how far the structure is from one shape, as one number, and it is blind to an entire class of displacements. A breathing distortion of benzene that moves every atom by 0.0200 ångström leaves the point group D6h intact and gives a measure of 1.3 × 10⁻²⁹ per cent — because the measure counts only what breaks the symmetry, which is everything except the totally symmetric part.

Tested in How far, and along which coordinate · the series on point group

“The displacement between a measured structure and an ideal one is the distortion.”

Six of the 3N directions translate and rotate the molecule without deforming it, and a measured structure in an arbitrary frame carries some of all six. In the displacement used here they hold 1.14 per cent of the total, and they are projected out before anything is resolved — because a moved molecule is not a distorted one.

Tested in How far, and along which coordinate · the series on point group

“Impurity levels sit in the gap at an energy characteristic of the impurity.”

They do at low concentration and nowhere else. At one impurity in a hundred and sixty the levels of a ring are all at −3.6056; at one in ten they span from −4.04 to −2.89; at three in ten they span 2.41 and reach to within 0.25 of the host band. The characteristic energy is the dilute limit of a distribution.

Tested in One defect is a level, many are a band · the series on defect

“Ring strain is angle strain plus torsional strain.”

Those two terms are functions of bond angles and dihedrals, and every conformer of one ring at one bond angle has the same value of the first. Twelve conformers of a ten-ring at 111 degrees have torsional energies from 44.5 to 89.0 kilojoules a mole and closest cross-ring approaches from 2.23 to 3.87 ångström, and the two-term account has no coordinate that distinguishes the second.

Tested in The atoms that meet across a ring · the series on strain

“The term can be neglected for small rings.”

It does not exist for small rings, which is a stronger statement. A cycle of n atoms has a pair separated by four or more bonds only when n is at least eight, so for every ring up to seven the two-term account is complete in form. It is missing a term from eight upwards, which is exactly where measured strains stop following the two-term prediction.

Tested in The atoms that meet across a ring · the series on strain

“The Weiss temperature measures the exchange coupling.”

It does at high temperature, which is where the Curie–Weiss law is an expansion. The 300–600 K fit gives −51.4 K against a mean-field −48.0, agreeing to seven per cent; the 80–150 K fit on the same sample gives −292.3, out by a factor of six. The conversion is a high-temperature statement and is used without the qualification.

Tested in The moment a fit invents · the series on magnetism

“A bigger basis gives a better wavefunction.”

It gives a lower energy, and that is the only guarantee. The mean radius of hydrogen is exact to eight decimal places in the one-function basis and 1.42 per cent low in the two-function one — a factor of 3.3 × 10⁵ in the error — while the energy error falls from 7.6 × 10⁻² to 1.4 × 10⁻² hartree over the same step.

Tested in The property that gets worse · the series on basis

“The bigger the electronegativity difference, the more polar the bond.”

True only if the second derivative is the same for both atoms, and it is not. Ranking thirty-five bonds by the difference and by the charge an equalisation model moves puts twenty-nine pairs in opposite orders. B–F has the second largest difference in the list, 6.12 eV, and comes sixth by transferred charge; K–Br is fourth by difference and first by transfer.

Tested in The quantity no scale prints · the series on electronegativity

“The Mulliken scale is a measurement.”

It is half the sum of two measurements, and reconstructing it from those two reproduces the tabulated value for fifteen of eighteen elements to within 0.021 eV. The three that fail — beryllium, magnesium, nitrogen — are exactly the three whose anion is not bound, so whose electron affinity is not a measured quantity at all. Beryllium is out by 0.498 eV.

Tested in The quantity no scale prints · the series on electronegativity

“The correlation energy grows with the repulsion.”

Against a restricted reference it grows without limit — 0.404, 1.481, 4.645, 12.11, 27.82 as U goes 2, 4, 8, 16, 32. Against an unrestricted one it peaks near U = 4 at 0.519 and then falls, reaching 0.107 at U = 32. Two definitions of the same quantity moving in opposite directions on one system.

Tested in The reference decides the correlation · the series on correlation

“An unrestricted reference is simply a better starting point.”

It has a lower energy, which is what variational means, and it is not a state of the system being described. Its ⟨S²⟩ rises from 0 to 1.988 across this range, where a singlet is 0 and a triplet 2, so the reference that reports little correlation is half a triplet — a description of a state the molecule is not in.

Tested in The reference decides the correlation · the series on correlation

“The level spacing going to zero is what makes a metal.”

It is necessary and it is not sufficient. The mean spacing at the band centre falls from 0.124 to 0.031 as the ring grows from fifty sites to two hundred at every disorder tested, disorder of four included — the same factor of four, on rings whose states have gone from covering two thirds of the system to covering a twentieth.

Tested in The third way to be an insulator · the series on metal

“A photoelectron spectrum measures orbital energies.”

It measures band positions and band shapes, and the shapes carry information the positions do not. Nitrogen's three bands sit at 15.58, 16.98 and 18.75 electronvolts and have 1, 5 and 2 resolvable vibrational lines — a count that follows from bond-length changes of +18.7, +77.2 and −23.7 thousandths of an ångström and from nothing about the energies.

Tested in The width of a band is a bond length · the series on photoelectron

“Benzene's delocalisation energy is 2β.”

It is 2β only when the neighbour overlap is set to zero. Across the family it is 2.000β, 1.330β, 0.997β and 0.797β — and 6.100 electronvolts in every case, because against three isolated double bonds the delocalisation energy is exactly twice the gap between the two measured ionisations, whatever the parameters are.

Tested in One spectrum, a line of models · the series on models

“The ninety per cent surface shows where the electron is, so it shows where the chemistry is.”

It shows where the electron is. The share of the overlap integral outside it rises from 8.3 per cent at a bond length to 36.9 per cent at a three-ångström contact and 53.4 per cent at seven bohr. The two statements come apart as the atoms separate, and non-bonded chemistry happens where they have come apart.

Tested in The tenth that is not drawn · the series on contour

“Drawing the contour further out would fix it.”

It would, and at a cost. To hold ninety per cent of the overlap at a three-ångström contact the contour has to enclose 97.28 per cent of the density, at a radius of 3.56 bohr; at seven bohr it takes 98.76 per cent. Those spheres are so large that they intersect everything else on the page, which is why nobody draws them and why no single contour can be a picture of both things.

Tested in The tenth that is not drawn · the series on contour

“A rotational spectrum determines a molecule's structure.”

It determines at most three numbers, and for a planar molecule only two of them are independent because the third moment is the sum of the other two. Formaldehyde has three structural parameters, so the spectrum leaves a curve of solutions: seven of them are computed here, all reproducing A and B exactly, with C=O lengths from 1.000 to 1.300 ångström.

Tested in Three numbers is not a structure · the series on rotation

“Adding an isotopologue adds two more numbers, so two isotopologues settle a three-parameter structure.”

One of the two is not new. Every member of the family predicts exactly the same rotational constant A for the doubly deuterated molecule — 4.7063 — because A is the moment about the C=O axis and depends only on the hydrogens' perpendicular distance, which the fit to the light molecule already fixed. It is B that separates the family.

Tested in Three numbers is not a structure · the series on rotation

“An antiaromatic ring has a large paramagnetic susceptibility.”

An undistorted one has none — not a large one, none. Its π energy has a corner at zero flux, with one-sided slopes differing by exactly 16π sin(πe/2n)/n, so the second derivative that a susceptibility is does not exist. A susceptibility appears only once a distortion opens a gap, and then it is inverse in that gap: a four-ring alternating by 0.01β has −977 against benzene's +59.

Tested in Two rules that share no arithmetic · the series on aromaticity

“The 4n + 2 rule is a rule about counting electrons into shells.”

It can be reached with no shells and no counting. Classifying eighty rings by whether their energy rises or falls when a flux is threaded through them separates exactly the same set, and that calculation contains no aufbau, no degeneracy tolerance and no notion of a closed shell — only a derivative.

Tested in Two rules that share no arithmetic · the series on aromaticity

“Coordination number determines the band width, and the band width determines the binding.”

The first half is exact: the mean of the squared levels is the coordination for any structure whatever, so five four-connected structures have second moments of 4.000000000. The second half is false: their bindings per site are 1.6363, 1.6315, 1.6093, 1.5886 and 1.2756, a spread of twenty-two per cent at identical coordination.

Tested in Two structures with the same neighbours · the series on cohesion

“The fourth moment fixes what the second leaves open.”

It separates the extreme case — the structure with ⟨x⁴⟩ = 48 is the least bound of the five, at 1.2756 against 1.63 for the rest. It does not settle the others: a wrapped six-by-ten lattice and a ring joined at one and seven have ⟨x²⟩ = 4, ⟨x³⟩ = 0 and ⟨x⁴⟩ = 36 alike, and bind at 1.63148 and 1.60929. They first differ at ⟨x⁶⟩.

Tested in Two structures with the same neighbours · the series on cohesion

“The errors cancel, so relative energies are reliable.”

The errors do cancel — from 83.4 per cent at weak repulsion to 96.9 at strong. The residue as a fraction of the answer goes the other way over the same range, from 1.1 per cent to 423, because the quantity being computed shrinks faster than what survives the cancellation does. Both halves are measured on the same pair of systems.

Tested in Two wrong numbers and a right difference · the series on approximation

“The shape of a density of states is set by the dimension.”

Only for hypercubic structures, where the coordination is 2d and cannot be varied separately. Three two-dimensional nets with three, four and six neighbours have normalised fourth moments of 1.667, 2.250 and 2.500, which is nearly the whole range one to three dimensions covers.

Tested in A band becomes a bell curve · the series on Bands in a solid

“A photoelectron band shows the vibrational structure of the ion.”

It shows a few of the modes and none of the others. Under a totally symmetric change of geometry, methane's nine vibrations get Huang–Rhys factors of 0.905 for one and between 10⁻²⁸ and 10⁻³⁵ for the other eight. The band reports one frequency out of four distinct ones the ion has.

Tested in A band is a filter on the modes · the series on photoelectron

“A redundant coordinate set is an inconvenience that a careful choice of coordinates removes.”

Removing methane's tenth angle removes the redundancy and breaks its tetrahedral symmetry, which is worse: the force field then gives symmetry-equivalent coordinates different constants and the computed mode species stop matching the group-theoretic ones. The redundancy is a property of the molecule's shape, not of anybody's choice.

Tested in More coordinates than motions · the series on normal mode

“A localisation returns the number of centres a bond has.”

It returns two numbers that agree only where the sharing is even. On two separate ethenes both are exactly two; on the methyllithium tetramer the count of atoms is four and the participation number is 2.540, because one carbon holds most of each pair and lends it to three lithiums.

Tested in One scale, from two centres to a cage · the series on multicentre

“A localisation has an answer.”

For seven of the eight systems here it has one answer, reached from every one of twenty-four random starting rotations. For the twelve-vertex cage it reaches ten distinct maxima spanning 0.0236 in the functional, against a convergence tolerance of 10⁻¹³ — so those are different answers rather than one answer computed badly, and the best found is a lower bound rather than the maximum.

Tested in One scale, from two centres to a cage · the series on multicentre

“A van der Waals radius is a measurement.”

It is a parameter fitted to close contacts in crystals, and a model with no length in it reproduces all three tested here to within twelve per cent. The repulsion is computed from the orbitals and the attraction from a polarisability and an ionisation energy, both measured; the only fitted constant is the Wolfsberg–Helmholz K, which is dimensionless.

Tested in The radius that was tabulated · the series on contour

“More overlap means more bonding.”

It depends entirely on whether the two orbitals are at the same energy. Two 1s orbitals at a fixed overlap of 0.25 are stabilised by 0.150 hartree and two 2s orbitals by 0.0375, the ratio of their atomic energies exactly; a 1s with a 2s is stabilised by 0.376 at an overlap of 0.10 and by 0.377 at 0.40, a change of 0.4 per cent for a fourfold change in overlap.

Tested in The same overlap, a different bond · the series on overlap

“A σ-overlap argument explains the series.”

It gets nine of the ten pairs in the right order and misses one, and the pair it misses is fluoride against chloride — the two ligands at the π-donor end, which is exactly where a σ-only calculation must fail. It also loses on size to a fitted power law on the donor's effective charge, 1.21 against 1.31, at an exponent of −1.38 that no model predicts.

Tested in The splitting against something structural · the series on ligand field

“Two ligands that split more must overlap more with the metal.”

Chloride's computed σ overlap with chromium is larger than fluoride's, 0.1757 against 0.1651, and chloride splits less: 13,600 wavenumbers against 15,200. The σ term is not the only term, and the missing one has a sign.

Tested in The splitting against something structural · the series on ligand field

“Running a localisation from several starting points and taking the best is enough.”

Twelve starts found seven of the ten. Which of the ten a single run returns is a property of where it started, and the best of a small sample is not the best.

Tested in How many descriptions a cage has · the series on multicentre

“A substitution structure is a measured structure.”

It is a measured structure with every sign supplied by hand. Formaldehyde's two hydrogens sit at b = +0.9348 and −0.9348 and the equations return +0.9348 for both; nothing in the moments distinguishes them, and the mirror image of any atom is an equally good solution.

Tested in The coordinate an isotope reports · the series on rotation

“The substitution method is more accurate than a fit because vibrational errors cancel in the difference.”

The cancellation is real and is worth a factor of 13.2 here, and it is not enough. The equations are driven by the change in a planar moment, which is a thousandth of the moment; a correlated error of one part in ten thousand leaves a coordinate 2.51 times worse than a length read straight off a moment.

Tested in The coordinate an isotope reports · the series on rotation

“A bond order is a robust quantity because it does not depend on the parameters.”

It is immune to four standard frame changes and not to a fifth. Letting each resonance integral depend on the bond order it produces moves butadiene's central bond from 0.4472 to 0.3676 — a fifth of its value — while leaving benzene's six at 0.6667 exactly.

Tested in The frame that was allowed to relax · the series on delocalisation

“A material is a metal or an insulator, and which one is a property of it.”

It is a property of two limits taken in an order. A uniform ring of forty-two at kT = 0.002 has the same carrier count as an alternating one to the last bit a double holds, and a gapped ring at kT = 0.1 has a quarter of a gapless one's — so at either end of the table the two are the same thing.

Tested in The metal a thermometer cannot find · the series on metal

“Fitting the right model instead of a Curie–Weiss law removes the ambiguity.”

It removes it when the model is right. The two-spin expression recovers J = −50.000 and g = 2.0000 from a two-spin sample over every window tried. Applied to a chain of eight it returns between −57.0 and −66.7 depending on the window, and nothing about the fit reports the difference.

Tested in The model is what is fitted · the series on magnetism

“A fitted g factor near 2 confirms a spin-only system.”

It is also what a wrong structural model returns. The chains here give 1.918 to 1.981 — all within the range a paper would call unremarkable — while the coupling they report alongside is out by between 6 and 33 per cent. The g factor is where the mismatch goes, and it goes there quietly.

Tested in The model is what is fitted · the series on magnetism

“An ionic radius is the size of the ion.”

It is one term of a sum. Across ten electrons the tabulated radius encloses 94.38 per cent of the density for Al³⁺ and 99.98 for neon — a spread of five and a half percentage points where three neutral atoms spanned three tenths of one — and the spread is not a property of the ions but of how the sum was split.

Tested in The surface a table draws · the series on contour

“Two bands separated by a gap are two independent one-band problems.”

Only while nothing connects them. With the inter-orbital hopping off, each band's kurtosis is the one-band value to nine decimal places; switch it on and the departure grows as the square of it, with a fitted exponent of 2.2007 over the small-coupling range.

Tested in Two bands, and the shape of each · the series on Bands in a solid

“Group theory settles which bands appear and says nothing about how strong they are.”

It fixes one intensity ratio exactly. The depolarisation ratio of a non-totally-symmetric Raman band is 3γ²/(45a² + 4γ²) with a — the mean polarisability derivative — identically zero by symmetry, so the ratio is three quarters for every such band of every molecule. Computed for eighteen bands across five molecules it is 0.750000000 in all of them, and it does not move when every bond parameter in the model is replaced by an absurd one.

Tested in The one intensity symmetry does fix · the series on spectrum

“A completely polarised Raman line means a very symmetric molecule.”

It means a cubic group. Methane's symmetric stretch has a depolarisation ratio of exactly zero because a cubic group leaves an isotropic quantity no anisotropy to have; boron trifluoride's, which is also totally symmetric and belongs to a group of order twelve, comes out at 0.031 — small, non-zero, and a property of the bond parameters rather than of the group.

Tested in The one intensity symmetry does fix · the series on spectrum

“Band intensities in a photoelectron spectrum count electrons in orbitals.”

The total counts electrons and nothing else does. The sum over every final state is 3.000000000 at every repulsion, exactly, because it is an anticommutator; how that total is distributed among lines is not a count of anything.

Tested in More bands than there are orbitals · the series on photoelectron

“A ghost basis is a bookkeeping device rather than a physical statement.”

It is an overlap. Put the ghost fifty bohr away and the atom's energy is unchanged to better than a nanohartree, so what the correction removes is the reach of the far centre's functions and nothing else.

Tested in The basis the other atom lent · the series on basis

“A displacement resolved into species is a distortion.”

Six of the 3N directions move the molecule without deforming it. Translating a molecule bodily comes back as more than 99.9 per cent rigid, and a decomposition that assigned it to a species would be reporting the orientation of a set of coordinates rather than anything about the structure.

Tested in The coordinate it was already soft along · the series on point group

“The assumption behind a composite method is untestable in practice.”

It is testable wherever the exact answer is available for both the reference and the target, which for a four-site Hubbard ring is a four-hundred-dimensional diagonalisation. Nothing about the shape of the test needs the system to be small.

Tested in The correction that was computed somewhere else · the series on approximation

“A finite chain has two degenerate dimerised structures, which is why a soliton between them is possible.”

Below a hundred and twenty-eight sites it has one. The long-bond-first phase does not pay for its own elastic cost at all: the search returns a distortion of 3 × 10⁻⁹, against 0.13434 for the other phase at ninety-six sites.

Tested in The distortion the ends decide · the series on Peierls distortion

“A ring's tendency to dimerise is a property of its size in the sense that larger rings distort less.”

It is a property of its size modulo four. A ring of eighteen distorts by 0.08304 and one of twenty by 0.13588, and the two families approach one another without meeting, because one is open-shell before it distorts and the other is not.

Tested in The distortion the ends decide · the series on Peierls distortion

“The gem-dialkyl effect is a steric effect and steric effects are about crowding at the reaction centre.”

It is about crowding along the chain. Making each extended rotation cost 3.8 kJ/mol — no more than a gauche interaction — raises the closable population by a factor of 10.8 with no bond angle changed at all.

Tested in The explanation with the wrong sign · the series on strain

“The rotamer account explains the observed accelerations.”

It has a ceiling. Only two of the eight sign patterns of three gauche rotations curl a chain the right way, so the closable population cannot exceed a quarter and the factor cannot exceed 36.5 — which the measured 250-fold for gem-dimethyl already exceeds, and the 11,000-fold for tetramethyl exceeds by three hundred times.

Tested in The explanation with the wrong sign · the series on strain

“The ceiling on the rotamer account is a limitation of the model.”

It is a closed form — ((1+2x)/2x)ʳ with x = exp(−g/kT) — and it therefore predicts. At the measured gauche energy it is 10.99, 36.46, 120.88 and 400.81 for two, three, four and five frozen rotations, and the numerical limit reaches each of them to a part in a thousand.

Tested in A ceiling that rises where the measurements fall · the series on strain

“The satellite intensity growing smoothly means the assignment gets steadily harder.”

The total grows smoothly — 3, 10, 20, 31, 46, 53, 66 per cent — and the distinguishability does not. It holds at a repulsion of 4, where three tenths of the intensity is already satellite, and both tests fail at 6.

Tested in A hundred lines and no way to sort them · the series on photoelectron

“Sanderson's geometric mean is the right rule, and the arithmetic mean is the naive one.”

It is the better of the two — a mean absolute error of 0.0706 eV against 0.1666 across twelve molecules — and it is the closest rule on only four of them, against the harmonic mean's five. Its advantage is that it sits below the arithmetic mean, and so does the exact answer, most of the time.

Tested in A mean that is low rather than right · the series on electronegativity

“A model with more parameters describes the data better, so the comparison between two models is a comparison of their fits.”

Adding the overlap to a Hückel fit of six ionisation energies moves the root-mean-square error from 0.5093 to 0.5029 eV — 12 per cent of the distance to the floor — while the resonance integral moves from −3.55 to −15.03 eV. The fit improves by one part in eighty and the parameter leaves every published range.

Tested in A parameter that never finds a value · the series on models

“A depolarisation ratio off three quarters means the assignment is wrong.”

It can equally mean the molecule is distorted. Stretching one N–H bond of ammonia by 0.1 Å moves a depolarised band from 0.750000 to 0.707613 — a departure of 0.042 — with the assignment unchanged and the force constants unchanged.

Tested in A ratio that squares what it measures · the series on spectrum

“A quantum defect measures how far an atom is from hydrogen.”

It measures how far its energies are. The dipole matrix element between the shell's own states is 3.000000 atomic units at every screening tested — unchanged — so what the screening moves is the splitting and not the coupling, and the crossover field is exactly the splitting divided by twice the dipole.

Tested in How nearly a broken symmetry survives · the series on representation

“Three arrangements agreeing with a rule is evidence for the rule.”

Three arrangements chosen at the extremes of a count will agree with any monotone function of that count. Searching all 1,820 arrangements of four raised sites in sixteen instead of three of them finds the counter-example directly, and the counter-example is not near the extremes.

Tested in The arrangement a count cannot pick · the series on cohesion

“A bent bond is what a double or triple bond has.”

That is one sense of the word — two or three equivalent hybrids none of which lies along the internuclear line. A small ring has bent bonds in a second and unrelated sense: single bonds whose hybrids point 22.75° outside the line joining the two carbons, computed from the measured H–C–H angle and orthogonality alone.

Tested in The hybrids that point outside the bonds · the series on hybrids

“The bending is a small correction that grows smoothly with ring strain.”

It falls very steeply: 22.75° in cyclopropane, 10.82° in cyclobutane, 4.48° in cyclopentane and 0.035° in cyclohexane. The last is a control rather than a result — a six-membered ring's hybrids point along its bonds to better than a twentieth of a degree.

Tested in The hybrids that point outside the bonds · the series on hybrids

“The π channel is the missing piece of an overlap account of the spectrochemical series.”

The integral is not the missing piece. A free two-parameter fit over five chromium complexes removes 19.7 per cent of the one-parameter error; supplying the sign from each ligand's known π character removes 45.0 per cent with the same two integrals. Most of what the second channel is worth is a fact about occupation.

Tested in The integral that cannot count electrons · the series on ligand field

“A composite correction that removes 99 per cent of the low level's error near its reference is doing 99 per cent of the work.”

The share removed is measured against a denominator that itself collapses. At a repulsion of 8 the transfer removes 39.2 per cent at a modulation of 6 and 81.4 per cent at 8 — improving — while the absolute error falls from 0.545 to 0.081 for the opposite reason: the low level has stopped being wrong.

Tested in The warning a cheap calculation gives · the series on approximation

“A correlation energy measures how strongly the electrons are correlated.”

Two rings matched to the same correlation energy per site — −0.084863 for both — differ by a factor of 2.4 in the probability of finding two opposite spins on one site: 0.2873 against 0.1183 of the uncorrelated value. The energy is one number and the correlation is a function.

Tested in Where the electrons are, without subtracting anything · the series on correlation

“Repulsion pushes the electrons apart.”

It moves them next door. What leaves the on-site pair reappears at separation one — the opposite-spin distribution there rises from 1.0000 at no repulsion to 1.5947 at 16 — so the total is conserved and only its arrangement changes.

Tested in Where the electrons are, without subtracting anything · the series on correlation

“Holding the confound fixed settles which of two candidates is responsible.”

Not on four points. Within each charge group the overlap ranks at 0.80 and the softness of the two ions at −1.00, which is a perfect ranking with the opposite sign. Two candidates that both track ionic size cannot be separated by four pairs that differ in size.

Tested in A control that outranked the mechanism · the series on contour

“The overlap repulsion is at least a well-posed test of the shortfall.”

As first computed it was not: the overlap was evaluated at the measured separation and the shortfall is measured against the same separation, so the two halves of the comparison shared an input. Recomputing the overlap at a distance every pair shares gives the same 0.8571, which is what makes the finding survive the objection.

Tested in A control that outranked the mechanism · the series on contour

“A result that changes sign inside a model's range is an artefact of the model's parameter.”

Where it changes sign is: the crossing sits between three and four rings at λ = 0.2 and 0.4 and between four and five at λ = 0.6. That it changes sign is not — every coupling tried gives a monotone falling sequence that passes through zero inside the range.

Tested in An anomaly that is not the first of a series · the series on delocalisation

“The ceiling is sensitive to the gauche energy, which was butane's rather than the tether's.”

It is sensitive, and the amount needed is not available. Solving the ceiling for the energy that reproduces 250 at three rotors gives 5.85 kJ/mol against butane's 3.8 — half again as large, which is not a value a hydrocarbon torsion has.

Tested in An estimate that can be wrong by two · the series on strain

“A parameter that comes back badly determined needs better data.”

It needs data from the right place. Sixteen thousand points from 20 K upward fix the monomer fraction to 0.90 per cent and forty points reaching 2 K fix it to 0.82 — four hundred times the data for no gain — while the temperature-independent term, which lives at the hot end both windows have, is bought back by a factor of ten.

Tested in How many parameters a curve is worth · the series on magnetism

“A redundant coordinate set leaves one flat direction in the force field.”

It leaves nc of them. Since Bᵀr = 0, adding r sᵀ + s rᵀ for any s changes the Hessian by nothing, and the nullity of the map from force constants to Hessians is 10 of 55 for methane and 7 of 28 for boron trifluoride — counted as a rank and agreeing with d·nc − d(d−1)/2.

Tested in Ten directions no frequency can see · the series on normal mode

“Projecting the force constant matrix onto the space orthogonal to the redundancy fixes the problem.”

It picks a point, and which point depends on the metric. Two formable conventions give methane a bend constant of 0.4941 and 0.4118, a stretch–bend constant of 0.3458 and 0.1729 — a factor of two — and a bend–bend constant of exactly zero and −0.0824, with every frequency identical to nine decimal places.

Tested in Ten directions no frequency can see · the series on normal mode

“The end effect is a fixed amount of energy at the two ends, so it has no length in it.”

It is a fixed amount of energy, which is the end-energy result and survives. It also has a length: the excess alternation decays over 1.51 bonds at a gap of 1.150 and 5.29 bonds at a gap of 0.143, going as the gap to the power −0.60 over the measured range.

Tested in The chain distorts hardest where it stops · the series on Peierls distortion

“The substitution method works because the zero-point errors of the parent and the substituted species nearly cancel.”

They cancel by less than assumed. The parent's and the deuterated species' fractional corrections differ by 20 to 65 per cent, so the correlation buys a factor of 2.5 rather than the thirteen an invented five per cent mismatch gave.

Tested in The correction that was invented · the series on rotation

“The identity fixing the mean ligand charge holds for any arrangement.”

It holds where every central orbital has a partner. A planar arrangement leaves the central p perpendicular to the plane unmatched and exactly degenerate with the orphan ligand combinations, and the mean charge comes back at 10⁻¹⁷ instead of the identity's −0.2 — the arithmetic saying it has been asked an ill-posed question.

Tested in Two models that disagree about the shape · the series on hypervalency

“Freezing the correction at one geometry costs only the change in its size.”

It costs the change in the division too. The share moves by more than half its range across the separations computed, so a frozen number is wrong about how much correction there is and wrong about which atom it belongs to.

Tested in A correction that is two functions · the series on basis

“A distortion's symmetry species tells you which vibrations soften it.”

Only where the species appears once, which is nowhere in this census. All seventeen molecules have a species appearing more than once, including water at 2A₁ + B₂ and methane at A₁ + E + 2T₂, and 71.4 per cent of their vibrations on average belong to such a species.

Tested in A label that prices nothing · the series on point group

“A highly symmetric molecule is the safe case for a species argument.”

It is the least bad case rather than a safe one. The two octahedral molecules have the smallest ambiguous share at 40 per cent, and the two with no symmetry at all have 100 — but 40 per cent is still two of every five distortions the label cannot price.

Tested in A label that prices nothing · the series on point group

“A correlation energy measures how strongly the electrons avoid each other.”

It measures how strongly they avoid each other where the interaction is. A Hubbard correlation energy is 100 per cent on-site as an identity — the interaction is exactly zero at every larger separation — so the enhancement at one site away, which is nearly as large as the depletion at zero, contributes nothing to it at all.

Tested in Half of it is given back at one bond · the series on correlation

“A finite ring cannot show a sharp transition, so its exponent cannot mean anything.”

What is computed is where a variational minimum leaves its boundary, which is a property of a free energy analytic in one order parameter, and that is obliged to give one half. It does: the closest pair of reduced temperatures gives 0.5017 to 0.5058 on every ring and stiffness tried.

Tested in The exponent was the window's · the series on metal

“A model with a good correlation against measurement has explained the measurement.”

It has explained at most what it can distinguish. VSEPR assigns NH₃ and PH₃ the same arrangement and they were measured 14.5° apart, out of a range of 15.7° across all four molecules — so 92.4 per cent of the variation is outside anything the model can see, whatever correlation it reports.

Tested in Two systems a model cannot tell apart · the series on models

“A model that gets a quantity right to a tenth of a unit is a good model of that quantity.”

The spin-only moment is right to about a tenth for the first five ions and it assigns Cr³⁺ and Co²⁺ the same number, which were measured 0.940 μB apart — 22.5 per cent of the range the nine ions span. The accuracy and the resolution are different quantities and only the first is usually quoted.

Tested in Two systems a model cannot tell apart · the series on models

“A response that falls with distance has a decay length.”

Only if there is a scale to set one. The bare acene's fitted length is 1.397, 1.669, 1.896, 2.104 and 2.302 rings at four to twelve rings, growing monotonically with no sign of a limit, while its own gap falls from 0.590 to 0.110.

Tested in A reach that has no length · the series on aromaticity

“The disagreement is a matter of the one constant, which could be refitted.”

Refitting it is a refit, and the constant carried over from the noble gases gives a mean error of 0.242 ångström where the radii give 0.183. What a sweep of the constant buys is reported rather than argued about, and it is not a prediction whatever it buys.

Tested in A size a confound cannot supply · the series on contour

“Choosing the best of the four tables would settle the matter.”

The best of them misses the halide series by 0.254 debye in the root mean square and the worst by 0.505, on a series whose largest member is 1.826. A factor of two between the tables sits inside a common error of the same size as the disagreement.

Tested in Four tables and one molecule to disagree about · the series on dipole

“Bent bonds are the localised description of a double bond.”

They are when |Δ| < 2|d|. With the π orbital 74.4 per cent on oxygen, the bent description wins for σ populations from 0.380 to 1.000 and loses below that — so a double bond polarised the wrong way round has σ and π as its localised orbitals, not bananas.

Tested in The angle that does not have to be searched for · the series on hybrids

“The undetermined parameter in a four-parameter susceptibility fit is the temperature-independent term.”

Over the usual window, the free direction is the temperature-independent term divided by the 0.40 power of the monomer fraction, with a largest component of 0.92 rather than 1. Over a window starting at 2 K it *is* nearly the term's own axis; over one starting at 80 K the monomer fraction dominates it instead.

Tested in The product a curve measures · the series on magnetism

“The second root of the cubic's fixed-point equation is either spurious or the ionic solution.”

It is neither, and which it looks like depends on the element. Carbon's sits at −6.396 electrons, which is not a state anything has; lithium's at −0.146, which is inside the range molecules use. What both are is the partner of a collision: the two roots meet at −η/3γ and past that there is no root at all.

Tested in Where a closed form stops being one · the series on electronegativity

“The number of ligand pairs with no central-atom partner is n + L − 4.”

It is, in twenty-three of the twenty-six arrangements computed here. It is not for methane in a square, phosphorus pentafluoride in a plane or sulfur hexafluoride in a plane: those give one, two and three orphans against a formula saying nothing, one and two.

Tested in The square that wastes an orbital · the series on hypervalency

“A length measured on a long enough chain is the chain's physics.”

At an elastic constant whose true length exceeds every chain here, the number returned is a fixed fraction of the chain: 0.0872, 0.0848, 0.0833, 0.0824, 0.0812, 0.0799 and 0.0786 of it at 64, 96, 128, 160, 224, 320 and 448 sites.

Tested in A decay that keeps slowing down · the series on Peierls distortion

“The correction is a property of the vibrations.”

It is the product of a vibrational quantity and a structural one, and across these four the structural one has the wider range: the moments span a factor of 55 and the softnesses a factor of 7.2, in opposite directions.

Tested in An expression for what was a warning · the series on rotation

“A single coefficient hides the differences between the molecules.”

The coefficients are 0.518, 0.647, 0.667 and 0.705 — a factor of 1.36, against a factor of 11.4 in the thing being predicted. What it hides is a real fifth of the answer for water, which has the fewest modes for a mean curvature to average over.

Tested in An expression for what was a warning · the series on rotation

“Each half of a counterpoise correction can be interpolated from a few computed points.”

The heavier centre's can — its second differences are 0.27 per cent of its value. The lighter centre's cannot: its second differences are 7.1 per cent at the median and 588 at the worst, because it is a difference of two energies of order a hartree whose difference is a millionth.

Tested in The half that cannot be computed · the series on basis

“A search that stops finding new answers has found them all.”

It has found them all with a probability its own basin sizes give. A search of 200 starts on this cage has a 63 per cent chance of missing at least one of the fourteen, and one of 480 has 12 per cent — both of which found the right number, and neither of which could have known.

Tested in Where the count stops being an effort · the series on multicentre

“A search that finds many maxima is finding structure in the problem.”

Only if a control finds one. Two isolated double bonds have exactly one localised description and 1,000 starts find exactly one, so the cage's fourteen are the cage's rather than the search's.

Tested in Where the count stops being an effort · the series on multicentre

“No real ligand comes near the threshold.”

Iodide is at eπ = 0.20 eσ, which is 80 per cent of the way to 0.25; bromide is at 72 per cent and chloride at 64. None crosses, so the protection holds for every ligand parameterised here — but the margin is a fifth rather than a factor.

Tested in The orbital a ligand cannot reach · the series on electron count

“Every orbit of internal coordinates contributes one totally symmetric combination.”

Only if the coordinate survives its own stabiliser. Boron trifluoride's out-of-plane coordinate is fixed by the horizontal mirror and sent to minus itself by it, so its orbit contributes nothing: two of its three orbits are symmetric, not three.

Tested in A formula that predicts minus eleven vibrations · the series on point group

“The sign of an overlap integral says which combination is bonding.”

It says so only in a model that ties the interaction to the overlap, which every extended-Hückel model does. The sign here is a difference between two regions — the lobes facing across the gap contribute negatively, the outer lobes positively — so it is set by a cancellation and not by anything happening between the nuclei.

Tested in A bond with nothing in the middle · the series on overlap

“Multiplying by the zero-point amplitude makes the soft coordinates dominate.”

It helps and is not enough. Boron trifluoride's ordering reverses — the soft pair departs 0.031 against 0.012 — and ammonia's does not: the stiff pair still departs 0.046 against 0.020.

Tested in One number was one direction · the series on spectrum

“The sign of the π term is all the denominator accounts for.”

The sign is exactly the sign of the denominator at every ligand in the series, and the denominator also over-accounts for the size — so it carries the sign and more than the whole of the trend, with the overlap required to give some back.

Tested in The gap that would have to be smaller · the series on ligand field

“Weighting a correlation hole with a longer-ranged interaction estimates what that interaction would cost.”

It does while the neighbour term is a perturbation: at V = 1 and V = 2 on a ring of six at U = 8 the weighted and solved give-backs are 10.99 against 10.92 per cent and 21.98 against 21.50. At V = 4 they are 43.97 per cent and −45.57 per cent, which is not a correction to an estimate but the opposite sign.

Tested in The give-back that turned into a saving · the series on correlation

“Defect levels become a defect band when there are enough defects.”

Not enough defects — enough chain. The typical spacing between runs of a given length does not depend on the chain length at all; what does is the closest pair, which is about 1/(Nρ²) apart and shrinks as the chain grows. The onset is where that closest pair's coupling overtakes the mean level spacing.

Tested in The length at which levels become a band · the series on defect

“A four-hundred-site chain is long enough to show what a random alloy does in the gap.”

It is long enough to show isolated box states, which is what a four-hundred-site chain shows and measures correctly. It is shorter than the onset length for every run length computed here, so nothing in it could have shown a band.

Tested in The length at which levels become a band · the series on defect

“What the boundary tracks is a property of the one-electron problem.”

It tracks the amount of correlation the filling admits. The satellite share of the removal weight at U = 64 is 0.21 to 0.33 below half filling and 0.55 to 0.68 at half filling or above, and the systems with no boundary are exactly the ones with the small shares.

Tested in A satellite that never loses its place · the series on photoelectron

“The gapless case was a special case of the gap ordering.”

It survives the change of filling and the ordering does not. Both systems with an exactly zero gap — a ring of four half filled and a ring of six two-thirds filled — have a contrast of 1.0 at every repulsion including the smallest, so a degenerate ground state has no distinguishability rather than a large boundary.

Tested in A satellite that never loses its place · the series on photoelectron

“It decays in the number of fused rings between them instead.”

Better, and not enough. The per-step decay length changes by six per cent between the linear and angular families where the per-unit length changes by twenty-two — but four pairs at two steps in one molecule span a factor of 1.74, which is a third of a step's worth of decay.

Tested in Neither of the two separations · the series on aromaticity

“The six-membered verdict gives way at 312.1 K.”

312.1 K is where the ceiling meets a horizontal line at ten. A rate ratio is not a horizontal line. With the substituted transition state lower by 1, 2, 3 and 4 kJ/mol the crossing moves to 315.7, 321.7, 334.3 and 388.6 K; at 5.308 there is no crossing at any temperature between 150 and 900 K.

Tested in One number decides which way it breaks · the series on strain

“Raising the temperature is what would refute the rotamer account for the six-membered closure.”

Only below 5.308 kJ/mol. Above it the measurement falls faster than the ceiling does, the crossing reappears below room temperature — 273.3 K at 8 kJ/mol, 288.0 K at 12 — and it is cooling that refutes the account. The direction of the test reverses at one number.

Tested in One number decides which way it breaks · the series on strain

“The excess gap divided by a band's shape departure is one constant times the separation of the two bands.”

True to 0.00% across a factor of four in the separation when both bands hop on a square net, and false by 128% on the same range when both hop on a triangular one. The quotient is a property of the net, not of the perturbation theory.

Tested in The constant that belonged to one net · the series on Bands in a solid

“A second measurement of the same kind is the one that fails to buy anything.”

The band shape and the excess gap are measurements of different kinds and invert at a condition number of 6.74 on one graph. On two they invert at 160.5, and one per cent measurements then fix the gap to 45% instead of 1.5%.

Tested in The constant that belonged to one net · the series on Bands in a solid

“Two independent localisations agreeing is evidence that the answer is well defined.”

It is evidence about the system. Two runs agree 95.32 per cent of the time on one of these cases and 3.51 per cent of the time on another, and the two are the same molecule shape with a different electron count.

Tested in Three shapes from one search · the series on multicentre

“The finding therefore does not survive.”

The threshold that misclassifies fewest on all 153 pairs is 0.05065 — the same line, to five figures, that the small set drew. It gets six pairs wrong. Over all 2,192 disputed-against-agreed comparisons the disputed pair has the smaller difference 97.45 per cent of the time. A boundary with a gap and a rule with exceptions are different claims, and the second is supported.

Tested in A gap that was a choice of bonds · the series on dipole

“A molecule adopts the arrangement that minimises the repulsion between its ligands.”

Four of the eight do, exactly, and all four have no lone pairs. The other four sit 4.20, 5.15, 10.52 and 26.47 per cent above it, and all four have lone pairs — which this energy does not count.

Tested in Expensive is not the same as unadopted · the series on hypervalency

“The alternation a Peierls chain settles at goes as exp(−π K/2), up to a constant.”

It is that expression's asymptote. Against the exact stationarity condition the expression is 10.46 per cent low at K = 1.0, 6.49 per cent at 1.2 and 0.0034 per cent at 4.0, and the error falls as the alternation to the power 1.72 — so it is an expansion in the distortion, and it is worst wherever the distortion is large enough to see.

Tested in The amplitude the collapse left behind · the series on metal

“A finite-size error in a lattice calculation is controlled by the number of sites.”

Forty-two rings across seven elastic constants and six sizes fall on one curve when the departure is plotted against the number of sites multiplied by the limiting alternation. A ring of 400 at K = 3.0 and a ring of 40 at K = 2.0 are the same distance from the limit, at ten times the site count.

Tested in The amplitude the collapse left behind · the series on metal

“A relation that holds between a molecule's moments need not hold between their corrections.”

For water it does, exactly: ΔI_c − ΔI_a − ΔI_b comes out at −9 × 10⁻¹⁶ amu Ų, because all three of its vibrations keep it planar. For boron trifluoride it fails by 0.046859 amu Ų, and the molecule has exactly one out-of-plane mode.

Tested in The axis that goes the other way · the series on rotation

“A better basis makes the counterpoise correction less important, so its assembly matters less.”

The correction does collapse — by a factor of 2,717 at a separation of two across the same sweep. But the fraction by which a pairwise assembly gets it wrong rises over the same range, so the regime where the correction is small is the regime where assembling it is proportionally worst.

Tested in The correction that gets harder to assemble · the series on basis

“A structural reach and a magnetic reach are the same length seen twice.”

Once the geometry is held rigid they agree to 0.7 per cent at the widest gap tested and to seven per cent everywhere, and their exponents against the gap are −0.6798 and −0.6803. Let the geometry relax and the structural one is 6 to 16 per cent longer, so the extra length belongs to the feedback rather than to the π system.

Tested in The floor was in the bookkeeping · the series on delocalisation

“Two crossovers in one shell differ because the levels they join differ.”

Two of these three join the same two levels — the p and the d — and have exactly the same gap, to twelve figures. What differs is the dipole: 5.196152 against 4.500000, a ratio of 1.1547007, which is 2/√3.

Tested in Three events, and a ratio of two dipoles · the series on representation

“A localisation over more orbitals is a localisation over more angles.”

It is a larger group, and the difference matters. Two orbitals have one angle and a functional quadratic in it with no linear term; three have a three-dimensional group and eighteen dipole matrix elements, of which the off-diagonal ones in two different components are what make an interior maximum possible.

Tested in An interior maximum a third orbital allows · the series on hybrids

“Running more starts would settle it.”

It would find more descriptions and would not change which is commonest. On the one cage where quality varies, more starts move the reported answer away from the commonest and towards a rarer, better one — so more effort changes the answer rather than confirming it.

Tested in Fifty descriptions of one molecule · the series on multicentre

“The pairwise assembly's error grows with the basis, so a better calculation makes the fragment method worse.”

It grows as a fraction and falls in hartree — by a factor of 793 across the sweep, from 6.27 × 10⁻³ to 7.90 × 10⁻⁶ at the closest separation. Both are true because the correction itself falls by 3,631, and the fraction grows by exactly the ratio of those two shrinks, 4.579.

Tested in One basis size where it is worth doing · the series on basis

“Whether the growing fraction matters depends on the accuracy target, which is a matter of judgement.”

It is decidable once the target is stated. At one kilocalorie a mole the correction is above the line at one and two Gaussians a centre and below it at three and above; the assembly's error is above the line at one and below it at two and above. Two is the only basis where the first holds and the second does, at every separation.

Tested in One basis size where it is worth doing · the series on basis

“The one usable basis size is a robust finding.”

Its assembly error is 1.582 × 10⁻³ hartree against a line at 1.594 × 10⁻³ — under it by 0.74 per cent. Any convention that put chemical accuracy one per cent tighter would leave no usable basis size at the two closest separations at all.

Tested in One basis size where it is worth doing · the series on basis

“A diagnostic that works on one axis of a model works on the other.”

It works, in the sense that the transfer fails exactly where the broken solution has collapsed on both axes. It does not work as one rule: a target at a polarisation difference of 0.12546 removes 96.5 per cent of the low level's error and one at 0.13945 removes 39.2 per cent, a factor of 46 in the error at the same diagnostic value.

Tested in The half of the square a ring of four cannot show · the series on approximation

“The rule of sixteen and the rule of eighteen are rules of the same kind.”

The sixteen-electron gap is between two d levels and closes to exactly zero by the time the coordination reaches six. Whatever a count of eighteen is about at six-coordinate, it is not this gap, because this gap does not exist there.

Tested in The ligand the rule was waiting for · the series on electron count

“The angle between a parent's principal axis and its daughter's is always a meaningful number.”

Not when the parent is a spherical top. Methane's three moments are equal, so every orthogonal triad diagonalises its tensor; the 54.74°, 45.00° and 35.26° a diagonaliser reports against CH₃D are the tetrahedral angles of whatever triad it happened to return.

Tested in Two moments about two different lines · the series on rotation

“Dropping the pairs that touch an end leaves a clean decay.”

It does on the two uniform chains, where the coefficient of determination rises from 0.95129 to 0.98986 and from 0.98954 to 0.99783. On the chain of two straight arms meeting at one angular ring it falls, from 0.84684 to 0.84376, so the scatter there is not at the ends.

Tested in An end effect with two signs · the series on aromaticity

“A pair whose overlap vanishes has the spectrum of two free atoms.”

One of the three levels sits at the free-atom energy exactly, at every third-orbital energy — the antisymmetric combination, which has nothing of its own symmetry to mix with. The other two move by up to twelve electronvolts.

Tested in A bond order between atoms that do not interact · the series on overlap

“A quantity that changes sign on an interval has a zero on that interval.”

The solved give-back on a ring of six at U = 8 changes sign at V = 3.8955 and again at V = 4.1595. Only the first is a zero; at the second the numerator is −4.404 against a scale of 19.18, and it is the denominator that has vanished.

Tested in A sign change is not always a zero · the series on correlation

“The give-back is a fraction of the on-site saving, so it is a fraction.”

The on-site saving it divides by passes through zero at V = 4.1595, where the neighbour repulsion has restored exactly as much double occupancy as the on-site repulsion removed. On either side of that repulsion the denominator has opposite signs, so the same numerator is reported as a give-back and as a gain.

Tested in A sign change is not always a zero · the series on correlation

“V = U/2 is where the two interactions balance on this ring.”

It is the limit of both crossings and neither of them. At U = 6, 8, 10 and 12 the root sits at 0.4741, 0.4869, 0.4934 and 0.4968 of U and the pole at 0.5304, 0.5200, 0.5140 and 0.5103, so U/2 falls strictly between them at every repulsion, closing as 1/U.

Tested in A sign change is not always a zero · the series on correlation

“The gap above eight electrons is a gap between d(z²) and d(x²−y²).”

Only while a symmetry keeps them apart. In C₂ᵥ they share a representation and mix: the purity of the less pure frontier orbital falls to 0.9678 at twenty degrees and 0.7500 at forty-five, so at large fold there are no two named orbitals for the gap to be between.

Tested in The distortion that opens the gap · the series on electron count

“The response is largest on the ring that carries the heteroatom.”

For an interior position two adjacent rings share the peak to within a per cent, because the carbon furthest from the molecule's centre in an interior ring lies on a bond that ring shares with its neighbour. Only at the end does one ring own the perturbed carbon.

Tested in The reach is the molecule's · the series on delocalisation

“The two measured negatives are weak evidence for a systematic sign.”

They are the only evidence there is, and nothing computed here strengthens or weakens them. What is established is that this route cannot add to them: the model's residue on the one block both can reach disagrees in sign, so its twenty other blocks carry no information about sign at all.

Tested in The residue is below its own noise · the series on contour

“The orphan count steps somewhere along a distortion, and where it steps relative to the energy's minimum decides what a slightly distorted molecule counts as.”

On the tetrahedron-to-plane path it does not step anywhere along the distortion. It is zero at the tetrahedron and zero at every angle up to and including 89.99°, and one at 90° exactly. There is one step and it is at an endpoint.

Tested in A count that changes at one point · the series on hypervalency

“A filled-shell result is at least independent of the arbitrary parts of the model.”

It is independent of every energy — the six-electron value does not move by 10⁻⁹ as the third orbital is swept over twenty-five electron volts — and it is a pure function of the overlaps, which are the part of the model a basis choice fixes. It has traded one dependence for another.

Tested in A filled shell is not an empty statement · the series on overlap

“Excluding systems with a degenerate level at the Fermi energy excludes the systems where the contrast is a ratio of a quantity to itself.”

A ring of six at half filling has a Fermi-level gap of 2.000 against the ring of four's 3 × 10⁻¹⁷, so it passes the exclusion, and it still reads exactly one at five of the eight repulsions measured. The degeneracy that produces the tie is in the ion, and a one-electron shell gap does not determine it.

Tested in A ratio of exactly one is a tie · the series on photoelectron

“A ring of four's contrast stops being exactly one above U = 32 because its degenerate ground state leaves the eigensolver free.”

Its many-electron gap is 0.332 at U = 8 and 0.014 at U = 0.5, and the contrast is exactly one at both — widest and narrowest alike. It is not exactly one at U = 64 and 256, where the gap is 0.062 and 0.016, in between. The gap does not predict the tie; the cut simply stops landing inside a degenerate pair between 16 and 64, which is a change in the weight ordering.

Tested in A ratio of exactly one is a tie · the series on photoelectron

“Tilting the field separates the coincident events, so the count rises with the tilt.”

It rises from three to six, and then falls back to five at three angles and to four at forty-five degrees. The count is not monotone in the tilt and is six almost everywhere rather than at the end.

Tested in Four angles the shell chooses · the series on representation

“A field at a general angle couples every pair, so the events are generic.”

Every pair is coupled at a general angle, but four angles are not general: at arctan 1, arctan 2/√3, arctan √3 and arctan 2 two crossovers coincide exactly, and those angles are fixed by the ratios 3√6 : 3√6, 3√3 : 9/2, 9/2 : 3√3/2 and 3√3 : 3√3/2.

Tested in Four angles the shell chooses · the series on representation

“A large slope floor is a strong statement about the model.”

The largest floor of the four predictors is 28.5, and it is a rise of 0.880 eV divided by a run of 0.0308 — one per cent of that predictor's range. A floor built on a near-tie in the predictor is a small denominator rather than a steep relation.

Tested in The error bar that would be needed · the series on models

“The composite graph's third moment is the quantity that decides.”

It is the right predicate and the wrong coordinate. Split into the part inside a band and the part crossing the gap, the second sorts all nine combinations and the first sorts none — and against the total, the two families' crossovers sit a factor of twenty-seven apart, against about two when the crossing part is used.

Tested in The triangles that were never in the bands · the series on Bands in a solid

“If most of the family is degenerate, counting descriptions as a measure of ambiguity needs restating.”

Thirty-seven of forty-eight pairs are degenerate, and thirty-five of those find exactly one description — where degeneracy is trivially true and means nothing. Of the thirteen that find more than one, eleven are genuinely different. The measure survives on the family.

Tested in Counting was right except where it mattered · the series on multicentre

“The four coincidence angles are properties of the shell.”

They are properties of the shell's matrix elements and therefore of the two-state estimates built from them. In the exact spectrum the one avoided crossing's field runs 1.68, 1.73, 1.89 and 1.95 × 10⁻⁵ at 45°, 49.107°, 60° and 63.435°, which lies smoothly between its values at neighbouring tilts. Nothing marks them.

Tested in None of the six was a crossing · the series on representation

“An avoided crossing is where two levels exchange character.”

Here the gap at its minimum is between 1.39 and 2.74 per cent below its zero-field value. Two levels exchanging character would close the gap to a fraction of it; a dip of one and a half per cent is a perturbation, and the two-state vocabulary of a crossover does not describe it. The second crossing found at 90° does close, to 2.1 × 10⁻⁷ against 5.0 × 10⁻⁴, which is what a real one looks like.

Tested in None of the six was a crossing · the series on representation

“The criterion's value of 1.732128 for the two lone pairs agrees with √3 to five figures, which is suggestive.”

It is √3. The criterion equals 1/√(p fraction) in closed form, with the radial dipole integral cancelling between numerator and denominator, and the departure from √3 falls by a factor of 197 — from 1.7 × 10⁻³ to 8.5 × 10⁻⁶ — as the quadrature is refined from forty points a dimension to a hundred.

Tested in The five figures were an identity · the series on hybrids

“The two ends of the twist have counts because they are the symmetric arrangements.”

They have counts because their groups happen to be tabulated. So do the two angles 0.01° and 59.99°, which are not those arrangements: the finder reports D3h and Oh for them because the worst residual, 1.4251 × 10⁻² Å per degree of twist, is still under its threshold there. The countable stretch is 0.0702° at each end of sixty — 0.234 per cent of the path — and on every bit of it the group being counted in is not the group the molecule has.

Tested in The group nobody wrote a table for · the series on hypervalency

“The displacement is a perturbation-theory result and therefore approximate.”

First-order perturbation theory puts the level at the mean of the two site energies, and the exact diagonalisation sits 1.06 × 10⁻⁴ eV from that mean at a detuning of 0.05. The departure is quadratic — a slope of 1.974 on logarithms — so first order is the whole of the small-detuning behaviour.

Tested in A symmetry holds or it does not · the series on overlap

“A low-temperature feature helps a four-parameter fit by making it better determined, and that is what it does.”

It does that — the worst relative uncertainty falls from 37.3 per cent on an open chain of eight to 21.9–23.9 per cent on the frustrated rings — and it also rotates the free direction through a sign change, which no amount of better conditioning would do.

Tested in The sign a frustrated ring changes · the series on magnetism

“A composition with one local optimum is one with nothing to get wrong.”

It is at half filling on a bipartite net, where the two-colouring is the unique arrangement with every bond unlike. Everywhere else a single basin is a statement about a search of twenty starts, and the smallest compositions have one because there is almost nothing to exchange.

Tested in The composition that is hard is not the full one · the series on cohesion

“The orphan count is constant along the Bailar twist, because the twist keeps D3 throughout.”

The conclusion holds and the reason given for it does not carry the weight. The count is 2 at all fifteen answerable geometries — but four of those are not D3, they are D3h and Oh, and the count is 2 there too. So the constancy is not the group staying the same, since the group does not stay the same. It is that six σ combinations and four matched valence orbitals are the same at every one of them.

Tested in The count the table was hiding · the series on hypervalency

“Writing the missing table closes the gap in the count.”

It closes eleven of the seventeen uncounted geometries. Six remain — three at each end, from 0.070° to about 6° and about 54° to 59.930° — and they are refused because the finder's own residual check fires, not because anything is missing from the tables. Answered geometries go from 4 to 15, and the two gaps were never the same kind.

Tested in The count the table was hiding · the series on hypervalency

“The ring's contrast of exactly one at half filling was an artefact of its degeneracy, so the true limit is unknown.”

An open chain of six has no exactly degenerate neighbouring pair in its removal spectrum at any of the eight repulsions or any of the three fillings — zero, against up to thirty on the ring. Its half-filled contrast converges to 0.999999, with residuals the guard permits. The degeneracy invalidated the ring's argument and not the ring's number.

Tested in The number the tie got right · the series on photoelectron

“The contrast falling below two at half filling is what makes satellites indistinguishable there.”

On the chain it falls below two between two electrons and four — 2.905 at two, 1.642 at four — and half filling is not where the crossing happens. Half filling is where the contrast reaches its floor, which is a different statement about a different repulsion.

Tested in The number the tie got right · the series on photoelectron

“A diagnostic that collapses the scatter onto one curve is the rule a composite method needs.”

The one that collapses it needs the target's exact energy, which is the diagonalisation the composite exists to avoid. Of the seven candidates computable from mean fields alone, the best leaves a factor of 3.67 between two points at the same diagnostic value and the worst leaves 45.7.

Tested in The second number is the error, rearranged · the series on approximation

“A finite upper end can be recovered by letting the two measurements differ in precision.”

It can, and it is useless as a bound: errors differing by ten per cent give ×14.14, by twenty-five per cent ×5.66, by half ×2.83, by a factor of two ×1.41. The ceiling exists only because of a precision ratio nobody has quoted, and it runs from 1.41 to unbounded as that ratio approaches one.

Tested in The other end of the bracket is not a number · the series on models

“Excluding the three functions the plane's mirror symmetry decouples is safe, since they genuinely do not couple.”

They do not couple, and excluding them leaves a subspace that is not closed under rotation — so the answer acquires a dependence on the field's direction that the whole problem does not have. The truncated crossing field runs 1.6435 to 2.2014 × 10⁻⁵ across ninety degrees; the whole shell's is 2.0876 × 10⁻⁵ everywhere.

Tested in The variation was the basis · the series on representation

“The depolarisation ratio reads the distortion's component outside the totally symmetric species.”

It reads a direction as well as a magnitude. At equal magnitude in the non-symmetric plane the departure from three quarters runs from 8.18 × 10⁻⁶ to 1.62 × 10⁻⁵, a factor of 1.978, so 'the component' is two numbers rather than one.

Tested in One number was a direction too · the series on spectrum

“A quantity quadratic in the distortion and invariant under a three-fold axis must be isotropic on a two-dimensional representation, so the reading should be constant on the circle.”

That is true of a single band. The reading is the largest departure over four depolarised bands, which the distortion splits, and a maximum of functions related by the three-fold is not invariant. Its period is sixty degrees rather than one hundred and twenty, and the six directions the group relates agree to two parts in a million.

Tested in One number was a direction too · the series on spectrum

“So the convention does not matter and the discrepancy can be left.”

The margins move by up to a factor of eleven. The six-membered ring clears its measurement by 9.9 per cent under the convention the decided verdict used, by 265 per cent under the torsion count, and by a factor of 12.1 under the n − 2 count. A verdict is convention-free and a margin is not, and margins get quoted.

Tested in The count that was never written down · the series on strain

“So the defect does not matter to the comparison.”

It sets every scale in the problem and it sets them together. The zero-field s-p gap, the exact crossing's field and all eight estimates shrink with it, so the ratios between them tend to constants — 0.04000 for the crossing over the gap, 0.02356 for the depth of the dip, 0.9067 for the nearest estimate over the crossing. The defect matters and cancels.

Tested in Consistently wrong is not a limit · the series on representation

“The magnitude decaying with ring size shows the frustration washing out.”

Open chains decay the same way and have no frustration to wash out: 0.792, 0.440, 0.272 at five, seven and nine spins against the rings' 0.857, 0.463, 0.287. At each odd count the two agree to within 0.065, 0.023 and 0.015.

Tested in It was the count, not the frustration · the series on magnetism

“Seven candidates scoring between 3.67 and 45.7 is seven measurements of the same thing.”

Two of the seven have no comparable pair at all and cannot be scored. The other five are judged on 2, 4, 8, 10 and 12 comparisons — a sixfold spread — so the scores are not equally well founded and a ranking among them reads more into the numbers than they carry.

Tested in Five failures in five different places · the series on approximation

“A census of failures is a property of the systems rather than of where the line is drawn.”

At a threshold of zero every comparable pair counts as a failure and the census implicates every point in the square. The answer moves with the threshold, which is what a census that measures anything must do.

Tested in Five failures in five different places · the series on approximation

“The five collapse cases differ in size and stiffness, so the collapse is tested across both.”

In the quantity the departure from the bulk amplitude actually depends on — the ring measured in its own alternations — they are 9.56, 4.89, 1.36, 7.33 and 9.77. That is a factor of 7.2, and three of them are rings of forty at three stiffnesses, which are three different ring sizes in those units.

Tested in Five rings that were five different sizes · the series on metal

“Matching the cases shows the collapse is exact to 1.6 per cent.”

A dimerised ring needs an even number of sites, so the target cannot be hit exactly: the matched set still differs by 1.09 per cent in n·δ∞. What is left is the same order as what the rounding leaves, so a tighter figure is not available on a lattice.

Tested in Five rings that were five different sizes · the series on metal

“A distortion mixing the two effects must pass through a direction in which the gap does not move to first order.”

It must, and there is more to say: the direction is the antisymmetric fold at every symmetric arrangement, and its blindness is exact rather than found. Folding one pair three degrees further and the other three less gives the same arrangement with the pairs swapped, so the two gaps agree bitwise, and an even function has a vanishing derivative at its centre.

Tested in The direction the gap cannot see · the series on electron count

“The depolarisation reading varies by a factor of two with the direction of the distortion, so it carries direction information.”

The largest departure over the four depolarised bands varies by 97.8 per cent. Their sum varies by 0.038 per cent — a factor of 2591 between the two departures — and a sum cannot be changed by reordering. The direction dependence is in the extremum.

Tested in The sum was flat all along · the series on spectrum

“Two of the four bands are nearly undeparted in the insensitive directions.”

They are exactly undeparted — zero to the last bit the arithmetic carries — in all six directions where one bond is stretched against another. Those distortions keep a mirror plane that fixes those bands' ratio at three quarters, so the zero is a selection rule and not a small number.

Tested in The sum was flat all along · the series on spectrum

“Then it is the third-order term instead, and the fitted exponent should have been 1.”

The cubic term is there — reversing a distortion changes the sum by 7.36 × 10⁻⁴ where the isotropy residual is 5.24 × 10⁻⁴, and only an odd power can do that — but it is not the whole residual either, since an exponent of 1.248 is 0.25 from the pure-cubic prediction and a single power fits to 7 per cent. Both terms are present and the cubic is the larger over the range swept.

Tested in The suspect that did not fit · the series on spectrum

“A cubic and a quartic term together account for it.”

They fit better — the worst misfit halves from 7.0 to 3.5 per cent — which is what a second parameter does whether or not it belongs. What says it does not is that the misfit keeps its shape: positive at both ends of the range and negative through the middle, two sign changes in nine points. A model with the right terms leaves residuals without a pattern.

Tested in The suspect that did not fit · the series on spectrum

“A coordinate set's count of totally symmetric vibrations is the molecule's count.”

Only when the set spans every vibration. Hydrogen peroxide's five coordinates have rank five against six vibrations and count three where the Cartesian displacements give four; both ferrocenes' sixty have rank forty-four against fifty-seven and count three where the molecule has four.

Tested in Five coordinates for six vibrations · the series on point group

“The plateau is flat.”

It is a broad maximum whose position moves with the stiffness. At K = 1.1 the exponent falls monotonically outward from six; at K = 2.8 it rises to 0.697 at fifteen bonds and falls to 0.664 at thirty. Both movements are small against the finding, and neither is noise.

Tested in The rule of thumb was on the flat part · the series on Peierls distortion

“The quarter rule is conservative, so the stiff cases are being cut short and their reaches would rise.”

Right for half the series and wrong for the other half. Five stiffnesses — 1.1 through 1.8 — end on the noise floor inside a quarter of the chain, and their exponents do not move by a part in a billion when the window is opened to the midpoint. The rule was free for those and binding for K ≥ 2, and the split at two is sharp.

Tested in The window that was not a plateau · the series on Peierls distortion

“The capacity is not a function of the second ionisation energy, and the other candidate inputs remain open.”

It is not a function of any of them. Every one of the six has two atoms adjacent in it whose capacities differ several-fold: the cubic coefficient ×7.9, the hardness ×10.4, the electronegativity and the first ionisation energy ×24.2 each, the electron affinity ×43.9, the second ionisation energy ×192.6.

Tested in The worst of the six was the one we asked about · the series on electronegativity

“A minimum of a gap between adjacent levels of the whole spectrum finds the places where levels come closest.”

It finds adjacent ones. The only gap minimum between two levels the field couples is between the two lowest m = 0 levels, at 0.0196, and both |m| = 1 levels lie between them there — so in the sorted spectrum those two are not neighbours and the search never looks.

Tested in The crossing nothing couples · the series on representation

“A shortfall of ten per cent in αₑ means the curve's asymmetry is ten per cent too small.”

αₑ goes as −(1 + a₁), where the 1 is the harmonic spread's contribution of the opposite sign. So the cubic coefficient needs to be only 5.9 per cent steeper for HCl to raise αₑ by 10.7 per cent — a difference of two terms amplifies the change in the larger one.

Tested in The cubic a Morse curve guesses · the series on rotation

“The direction the gap cannot see is the one a complex is softest along without paying for it.”

On the symmetric line it is better than that: the repulsion falls by about a tenth of a per cent along the antisymmetric fold at every amplitude tried, so the distortion is paid to happen rather than merely free. Off the symmetric line it is worse: at fifteen and five degrees the same direction costs 1.30 per cent while the gap moves by 0.44.

Tested in A blindness that is inherited · the series on electron count

“One target is enough to establish that matching tightens the collapse.”

It tightens it at all three, by factors of 1.36, 1.65 and 2.10 — and the factor rises with the target. The single-target factor of 2.10 is the largest of the three rather than a typical one, and reporting it alone would overstate what matching generally buys.

Tested in Three points, and they all go down · the series on metal

“Three points make a curve of residual against target.”

They make a direction. The residual is 2.515, 2.070 and 1.621 per cent at targets of 5, 7 and 9.6 — monotone and still falling at the largest target that could be afforded, which is a finite-size effect that keeps shrinking rather than a limit approached.

Tested in Three points, and they all go down · the series on metal

“So the classification has genuine exceptions and the rule is approximate.”

The three are consistent with chance. The rule flags 19 pairs; four independent signs all fall the same way one time in eight, so 2.375 of those 19 are expected to look agreed for no reason, against 3 observed — a ratio of 1.26. And the tables are correlated, which makes agreement more likely still, so 2.375 is a lower bound. The boundary may be exact and the panel too small to show it.

Tested in Two numbers caught what one could not · the series on dipole

“The improvement is a smaller version of the same thing.”

It changes the kind of mistake as well as the number. The one-variable rule misses disputed pairs and over-flags agreed ones; the two-variable rule misses no disputed pair at all and over-flags three. As a screen for where a computed sign cannot be trusted, one direction of error is harmless and the other is not.

Tested in Two numbers caught what one could not · the series on dipole

“The chance calculation's 2.375 spurious agreements is a lower bound, and three observed sits comfortably above it.”

Measured near the boundary, the tables correlate at 0.325, and four signs with that correlation all agree 29.5 per cent of the time rather than 12.5. The expectation among the nineteen flagged pairs is 5.6, so the three exceptions sit below chance, not above it.

Tested in No panel of this kind can find an exception · the series on dipole

“A correlation between tables inferred from a model of their signs is only a model.”

It is a model, and it can be checked. The correlation from the tables' values and the one from their sign agreement agree to 0.003 near the boundary, and the fraction of pairs on which all four signs agree matches the prediction from pairwise agreement to within 0.02 in every window from nineteen pairs to sixty.

Tested in No panel of this kind can find an exception · the series on dipole

“The coupled minimum is a robust feature of the shell.”

It exists because the d level sits a fifth of the s–p gap above p, inside the quarter at which the f² coefficient of the separation, 2·54 − 27/δ, changes sign. At a quarter it is gone, and at the shell's own offset it is 0.436 per cent deep.

Tested in Two levels cannot make a minimum · the series on representation

“The inversion barrier is what decides how fast a pyramidal molecule turns inside out, so quoting it settles the question.”

The barrier enters through the action ∫√(2μ(V−E))dx, which is an area. Two wells with the same minima and barriers 12 per cent apart — 2020 and 2262 wavenumbers — give ground splittings of 1.3508 and 0.7935, a factor of 1.70. The height moves by a tenth and the answer by a factor.

Tested in A barrier is not what a splitting measures · the series on inversion

“The splittings are too large because the barrier used was too low.”

Raising it to 2329.9 reproduces the ground splitting exactly and leaves the excited one at 47.30 against 35.81, still 32 per cent out. And the barrier the excited splitting alone asks for is 2559.8 — 9.9 per cent higher again. One height cannot satisfy two lines.

Tested in A barrier is not what a splitting measures · the series on inversion

“There is exactly one basis size at which the three-body correction is worth computing and can be assembled pairwise.”

True at a kilocalorie a mole and at no other line tried without qualification. Over two decades the answer is basis 4, then nothing, then 3, then nothing, then 2, then nothing, then 1, then nothing. Four different sizes are the answer somewhere, and more of the range has no answer than has one.

Tested in The line was holding the answer up · the series on basis

“The published result is at least robust upward, since a looser line can only make the assembly adequate more often.”

It is robust upward by 72 per cent and fragile downward by 0.7. A looser line does make the assembly adequate more often and also stops the correction mattering, so the window closes at both ends — just very asymmetrically about the value that was chosen.

Tested in The line was holding the answer up · the series on basis

“The far end is limited by the noise floor.”

For the five softest it is: the reaches are 23, 30, 39, 54 and 74 bonds and each is where the excess falls below the relaxation's convergence. For the five stiffest all five reaches are exactly 77, which is a quarter of the 320-site chain — an arbitrary reading rule rather than a property of the profile.

Tested in The other window was a plateau too · the series on Peierls distortion

“A flat sweep means a robust quantity.”

It means one only where the sweep is doing something. The softest chain has exactly one far end inside its reach, so its perfectly flat curve is a statement about a single fit repeated eleven times, and its zero per cent carries no information at all.

Tested in The other window was a plateau too · the series on Peierls distortion

“The sign is a property of the ground state's total spin.”

It is right on fifteen of sixteen clusters at the coupling every earlier reading used, and wrong on a star of six, whose exponent is +28.893 because its pole lies at 47.03 cm⁻¹, three wavenumbers short of that coupling. It is right on all sixteen at once only between 20.1 and 47.0 cm⁻¹. The ground spin places the poles; it does not fix the sign.

Tested in The sign rule holds between two poles · the series on magnetism

“Which kinds of constant a spectrum determines is a property of the particular molecule.”

On the two available it is the same: every stretching constant is fixed on its own in both, no bending constant is fixed on its own in either, and the mixed stretch–bend kinds are partly determined in both. Two molecules is not a rule, and the pattern holding across a tetrahedral centre and a planar one is more than nothing.

Tested in The second molecule with a blind spot · the series on normal mode

“The quarter is a number about the n = 3 shell, found by bisecting a numerical search.”

It is 2(n² − 4)/(5(n² − 1)) for every shell, from the closed forms of two dipole matrix elements. At n = 3 that is exactly 27/108, which is the quarter the bisection found at 0.24999886; at n = 4 it is 8/25 and it rises to two fifths.

Tested in The quarter, generalised · the series on representation

“Correlated measurement errors cannot change which of this collection's claims is the most fragile.”

True for one common correlation, which is what the common-factor result proves: a common factor multiplies every price equally and the ordering slides rigidly. False in general — with a correlation per predictor the factors form a ratio, and three of the four adjacent pairs in the ranking can be swapped, the easiest at ρ = 0.154.

Tested in The ranking moved and the headline did not · the series on models

“A net that cannot be two-coloured behaves differently from one that can.”

Not at the smaller contrast. There the frustrated net's half filling has one basin and every start reaches it, exactly as the bipartite net's does. The difference appears only at the larger contrast, where the bipartite net loses four points of hit rate and the frustrated one loses thirty-six — so what the triangles cost is invisible until the site energies dominate the band term.

Tested in A net with no two-colouring · the series on cohesion

“Dropping one of the two runaway constants is the repair.”

It is a repair to the search. The converged field and the field moved ninety times its own norm project to the same point, agreeing to 5.8 × 10⁻¹³ in every one of fifty-five components — so the projection repairs the answer regardless of where the fit stopped, and dropping a constant is a choice about which coordinate to lose.

Tested in Adding data made it worse · the series on normal mode

“The clause about hydrogen repaired the bond lists.”

It repaired hydrogen peroxide, which is what it was written for. Across the twenty-three molecules the rule still gets seven wrong, and five of those come back with no bonds at all: both xenon fluorides and all three complex ions, every bond in which is longer than the cutoff.

Tested in No length separates them · the series on point group

“The reduced mass barely changes, which is why it does so little.”

It changes by 10.8 per cent, and the splitting's sensitivity to it is steeper than to the barrier — the mass exponent is exactly one below the barrier exponent, so 10.8 per cent of mass is worth more than 10.8 per cent of barrier. It does little because the change is small, not because the dependence is weak, and the same 10.8 per cent applied to the pyramid height would cost 2.5 times as much.

Tested in It was never the mass · the series on inversion

“A force field fitted exactly to every observed frequency, with its flat direction removed, is determined.”

Boron trifluoride's E′ block has three combinations of constants — r − rr, rt and t − tt — and two frequencies. Its exact fits form a closed conic on which the B–F stretch constant spans 3.94 to 12.11 millidyne per ångström, the stretch–stretch coupling changes sign and the stretch–bend coupling runs from −0.36 to 6.71, with all six frequencies fixed.

Tested in The residual was a loop · the series on normal mode

“A second isotopologue settles an underdetermined block.”

Boron-10 frequencies predicted by the first fit, fed back as data, agree with exactly two points of the boron-11 curve: the fitted field, with a stretch constant of 7.43, and a different one at 3.97 with a bend combination four times larger. Known to half a wavenumber they leave a stretch constant of 7.32 to 7.52 near the fitted field — a wider range than the four fits happened to agree to.

Tested in The residual was a loop · the series on normal mode

“The arrangement that binds best at half filling is the one that opens the widest gap at the Fermi level.”

True in eighteen of twenty composition–net–contrast cases. On the triangular net at half filling and a contrast of 4, the winner's gap is 4.947 and a runner-up's 5.088. On the square net at a contrast of 1 with six sites raised, the winner's gap is 0.806 and a runner-up's 0.898.

Tested in The gap follows the winner late · the series on cohesion

“A pairwise assembly of the counterpoise correction over-corrects.”

With the heavy centre inside it over-corrects on all twenty-four cells, with three centres and with four. With every centre alike it under-corrects on seventeen of twenty-four, and with the heavy centres outside on sixteen and eighteen. The overshoot is a property of one arrangement.

Tested in The overshoot was one arrangement · the series on basis

“Comparing the assembly's error with the accuracy line is a test of whether the assembly is adequate.”

Only if the error is taken by its size. With the heavy centres outside, one Gaussian a centre under-corrects by 5.17 kilocalories a mole, and a test on the signed error would call that adequate.

Tested in The overshoot was one arrangement · the series on basis

“Dunham's relations fix a potential's cubic and quartic, and the higher coefficients only refine the result.”

The Morse curve satisfies the relations exactly, and its own series cut after the quartic — with the same cubic and quartic — moves its αₑ by 0.7 per cent for CO, 2.8 for DCl, 4.9 for HCl and 6.3 for HF, which is between a sixth and a half of each Morse shortfall. Kept to the octic, every molecule is back within half a per cent.

Tested in Two coefficients are not a potential · the series on rotation

“A change of sign in the third direction's exponent is a pole.”

At two per cent every change of sign below 150 cm⁻¹ is a pole. At four per cent every doublet's lower separation is a zero of the same exponent instead, at a coupling moved by two per cent, because the third and fourth singular values have traded places. By sixteen per cent every cluster's upper separation is a zero.

Tested in Purity renames the poles · the series on magnetism

“The linear per-decade rise ρξ ln 10 predicts the growth wherever the run density is small.”

Where ρ is below 10⁻³ it does over the lengths computed: at x = 0.3 the rise over 10²⁰–10³⁰ is 0.97 of that over 10³–10⁶. At x = 0.75, where runs of six are most common at ρ = 0.011, it is 0.35 of it. What decides is ρ times the separation, not ρ alone.

Tested in The share that was read as a line · the series on defect

“The supply argument is the refutation without a margin in it, since it compares against bonds rather than against a convention.”

It compares a requirement against bonds and computes the requirement from the ceiling, which contains the gauche energy. The 250-fold acceleration needs seven rotors at 2.5 kilojoules a mole, five at 3.8 and four at 4.5, against a supply of four — so the argument holds at the bottom of the range and fails at the top.

Tested in The lever that was supposed to be smaller · the series on strain

“The fragility of a pairwise counterpoise assembly — one usable basis size, a window a hundredth wide — is a property of the method.”

It is a property of stopping at pairs. With the three-body increments added, every four-centre arrangement has a usable basis size at a kilocalorie a mole, two of them have two, and more than one size is usable over 23, 60 and 38 per cent of three decades of line where pairs gave 0, 0 and 3.

Tested in A repair that costs more than the whole · the series on basis

“Taking a many-body expansion one order further improves the assembly.”

It improves the usable range in every arrangement and the error on most cells. It makes the error larger on five cells of twenty-four with every centre alike and on four with the heavy centres outside, and it raises two of the eighteen window floors.

Tested in A repair that costs more than the whole · the series on basis

“The triples whose fragments are not consecutive can be dropped from a chain's assembly.”

With the heavy centres inside or outside, dropping them leaves the covered share exactly where every triple put it, 75 and 80 per cent. With every centre alike it falls to 52 per cent, below the 58 that pairs alone give, because the triples with a gap carry a median of 42 per cent of each cell's three-body increment there.

Tested in A repair that costs more than the whole · the series on basis

“Changing an overlap puts two first-order terms into the level's shift, which will add or nearly cancel.”

Both terms are there and both multiply the level's amplitude on C, which is zero, so the first-order shift vanishes without any cancellation — and so does every higher order, because the pair's two rows of H − αS stay proportional whatever their overlaps to C are.

Tested in A level no symmetry was protecting · the series on overlap

“The spreads are properties of the cages rather than of the search.”

A spread is a difference between the best and worst maxima found, so more starting points can only raise it. The ten-vertex cage's Boys spread is 7.17 × 10⁻⁴ at twenty starts and at sixty, and 1.07 × 10⁻³ at a hundred and twenty. Every spread quoted here is a lower bound, and the zeros are the only entries a longer search cannot move.

Tested in The cage is on both sides · the series on multicentre

“The count of third-row ions is the variable the model's error runs with.”

It is a coarse version of one. The sum of the two exponents correlates with the error at −0.986, matched by 4 of 720 orderings where the count is matched by 24, fits a line through all six at r² = 0.974, and orders the three pairs with one third-row ion, to which the count gives one value.

Tested in The sum of the exponents, not the softer ion · the series on contour

“The slope the sum gives is the slope a correction would need.”

A line through the three pairs outside the middle group predicts the middle group's order and a slope of −23.8 points per unit of exponent, where those three have −11.5 among themselves. The sum is the right direction and not the whole variable.

Tested in The sum of the exponents, not the softer ion · the series on contour

“The chain of six has no pair of removal lines of identical weight, so its contrast of one cannot be a tie.”

At the repulsions sampled it has none. At U = 28.916 its third and fourth strongest lines, at removal energies +0.664 and −1.666, carry weights equal to eight figures and the contrast is 1.00000004. The lines are not degenerate; their weights cross.

Tested in The limit of one is a parity · the series on photoelectron

“A half-filled chain's intensity contrast converges to one.”

At large repulsion the lines sit at 2cos(kπ/(n + 1)) in pairs at ±E whose weights agree as 1/U. With three up electrons on the chain of six, and one on the chain of two, the cut falls inside a pair and the contrast goes to one. With two on the chain of four it falls between pairs and the limit is 1.3125.

Tested in The limit of one is a parity · the series on photoelectron

“The eighteen-electron rule's t2g set is a feature of octahedral symmetry.”

It survives the whole twist as a three-dimensional set, becoming A1 + E in the D3 interior and A1' + E' at the prism. A d6 centre therefore has the same closed shell at every geometry on the path, so the electron count cannot be what resists a Bailar twist.

Tested in The leftover changes sides · the series on hypervalency

“Contracting the model's p functions tests the range explanation.”

Not if every shell is contracted alike. At a factor of 1.1 every separation shrinks by between 9.7 and 10.2 per cent, the spread of the six errors narrows from 32.7 points to 30.1 — less than rescaling every distance by the mean change would give — and the root-mean-square error rises from 11.4 per cent to 14.4.

Tested in One contraction for two conditions · the series on contour

“Correcting the third-row shells' range would make the ionic model accurate.”

It removes the pattern and not the error. At the aligning contraction the spread of the six errors falls from 32.7 points to 5.2, and the root-mean-square error rises to 14.7 per cent, because every pair is brought to the second-row pairs' error of about −14 per cent rather than to zero.

Tested in One contraction for two conditions · the series on contour

“A family of orderings that brackets the measurement makes the improvement meaningless.”

The family is unbounded, so it brackets every number. The ordering that lands on the measured 0.7935 has exponents of +3.04 and −3.04, and the one that undoes the mass construction has α = γ = −4.63 and β = +8.26 — three and five times further out than any ordering proposed.

Tested in An ordering worth half a per cent · the series on inversion

“The ordering ambiguity is small because ammonia's mass varies only by a fifth.”

It is small because the ordering's potential carries ħ² and a reciprocal mass. Multiplying every nuclear mass by λ divides that potential by λ exactly, and the spread falls from 0.63 to 0.25 per cent between λ = 0.5 and λ = 4 while the mass construction's effect rises from 34 to 70 per cent.

Tested in An ordering worth half a per cent · the series on inversion

“The D3 gap found by accident on the Bailar twist was likely to be the only one.”

It was not, and there is exactly one more. Eighteen paths sampled at fifteen geometries each give 270, of which 243 are named by a tabulated group, 25 are refused by the finder's tolerance and 2 are a finite group of order ten that no table here reaches. Two of 270 is a small answer and it is not zero.

Tested in The gap found on purpose · the series on hypervalency

“ND₃'s measured splitting is a sharp test of the bond-conserving reduced mass, because that mass changes with deuteration in a way no constant mass can.”

The mass does change differently — 23.7 per cent across the coordinate for ND₃ against 20.1 for NH₃ — but fitted to NH₃ and carried to ND₃ the two constructions predict isotope ratios of 19.26 and 19.00 in one well and 17.63 and 18.49 in another, against a measured 14.94. They differ by less than either misses, and which is nearer changes with the well.

Tested in Deuterium cannot tell the masses apart · the series on inversion

“One contraction of the third-row shells accounts for the pattern, since its two conditions agree to 2.2 per cent.”

Taken per ion, chloride needs 1.302, potassium 1.387 and calcium 1.441 — a spread of 10.7 per cent. The single factor's two conditions, 1.3765 and 1.3463, are within 0.03 and 0.11 per cent of the means of the per-ion factors of the ions each condition contains. They agreed because they averaged overlapping sets of ions.

Tested in Three contractions for one shell · the series on contour

“The four predictions can be tested against the measured splittings of the mixed molecules.”

Both are measured and neither is quoted here. What is computed is a prediction from a barrier fitted once to NH₃ — 2116.9 wavenumbers under the bond-conserving construction — and carried to the other three with only masses changed. The prediction for ND₃ is 0.0418 against a measured 0.0531, so the model is already a fifth low at the far end, and the two middle points inherit that.

Tested in The two that are not on the line · the series on inversion

“Making the two ends of the radical inequivalent is what shifts its spin off an even split.”

An overlap change does, at 2.0 per unit of overlap. A site-energy change that moves the level by a tenth of an electronvolt moves the spin by only 0.012, and a coupling between the ends — which moves the level by 0.54 eV at an overlap of 0.05 — moves the spin by nothing, because it leaves the ends equivalent.

Tested in The spin the count does not hold · the series on overlap

“The missing zero-point term cannot be assessed without a force field along the inversion path.”

Its value cannot; its required size can. The gap between the barrier each construction wants for NH₃ and for ND₃ is 97 wavenumbers under the bond-conserving mass and 109 under the rigid-triangle one. The transverse zero-point energies differ between the two molecules by 1783 wavenumbers, from eight measured bands, so the correction would have to be 5.4 or 6.1 per cent of that difference — an ordinary amount for five frequencies to change between a pyramid and a plane.

Tested in The ordering a manifold picks · the series on inversion

“The scatter within a separation class is an effect of where a pair sits relative to the ends.”

Over 702 pairs in 27 chains, whether a pair touches an end raises R² from 0.780 to 0.786. Counting the angular rings between a pair's two rings and under them raises it to 0.935, and adding the end term back changes nothing. Divided by those counts, no chain's worst spread exceeds 2.27, against measured spreads up to 5.95.

Tested in The scatter counts angular rings · the series on aromaticity

“A momentum measurement along a bond reads the bond length off the first zero of the profile.”

At π/R the profile per electron is the antibonding occupation times a number the atom fixes, exactly. Two hundredths of an electron in the antibonding orbital turn the zero into a minimum at 1.608 that reads the separation as 1.954 bohr instead of 1.997; past 0.104 there is no minimum at all; and the Heitler–London bond's occupation is above that threshold at every separation from 1.4 to 4 bohr.

Tested in The zero belongs to one determinant · the series on orbital

“A longer bond goes to the axial site of a bipyramid.”

At five sites it does, by 2.8 × 10⁻³ at a length ratio of 1.1 and 1.4 × 10⁻² at 1.5. At seven it does the reverse: a bond of 1.1 prefers the equator by 7.7 × 10⁻³ and one of 1.2 by 1.7 × 10⁻². What the long bond follows is the more crowded site, which is axial in the trigonal bipyramid and equatorial in the pentagonal one.

Tested in The long bond goes to the crowded site · the series on VSEPR

“Triples with a gap in them matter less the wider the gap.”

Nothing more than four separations away carries more than about a per cent. But the triples with a gap carry a median of 0.42 of the three-body increment by size at four fragments, 0.82 at five and 1.13 from six on, because two one-gap shapes seen from one side carry +20.8 and −11.0 per cent and a two-gap shape carries −16.5 — the gap triples do not fall away, they take back what the consecutive ones overshoot.

Tested in Length did not rescue the consecutive triples · the series on basis