Simply false

“Solid-state physics and molecular bonding are separate subjects with separate methods.”

The band of a two-hundred-atom chain is produced here by exactly the calculation that gives ethene its bonding and antibonding pair: build the adjacency matrix, diagonalise it, fill the levels. Nothing is added at any size. The figures for n = 2 and n = 200 are the same calculation done twice.

Tested in A solid is a molecule that did not stop · the series on Bands in a solid

“A larger molecule has a wider spread of orbital energies.”

Every level of a chain of n sites lies in the interval from −2β to +2β for every n whatever. Measured across n = 2, 4, 8, 16 and 40, the band width stays at 4β to within a part in a thousand; what changes is the spacing between levels, which falls as 1/n.

Tested in A solid is a molecule that did not stop · the series on Bands in a solid

“Sulfur hexafluoride expands its octet by hybridising in two 3d orbitals.”

Sulfur's 3d lies about ten electron volts above its 3p and is far more diffuse, so the mixing that overlap over energy difference allows is negligible; calculated d participation is a few per cent. The three-centre four-electron account needs no d orbitals and gives the geometry.

Tested in Hypervalency without d orbitals · the series on hypervalency

“Each bond in a molecule has its own stretching vibration, and the spectrum shows one line per bond.”

Diagonalising water's mass-weighted Hessian gives two stretching modes, and the share of the motion in each O–H bond is 50% in both of them. Neither mode is a motion of one bond. The same computation gives methane four stretching modes, every one of them a quarter in each of the four C–H bonds.

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

“A ligand field lowers the energy of the d orbitals, which is why complexes form.”

The five energies sum to five times the spherically averaged potential whatever the arrangement — Unsöld's theorem, reproduced to ten decimal places by the point-charge integral for octahedral, tetrahedral, cubic and square-planar fields. A field redistributes the levels about a centre of gravity it cannot move. Complexes form because of the metal–ligand bonding, which is a different calculation.

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

“An orbital is the region of space in which the electron is found.”

A contour is a surface of constant |ψ| at a level somebody chose. The ninety per cent surface leaves a tenth of the density outside it, and the same function drawn at ninety-nine per cent is a different surface of the same orbital.

Tested in What an orbital is · the series on orbital

“Benzene resonates between two Kekulé structures.”

The Kekulé forms are basis functions in a description, not states the molecule visits. The computed π bond order is exactly 2/3 on all six bonds, which is one structure rather than an average of two.

Tested in Delocalisation · the series on delocalisation

“An overlap that symmetry forbids is very small.”

Computed, the 1s with 2px overlap at 2.8 bohr is −3.7e-17: arithmetic noise, not a small physical quantity. The same integrator reproduces the closed-form 1s–1s overlap to 6e-5, so the vanishing is not a defect of the method.

Tested in Exactly zero · the series on overlap

“The five positions of a trigonal bipyramid are equivalent.”

The angle spectrum of the minimised five-point arrangement has three distinct angles where four and six sites have one and two. That inequivalence is measured, not assumed, and it is what Berry pseudorotation exchanges.

Tested in Five sites are not alike · the series on VSEPR

“A force field has to be refitted for each isotopologue.”

Force constants are properties of the electronic energy surface, which the Born–Oppenheimer separation makes independent of nuclear mass. Water's field, fitted to H₂O and D₂O, gives HOD's three frequencies at 1445, 2824 and 3890 with nothing refitted — against quoted harmonic values of 1440, 2824 and 3890.

Tested in The isotope shift is arithmetic · the series on normal mode

“A band gets wider as the crystal gets larger.”

⟨x²⟩ is the mean coordination, which is bounded by how many neighbours an atom has and is unaffected by how many atoms there are. Measured on chains and rings of 40, 60 and 80 sites, the second moment agrees with the counted coordination to machine precision at every size.

Tested in The width of a band is a count of neighbours · the series on Bands in a solid

“A tetrahedral complex has a smaller splitting because it has fewer ligands — four instead of six, so two thirds as much.”

Two thirds is 0.667 and the computed ratio is 0.4444, in both models, to eight decimal places. The ligand count enters through the trace, which is exactly N times the σ parameter and is not the splitting; the splitting is a difference between eigenvalues and depends on the directions. A cube has eight ligands, more than an octahedron, and splits by eight ninths of one.

Tested in Two models, one ratio · the series on ligand field

“The density of states is what an absorption spectrum measures.”

An absorption spectrum measures transitions, which need an initial state, a final state and a non-zero moment between them. The density of states counts states at one energy and knows nothing about pairs. Two systems with identical densities of states and different symmetry give different spectra, and one of them can give no spectrum at all.

Tested in A density of states is not a spectrum · the series on Bands in a solid

“Selection rules stop mattering once a molecule is large enough to have bands.”

A vanishing transition moment vanishes for the same reason at any size — an integrand odd about a plane integrates to zero whether the molecule has four atoms or four thousand. What changes is that a forbidden transition in a solid removes a contribution from a continuum rather than removing a visible line, so the failure is a missing shoulder rather than a missing peak.

Tested in A density of states is not a spectrum · the series on Bands in a solid

“A 2s orbital is a single sphere, slightly larger than a 1s.”

n − l − 1 = 1 radial node, so the isosurface is a shell inside a shell. The shells are counted on the drawn surface and required to equal the radial node count plus one, and a 2s surface claimed as one shell is refused.

Tested in Nodes · the series on orbital

“Bands come from periodicity.”

Every band in this field is computed from a chain or ring with no periodicity assumed anywhere, and amorphous solids — which are periodic in no direction — have bands, gaps and densities of states much like their crystalline counterparts. What periodicity buys is a labelling of the states, not their existence.

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

“Eight-coordinate structures are cubic.”

Minimised, eight points on a sphere give the square antiprism — one face of the cube twisted by forty-five degrees — at a lower repulsion energy than the cube. Both arrangements are computed here and compared.

Tested in The shapes above six coordination · the series on VSEPR

“Electronegativity is a property of an atom, and the scales agree well enough.”

Four definitions in four sets of units. Measured across the table: twelve discordant pairs between Pauling and Mulliken out of a hundred and fifty-three, and six ordinary bonds — C–H among them — whose polarity points in opposite directions depending on the table consulted.

Tested in Electronegativity is not one quantity · the series on electronegativity

“A molecule with 3N−6 vibrations shows 3N−6 bands.”

Two separate reductions stand between the count and the spectrum. Degeneracy collapses benzene's thirty modes into twenty frequencies and methane's nine into four; selection rules then cut benzene's twenty to eleven observable in either experiment, with nine visible in neither. The count 3N−6 is an upper bound on a number of lines, and for a symmetric molecule it is a bad one.

Tested in How many frequencies, not how many modes · the series on spectrum

“Anything a molecule does vibrationally can be seen in the infrared or the Raman.”

A mode is infrared active when its species carries x, y or z and Raman active when it carries a quadratic function, and a species can carry neither. Nine of benzene's twenty frequencies belong to such species, as does one of sulfur hexafluoride's six and one of ethene's twelve. These are silent modes, forbidden in both experiments by the same vanishing-integral theorem that permits the others.

Tested in How many frequencies, not how many modes · the series on spectrum

“Molecular orbital theory and valence bond theory are rivals, and one of them is right.”

Each turns into the other when pushed: configuration interaction on one side, ionic structures on the other. They are two bases for one description, and the observable they are meant to disagree about is the same in both.

Tested in Molecular orbital and valence bond · the series on models

“A metal is a very large molecule, so its levels are very spread out.”

Every level of a ring of n atoms is 2cos(2πk/n), so all of them lie in [−2, 2] for every n. Adding atoms does not widen the range; it puts more levels inside a range that never changes. The width of a band is set by the neighbour interaction, not by the size of the crystal.

Tested in The band limit · the series on multicentre

“The density of states in a band is roughly uniform.”

Computed for a ring of 2000 and compared with the closed form 1/(π√(4−x²)), the density piles up at both band edges and diverges there. The histogram agrees with the closed form to better than 0.01 across the interior, which is the check the figure makes.

Tested in The band limit · the series on multicentre

“A molecular geometry has to be settled by diffraction; a spectrum can only confirm it.”

Linear XY₂ has four vibrations giving three frequencies, of which two are infrared active and one Raman active with no coincidence; bent XY₂ has three, all active in both, with three coincidences. Those predictions differ in the count alone. Carbon dioxide shows the first pattern and sulfur dioxide the second, and neither conclusion needs a frequency to be assigned or a force constant to be known.

Tested in Two structures, two spectra · the series on spectrum

“Both isomers would show the same number of bands, since both have the same number of atoms.”

3N−5 for a linear molecule against 3N−6 for a bent one is already a difference of one, and degeneracy and selection rules widen it: the linear case has one of its four modes doubly degenerate and its infrared and Raman spectra disjoint, so three frequencies give three observable bands split two and one across the two experiments, against the bent case's three and three.

Tested in Two structures, two spectra · the series on spectrum

“A double bond is two bonds, so it is twice as strong.”

Measured, C–C is 348 kJ/mol and C=C is 614 — a ratio of 1.76, not 2. Computed, the σ and π overlaps of two carbon 2p orbitals at the C=C bond length are 0.325 and 0.270: different integrals over different orbitals, with no reason to be equal and no reason for their contributions to add.

Tested in A double bond is not two single bonds · the series on overlap

“A strong ligand strengthens the whole complex, so a good donor makes all the metal–ligand bonds shorter.”

Two ligands trans to one another compete for one metal orbital. In a three-level model solved exactly, raising one ligand's interaction from 1.0 to 1.6 raises its own bond order from 0.6155 to 0.7807 and lowers the bond order to the ligand opposite from 0.6155 to 0.4880. The measured counterpart is a bond length: a Pt–Cl bond trans to a strong σ donor is longer than one trans to a chloride by up to a tenth of an ångström.

Tested in The trans influence is an overlap argument · the series on overlap

“A three-centre four-electron bond puts extra electrons on the central atom.”

Solved as a three-by-three matrix, the occupied non-bonding orbital has a coefficient of exactly zero on the central atom — required by the mirror symmetry of the arrangement, not by the parameters. The computed charges are 1.92 on each ligand and 0.17 on the centre: the centre has given charge away rather than taken it.

Tested in Three-centre bonding, computed · the series on hypervalency

“The band is missing, so that vibration is not there.”

Nine of benzene's twenty vibrational frequencies belong to species that carry neither a linear nor a quadratic function, so they cannot appear as fundamentals in an infrared or a Raman spectrum at all. The vibrations exist, at well-defined energies, and are reached by inelastic neutron scattering, by combination bands and by lowering the symmetry.

Tested in What an absence proves · the series on spectrum

“A symmetry-forbidden band is very weak.”

The integral that would give it vanishes identically: every contribution is paired with its own negative by an operation of the group, so the sum is zero rather than small. The same distinction is what makes an overlap forbidden by symmetry come out at arithmetic noise rather than at a small number — see the computation in `exactly zero`. What is weak, and what does appear, is the intensity a lower-symmetry environment or a vibronic mechanism restores.

Tested in What an absence proves · the series on spectrum

“A conjugated chain has equal bond lengths, like benzene.”

Benzene's equal bonds are protected by a closed π shell; a long chain's are not. Computed on a hundred-site chain, alternating the bonds by twelve per cent gains 0.0078β per site more than a spring constant of 1.6 costs, and the best alternation is at a non-zero value for every spring constant tested.

Tested in A chain cannot stay even · the series on Peierls distortion

“A small enough distortion always costs more than it gains, because the elastic term is quadratic and the electronic term is too.”

The electronic gain is not quadratic. Dividing the computed gain by δ² gives a ratio that rises as δ falls — 2.564, 3.194, 5.330 at δ = 0.08, 0.05 and 0.02 on a hundred-site chain — which is the logarithm showing itself. A quadratic gain would give the same number three times, and the same test applied to a genuinely quadratic sequence returns one.

Tested in A chain cannot stay even · the series on Peierls distortion

“A molecule with five equivalent-looking bonds gives one signal because the five atoms are equivalent.”

The five fluorines of phosphorus pentafluoride fall into two orbits under its D3h group — two axial and three equatorial — and orbit sizes 2 and 3 with stabilisers of order 6 and 4 both satisfy the orbit–stabiliser relation against a group of order 12. The atoms are not equivalent. One signal is observed because the molecule exchanges them faster than the measurement can resolve.

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

“A copper(II) complex with six identical ligands is octahedral, because six identical ligands arranged by repulsion give an octahedron.”

The repulsion argument fixes the arrangement and the electrons overrule it. A d⁹ ion has three electrons in the doubly degenerate upper pair, and the energy gained by elongating along one axis is linear in the distortion while the elastic cost is quadratic — computed slope −7.32 against a cost that starts at zero curvature, so the best distortion is at a non-zero value for any stiffness. Copper(II) hexaaqua has four bonds near 1.97 Å and two near 2.30.

Tested in Copper is never quite octahedral · the series on Peierls distortion

“Any partly filled degenerate level makes a complex distort.”

Only an unequal occupation does. Computing the initial slope for six configurations gives −7.32 for d⁹ and high-spin d⁴, and 0, −4×10⁻¹⁴, 0 and −9×10⁻¹⁴ for d³, high-spin d⁵, low-spin d⁶ and d¹⁰ — arithmetic zeroes, because the level shifts cancel term by term over a symmetric occupation. A half-filled t2g set is degenerate and gains nothing.

Tested in Copper is never quite octahedral · the series on Peierls distortion

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

Computed from Slater's screened charges, a scandium 3d electron feels 3.00 and a 4s electron feels 3.00 — but the hydrogenic energy goes as Z²/n², so the 3d comes out at −13.6 eV and the 4s at −7.7. By iron the gap is −59.1 against −12.0. Every one-electron estimate puts the 3d lower, at every element in the series.

Tested in The aufbau order is not a property of the atom · the series on approximation

“Once the 4s has filled, the 3d fills, so ionisation removes a 3d electron.”

Every transition metal ionises from the 4s. Fe²⁺ is [Ar]3d⁶ with no 4s electrons at all. A filling order that is reversed by removing one electron is not describing a fixed order of orbital energies.

Tested in The aufbau order is not a property of the atom · the series on approximation

“Seven-coordinate geometry follows from the same rule as six.”

Minimised at 1/r the seven-point arrangement is a pentagonal bipyramid with four distinct angles; at 1/r⁶ it is a different arrangement with six. There is no maximally symmetric arrangement of seven points on a sphere, so nothing pins the answer and the unstated force law decides it.

Tested in Which angles are symmetry and which are the model · the series on VSEPR

“A magnetic moment measures how many d electrons a metal has.”

It counts the unpaired ones. Iron(II) and iron(III) differ by one d electron and their high-spin moments are 5.4 and 5.9 Bohr magnetons, while iron(II) high-spin and low-spin have the same six d electrons and moments of 5.4 and zero. The formula √(n(n+2)) takes the unpaired count and nothing else, and reproduces five of nine measured first-row values to a hundredth.

Tested in A moment counts electrons, not orbitals · the series on magnetism

“A bigger orbital is a more loosely bound one.”

The 4s orbital of scandium is more than twice the size of its 3d by every radial measure and is filled first. What binds it is a peak at 0.73 bohr holding a few per cent of its density, which no measure of overall size records.

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

“Hydrogen sulfide and phosphine are sp³ centres with badly distorted angles.”

At 92.1° the relation gives an s fraction of 0.0353, which is sp²⁷: the S–H bonds are all but pure p. Phosphine at 93.3° gives 0.0544 per bond, so its three bonds spend 0.163 of the s function and its single lone pair holds 0.837 — eighty-four per cent s. Calling either an sp³ centre misdescribes the bonds by a factor of seven in s character and misplaces almost the whole s orbital.

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

“A carbon atom could hold four bonds at 120° if the geometry demanded it.”

Four bonds at 120° require an s fraction of 1/3 each, which is 4/3 of an s orbital, and there is one. The budget refuses the arrangement rather than costing it, and the refusal is checked, not assumed.

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

“More force constants give a better description of the molecule.”

Methane's six bond angles are not independent — five bending coordinates span every deformation there is — so a fit with both a bending and a bend–bend constant has a direction it can move along freely. Allowed to, it did: both came back at −97,596 mdyn Å⁻¹, equal to four figures and opposite in effect, reproducing all eighteen frequencies of CH₄ and CD₄ to half a per cent.

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

“A wide band means a small gap and a narrow band means a large one.”

Alternating a chain's bonds by fifteen per cent opens a gap of 0.60β while leaving the second moment — the exact measure of the band's scale — essentially unchanged at 2 + 2δ², a shift of four per cent in the wrong direction. Width and gap are set by different features of the same matrix and moved by different changes to it.

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

“Canonical molecular orbitals and localised bond orbitals describe different distributions of the electrons.”

The transformation is applied to four Löwdin-orthogonalised bond orbitals and the total density is unchanged to 1e-16, at four values of the free mixing parameter. A mixing that is not orthogonal changes it by a fifth, so the test can fail.

Tested in The localisation transformation, demonstrated · the series on hybrids

“A rotational spectrum tells you about the bonding in a molecule.”

Every line position follows from the masses and the coordinates alone: B = h/8π²cI with I a sum of mass times distance squared. Two molecules with identical mass distributions and completely different bonding would give identical spectra. What a rotational spectrum determines is a geometry, which is why it determines one so well.

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

“Every molecule has a rotational spectrum; it is just a matter of sensitivity.”

A pure rotational transition needs a permanent dipole, and whether a molecule has one follows from its point group with no reference to bonding at all. Carbon dioxide, benzene, methane and sulfur hexafluoride each have a perfectly well-defined set of rotational levels computed here, and none of them can show a line of it.

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

“A subgroup gives the same answers as the group, just with different labels.”

A subgroup has fewer operations, so it forbids less. Computed for D∞h in its working subgroup D2h: the species Δu is infrared forbidden in the full group and correlates onto Au ⊕ B₁u, of which B₁u carries z. The subgroup therefore calls infrared-active a mode the real group forbids, and the error is in the permissive direction.

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

“The octet rule and band filling are different ideas from different subjects.”

Both say a system is stable when no level is partly occupied. An octet fills the four valence orbitals of one centre; a filled band fills every level of an extended structure. The 8 − N rule connects them explicitly by predicting how many neighbours an element takes in order to reach the closed case, and it is right for groups four, five, six and seven.

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

“A molecule with a rotation axis cannot be chiral.”

Chirality is forbidden by improper operations — mirrors, inversion, rotoreflections — and not by proper ones. Hydrogen peroxide belongs to C2 at every dihedral strictly between 0° and 180°, which is a genuine twofold rotation and no improper operation at all, and it is chiral throughout that range. The two angles at which it is achiral are the two at which a mirror plane or an inversion centre appears.

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

“A set of orbitals drawn at the same contour value is drawn on a common footing, so their sizes can be compared.”

At the level that encloses 90% of a 1s — |ψ| = 3.94×10⁻², computed by bisection on the integral — a 2s encloses 4.44%, a 3s 0.80%, a 4s 0.14% and a 3d nothing at all. A plate at that one level compares one orbital's ninety per cent with another's four, which is not a comparison of sizes but of how sharply each function peaks.

Tested in One level is not one comparison · the series on contour

“Drawing an orbital at a lower contour level shows more of its structure, so the shells and nodes that appear depend on how generously the picture is drawn.”

The number of closed shells the surface has is n − l at every fraction tested, from 30% to 99%: two for a 2s, three for a 3s, four for a 4s, one for every p and d here. A radial node is a sphere on which the wavefunction is exactly zero, and no contour at a positive level can cross it, so the topology of the picture is fixed and only its size moves.

Tested in One level is not one comparison · the series on contour

“In a two-orbital interaction, the bonding level goes down by as much as the antibonding level goes up.”

Solving the 2×2 secular determinant with the overlap kept at S = 0.25 gives a bonding level 0.800|β| below the mean and an antibonding level 1.333|β| above it. The equality is what happens when S is set to zero, and S is not zero — it is the reason the two orbitals interact at all.

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

“Two bonds of the same order between the same elements have the same stretching frequency.”

Carbon dioxide's two C=O bonds are related by symmetry, so they are the same bond by any definition, and they carry one force constant in the fitted field. The molecule has two stretching frequencies a thousand wavenumbers apart, because a frequency is a property of a mode and a mode involves both bonds and the atom between them.

Tested in The frequency is not the bond strength · the series on normal mode

“Low-spin tetrahedral complexes do not exist because a tetrahedral field offers no spin choice.”

The enumeration says otherwise. With two orbitals below three, four configurations — d³ to d⁶ — have distinct high-spin and low-spin fillings, being the octahedral four reflected by the particle–hole map. What makes them unreachable is a magnitude: a tetrahedral splitting is four ninths of an octahedral one, so even carbon monoxide's 34,000 cm⁻¹ becomes about 15,000, below every pairing energy in the table.

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

“A delocalised electron pair is spread thin, so it binds the molecule less than a localised pair would.”

One pair in a ring of n gives a total π bond order of exactly 2 and a π energy of exactly 4β, for every n from three to eight and beyond, because a ring's lowest level is 2β at any size. The per-bond order falls as 2/n and the total does not move. What is divided is the bonding among the bonds, not the amount of it.

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

“A single impurity in a crystal of 10²³ atoms cannot change anything measurable.”

It changes the states, not just their number. One modified site in a hundred-and-sixty-site chain pulls a level out of the band entirely and binds a state to itself; that state is at an energy no state of the pure chain occupies, so it absorbs light the pure material does not. One part in 10⁷ of a dopant changes silicon's conductivity by orders of magnitude for exactly this reason.

Tested in A defect is a level in the gap · the series on defect

“A defect has to be strong enough to bind a state, like a potential well needing a minimum depth.”

In three dimensions, yes. In one dimension, no — measured by bisection on chains of 40, 80 and 160 sites, the apparent threshold for a mid-chain defect is 0.098, 0.049 and 0.025 β, halving with every doubling. It is the finite chain's own level spacing, and it goes to zero: an arbitrarily weak one-dimensional defect binds a state.

Tested in A defect is a level in the gap · the series on defect

“Sulfur's 3d orbitals are the wrong symmetry to help, so d participation is forbidden.”

Sulfur's 3d functions span eg ⊕ t₂g in Oh, and eg is exactly the species the ligand σ set has spare. Symmetry permits the interaction; what makes it negligible is size and energy, not symmetry. A free-atom 3d at an effective charge near one has a mean radius of 5.6 Å against an S–F distance of 1.56, and how much a molecular field contracts it is beyond a symmetry argument.

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

“Nitrogen trifluoride has a larger dipole than ammonia, because N–F bonds are more polar than N–H bonds.”

The bond-vector sum is larger too — three N–F bonds at 102.2° sum to 1.316 of a bond unit against 1.080 for three N–H bonds at 107.8° — so a bond-dipole model predicts NF₃ to beat NH₃ by about a third. The measured moments are 0.24 D and 1.47 D, the wrong way round by a factor of eight, because the lone pair contributes a moment that adds in one molecule and subtracts in the other.

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

“Two molecules with the same dipole moment have similar charge distributions.”

A dipole is three numbers taken from a function of three variables, and the map is enormously many-to-one. Methane has no dipole and no quadrupole either — Td forbids both — while boron trifluoride has no dipole and a quadrupole of 0.437 in the same model. Zero dipole is compatible with distributions that differ at every higher moment, and the quadrupole is only the first place they part.

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

“Two magnetic ions interact because each one's magnetic field is felt by the other.”

The magnetic dipole–dipole energy of two moments of about two Bohr magnetons three ångströms apart is roughly 0.06 cm⁻¹. Measured exchange couplings in bridged copper and iron complexes run from tens to several hundred wavenumbers — three to four orders of magnitude larger. The interaction that produces them is electrostatic and kinetic: an exactly solved two-site model with no magnetic term in it at all gives a singlet–triplet splitting approaching −4t²/U.

Tested in What couples two spins · the series on magnetism

“The lower-energy orbital of a shell is the smaller one.”

At any shared effective charge the closed forms give ⟨r⟩ = 6/Z for a 2s and 5/Z for a 2p, so the 2s is larger by exactly a fifth at every element in the period — 2.308 against 1.923 bohr at boron, 1.026 against 0.855 at neon — and it is the 2s that lies lower. Size and energy are answers to different questions, and neither settles the other.

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

“d–d transitions are weak because the d orbitals overlap poorly with the light.”

The intensity of an electric dipole transition is fixed by whether the product of the two states' representations with the dipole operator contains the totally symmetric representation. In Oh both d levels are even and the dipole is odd, so the product is odd and the integral is exactly zero — computed as a character product and a reduction, with nothing about overlap or size entering. The same d orbitals in a tetrahedral complex give an allowed transition and absorb about a hundred times more strongly.

Tested in Why a d–d band is weak · the series on representation

“The extra skeletal pair is a feature of the icosahedral geometry of B₁₂H₁₂²⁻.”

It is a property of any connected graph. The largest eigenvalue of a non-negative connected adjacency matrix is simple and its eigenvector has one sign throughout — Perron's theorem — so there is exactly one nodeless radial combination for every cage whatever its shape, and every other radial combination has a node. The octahedron, the pentagonal bipyramid and the icosahedron all give one, and so would a cage nobody has made.

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

“A neon atom has eight electrons in dumbbell-shaped orbitals, so its electron density has lobes.”

The angular densities of the three 2p functions sum to 3/4π in every direction, and the departure from that constant over a grid of 768 directions is 1.1×10⁻¹⁶. Neon's density is exactly spherical. The lobes exist in one choice of basis for the shell and cancel completely when the shell is filled.

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

“The failure can be repaired by computing the band more accurately.”

The failure is not in the band. Every one-electron scheme, however carefully the average potential is constructed, assigns each electron a single orbital and lets it ignore where the others are. What is missing is a correlation between electrons on different sites, which no single-determinant description contains at all.

Tested in A half-filled band is not always a metal · the series on metal

“Isotopic substitution shifts the bands of a spectrum without changing how many there are, since it does not change the bonding.”

Methane's nine vibrations give four distinct frequencies and CH₂D₂'s give nine, from the identical force field. The count is set by the degeneracies, the degeneracies are set by the point group, and the point group is set by which operations permute atoms of equal mass — 24 for CH₄, 6 for CH₃D, 4 for CH₂D₂. Substitution changes the group without moving a nucleus.

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

“Lowering the symmetry of a molecule with a threefold axis can leave it with a fivefold one.”

Lagrange's theorem: the order of a subgroup divides the order of the group. Ammonia's C3v has order six, so its subgroups have orders 1, 2, 3 and 6 and nothing else, and the search finds exactly one of order three, three of order two, and the two trivial ones. A fivefold axis would need a subgroup of order five in a group of order six.

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

“A molecule that is not chiral cannot be made chiral by distorting it.”

Chirality is the absence of any improper operation, and it is a property of the subgroup the distortion leaves rather than of the parent. Methane's Td contains a subgroup of order twelve with no improper operation in it, and three more of order four, three of order three and three of order two. A distortion that lands in any of them gives a chiral structure from an achiral parent.

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

“Orthogonality of hybrids is a mathematical nicety with no physical content.”

Every quantity computed from a set of orbitals — a population, a bond order, an energy from a secular determinant — assumes either that the set is orthogonal or that the overlap has been kept. At an overlap of 0.625 the two assumptions differ by more than the quantities being computed. Löwdin orthogonalisation repairs the set and changes the directions of the orbitals while doing it, which is the price.

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

“A sixteen-electron complex is two electrons short and therefore unstable.”

It has the same number of filled orbitals its geometry offers, so it is closed by its own arithmetic. Wilkinson's catalyst, Vaska's compound, tetrachloridoplatinate and Zeise's salt are all sixteen-electron and all isolable. What is true is that the empty perpendicular orbital gives them a route to add a ligand that an eighteen-electron octahedron does not have, which is a statement about a rate rather than about stability.

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

“A surface atom is the easiest place to trap a state, because it has a neighbour missing.”

Measured by bisection: a mid-chain site binds a state for any energy difference at all, while an end site needs its energy to differ by more than one β. The thresholds on chains of 40, 80 and 160 are 1.025, 1.013 and 1.006 at the end and 0.098, 0.049, 0.025 in the middle — one converging on one, the other on zero.

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

“A band gap of 1.1 eV means it takes 1.1 eV to break a bond in silicon.”

Silicon's Si–Si bond enthalpy is about 2.3 eV, twice the gap. The gap is the cost of moving one electron from a filled level to an empty one with the atoms staying where they are; the bond energy is the cost of removing a pair from a bonding level and letting two atoms separate to infinity. The two differ by the antibonding level's position, by the relaxation of the structure, and by every repulsion term the one-electron picture omits.

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

“Carbon monoxide is high in the spectrochemical series because it is a strong sigma donor.”

Its sigma donor orbital is a weakly bonding lone pair on carbon and it is a mediocre donor by any measure. It is high in the series because it is a strong pi acceptor: its empty pi* orbitals lie close in energy to the metal t2g set, so the gap in the denominator is small and the interaction is large. This site orders the whole series by the pi parameter and by charge, and the pi parameter orders all of it.

Tested in Overlap is not interaction · the series on overlap

“Ring strain is a matter of bond angles.”

Cyclopentane's planar angle of 108° is within a degree and a half of tetrahedral, closer than cyclohexane's chair, and cyclopentane carries about 26 kJ/mol of strain against cyclohexane's nothing. The angles are not the problem there; the eclipsed hydrogens are. Two different quantities travel under one word, and the geometry computed here bounds only the first.

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

“Contracting an orbital always lowers the energy, since the electron gets closer to the nuclei.”

Kinetic energy goes as the square of the exponent and nuclear attraction goes as the exponent, so contracting always costs more kinetic energy than it buys attraction at large enough exponents, and there is a minimum. Past about five bohr the balance tips the other way and the molecule's best exponent falls slightly below one, reaching 0.9951 at six bohr — an expansion, not a contraction.

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

“Electrons avoid each other because they repel, so a calculation without repulsion has them moving independently.”

At U = 0 the exact ground state of a two-site model has a neighbouring spin correlation of −0.125000000, not zero. The same number comes out of the one-electron density matrix as −½ρ², with no repulsion anywhere in the derivation. Two electrons of the same spin cannot occupy the same site whatever the interaction is, and that alone anticorrelates the spins.

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

“Dispersion forces are negligible.”

Computed for argon from the London expression at its nearest-neighbour separation, one pair is bound by 0.0099 eV — small against a covalent bond by a factor of 251, and large enough that argon is a solid below 84 K. Every molecular crystal, every protein fold and all of adhesion runs on the quantity being called negligible.

Tested in What holds a solid together · the series on cohesion

“A character table lists the representations somebody found useful; a group could have more of them.”

The number of inequivalent irreducible representations equals the number of conjugacy classes, exactly. C2v has four classes and four representations, and the sum of the squares of their dimensions is 1+1+1+1 = 4, which is the order of the group. A fifth representation of any dimension whatever would push that sum past four, and the sum is an identity rather than a bound.

Tested in Why a character table stops where it stops · the series on representation

“The characters in a table are measured or fitted quantities of some kind.”

They are traces of matrices, and the table's shape is settled before any of them is written down. Reducing the regular representation — character h at the identity and zero everywhere else, because multiplying by anything else moves every group element — returns each representation exactly as many times as its own dimension, in all nineteen tables here, with the multiplicities coming out as whole numbers or the reduction refusing.

Tested in Why a character table stops where it stops · the series on representation

“A photoelectron spectrum shows one band per bonding electron pair, so a molecule with four equivalent bonds shows four bands.”

Methane has four equivalent C–H bonds and two valence bands, at 12.7 and 23.0 eV. The four hydrogen 1s functions span a₁ ⊕ t₂ in Td, so the eight valence electrons occupy two symmetry species — one non-degenerate and one triply degenerate — and a spectrum counts species rather than pairs.

Tested in Fewer bands than electrons · the series on photoelectron

“Lowering a molecule's symmetry does not change how much information its spectrum carries, since the same electrons are there.”

Methane, ammonia and water all have eight valence electrons and give two, three and four bands. The valence basis spans A1 ⊕ T2 in Td, 2A1 ⊕ E in C3v and 3A1 ⊕ B1 ⊕ B2 in C2v — four occupied orbitals throughout, falling into two, three and four species. Every degeneracy the group permits is one band fewer.

Tested in Fewer bands than electrons · the series on photoelectron

“A lattice energy is a sum over neighbours, so it can be computed by adding up shells until the terms get small.”

Summing rock salt shell by shell gives partial sums of 6.000, −2.485, 2.134, −0.866, 9.867 and 0.069 for an answer of 1.7476 — oscillating with an amplitude larger than the answer and getting worse at the fifth shell. The terms do not shrink: a shell at distance r holds about r² ions each contributing 1/r, so every shell contributes about as much as the last. The series converges only conditionally, and a different order of summation gives a different sum.

Tested in The lattice sum that depends on the order of adding · the series on cohesion

“The tight-binding picture could be extended to ionic solids by including more neighbours.”

The tight-binding matrix works because the covalent interaction decays exponentially, so second neighbours contribute a few per cent and third essentially nothing. A Coulomb interaction decays as 1/r and never becomes negligible; there is no distance at which the list of neighbours can be truncated, and the whole structure of the method fails rather than becoming inaccurate.

Tested in The lattice sum that depends on the order of adding · the series on cohesion

“Drawing a set of orbitals at the same isovalue makes them comparable.”

It makes them incomparable in a particular way. At ψ = 0.02 atomic units, hydrogen's 1s encloses 96.2 per cent of its density, its 2s 51.8, its 3s 1.3 and its 4s 0.4. The picture of the 4s is a surface containing four electrons in a thousand and looks like an orbital; the picture of the 1s is a surface containing almost all of one and looks like an orbital too.

Tested in The isovalue nobody chose · the series on contour

“A ring with fixed bond lengths and fixed bond angles has a fixed shape.”

The rank of the closure constraints says otherwise. For six atoms the chair's constraint matrix has rank 9 out of 12, so its only zero modes are the three rotations and it is rigid; the twist-boat's has rank 8, leaving one direction that changes no bond and no angle. Following that direction traces thirty-six distinct conformers, every one with the same six bonds and the same six angles to a residual below 10⁻¹¹, and torsions spanning 120°.

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

“Ligands split a d shell in proportion to how strongly they repel the d electrons, so more charge means a bigger splitting.”

Of the twenty pairs in the tabulated series that differ in charge, three are in the order a point-charge argument predicts and seventeen are not. The two ligands with the largest measured splittings, carbon monoxide at 34,000 cm⁻¹ and ammonia at 21,600, are electrically neutral, and iodide with a full negative charge is at the bottom of the list at 7,000.

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

“Loosening the tolerance can only move the answer up a chain of subgroups.”

A benzene stretched by 2 per cent along one axis is D2h up to a tolerance of 0.04 Å and C6h at 0.08. D2h has order 8 and C6h has order 12, and 8 does not divide 12 — so by Lagrange the first is not a subgroup of the second. What is FOUND does grow monotonically: the count of accepted operations rises from 7 to 15 to 19 across the same sweep. It is the classification of what was found that is not monotone.

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

“Molecular orbital theory and valence bond theory are two languages for the same answer, so the choice between them is a matter of taste.”

On an exactly solvable two-site, two-electron model they give different energies at every repulsion strength but one. At U = 0 the molecular orbital answer is exact and valence bond is 2t too high; at U = 32t valence bond is 0.12t too high and molecular orbital is 14.12t too high. They are equally wrong at exactly U = 4t, where both miss by 0.8284t, and the errors run in the same direction and at wildly different rates.

Tested in Where molecular orbital theory dissociates · the series on models

“A molecular orbital wavefunction describes a bond that weakens smoothly as the atoms separate.”

The doubly occupied bonding orbital gives a total double occupancy of exactly 0.5 at every value of the repulsion and therefore at every bond length — the two electrons are on the same atom half the time whatever the separation. The exact ground state gives 0.5 at U = 0, 0.1464 at U = 4 and 0.0039 at U = 32. The failure is not a matter of degree: the description contains a fixed fifty per cent ionic character it has no way to remove.

Tested in Where molecular orbital theory dissociates · the series on models

“Benzene's bonds are equal because delocalisation stabilises the π system.”

The π energy is LOWERED by alternating the bonds, at every ring size computed here. Fitting the gain against the distortion gives a coefficient of 5.96 for benzene, 13.19 for a ten-ring and 45.97 for cyclobutadiene, all positive — so a π system left to itself would alternate. What keeps the ring regular is the σ frame, which pays for every bond it stretches.

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

“A material with a half-filled band is a metal.”

The one-electron gap of a half-filled four-site ring is exactly zero at every value of the repulsion, because the repulsion is not in that matrix. The exact charge gap of the same ring — E(N+1) + E(N−1) − 2E(N), three separate diagonalisations — is zero at U = 0 and rises to 5.99 t at U = 8, with no sign of stopping. Nickel oxide has a half-filled d band and a gap of about four electronvolts.

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

“Adding a lone-pair moment repairs the additive bond-dipole model.”

For phosphorus trifluoride the bond sum comes out at −6.14 D against a measured total of 1.03, so the leftover would have to be at least 5.1 D — larger than the dipole of any small molecule, and far larger than any lone pair could supply. The additive model has failed before the lone pair is reached, and a correction refitted to every molecule it is applied to is not a correction.

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

“The symmetry-adapted combinations of a degenerate representation are determined by the symmetry.”

The projector onto E1g applied to the p function on atom 1 gives (0.577, 0.289, −0.289, −0.577, −0.289, 0.289); applied to atom 2 it gives (0.289, 0.577, 0.289, −0.289, −0.577, −0.289). Both belong to E1g, both are correct, and they overlap by exactly 0.500 — they are two bases of one subspace, related by a rotation nobody chose. What is determined is the subspace, and the Hückel eigenvalue computed in either is 1.000000 to twelve decimal places.

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

“Adding electrons to a conjugated system increases its π binding energy, since there is more bonding to go round.”

The cyclopropenyl cation's π energy is 4.0000β and the anion's, with two more electrons, is 2.0000β. The extra pair goes into a degenerate antibonding level at −1β apiece, so the total falls by 2β. Cycloheptatrienyl falls from 8.9879β at six electrons to 8.0978β at eight by the same mechanism.

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

“A ring with a delocalisation energy of zero has no π bonding in it.”

Square cyclobutadiene has a π energy of 4.0000β at two, four and six electrons alike, and a delocalisation energy of exactly zero at all three — because the reference state of isolated double bonds has the same energy, not because the π system is absent. The delocalisation energy is a difference against a stated reference and can be zero while both terms are large.

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

“Square-planar complexes are square planar because that arrangement minimises ligand repulsion.”

It maximises it among four-coordinate arrangements. A tetrahedron's angles are all 109.47° and a square plane has four at 90° and two at 180°, so the square plane is worse on every repulsion measure. The arrangement is adopted in spite of the repulsion, which is why the effect appears only where a ligand field term is large enough to pay for it.

Tested in VSEPR does not reach a transition metal · the series on VSEPR

“If a molecule's infrared and Raman spectra have no band in common, it has a centre of inversion.”

Eclipsed ferrocene is D5h, which contains no inversion. Its 57 vibrations span 4A1′ ⊕ A2′ ⊕ 6E1′ ⊕ 6E2′ ⊕ 2A1″ ⊕ 4A2″ ⊕ 5E1″ ⊕ 6E2″; the infrared-active species are E1′ and A2″, the Raman-active are A1′, E2′ and E1″, and the two lists share nothing. The rule runs one way only, and the counterexample needs a fivefold axis: the coordinates go to E1 and their products to E2, which cannot happen with a threefold or fourfold axis.

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

“Two identical impurities in a crystal give two identical levels at the same energy, since each sees the same environment.”

Two impurities of strength −2β in a chain of 61 give levels split by 0.535β at two sites apart, 0.0143 at six and 1.2×10⁻⁵ at fourteen. The splitting is never zero at any finite separation, because the two bound states overlap, and it falls geometrically with a ratio of 0.41421 per site — the same κ the isolated bound state decays with.

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

“The correlation energy tells an attraction from a repulsion.”

At U and −U the correlation energy is identical to the last bit a double holds, and so are the natural occupations. What differs is the sign of the mixing coefficient: a repulsion of 4 gives an ionic weight of 0.146 and an attraction of 4 gives 0.854, and the two weights sum to exactly one. Every scalar measure of correlation is blind to a qualitative difference between two states.

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

“The difference between them is a matter of interpretation.”

It is one number. The weight of the configurations with both electrons on the same atom is fixed at exactly 1/2 by the molecular-orbital function and at exactly 0 by the valence-bond function, whatever the repulsion. The exact value runs 0.500, 0.379, 0.276, 0.146, 0.053, 0.015 as U/t goes 0, 1, 2, 4, 8, 16 — never reaching either.

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

“A calculation whose energy is accurate to a tenth of a per cent has a wavefunction accurate to about that much.”

The energy error is second order in the wavefunction error and an observable's is first, measured here at exponents of 2.00 and 1.01. A trial state overlapping the exact one by 0.9806 — a two per cent error in the wavefunction — has a double occupancy out by 53 per cent. Halving the wavefunction error divides the energy error by four and the observable's by two, so the ratio doubles at every halving without limit.

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

“A method that is accurate for one molecule is accurate for two of them.”

Truncating the expansion at one doubly occupied site is exact for one Hubbard dimer, because two electrons cannot doubly occupy two sites. For two independent dimers at U = 4t it misses 0.1928 of a hopping unit, and for three it misses 0.4853 — an error PER DIMER of 0, 0.0964 and 0.1618. The exact energies are additive to 10⁻⁹: −0.828427, −1.656854, −2.485281.

Tested in A method that is not additive · the series on correlation

“The error of a truncated method is a fixed percentage of the correlation energy.”

The fraction recovered falls as the system grows, and it falls faster where there is more to recover. With the repulsion switched off — the most correlated case in this model, since the exact state is then maximally ionic — the same truncation misses 1.1716 for two dimers and 2.5359 for three, against 0.1928 and 0.4853 at U = 4t. Both the error and its growth depend on the repulsion, so no single percentage describes the method.

Tested in A method that is not additive · the series on correlation

“A solid with more valence electrons is more strongly bound, since there is more bonding to go round.”

The occupied-level sum per site of a ring of 60 rises to 1.2721β at half filling and falls symmetrically to exactly 0 when the band is full. Beyond half filling every added electron enters an antibonding level and subtracts as much as an electron below it added. The measured cohesive energies of the transition series peak near the middle for the same reason.

Tested in Half filled is as bonded as it gets · the series on Bands in a solid

“A filled band still holds the solid together — the electrons are in bonding orbitals lower than the atomic level.”

The sum of every level of a symmetric band is exactly zero, because the trace of the hopping matrix is zero: each bonding level below the atomic energy is matched by an antibonding one exactly as far above. Computed here the full-band value is 0 to 10⁻¹⁵β per site, which is arithmetic noise rather than a small residue.

Tested in Half filled is as bonded as it gets · the series on Bands in a solid

“A character table determines the point group.”

Comparing character matrices and class sizes rather than names finds four sets of point groups sharing a table: {Cs, Ci, C2} at order 2, {C2v, C2h, D2} at order 4, {C4v, D4, D2d} at order 8, and {D5h, D5d} at order 20. Every entry agrees; only the class labels differ.

Tested in One table, three groups · the series on representation

“Two groups with the same character table make the same physical predictions.”

Of {C2v, C2h, D2}: one may be polar, one is centrosymmetric, one is chiral, and no two of them are the same. A z-polarised transition belongs to A1 in the first, Au in the second and B1 in the third. Every number in the three tables is identical.

Tested in One table, three groups · the series on representation

“A quenched orbital moment contributes nothing to a magnetic measurement.”

Spin-orbit coupling mixes the quenched ground state with excited d states, and the g-value shifts by −2λ times a sum of squared matrix elements over energy denominators. For copper(II) that is 8|λ|/Δ along the axis and 2|λ|/Δ across it, giving g∥ = 2.417 and g⊥ = 2.095 against measured 2.39 and 2.08.

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

“A quadrature that agrees with itself at higher resolution has converged.”

A product rule over a box fourteen bohr across, checked by raising the number of points, is 9 per cent low at one bohr and 31 per cent high at six for two 2p orbitals in σ — and agrees with itself to five decimal places throughout, because the error is in the domain and the check varies the resolution.

Tested in Closer is not more overlap · the series on overlap

“A polyiodide is an iodide ion loosely holding some iodine molecules.”

That picture predicts short bonds where the I₂ units are and long ones between. The chain predicts strong bonds at the ENDS and weak in the middle — orders 0.789, 0.577, 0.577, 0.789 for five centres — and I₅⁻ is measured at 2.81, 3.17, 3.17, 2.81 Å. Short outside, long inside, which is the chain's order and the reverse of the other.

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

“A self-consistent field gives one answer for a molecule.”

It gives whichever answer the symmetry it was told to look for allows. Above a repulsion of about 2t the same iteration on the same four-site system finds a second solution with up electrons on alternate sites, and it is LOWER: at U = 16t the symmetric solution gives +11.53 and the polarised one −0.374, against an exact −0.583. The better mean field is the one describing a spin arrangement the exact ground state does not have.

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

“A structure with more neighbours per atom is more strongly bonded, in proportion to the number of neighbours.”

Binding per site goes 1.272, 1.611, 1.979 for coordinations of two, four and six — a rise of 56 per cent while the coordination triples. Binding per bond therefore falls from 0.636 to 0.330. More neighbours means more binding in total and less binding per bond, which is why a close-packed metal has bonds worth a fraction of a covalent one and a great many of them.

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

“A molecule with no dipole moment has no long-range electrostatic interaction.”

Benzene's quadrupole moment is −8.7 D Å. Two benzenes 5 Å apart interact by +8.8 kJ/mol stacked and −4.6 kJ/mol edge-to-face, from the quadrupole alone with no dispersion in the calculation at all. The sign changes between 50° and 60° of relative tilt, so the two arrangements are not merely different in strength.

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

“The momentum-space wavefunction contains different information from the position-space one.”

It is a unitary transform: nothing is lost and nothing is gained. Every moment here is computable from either. What changes is which facts are obvious — the kinetic energy is a curvature in position and a width in momentum, and ⟨p²⟩ computed by transforming agrees with 2⟨T⟩ from the virial theorem to three parts in a thousand for every orbital tested, which is the same number arrived at two ways.

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

“A function that is nearly a duplicate is a harmless waste.”

A fifth Gaussian at 1.01 times an existing exponent lowers the energy by 3.4×10⁻⁹ hartree against a well-placed fifth function's 5.3×10⁻⁴ — a factor of 160,000 — while the smallest overlap eigenvalue falls to 4.6×10⁻⁶. At 1.00003 times, the set is refused: it is no longer a basis.

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

“A surface costs one broken bond per surface atom, at the bulk value of a bond.”

A surface site keeps 91.24 per cent of the binding a bulk site has while keeping 83.33 per cent of its bonds. The one broken bond therefore costs 0.526 of what a bond is worth in the bulk — the estimate is nearly double the truth, at a coordination of six.

Tested in A surface is not a count of broken bonds · the series on cohesion

“The error is a small correction to a sound argument.”

The computed fraction is 91.24 per cent against √(5/6) = 91.29 per cent, agreeing to five hundredths of a point, and against 5/6 = 83.33 per cent, differing by eight points. The bond-counting estimate is not a first approximation to the answer; a different rule is, and it is not more complicated.

Tested in A surface is not a count of broken bonds · the series on cohesion

“A rotational spectrum determines a molecule's moments of inertia.”

For a symmetric top the transition energies are 2B(J+1) for every J and every K, to 10⁻¹², so the constant of the unique axis appears in no line. Ammonia's A is 6.3406 wavenumbers against B of 9.8601 — a difference of 36 per cent of the larger — and the spectrum reports only the second.

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

“An accidental degeneracy is an accident.”

It survives no departure from 1/r whatever. ⟨1/r²⟩ is 0.2500 for the 2s and 0.0833 for the 2p, in the ratio 3:1 exactly, so a correction going as 1/r² splits the shell at first order — and 5:1 across the n = 3 shell. Classically, the orbit closes for an inverse-square force to 0.003 degrees a turn and precesses by 17 to 20 degrees a turn for exponents of 1.9 and 2.1.

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

“The shape of a density of states is a fact about the chemistry.”

Every structure here is a graph with no chemistry in it at all: the matrix has ones where two sites are joined and zeroes elsewhere. The three shapes differ because the three dimensions differ, and nothing else varies between them.

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

“The spread is a thermal effect and vanishes on cooling.”

Every number here is the ground state of the oscillator, at no temperature. The mean square displacement of a normal coordinate is ħ/2ω, which is what it is at absolute zero; warming the molecule only increases it. Hydrogen's own molecule is spread by 0.087 Å against a bond of 0.741 — 11.8 per cent — with no thermal contribution at all.

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

“Repulsion is a correction to the two-level result.”

At a difference of 8 and no repulsion the transfer is 0.9701 of an electron; at the same difference and U = 16 it is 0.0247. A correction changes an answer by a fraction of itself. This changes it by a factor of thirty-nine, and the limit at large repulsion is zero transfer at every difference.

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

“A molecule needs a dipole moment to have an infrared spectrum.”

Carbon dioxide's dipole is exactly zero at every origin and three of its four modes have a non-vanishing dipole derivative — the bend at 673 cm⁻¹ twice over and the antisymmetric stretch at 2396. What a band needs is a dipole that changes as the molecule moves, which is a different quantity.

Tested in A dipole is not what an infrared spectrum sees · the series on dipole

“A more polar molecule has a stronger infrared spectrum.”

Of five molecules here, the three with no dipole at all — carbon dioxide, methane and boron trifluoride — account for fourteen of the nineteen active modes. Boron trifluoride's largest dipole derivative is 0.1676 against water's 0.1751, on molecules whose dipole moments are 0 and 0.1697.

Tested in A dipole is not what an infrared spectrum sees · the series on dipole

“Deciding which bands appear needs a calculation.”

It needs the point group. Every mode whose species carries none of the coordinate functions has a dipole derivative that vanishes — computed here from displacements and charges, with no group anywhere in the calculation, at 10⁻¹⁶ for exact coordinates and 10⁻¹⁰ for coordinates quoted to four decimal places. The two routes agree on all twenty-eight modes.

Tested in A dipole is not what an infrared spectrum sees · the series on dipole

“Cyclooctatetraene is aromatic if it is planar, since its delocalisation energy is 1.66β.”

1.6569β is what the isolated-bond reference gives, and it is 83 per cent of benzene's. Against the ring cut open the same planar structure gives 0.1393β, an eighth of benzene's 1.0121β and a fortieth of a beta per electron. The first number is measuring the four double bonds it was allowed to count, not the ring.

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

“A vibrational assignment's percentages are properties of the mode.”

Boron trifluoride's 1454 cm⁻¹ mode is 36.7 per cent B–F stretch by squared displacement, 49.9 per cent with the angles measured in radians instead of ångström, 97.9 per cent by the potential-energy distribution, and 94.6 per cent by its diagonal part. The mode is the same eigenvector in all four.

Tested in How much of a band is a bond stretch · the series on normal mode

“A potential-energy distribution is a distribution, so its entries are between nought and one.”

The entries include the off-diagonal terms of the force matrix, which belong to no single coordinate. Water's bend at 1649 cm⁻¹ comes out at 100.1 per cent of the energy with each of its two stretches at −0.03 per cent — small here, and negative, which no share of anything can be.

Tested in How much of a band is a bond stretch · the series on normal mode

“The conventions disagree, so the numbers mean nothing.”

Where a mode really is one coordinate every convention says so and says the same one: boron trifluoride's 888 cm⁻¹ band is 100.0 per cent B–F stretch under all four, water's 3833 is 100.0 per cent stretch under all four. The disagreement is confined to mixed modes, which is exactly where the percentage is being used to decide something.

Tested in How much of a band is a bond stretch · the series on normal mode

“A ligand's charge tells you how it behaves in this respect.”

Of the anionic ligands here, iodide, bromide, chloride and fluoride close at twelve and cyanide closes at eighteen. Of the neutral ones, water closes at twelve, carbon monoxide at eighteen, and ammonia at neither. The sign of eπ sorts all nine perfectly and the charge sorts none of them.

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

“The period of the distortion is a property of the lattice.”

The same 120-site ring, with the same hopping and the same elastic cost, chooses period two, three or four according to how many electrons are in it. Nothing about the structure changes between the three cases; only the electron count does.

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

“A basis set that gives the energy to five figures is accurate to five figures.”

The best six-Gaussian fit to hydrogen's 1s gives −0.499946 against −0.5, an error of 0.011 per cent. Its Compton profile at zero momentum is 0.84653 against 8/3π = 0.84883, an error of 0.27 per cent — twenty-five times larger, and larger still at other momenta.

Tested in The measurement a basis was not fitted to · the series on basis

“Improving a basis improves every property it computes at the same rate.”

Going from one Gaussian to six divides the energy error by 1,389 and the profile error by 43. The ratio between the two errors runs 0.8, 4.8, 24.9 across the three bases, so the two do not merely differ — they diverge.

Tested in The measurement a basis was not fitted to · the series on basis

“The error a fit makes is spread evenly over the property it computes.”

The relative error in the three-Gaussian profile crosses zero several times between q = 0 and q = 4 and reaches 16.7 per cent at its worst. A fit is a compromise, and a compromise buys agreement in one place by paying for it in another.

Tested in The measurement a basis was not fitted to · the series on basis

“Hydrogen's l-degeneracy is accidental.”

Six operators built from the shell's own dipole integrals close into an algebra — [Aᵢ, Aⱼ] = −εᵢⱼₖ Gₖ to 1.2 × 10⁻⁸, which is the integrator's noise — and their Casimir comes out at 3.0000000 against n² − 1 = 3. A degeneracy that follows from an algebra that closes is not an accident; it is the signature of a symmetry that is not a rotation of ordinary space.

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

“A picture of a σ bond is two atomic orbitals overlapping.”

At two bohr the surface enclosing ninety per cent of the molecular orbital's density is a single closed surface with both nuclei inside, at |ψ| = 0.0359. The pair of atomic surfaces at the same stated fraction sits at 0.0394, has a waist where the molecular one does not, and encloses 91.70 per cent of the molecular orbital rather than ninety.

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

“Whether a bonding orbital looks like one object or two is a fact about the bond.”

The contour divides in two at 4.00 bohr if it encloses half the density, 6.76 bohr at ninety per cent, and 9.82 bohr at ninety-nine — a factor of 2.45, on the same orbital at the same separations. Nothing happens to the orbital at any of these; what changes is the level the caption asked for.

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

“A molecule with a non-degenerate ground state keeps its symmetric structure.”

The second-order lowering from mixing in an excited state goes as λ²Q²/Δ and the elastic cost as ½kQ². Both are quadratic, so the symmetric structure survives only while Δ exceeds 2λ²/k. At λ = 0.5 and k = 0.5 that is a gap of 1.00, and a molecule with a gap of 0.7 distorts to Q = 0.85 with nothing degenerate anywhere.

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

“Any low-lying excited state will do.”

The triple product of the two states and the distortion must contain the totally symmetric species. In an octahedron a T1u distortion acting on an A1g ground state can mix in exactly one of the ten species — T1u — so a molecule with no T1u excited state cannot distort along that mode at any gap whatever.

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

“Correlation energy is a small correction.”

For four electrons on four sites the mean field's error is 0.10 at U = t, 1.48 at U = 4t, 12.11 at U = 16t and 27.82 at U = 32t, while the exact energy over the same range shrinks from −3.58 to −0.29. At U = 32t the correction is ninety-four times the quantity it is correcting.

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

“How much correlation energy a system has is something only a calculation can say.”

The rate at which the error grows with the repulsion is n²/4N — 0.25, 1.00 and 1.50 for two electrons on four sites, four on four, and six on six. The measured slopes are 0.2477, 0.9820 and 1.4715. Nothing in the prediction is an energy: it is the number of times a uniform density puts two opposite spins on the same site.

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

“A valence bond wavefunction is so many per cent covalent and so many per cent ionic.”

The structures are not orthogonal — at hydrogen's bond length the covalent and ionic ones overlap by 0.873 — so squaring a coefficient is not a weight. The three standard repairs give 18.73, 34.74 and 5.88 per cent for one wavefunction, and each of them sums its weights to one.

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

“The disagreement is a technicality about how the numbers are normalised.”

Take the wavefunction with no ionic structure in it at all. Chirgwin–Coulson and the inverse-overlap convention both report zero ionic weight; Löwdin reports 25.59 per cent, because symmetric orthogonalisation moves ionic character into the covalent structure before anything is squared.

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

“Two structures that a point-group search calls by the same name are equally symmetric.”

Benzene with one atom moved 0.06 Å and benzene with alternating bonds both first come out as D6h at a tolerance of 0.08 Å. Measured against D6h itself, the alternating structure is 0.00962 and the one-atom structure 0.00558 — 72 per cent further, at the tolerance the search treats as identical.

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

“Finding the nearest symmetric structure means searching over structures.”

It is an average. For a group of order h with the permutation each operation induces, the nearest structure with that symmetry is qᵢ = (1/h) Σ g⁻¹ r[g(i)], and applying any operation of the group to that returns it exactly — checked to 10⁻⁹ rather than argued for. No minimisation is involved.

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

“A molecule has a bond length.”

Hydrogen chloride's potential has its minimum at 1.27455 Å. The average separation in its ground vibrational state is 1.28985, and the length implied by its rotational constant — which measures ⟨1/r²⟩ rather than ⟨r⟩ — is 1.28299. Three numbers from one potential, spanning fifteen milliångström.

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

“The zero-point motion smears a bond length out symmetrically.”

A harmonic well would. A real one is steeper inward than outward, so the average is displaced outward — 15.30 mÅ for HCl, 3.95 for carbon monoxide — and the displacement scales as the square root of the anharmonicity constant, which is checked here by changing that constant by a factor of four hundred and watching the shift change by twenty.

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

“Ring strain is angle strain.”

A flat cyclopentane has interior angles of 108.0°, which is 1.5° from tetrahedral, and an angle strain of 0.4 kJ mol⁻¹ by an ordinary bending constant. It is strained by 26 kJ mol⁻¹. The difference is torsional: a flat ring has every bond eclipsed, which costs one ethane barrier — 12.1 kJ mol⁻¹ — per bond, whatever the ring size.

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

“The chair is the shape the bond angles pick out.”

Bond angles and ring closure leave a continuous family of six-rings — a result of counting constraints against freedoms — and every member has the same angle strain by construction. The best member a search over those constraints returns has a torsional energy of 23.9 kJ mol⁻¹ against the chair's 1.41. What picks the chair out is the torsional term and nothing else.

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

“A distortion lowers a molecule to a subgroup that depends on the molecule.”

It depends on the species. Applying each of benzene's twelve species as a distortion on its own and recovering the point group from the coordinates gives D6h, C6h, D3d, C2h, C6v, D3h, Cs and C2 — one per species, and two species that describe no displacement of twelve atoms at all.

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

“Counting levels below the band counts the defects.”

It does exactly, up to one impurity in ten, and then stops: at fifteen per cent a ring of a hundred and sixty carries 23.25 levels below the band for twenty-four impurities, and at twenty per cent 30.50 for thirty-two. Two impurities on neighbouring sites interact strongly enough to push one of their two levels back into the host band, and the shortfall is a measure of how often that happens.

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

“There is a concentration at which impurity levels start interacting.”

There is not. The width of the impurity band grows smoothly from zero — 0.014 at two and a half per cent, 0.29 at five, 1.45 at ten — so the levels were always interacting and the question is only when the interaction exceeds whatever resolution is being used to look.

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

“The lowest-energy conformer of a ring is the one with the best torsion angles.”

Of the twelve conformers here, the one with the lowest torsional energy — 44.5 kJ/mol, ten below its nearest rival — brings two carbons to 2.639 ångström of each other. The two-term account's favourite is not the roomiest structure, and on the quoted contact potential its cross-ring term alone is 122 kJ/mol.

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

“A magnetic moment is what a susceptibility measurement returns.”

A susceptibility measurement returns χ at each temperature. A moment is extracted by fitting a model, and for a coupled pair the model is wrong: fitting the same exact data over 80–150 K gives 3.590 Bohr magnetons and over 300–600 K gives 2.471, a spread of 45 per cent.

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

“A good fit means the parameters are right.”

The 150–300 K fit reproduces its own data with r² = 0.99950 and returns a Weiss temperature of −78.1 K for a system whose coupling corresponds to −48.0 K in the mean-field conversion. The 300–600 K fit has r² = 0.99999 and returns −51.4. The quality of a fit measures agreement over the range fitted and says nothing about the extrapolation the parameters are.

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

“The virial ratio checks that a calculation has converged.”

Scaling every exponent is a variation the optimiser is free to make, so at any stationary point the ratio is two identically. It is 2.0000000 for the six-function basis whose energy is right to a hundredth of a per cent and 2.0000001 for the one-function basis whose energy is 15 per cent wrong. A quantity that is true of a bad answer as well as a good one is not a diagnostic.

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

“Agreement with an exact property is evidence that a wavefunction is right.”

The optimal single Gaussian has ⟨r⟩ = 3/2Z exactly, which is hydrogen's exact value, because the stationarity condition in the exponent is that equation. The same function has ⟨1/r⟩ = 8Z/3π, which is 15.1 per cent below the exact Z, and a density at the nucleus 76 per cent too small.

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

“Electronegativity is the atom's pull on shared electrons, and that is all a bond needs.”

The pull is the first derivative of the atom's energy with respect to its electron count. Moving charge changes that pull, and the rate at which it changes is the second derivative — the hardness, which runs from 1.92 eV for potassium to 7.30 for nitrogen. The charge that flows is the ratio of the two, not the first alone.

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

“The correlation energy is a property of a system.”

It is the difference between the exact energy and a mean-field one, and the second term is a choice. For four electrons on four sites at U = 32 the restricted reference gives −27.82 and the unrestricted one −0.107. The exact energy in both subtractions is the same number.

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

“A material with states at the Fermi level is a metal.”

The states are there and each of them lives on ten sites out of two hundred. Growing the ring by a factor of four grows the clean chain's states by 3.8 and the disordered chain's by 1.27, so at that disorder a state has stopped knowing how large the system is — which is what makes it unable to carry anything from one end to the other.

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

“Disorder makes a gap.”

It does not. The band widens rather than splitting — from 4.00 to 6.80 in units of the hopping as the disorder goes from zero to four — and the density of states at the middle stays finite. Nothing is removed from the Fermi level; what changes is what the states there look like.

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

“A broad band means a poorly resolved one.”

The breadth is computed here from two measured potential curves and comes out as a distribution over the ion's vibrational states, with 0.263 of the intensity in the lowest and 0.318 in the next. The band is spread because the geometry changed, and its width in quanta is √S, the square root of the displacement parameter.

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

“Bonding and antibonding cannot be told apart in a spectrum.”

The sign of the bond-length change separates them and the vibrational spacing measures it. Removing nitrogen's 1πu electron lengthens the bond and lowers the vibrational frequency from 2359 to 1904 wavenumbers; removing the 2σu electron shortens it and raises the frequency to 2420. Both are read off the same spectrum.

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

“Hypervalency happens when an atom uses more than four orbitals.”

In every geometry here the number of ligand σ combinations that transform as one of the central atom's s and p orbitals is at most four — two for a linear ligand set, three for a trigonal planar one, four for tetrahedral, trigonal bipyramidal, square planar and octahedral. Six ligands and four matches leaves two combinations with no partner, and the electron count exceeds eight by exactly four.

Tested in Four is all that s and p can match · the series on hypervalency

“Hypervalent molecules are the ones with more than four ligands.”

Xenon difluoride has two ligands and is the archetype. The census makes hypervalency the condition n + L > 4, counting lone pairs as well as ligands, and the two mechanisms are different: sulfur hexafluoride's leftover pair has no central orbital of its symmetry, xenon difluoride's has one that is already holding a lone pair. Xenon tetrafluoride has one of each.

Tested in Four is all that s and p can match · the series on hypervalency

“The octet is broken in these molecules.”

The electron count around the centre exceeds eight and the electrons on the centre do not. In every row of the census the centre holds at most eight, in at most four orbitals, and the excess sits in ligand combinations with no central partner — which is why the ligands in a hypervalent molecule carry the negative charge.

Tested in Four is all that s and p can match · the series on hypervalency

“β is measured from a spectrum.”

A spectrum with two lines in it constrains two combinations of three parameters. Fitting benzene's two π ionisation energies exactly at neighbour overlaps from 0 to 0.30 gives resonance integrals from −3.05 to −7.66 electronvolts, a factor of 2.5, and every member reproduces both measurements to the last digit the solver carries.

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

“The bond orders are a structural prediction, so they do not depend on the parametrisation.”

The orbitals do not depend on it — the same six vectors come out at every overlap, pointing in the same directions to nine decimal places. The bond orders computed from them do: butadiene's terminal bond order is exactly twice its central one at zero overlap and 2.671 times it at 0.30, because each orbital is normalised against a different metric.

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

“A strong infrared band is a big motion.”

Across the active modes of five molecules the rank correlation between band strength and root-mean-square zero-point displacement is +1.00, +0.50, +0.20, −0.50 and −0.66. Two of the five run backwards, and boron trifluoride's strongest band has the third smallest amplitude of its five.

Tested in The mode that moves least radiates most · the series on dipole

“Heavier atoms moving means a stronger band.”

Every normal mode moves exactly the same weighted amount of mass — the sum of m|d|² is one for all of them, by the mass-weighted normalisation — so that quantity distinguishes nothing. What does vary is the reduced mass, from 1.03 to 20.3 across these molecules, and for boron trifluoride it ranks the intensities exactly backwards, at −1.00.

Tested in The mode that moves least radiates most · the series on dipole

“Infrared intensities are the part of a spectrum a simple model gets roughly right.”

They are the part it gets least right. Which bands exist is exact from the group alone and agrees with the computed derivatives on all twenty-eight modes; the positions come from a fitted force field and are good; the intensities need the electron density's response to a nuclear displacement, which a fixed-charge model does not contain at all.

Tested in The mode that moves least radiates most · the series on dipole

“The electron count depends on how the ligands are counted.”

It does not. Ten complexes counted both ways give the same total in every case — an anionic ligand brings one electron and leaves the metal neutral, or brings two and takes one from the metal, and the sum is unchanged by construction. The invariance is the useful part of the rule.

Tested in The same count, two oxidation states · the series on electron count

“The oxidation state is a property of the compound.”

Hexaamminecobalt(III) is cobalt(III) on the ionic convention and neutral cobalt on the other, and both counts give eighteen. For a nitrosyl the ionic convention itself does not settle it: counting NO as a cation, a neutral radical or an anion puts cobalt at −1, 0 or +1 in the same molecule, with a total of eighteen in all three.

Tested in The same count, two oxidation states · the series on electron count

“Since the total is the same, the choice of convention does not matter.”

Everything except the total differs, including the d count that a ligand-field diagram is labelled with. For three complexes with measured moments the ionic d count predicts 0.00, 4.90 and 2.83 Bohr magnetons against measurements of 0.00, 5.40 and 3.20; the group-number count predicts 1.73, 2.83 and 0.00, and is wrong by more than a Bohr magneton in each.

Tested in The same count, two oxidation states · the series on electron count

“If two atoms' orbitals do not overlap in the picture, they do not interact.”

Two ninety per cent contours of hydrogen 1s orbitals stop touching at 5.322 bohr. The overlap integral there is 0.0768 — larger than the overlap that produces a hydrogen bond — and it is still 0.0222 at seven bohr, where the two spheres are separated by a bohr and a half of apparently empty space.

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

“The structure that fits all the measured constants is the right one.”

Fitting the deuterated molecule's B as well picks the member at a C=O length of 1.2495 ångström, a C–H of 1.0222 and an angle of 134.6 degrees, against an accepted structure of 1.2033, 1.1005 and 116.3. The measured constants are vibrational averages and no rigid structure reproduces them: the measured moments fail the planarity relation by 0.057 u Ų, which every structure in the family satisfies exactly.

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

“A Möbius ring is a different kind of system from a flat one.”

It is a flat ring at exactly half a flux quantum. The twisted eigenvalues 2cos((2k+1)π/n) and the flux expression 2cos(2π(k+½)/n) are the same list for every ring from three to ten, to the last digit the solver carries, because a twist and a half-quantum put the same half-integer into the label.

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

“The binding energy per bond is a transferable quantity.”

Divide the bindings above by four and the same spread remains. A bond in a structure with many short closed walks through it is worth less than the same bond elsewhere, because the binding is a sum over a whole distribution of levels and the distribution is not fixed by counting neighbours.

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

“Error cancellation is a property of the method.”

It is a property of the pair. The same mean field applied to the singlet and the triplet of one four-site chain makes errors of 12.11 and 0.17 at the same repulsion — a factor of seventy — because one of the two states is nearly a single determinant and the other is not.

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

“A method that gets total energies badly wrong can still order two states correctly.”

From a repulsion of four times the hopping the mean field puts the triplet 0.53 below the singlet, where the exact answer has it 0.54 above. Not an inaccurate gap: the wrong ground state, and it gets worse as the repulsion rises.

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

“More neighbours means a more Gaussian band.”

A chain reaching to its third neighbour has six neighbours like a cubic structure and a fourth moment of 3.389 against the cubic 2.500 — on the far side of a Gaussian's 3, where no hypercubic structure of any dimension can go. A hypercubic band approaches 3 from below and never reaches it.

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

“Two structures with the same coordination have the same band.”

A chain touching its second neighbours and a square net both have four neighbours and the same normalised fourth moment, 2.250 to nine figures. Their sixth moments are 6.719 and 6.250, and their binding per unit band width differs by 27 per cent.

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

“Which modes appear depends on how big the geometry change is.”

It depends on the group. The zeros above are zeros of a projection between a symmetric displacement and a non-symmetric mode, and they stay zero at any size of change. An unsymmetric change of the same size activates them: the same modes get factors up to 0.34.

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

“A molecule with several progressions in one band has a low-symmetry ion.”

It has more than one totally symmetric vibration. Sulfur dioxide is C2v and has two, and a symmetric change of geometry gives both of them intensity — 0.145 and 2.089 — while boron trifluoride, of much higher symmetry, has one and shows one.

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

“Every molecule's internal coordinates span its vibrations.”

Carbon dioxide's do not. Its bending coordinate has an identically zero row in the matrix that turns a displacement into a coordinate change, because an angle at exactly 180° is stationary, so three coordinates span two directions where the molecule has four vibrations. Both bending modes have no coordinate at all.

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

“A force field fitted to every observed frequency of a molecule is determined by them.”

For a redundant coordinate set it is determined only up to a whole direction. Adding λ r rᵀ to the force constant matrix, for a redundancy r, changes the Cartesian Hessian by exactly zero — measured at 5.5 × 10⁻⁸ for methane at λ = 25, against 135 per cent for the same nudge along an ordinary coordinate.

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

“A symmetric cage localises to symmetric bonds.”

The six-vertex cage's four skeletal pairs come out as two two-centre orbitals and two spread over five atoms, reaching a functional of exactly 11/6 from every start tried. The maximum breaks the octahedron's own symmetry, and it is the unique maximum.

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

“A ninety per cent contour is a picture of how big an atom is.”

The surface two closed shells actually stop at encloses 99.73, 99.75 and 99.38 per cent of the outermost orbital's density for helium, neon and argon. A ninety per cent contour is far inside where the next atom is, and the two conventions differ by an amount that a picture cannot make small.

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

“Slater's rules and a Hartree–Fock fit are two ways of writing the same effective charge.”

They disagree by thirty-one per cent for neon's 2p — 5.85 against 4.4532 — and the disagreement is the difference between a contact distance 4.7 per cent too long and one 27.7 per cent too short. Helium, where the two agree to one part in a hundred, does not move.

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

“The secular determinant's overlap denominators move the maximum of the bonding away from the maximum of the overlap.”

They do not, and the calculation refused the expectation. At fixed atomic energies the stabilisation is a function of the overlap alone and a strictly increasing one, so it peaks where the overlap peaks: 4.16818 against 4.16817 bohr for a 1s with a 2s, and 7.58281 against 7.58281 for a 2s with a 2p.

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

“Doubling the overlap doubles the bonding.”

For two orbitals at the same energy the stabilisation is exactly 2(1−K)αS/(1+S), which saturates. Going from 0.05 to 0.1 buys 1.909 times the bonding; going from 0.4 to 0.8 buys 1.556.

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

“The spectrochemical series has no structural explanation.”

One computed overlap integral, squared, accounts for most of it. Across five chromium(III) complexes whose splittings vary by a factor of 1.96, the ratio of splitting to squared overlap varies by 1.31 — with the exponent imposed by the model rather than fitted, and with the bond lengths taken from structure determinations rather than adjusted.

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

“A triple bond's bent components are a different kind of hybrid from a double bond's.”

They are the same hybrid. Both have exactly one sixth s character and both make an angle of 101.53696 degrees with one another. A double bond's carbon is sp², shares a third of an s orbital with the C–C region and divides it between two components; a triple bond's is sp, shares a half, and divides it between three. One third over two is one half over three.

Tested in Three bent bonds, and the same hybrid · the series on hybrids

“The angle between two equivalent hybrids depends on the molecule.”

It depends on their s character and on nothing else. Two hybrids √s|s⟩ + √(1−s)|p⟩ are orthogonal exactly when s + (1−s)cos θ = 0, so cos θ = −s/(1−s). At a quarter that is 109.47122 degrees, at a third 120, at a sixth 101.53696 — and the computed angles here reproduce all of them.

Tested in Three bent bonds, and the same hybrid · the series on hybrids

“The bent-bond and σ–π descriptions of a triple bond are rival accounts.”

They are two bases for one three-dimensional occupied space, related by an orthogonal mixing. Every mutual overlap between the three bent bonds is zero to 10⁻³, which is the quadrature's limit rather than the algebra's, and the charge each puts off the axis is the same 0.31519 ångström for all three.

Tested in Three bent bonds, and the same hybrid · the series on hybrids

“A dimensionless ratio between two molecules is a safe way to quote a comparison.”

It is immune to both exact symmetries of the model — moving the zero of energy and changing the unit leave it alone to 10⁻¹⁵ — and it is the single most reference-sensitive quantity in the table. Benzene's delocalisation energy over cyclobutadiene's is infinite against one reference, because cyclobutadiene's is exactly zero, and 2.14 against the other.

Tested in Which numbers carry a frame · the series on delocalisation

“An orbital energy is a quantity.”

It moves under both exact symmetries. Adding 0.37 to the zero of energy moves benzene's highest occupied level by 0.37 β and changing the unit by 1.83 moves it by 0.83 β, and neither change alters anything about the molecule. It is a coordinate, and only differences of coordinates are quantities.

Tested in Which numbers carry a frame · the series on delocalisation

“Every quantity a Hückel calculation prints depends on the parameters.”

Three of the nine tested do not depend on any of the four changes: the spread of bond orders, the ratio of two bond orders, and the largest charge on an atom. All three are properties of the eigenvectors, which are unchanged by any shift or scaling of the eigenvalues.

Tested in Which numbers carry a frame · the series on delocalisation

“The localised description of a molecule is a well-defined object.”

A twelve-vertex cage has ten of them, found by a search that then goes dry, and their functional values span 0.0236 — with the two closest separated by 9.89 × 10⁻⁶, which is seven orders of magnitude above the tolerance the sweep converges to.

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

“Multiple localisation solutions are the same solution reached by different routes.”

A symmetry of a structure permutes its sites and therefore cannot change the multiset of participation numbers of a localised set. All ten multisets here differ, so no two of the ten are related by any symmetry of the cage, whatever that symmetry is.

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

“A coordinate near a principal plane is merely determined with lower precision.”

Its square comes out negative. Every one of formaldehyde's four out-of-plane coordinates is imaginary at a fractional moment error of 10⁻⁵, and there is no precision at which the method returns a small number rather than an impossible one.

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

“Allowing the geometry to relax always increases the alternation, because a stronger bond becomes stronger still.”

Naphthalene's bond-order spread narrows, from 0.206330 to 0.205622, because its weakest bond is the one the two rings share and relaxation raises it from 0.5182 to 0.5386. A shared bond is fed by two rings and is not an isolated bond with a feedback loop on it.

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

“A self-consistent calculation of this kind could break benzene's symmetry.”

It cannot. Benzene's six bond orders stay equal to better than 10⁻⁹ and its π energy does not move by 10⁻⁹ either, because the feedback acts on a matrix whose symmetry it cannot lower.

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

“A finite piece of metal is a metal.”

Its level spacing is a gap. A uniform ring of forty-two has a spacing of 0.29892 against an alternating one's gap of 0.6688 — a factor of two, not of infinity — and at a temperature below either of them neither carries anything.

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

“The number of thermally excited carriers depends smoothly on the size of the system.”

A half-filled ring of 4m has levels exactly at the Fermi level and carries exactly one electron pair at every temperature above zero. A ring of 4m + 2 four sites away carries nothing at all below kT = 0.001. The two families never converge.

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

“A good residual means the model describes the sample.”

Every wrong-model fit here has R² above 0.990, and the worst of them reports a coupling a third too large. The residual measures how well a two-parameter curve follows a smooth monotone function of temperature, which almost any two-parameter curve does.

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

“Two atomic radii add up to the distance between the atoms because each is a surface the atom stops at.”

Radii set so that every ion's surface encloses the same fraction of its own density miss eight measured rock-salt separations by between −6.33 and +37.27 per cent. The fraction that would reproduce each one exactly runs from 91.26 to 99.42 per cent, so there is no such surface.

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

“A van der Waals radius and an ionic radius are two entries in the same kind of table.”

They are fitted to different measurements and behave differently. The enclosed fraction peaks at the neutral atom in both isoelectronic series here — 99.98 per cent at neon against 99.66 at Na⁺ and 99.31 at F⁻ — which is a discontinuity at exactly the row where the tabulation changes.

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

“A band's shape can be read off a density of states without asking where the band sits.”

The fourth moment about zero of a band centred at −Δ/2 is dominated by the offset: for a square net's band it returns 1.6900 at Δ = 8 and 1.0176 at Δ = 60, against a true shape of 2.2500. The number that comes back is plausible, stable and about the offset rather than the band.

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

“A band shape measures how strongly two bands are mixed.”

The lower band's shape is pushed one way and then the other and passes back through its own uncoupled value between mixings of 0.4 and 0.6, so one measurement of it is consistent with two very different couplings.

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

“A molecule that is not second-order Jahn–Teller active is not affected by second-order coupling.”

Activity is a statement about the symmetric structure. With the gap at 1.5 times the critical value the second-order term alone leaves the symmetric structure exactly where it was, and added to a first-order effect it moves the minimum from 0.600 to 1.075 — a 79 per cent larger distortion — and doubles the stabilisation.

Tested in Two distortions in one coordinate · the series on Peierls distortion

“The two effects can be added: work out each and take the larger.”

The stabilisation of the two together, 0.18031, exceeds the sum of the two computed separately, 0.09000, by 0.09031. They reinforce, so neither one alone is a lower bound on the answer and the sum is not an upper one.

Tested in Two distortions in one coordinate · the series on Peierls distortion

“A second-order term is largest where the gap is smallest, so a first-order distortion helps it by closing the gap.”

The branch a first-order term pushes down moves away from the excited state, so at the distorted geometry the gap has opened from 1.5 to 1.8225 — by twenty-two per cent. The two effects reinforce anyway, because the coupling grows as λ₂Q and the gap only as λ₁Q.

Tested in Two distortions in one coordinate · the series on Peierls distortion

“A depolarisation ratio measures how much the polarisability changes along a mode.”

It measures the ratio of two invariants and is blind to the size of either. Multiplying every bond polarisability derivative by ten multiplies every intensity by a hundred and moves no depolarisation ratio at all.

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

“A mixture's cohesive energy is the composition-weighted average of its components'.”

Never, in this model. Every intermediate composition binds more strongly than the line between the two pure ends, by up to 0.42987 per site at half and half, and the departure vanishes only when the two components are identical.

Tested in A mixture is not the average of its ends · the series on cohesion

“A bowing parameter is a property of a pair of components.”

It is a property of the pair, the arrangement, the contrast and the dimension. At one composition three arrangements give 0.42987, 0.26968 and 0.23690; and as the contrast grows the departure runs from a third of it to all of it.

Tested in A mixture is not the average of its ends · the series on cohesion

“An ordered compound and a random alloy of the same composition differ only in entropy.”

They differ in the band. At half and half, ordering opens a gap of exactly 2δ and random arrangement leaves 0.0997, and the ordered structure is more bound by 0.16 per site before any entropy is counted.

Tested in A mixture is not the average of its ends · the series on cohesion

“A photoelectron spectrum has one band per occupied orbital.”

At no repulsion it does: three bands for three occupied orbitals, and the whole of the intensity in them. At U = 8 the same molecule has a hundred lines above a millionth of the total, on six orbitals, and the strongest three hold 46.78 per cent of the intensity.

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

“Extra lines in a photoelectron spectrum are weak features that do not affect the assignment.”

By U = 8 they hold 53.22 per cent of the total intensity — more than the main lines — and the largest single satellite is stronger than two of the three main lines.

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

“Counterpoise correction makes a binding energy more accurate.”

It makes it less accurate here, at every basis size. A finite basis underbinds by incompleteness and overbinds by superposition, the two have opposite signs, and the first is between 57 and 185,855 times the second — so removing the second alone moves the answer further from the exact 0.102634 hartree at every one of six basis sizes.

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

“The superposition error is largest where the atoms are closest, because that is where the functions overlap most.”

It peaks at 4.19 bohr and falls away on both sides. Close in, the two atom-centred sets are so nearly the same set that the far one lends almost nothing new; at 1.4 bohr it is 0.03 microhartree against 235.7 at the maximum.

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

“A structure's departure from its ideal shape says what caused the departure.”

It says which coordinates are involved and in what proportion. What decides whether the molecule fell down its own slope is the force field, and a unit distortion along ammonia's stiffest coordinate costs 3.4965 times what one along its softest does — a factor computed from masses and force constants, which the species decomposition knows nothing about.

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

“The cost of a distortion can be read off its symmetry species.”

Methane's softest and stiffest vibrations are both of species T₂, at 1362.7 and 3173.3 cm⁻¹. A species is a label on a subspace, and a subspace of dimension three can contain modes of very different steepness.

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

“A correlation correction is a property of the method pair rather than of the system.”

The exact correlation energy of a four-site ring falls from −0.339451 to −0.000472 as its two kinds of site are pulled apart, while the transferred number does not move at all. By the far end the recipe is adding a correction seven hundred times the one the system needs.

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

“A composite scheme is at worst no better than its low level.”

It is fifteen times worse at ε = 8: the mean field's error is 0.020578 and the composite's is 0.318873, both against the exact ground state, so the correction has overshot by more than the whole of the error it was removing.

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

“The ligand charge in a hypervalent molecule grows with the number of orphan pairs.”

It grows with the orphan count divided by the ligand count. Xenon difluoride has one orphan pair and charges each fluorine by 0.5; sulfur hexafluoride has two and charges each by 0.3333. The prediction and its refusal come from the same census.

Tested in The count is the population · the series on hypervalency

“The charge a three-centre four-electron bond puts on its ligands is a consequence of their electronegativity.”

It survives the electronegativity being set to zero. With the ligand and the centre at the same Coulomb integral, sulfur hexafluoride still puts −0.3333 on each fluorine and methane still puts exactly nothing on each hydrogen.

Tested in The count is the population · the series on hypervalency

“All five fluorines of phosphorus pentafluoride are alike in a molecular-orbital description.”

The model finds two sets without being told there are two: three at −0.1333 and two at −0.3000, with the mean over all five exactly −0.2, which is one orphan pair over five ligands.

Tested in The count is the population · the series on hypervalency

“The two dimerisations of a chain are the same structure translated by one site.”

Only in an infinite chain. A chain of an even number of sites has an odd number of bonds, so one phase has a short bond at each end and the other a long one; held at the same distortion they differ by 1.08715 in units of the hopping, and the difference does not shrink.

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

“Two substituents on a carbon accelerate ring closure by compressing the internal angle.”

Computed from a bending term, that compression is worth +21.3 kJ/mol to a three-ring and −6.20 to a six-ring. The measured accelerations are for the five- and six-rings, and both are rings the angle account slows: 0.754-fold predicted against 250-fold measured for the five.

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

“An element has an electronegativity.”

It has a starting value. Solved for equalisation, carbon's charge runs from +0.0692 in methane to +0.3074 in tetrafluoromethane, and the value it ends up sharing with its neighbours runs from 6.954 to 9.334 eV — a spread of 2.38 eV where the table prints 6.2615.

Tested in The value that only exists in the bond · the series on electronegativity

“The polarity of a bond is a property of the two atoms in it.”

Hydrogen is −0.0173 in methane and +0.0268 in fluoromethane on the same scale, with the same carbon. Adding one fluorine four bonds away reverses the C–H dipole.

Tested in The value that only exists in the bond · the series on electronegativity

“Sanderson's geometric mean is the equalised electronegativity.”

The equalised value is a hardness-weighted mean, and the two differ by up to 0.21 eV across twelve molecules. The disagreement tracks how far apart the hardnesses are — a rank correlation of 0.62 — which is the only quantity a weighting can respond to.

Tested in The value that only exists in the bond · the series on electronegativity

“A rotamer account that fails on one ring fails on rings generally.”

It fails on the five-membered ring, where 250-fold is measured against a ceiling of 36.5, and it is more than sufficient for the six-membered one, where tenfold is measured against a ceiling of 120.9. The ceiling rises with ring size and the measurements fall, so they cross between the two.

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

“Computing the counterpoise correction once at the equilibrium geometry and applying it across the surface is a reasonable approximation, because the correction is roughly constant.”

It runs from 10.1 microhartree at 1.75 bohr to 236.0 at 4.25 — a factor of 23.4 — over exactly the range a well and its outer wall occupy, in two Gaussians a centre on H₂⁺.

Tested in A correction computed at one length · the series on basis

“The reference geometry for a frozen correction is a detail.”

It decides the whole of the error. The depth error runs from −77 to +135 microhartree across seven references and changes sign twice, and the error left at the dissociation limit — where the true correction is exactly zero — is whatever value was frozen, from 18 to 230 microhartree.

Tested in A correction computed at one length · the series on basis

“Satellites are weak, so a strong line is a fundamental.”

The weakest fundamental is 23.8 times the strongest satellite at a repulsion of 1 and 1.15 times it at a repulsion of 6, on the same ring. There is no height at which a cut can be drawn.

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

“Satellites appear at higher energy than the fundamentals, so they can be recognised by where they are.”

Up to a repulsion of 4 every satellite is outside the range the six fundamentals span. From 6 onwards ten of them are inside it, carrying 66 per cent of the satellite intensity, and a line between two bands cannot be recognised as an extra.

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

“Some unweighted mean of the free atoms' electronegativities gives the molecule's.”

Every power mean of order at or below one is bounded above by the arithmetic mean. The exact answer for hydrogen chloride is 7.8205 eV and the arithmetic mean is 7.7330, so no rule on that side of the family can reach it at any parameter. Water is the same case at +0.0045 eV.

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

“The molecule's electronegativity is fixed once the atoms' electronegativities are known.”

Holding two electronegativities at 6.2615 and 8.2900 eV and sweeping only the ratio of the two hardnesses over a factor of sixty-four moves the exact answer from 8.0646 to 6.4869 eV. All four unweighted rules are constants across that sweep, spanning 0.212 eV between them.

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

“An oxidation state is a rough estimate of the charge on the metal.”

It is not in the range. In a σ-only octahedral d⁶ model the metal's charge runs from −3.040 when it keeps every shared electron to −0.979 when it keeps none, and the oxidation state is 0 — outside the family at the positive end, not a member of it.

Tested in An integer nobody measured · the series on electron count

“Adding a bonding interaction to a model changes what the electron count is.”

Adding twelve empty π* orbitals and the metal's donation into them moves the Mulliken charge from −2.009 to −0.001, more than two electrons, and leaves the count at eighteen — because the count is a count of filled orbitals and no term in the Hamiltonian can change how many there are.

Tested in An integer nobody measured · the series on electron count

“The residual of a fitted model is error, to be reduced by fitting better.”

The six residuals of the two-parameter fit sort the molecules perfectly into open chains and rings: −0.605, −0.531 and −0.355 eV for the chains, +0.665, +0.409 and +0.416 for the rings, with 0.764 eV between the nearest pair across the divide. No value of α, β or S can reach that, because none of the three knows whether a molecule closes on itself.

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

“A localisation search finds the best description, so a published localised picture is the best one.”

On a twelve-vertex cage the description that maximises the functional is reached by 37 of 480 starting points — 7.7 per cent — while the commonest is reached by 65 and has a smaller functional. A single run is more likely to return a worse answer than the best one.

Tested in The answer a search is most likely to give · the series on multicentre

“The literature agrees about a cluster's localised picture because one solution dominates the search.”

Two independent searches return the same description 9.86 per cent of the time, against 7.14 for perfectly equal basins. The effective number of answers is 10.93 out of 14 — the basins are uneven and nowhere near uneven enough.

Tested in The answer a search is most likely to give · the series on multicentre

“A ring current in a fused system divides between the rings the way a current divides between parallel resistors.”

The response is a matrix and not a set of independent loops. In a four-ring acene the largest off-diagonal entry is 0.06390 against a smallest diagonal of 0.14082 — a neighbouring ring's flux drives nearly half the current a ring's own flux does.

Tested in The current does not divide · the series on aromaticity

“Rings of the same size and the same area carry the same current.”

Anthracene's three rings are identical hexagons and its middle one carries 0.28431 under a uniform field against 0.24098 for each outer one, a ratio of 1.1798. In tetracene the ratio is 1.2223, and it grows with the length of the acene.

Tested in The current does not divide · the series on aromaticity

“Fusing rings shares out a fixed response between more places.”

It increases the total. Four fused rings respond at 1.05482 against 0.88889 for four separate benzenes — an enhancement of 1.187 — and the enhancement grows monotonically with the number of rings fused.

Tested in The current does not divide · the series on aromaticity

“An alloy's band is the average of its two components' bands, shifted and broadened.”

The mean and the variance are the composition's, and nothing else about the band is. One composition arranged three ways on a chain of 240 sites gives the same variance to ten decimal places — 4.241666666667 for all three — and fourth moments of 1.1107, 1.5859 and 2.1028. Two of the three have a gap at a contrast where the third does not.

Tested in Two bands, if the chain is short enough · the series on defect

“Disorder smears bands into one another, so a random arrangement is the least likely of the three to show a gap.”

It is the second most likely, and above the threshold its gap is the widest of the two that are not ordered. At a contrast of 2.5 on a chain of 2,048 sites the random arrangement's gap is 1.093 against the segregated one's 1.000, and at a contrast of 4 it is 4.091 against 4.000. A random chain has no long run of like sites, and a short run is a narrow sub-band.

Tested in Two bands, if the chain is short enough · the series on defect

“Whether an alloy splits into two bands is a property of the alloy.”

It is a property of the alloy and of how much of it there is. The random arrangement's gap above the segregated closed form falls from 0.1849 on a chain of 512 sites to 0.0876 on one of 32,768, and the longest run of like sites in those chains rises from 9.75 to 14.5 — which accounts for the fall to within twelve per cent through an isolated-run band edge.

Tested in Two bands, if the chain is short enough · the series on defect

“A depolarisation ratio is therefore a structural measurement with no model in it.”

Symmetry decides that the ratio moves and says nothing about how far. The same 0.1 Å distortion computed with bond polarisabilities chosen to be absurd gives a departure of 6.5 × 10⁻⁵ instead of 4.2 × 10⁻², a factor of 650 — so the size of the effect is a property of the parameters.

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

“A small distortion gives a small but proportionate signal.”

The departure goes as the square of the distortion, at a fitted exponent of 1.9994 over a factor of fifty. A tenth of an ångström gives 0.042 and a hundredth gives 0.00045, so a ratio measured to three decimal places locates a length only to about a hundredth of an ångström.

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

“Hydrogen's linear Stark effect is a general property of atoms.”

It depends on a degeneracy no other atom has. A screening of λ = 0.01 in −λ/r² splits the n = 2 shell by 1.75 × 10⁻³ hartree, and a field of 1.5 × 10⁶ volts a centimetre is needed before the shift is linear again.

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

“The arrangement of a binary composition that binds most strongly is the one with the most unlike neighbours.”

True at every composition on a square net at a contrast of 1, and false at four compositions on the same net at a contrast of 4. With four of sixteen sites raised the winner has 12 unlike bonds where 16 are available, and it binds at 1.103953 per site against 1.080031 for the best arrangement that does have 16.

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

“Cyclopropane's carbons are sp³ hybridised, with a strained angle.”

No set of equivalent s–p hybrids makes a 60° angle at any s character, because the orthogonality angle runs from 90° to 180°. With cyclopropane's measured H–C–H of 114° the ring hybrids come out at sp³·⁷⁴ and meet at 105.5°, which is 45.5° away from where the carbons are.

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

“A better model of the same kind fits everything better.”

The model with the lower total error puts the two halides in the wrong order — 14,480 and 14,188 cm⁻¹ against measured 13,600 and 15,200 — while the free fit, whose total is worse, puts them right at 14,428 and 15,185. No one of the three fits gets both the halides and cyanide.

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

“A sideways overlap is small enough to neglect.”

It runs from 0.55 to 0.97 of the head-on overlap across the five ligands, computed from the same exponents at the same separations. Chloride's diffuse 3p makes its π overlap 97 per cent of its σ one.

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

“A fixed enclosed fraction is a shaky definition of an atom's size because a neighbouring ion polarises it.”

The correction to the density is odd in the angle to the field, so it integrates to nothing over a sphere: what a sphere encloses is unchanged to first order, checked to twelve decimal places at five effective charges. It is the radius *along a direction* that moves, and the two have been used interchangeably.

Tested in The surface a neighbour moves · the series on contour

“The failure of radius additivity is polarisation — the ions deform towards each other, so the radii overlap.”

Computed at the fields the measured separations produce, what the eight rock-salt pairs need and what the field supplies rank at −0.9524. Sodium fluoride needs 0.837 Å and gets 0.031; calcium sulphide needs 0.079 Å and gets 0.601, which is seven and a half times too much.

Tested in The surface a neighbour moves · the series on contour

“The four scales are close enough that the choice rarely matters.”

Anchored so that hydrogen and fluorine map to the same charge separation on each, they give hydrogen cyanide dipoles of 0.331, 0.029, 0.322 and 0.284 — a factor of 11.4 — and hydrogen's charge is positive on three of them and negative on Mulliken's, because Mulliken puts hydrogen above carbon.

Tested in The table that could not have mattered · the series on dipole

“Whether a composite correction will transfer can only be judged after the fact.”

The mean field's own spin polarisation says so beforehand. Over thirty-two systems the change in polarisation between reference and target ranks against the transferred correction's error at 0.9025, and every target whose polarisation is within 0.01 of the reference's has at least 93.2 per cent of the low level's error removed while every one past 0.05 has at most 76.4.

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

“Symmetry breaking is a matter of degree, so a diagnostic built on it is a soft one.”

The broken solution does not fade; it stops. The polarisation is positive below a critical modulation and exactly zero above it, and the critical value rises from 1.05 at a repulsion of 2 to 11.25 at a repulsion of 12 — a bifurcation, located here to a bracket of a tenth.

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

“Correlation is what a mean field leaves out, so it can only be defined by comparison with one.”

The pair distribution is a property of the exact state alone. At no repulsion it is exactly one at every separation, to nine decimal places; at a repulsion of 16 the chance of two opposite spins on one site is 0.0430 of the uncorrelated value, and no determinant appears anywhere in the calculation.

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

“A strong rank correlation over eight cases identifies a mechanism.”

The closed-shell overlap ranks at 0.8571 against the additivity shortfall, and the cation's formal charge — an integer that cannot be a mechanism — ranks at 0.9524 on the same eight pairs. The set is confounded, and both numbers are about the confound.

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

“Naphthalene's shared bond is anomalous because two rings feed it, so three rings would do more.”

The relaxation's move on the most interior cross bond falls at every step and changes sign: +0.02036, +0.00678, −0.00029, −0.00603 for two to five rings at λ = 0.4. Three rings do less than two and four rings do the opposite.

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

“Allowing the geometry to relax makes a conjugated molecule's bonds more alike.”

It makes them less alike from three rings onwards. The spread of bond orders goes 0.20633 → 0.20562 for naphthalene, which is a narrowing, and 0.25254 → 0.27411, 0.28271 → 0.31519 and 0.29148 → 0.33137 for the next three, which are widenings. Naphthalene is the exception again.

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

“The rotamer account of the gem-dimethyl effect might survive at five rings if the rotor count were slightly wrong.”

It would have to be wrong by two. The ceiling is 36.46 at three rotors, 120.88 at four and 400.81 at five, and the measurement is 250 — so a closure with three rotations to freeze would need five, which is more rotations than the tether has.

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

“An ester tether's hindered rotor could rescue the account by changing the count.”

It changes it in the direction that makes things worse. A bond that is not a free three-state rotor is a rotation the closure never had to freeze, so removing it lowers the ceiling: 36.46 to 10.99 to 3.32 as one and two are taken out of the five-membered case.

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

“A four-parameter fit to a hundred susceptibility points is well determined because there are far more points than parameters.”

Counting points against parameters measures nothing. The four directions of the design span a factor of 500 — singular values 12.411, 2.026, 0.149 and 0.025 — so the same one per cent data fix the coupling to 2.95 per cent and the temperature-independent term to 37.34.

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

“Adding a monomeric impurity and a temperature-independent term to a fit is harmless if they turn out small.”

Adding both takes the condition number from 6.2 to 500 and the coupling's own uncertainty from 0.47 per cent to 2.95, a factor of six — because an undetermined direction is a direction in the whole parameter space and not a property of one parameter.

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

“The natural metric for such a projection is the G matrix, since that is what vibrational analysis is written in.”

It cannot be formed. A redundancy is by construction a null vector of G, so rᵀGr is 1.8 × 10⁻¹⁶ for methane and −4.7 × 10⁻²⁷ for boron trifluoride, and the projector divides by it. Forcing the division through moves the frequencies by 506 per cent.

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

“A force field whose fit reaches a residual of 10⁻⁹ has determined its constants.”

A fitted boron trifluoride field has not. Refitting from four starting points that differ only in where the bend and bend–bend constants began returns bend constants from −0.2191 to +0.6622 mdyn/Å, one of them negative, with their difference fixed at 0.5106 and the residual unchanged at 1.4 × 10⁻⁹.

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

“A degenerate pair at the Fermi level is a metal's signature.”

It is the condition for a distortion. The gain from opening a gap at a degenerate pair is first order in the alternation, so no elastic constant stops it: at stiffnesses of 3, 6, 12, 24 and 48 the ring of forty still distorts, while a ring of forty-two stops distorting between 1.6 and 3.

Tested in The carriers a distortion was hiding · the series on metal

“The distortion is undone abruptly, so the carrier count jumps.”

The alternation falls continuously — 0.0975 at kT = 0.10, 0.0163 at 0.135, zero at 0.14 — as a power of the reduced temperature with an exponent of 0.44, and the carrier count is already at 1.75 of an eventual 1.81 before the distortion goes.

Tested in The carriers a distortion was hiding · the series on metal

“A dimerised chain has one alternation, and an end is a boundary condition on it.”

Relaxed bond by bond, the end bond alternates more than the middle at every elastic constant — by 1.11 times at a stiff one and 3.39 times at a soft one — and the excess decays over a length that is a property of the chain rather than of the end.

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

“The difference between a measured rotational constant and a rigid molecule's is a few tenths of a per cent.”

Computed from water's own force field it is 1.88 per cent on the largest of the three moments — five times that — and it is 1.68 for methane and 1.47 for ammonia. Only boron trifluoride, whose atoms are heavy, comes in under a fifth of a per cent.

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

“The zero-point motion inflates every moment of inertia.”

One of water's three shrinks: its three fractional corrections are −0.46, +1.88 and +1.07 per cent. The motion moves mass towards one principal axis and away from another, and no model that multiplies every moment by one plus a positive fraction can produce it.

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

“A hypervalent molecule adopts the arrangement that shares its ligand charge most evenly.”

That arrangement is the pentagonal plane, whose five ligands carry exactly the same charge and which costs 6.3 per cent more in repulsion than the trigonal bipyramid. The bipyramid is what phosphorus pentafluoride adopts and it has two kinds of ligand.

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

“The two accounts of hypervalent geometry — charge sharing and repulsion — are two views of one thing.”

Along the Berry interchange the repulsion changes by 0.150 per cent and the spread of the ligand charges by 53.6. One model sees a nearly flat coordinate and the other a large excursion, on the same path between the same two structures.

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

“The counterpoise correction is one quantity with one shape.”

It is two, and they differ by a hundredfold where a bond would be. For a pair with charges 1 and 2, the lighter centre's share of the correction runs from 0.03 per cent at short separation to 67 at nine bohr, and the two change places at 6.29.

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

“A symmetric pair's correction splits in half because that is a reasonable approximation.”

It splits exactly in half, at every separation, to the last digit the solver carries — which is a symmetry statement rather than an approximation, and is why it fails completely the moment the symmetry goes.

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

“Repeated species happen because large molecules have more modes than their group has boxes.”

For nine of the seventeen they do not. A group can hold Σ dim(Γ) modes before repeating; sulfur hexafluoride has 15 against a capacity of 20, methane 9 against 10, water 3 against 4 — and every one of them repeats.

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

“The structure a correlation hole has beyond the on-site depletion is a small correction.”

It is 1.1126 pairs against a depletion of 1.2653 — nearly all of what was removed. Weighted by an interaction reaching one neighbour it gives back 44.0 per cent of the on-site saving, and by a truncated Coulomb tail 46.9.

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

“A correlation energy computed with an on-site model transfers to a system with a longer-ranged interaction.”

It overstates it by nearly a factor of two, and the factor is stable: the give-back is 55.9, 51.3, 45.7, 44.0, 44.2 and 44.4 per cent at repulsions from 1 to 32, so it is not a small-repulsion artefact.

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

“The repulsion at which satellites stop being tellable from fundamentals is a property of the repulsion.”

It is 8 for a ring of six, 4 for a chain of four and 2 for a chain of six — a factor of four across three systems that differ only in size and topology, at the same filling and with the same interaction.

Tested in The boundary belongs to the gap · the series on photoelectron

“It is the repulsion measured against the band width.”

A ring and a chain of six have band widths of 4.00 and 3.60 in units of the hopping — within ten per cent — and boundaries of 8 and 2, a factor of four apart.

Tested in The boundary belongs to the gap · the series on photoelectron

“Every system has such a boundary somewhere.”

A ring of four does not. Its half-filled ground state is degenerate, its one-electron gap is zero, and the weakest fundamental is the same strength as the strongest satellite at every repulsion tried including none at all — so there is nothing for a boundary to separate.

Tested in The boundary belongs to the gap · the series on photoelectron

“A strong enough π channel could put a ligand orbital below a metal one and break the eighteen-electron count.”

Nothing can move the combinations that would have to move. An octahedron's twelve π functions span T₁g + T₂g + T₁u + T₂u and the metal has no orbital of T₁g or T₂u species at all, so those two sit at the free ligand energy at every coupling and cannot fall below anything.

Tested in The count that cannot be broken by strength · the series on electron count

“Strong back-donation makes the counted orbitals mostly the ligands'.”

It cannot. The lower eigenvector of a two-level problem always carries more of the lower basis function, so the filled T₂g orbital's metal share falls from 1 to 0.5467 across a coupling range of 80,000 cm⁻¹ and approaches a half from above without reaching it.

Tested in The count that cannot be broken by strength · the series on electron count

“The eighteen-electron rule counts electrons on the metal.”

It counts filled orbitals matched by symmetry species, and a species does not depend on a mixing coefficient. The count is 18 at every π coupling and on both sides of the level crossing where the counted orbital changes from mostly the metal's to mostly the ligands'.

Tested in The count that cannot be broken by strength · the series on electron count

“A rotation a closure freezes is either free or already frozen.”

The factor one rotor contributes to the ceiling is 3.316 when free, 2.033 at a hindrance of 2 kJ/mol, 1.461 at 4 and 1.092 at 8. Between the two rows of the usual table there is a continuous curve, and no measured energy has to sit at either end of it.

Tested in The curve between two rows · the series on strain

“Taking a hindered rotor out of the count is an approximation whose size is unknown.”

Its size is 1.0007. At the 20 kJ/mol an ester linkage costs, a hindered rotor contributes 1.000726 rather than 1, so the shortcut is right to a part in fourteen hundred for the molecules it was used on.

Tested in The curve between two rows · the series on strain

“A stiff tether could rescue the rotamer account of the five-membered closure.”

Hindering lowers the ceiling at every step — 120.88, 42.09, 17.10, 4.56, 1.42, 1.01 as four rotors are hindered by 0, 1, 2, 4, 8 and 16 kJ/mol. The five-ring's ceiling of 36.46 is already below its measured 250, and every hindrance takes it further below.

Tested in The curve between two rows · the series on strain

“The six-membered closure's sufficiency verdict is as robust as the five-membered refutation.”

It gives way at 4.09 kJ/mol of hindrance on three of its four rotors, or 2.70 on all four. With two hindered no hindrance is enough — but the free part alone is 10.99 against a measured 10, which is a margin of nine per cent.

Tested in The curve between two rows · the series on strain

“An exponent fitted over the last decade before a transition is the transition's exponent.”

The local slope on the same ring runs 0.4115, 0.4745, 0.4928, 0.4995, 0.5053 as the transition is approached. A fit over the original window returns 0.4436; over the innermost decade tried it returns 0.5022. There is no window at which the answer stops depending on the window.

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

“An exponent that does not change when the system is made larger is a property of the system.”

It is stable at 0.4413, 0.4439, 0.4478, 0.4431 and 0.4431 across a factor of two in ring size and a factor of two in stiffness — because the scaled alternation is one curve to within 3.41 per cent for all five, so a slope fitted over a stated window is the same number for all of them. Stability across systems is evidence that the number is the window's.

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

“Two quantities of the same order in a small parameter carry the same information about it.”

The shape departure goes as t⊥²/Δ² and the excess gap as t⊥²/Δ — exponents of (1.989, −2.038) and (2.000, −1.036) by central differences. Both are second order in the coupling; the pair inverts at a condition number of 6.38 because they differ in the separation.

Tested in Two ways of being second order · the series on Bands in a solid

“A measurement gets more informative as the model it is interpreted in gets more accurate.”

For the same-kind pair the opposite holds exactly. Its condition number runs 12.5, 27.3, 42.2, 106.2, 292.6 as the coupling falls from 0.15 to 0.03, because the two shapes become the same function of the same ratio precisely as second-order perturbation theory becomes exact.

Tested in Two ways of being second order · the series on Bands in a solid

“Testing a model needs data it has not been fitted to.”

A tie needs no data at all beyond what is already in hand and no fitting whatever: two systems assigned the same predictor value get the same answer from every model of that form, so the gap between their measurements is a floor on the error of all of them. Three of this collection's four predictors are refuted this way from data they were built on.

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

“The states at the edge of a disordered alloy's gap are band states like any others.”

They occupy a run rather than the chain. The eight levels nearest the gap centre of a 400-site chain occupy 3.88 to 9.20 sites; the band-edge state of the ordered arrangement of the same composition occupies 133.7 of the same 400.

Tested in A particle in a box the alloy made · the series on defect

“A localised state's size is a property of the disorder and has to be measured.”

It is a closed form. A run of L like sites holds a ground state proportional to sin(πj/(L+1)), whose participation ratio is 2(L+1)/3 exactly. Thirty-eight of forty gap states match it to within 3.1 per cent, with nothing fitted.

Tested in A particle in a box the alloy made · the series on defect

“A deeper contrast between the two components binds the gap state more tightly.”

It does not move it at all. The weight a gap state keeps inside its own run is 91.3 per cent at δ = 2.5 and 91.0 at δ = 8, across a gap that grows from 1.11 to 12.10. The barrier is the run's ends, and it is already complete at the smallest contrast that opens a gap.

Tested in A particle in a box the alloy made · the series on defect

“A growing measured length means the method is failing on larger systems.”

The same method on the same molecules with a gap held open by a staggered site energy returns 0.678, 0.642, 0.636, 0.636 and 0.639 — settled by the third molecule and moving in the fourth decimal place after it. The method is sound and the bare acene has no length to find.

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

“A repulsion that ranks with the shortfall is the mechanism behind it, once the constant is put in.”

With the constant put in, the computed equilibrium separations miss the measured ones by 0.242 ångström on average and the sum of two tabulated radii misses by 0.183. The displacement the repulsion produces has a rank correlation of 0.14 with the shortfall it is supposed to explain, and the wrong sign on three of the six pairs.

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

“The computed repulsion is too small to matter at these separations.”

It runs from 0.106 eV for sodium fluoride to 6.72 eV for calcium oxide at their own measured separations, and produces displacements of −0.35 to +0.38 ångström — the same order as the shortfalls it is being compared against, which is what makes the comparison worth making.

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

“Summing the repulsion over the whole valence shell rather than the facing orbital is a detail.”

With the facing orbital alone, magnesium oxide's equilibrium comes out at 0.79 ångström — inside the cation. The same failure is recorded for a noble-gas contact computed the same way, where it put argon at a quarter of the observed distance.

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

“The four electronegativity tables agree well enough that a trend down a group is safe on any of them.”

Anchored to a common hydrogen–fluorine separation, the hydrogen halide bond dipoles run 1.372, 1.029, 0.904, 0.622 on Pauling's and 1.372, 0.661, 0.269, −0.313 on Mulliken's. Three of the four tables have the series falling in magnitude; the fourth takes it through zero.

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

“The tables disagree about which molecule breaks a trend.”

They agree completely. Every one of the four has its largest residual against the measured halide dipoles at hydrogen iodide — 0.380, 0.865, 0.431 and 0.348 debye — with each table scaled to reproduce hydrogen fluoride exactly. What they disagree about is what that molecule is, by 1.245 debye.

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

“A partial charge computed from an electronegativity difference is a rough number with the right sign.”

Of twelve hydrogen bonds here, four have hydrogen at the negative end on all four tables, three have it there on Mulliken's alone, and five have it at the positive end on all four. The sign is not agreed on for a quarter of them, and H–C is among the disputed.

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

“An unsymmetrical double bond's best bent description has to be found by searching a mixing angle.”

There is nothing to search. Written in v = cos²θ − ½, the Boys functional is 2d² + 2m² + v²(2Δ² − 8d²) — a quadratic with no linear term — so it is stationary at v = 0 and extremal only there or at the ends. Checked against a direct evaluation at 181 angles, agreeing to 10⁻¹⁰ at every one.

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

“The two components of an unsymmetrical double bond have different s characters.”

Only if the mixing is intermediate, and it never is. At the bent solution both carry exactly half the σ hybrid's s character; at the canonical solution one carries all of it and the other none. The intermediate case the question was about is not a solution of any localisation criterion of this form.

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

“An underdetermined force field means every constant in it is uncertain.”

All four of methane's stretching constants and all three of boron trifluoride's are fixed on their own, to machine precision, in fields whose flat spaces are ten- and seven-dimensional. The freedom is confined: it lives entirely in the bending and coupling constants.

Tested in The forty-five that are fixed · the series on normal mode

“An underdetermined fit cannot be made to give one answer.”

Projecting onto the determined subspace does. Boron trifluoride's bend constant fitted from four starting points spans 0.8813 mdyn per ångström, one of them negative; the four projections are 0.2559, 0.2560, 0.2553 and 0.2533, and every projected field reproduces every observed frequency to a part in 10⁸.

Tested in The forty-five that are fixed · the series on normal mode

“The part removed by the projection is small enough to ignore.”

It shrinks methane's field by only 0.46 per cent in length, and it moves ten individual constants by more than a tenth of a mdyn per ångström — including the bend constant, from 0.4941 to 0.3530. A small direction in a large space can be a large change to a quoted number.

Tested in The forty-five that are fixed · the series on normal mode

“Four parameters with four uncertainties is a fair report of what the curve fixed.”

It is pessimistic and optimistic at once. The best-fixed combination — g divided by the cube root of the coupling — is determined to 0.08 per cent where the best single parameter is 2.0; and the monomer fraction's 16 per cent and the temperature-independent term's 37 hide that their product is fixed to 6.7.

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

“A direction the curve does not fix is one the curve barely notices.”

It is one the curve does not notice at all. Moving all four parameters along it — the coupling by 2.4 per cent, the g factor by 1.7, the monomer fraction by 12 and the temperature-independent term by 38 — changes the computed curve by at most 0.87 per cent, which is under the precision it is supposed to be measured to.

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

“Which combination is free is a property of the model.”

It is a property of the temperature window. Starting the measurement at 2 K makes the free direction essentially the temperature-independent term at 11 per cent; at 20 K it is a mixture at 40 per cent; at 80 K it is the monomer fraction at 241, and the condition number runs from 132 to 2,922.

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

“A cubic term is a refinement of the quadratic energy model.”

It replaces the electronegativity rather than correcting it. The slope at zero charge becomes χ − γ, and γ = (I₂ − 2I₁ + A)/6 is one sixth of a second difference — 10.912 eV for lithium, whose cubic electronegativity is therefore −7.907 eV against a quadratic 3.005.

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

“A model that fails does so by giving a wrong answer.”

This one gives none. A molecule of two alkali metals needs one of them to accept charge, and neither can accept as much as equalisation asks — lithium's capacity is 0.073 of an electron — so the bisection finds no chemical potential at which the charges sum to zero.

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

“A central atom cannot have both a spare orbital and an unmatched ligand pair.”

It has both in exactly the three flat arrangements above. The spare orbital points out of the plane and the unmatched combination lies in it, so neither can pair with the other and both are left over at once.

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

“The orphan count is a property of the molecule's composition.”

Four of the ten formulas give two different counts depending only on where their ligands are put — methane nothing or one, phosphorus pentafluoride one or two, sulfur hexafluoride two or three, carbon dioxide nothing in either arrangement it can be worked in.

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

“A square-planar arrangement always breaks the formula.”

Xenon tetrafluoride is square planar and the formula is right for it. Its two lone pairs occupy the orbital pointing out of the plane, so the orbital a square wastes is one the molecule had no use for — which is why xenon tetrafluoride is square planar and methane is not.

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

“A profile that decays away from an end has a decay length.”

The local decay length of the excess alternation on one relaxed chain of 320 is 4.12 bonds at the third bond, 8.11 at the tenth, 11.99 at the twenty-fifth and 14.67 at the fifty-eighth, and it is still rising. The rate falls as the reciprocal of the distance from the end, which is a power law times an exponential rather than an exponential.

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

“The healing length goes as the gap to the power −0.60.”

Extrapolating the local decay rate to a bond infinitely far from the end gives a length with no window in it, and that length goes as the gap to the power −1.029 over a factor of twenty in the gap, with every neighbouring pair of points between −1.06 and −1.02.

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

“Heavy molecules are safer from the zero-point correction because their vibrations are slower.”

Boron trifluoride has the softest modes of the four — the largest sum of reciprocal wavenumbers, 8.06 × 10⁻³ against water's 1.12 × 10⁻³ — and the smallest correction. Its safety is its moment of inertia, 65.2 amu Ų against water's 1.18, and its frequencies work against it.

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

“The size of the correction has to be computed molecule by molecule.”

One expression with one dimensionless coefficient gives 1.02, 1.59, 1.60 and 0.13 per cent against computed values of 0.83, 1.63, 1.68 and 0.15 — the worst error is 22 per cent on corrections that span a factor of 11.4.

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

“How much two orbitals are bonded by is a property of those two orbitals.”

The pair's response to its own overlap — the ratio of what it is bonded by at an overlap of 0.4 to at 0.1 — is 11.83 alone and 1.46, 0.92 and 0.74 with a third orbital at three different energies. Two of those are below one, so a fourfold increase in overlap buys less bonding rather than more.

Tested in A regime that belongs to the neighbours · the series on overlap

“Adding an interaction between two orbitals lowers the energy.”

Not in a closed shell with a third orbital present. Removing the A–B term from the trio and putting it back costs between 1.14 and 3.10 eV at every third-orbital placement tried, against a gain of 0.539 eV when the third orbital is decoupled.

Tested in A regime that belongs to the neighbours · the series on overlap

“A three-centre problem is a small correction to the two-centre one.”

It changes the sign of what the pair is bonded by and inverts its dependence on overlap. And the correction is not small where the third orbital is far away either — the Wolfsberg–Helmholz coupling grows with the mean of the two energies, so a deeper third orbital is a more strongly coupled one.

Tested in A regime that belongs to the neighbours · the series on overlap

“A computed ceiling nine per cent above a measured ratio is a sufficiency verdict.”

The computed side is built on a gauche energy quoted as 3.8 ± 0.4 kJ/mol, and that band alone runs the ceiling from 8.825 to 13.848 — 22.8 per cent either way. The margin is 9.9 per cent and sits inside it.

Tested in A verdict inside its own error bar · the series on strain

“The uncertainty that matters is the measurement's, because the computation is exact.”

The computation is exact and its input is not. A rate ratio assumed good to a fifth gives a band of 8 to 12; the gauche energy gives the ceiling a band of 8.8 to 13.8. The computed side is the wider of the two.

Tested in A verdict inside its own error bar · the series on strain

“The verdict would need a large revision of the input to overturn.”

It turns at a gauche energy of 3.6300 kJ/mol, which is 0.425 of one standard deviation below the quoted value — or at a temperature of 312.1 K, fourteen degrees above the one computed at and inside the range ordinary kinetics is run over.

Tested in A verdict inside its own error bar · the series on strain

“Giving both verdicts error bars weakens both of them.”

It sharpens the contrast. The five-membered ceiling stays below its measured 250 across the whole band — 77.9 to 191.8 — so the refutation is untouched, while the sufficiency verdict is not decided at all.

Tested in A verdict inside its own error bar · the series on strain

“An interpolation that stops improving with more points has reached the function's complexity.”

It has reached the nodes' noise. The heavier half's interpolation falls from 48 per cent at two points to 1.4 at six and then stalls, and the half's own second differences say it is known to 0.27 per cent — an interpolant through noisy values amplifies that by its Lebesgue constant.

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

“A counterpoise correction computed the usual way is as accurate as the energies it comes from.”

It inherits their cancellation. At a bonding separation the heavier centre's half needs three figures of cancellation and the lighter centre's six, from a variational solve that reports both the same way.

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

“There is no cheap check on a counterpoise calculation.”

A symmetric pair's halves are equal by symmetry, and computed here they agree to 7 × 10⁻¹⁶. Leaving one ghost function out gives a share of 0.4628 instead of 0.5 — while the total moves by 6.9 per cent, which is a size an ordinary basis-set effect has.

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

“A predictor whose values are all distinct cannot be audited without fitting a model to it.”

Every pair bounds the derivative: any model whose slope is at most L satisfies |Δy| ≤ L|Δx|, so the largest ratio over the pairs is a floor on L. The computed angle strain of a ring demands a slope 14.0 times the one the whole set has, established with no fit.

Tested in The pair that is not a tie · the series on models

“The ring-strain predictor is the well-behaved one, because the tie instrument found nothing wrong with it.”

It found nothing because the predictor never repeats a value, which is a property of the predictor and not evidence about it. Under the near-tie instrument the same predictor is the worst of the four: a slope floor of 14.0 times its own slope, and three of its fifteen pairs ordered the wrong way round.

Tested in The pair that is not a tie · the series on models

“One measure of how badly a predictor performs will rank a set of predictors the way any other would.”

The exact instrument ranks VSEPR worst at 92.4 per cent unexplained and is silent about ring strain; the near-tie instrument ranks ring strain worst at 14.0× and cannot speak about VSEPR at all. The worst predictor under each is invisible to the other.

Tested in The pair that is not a tie · the series on models

“The rarest descriptions might be arbitrarily rare, so the count is only a lower bound.”

The rarest holds 0.60 per cent of the starts — 24 of 4,000 — and the basins fall into two groups with a factor of five between them and nothing in the gap. A population continuing downward would show a tail; this shows a gap.

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

“The description a localisation program returns most often is the best one it can find.”

The best description by the Boys functional takes 8.65 per cent of four thousand starts and the largest basin takes 14.6, and they are different descriptions. What a single run most often returns is the one with the widest catchment, which is a property of the optimisation landscape.

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

“The sixteen-electron rule is the one a π channel should break, because it rests on an energy rather than on symmetry matching.”

The gap above sixteen electrons is 2eσ exactly — the same number to the last bit — at every π strength between −0.4 and +0.25 in units of eσ, including strengths larger than any ligand offers. The octahedron's eighteen-electron gap moves at every one of them.

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

“A square-planar complex's d orbitals all feel the ligand π channel.”

d(z²) does not, at any strength. It is symmetric about the ligand plane and about the axis, and the π set of a square-planar ML₄ contains nothing of that symmetry — so its energy is eσ whatever eπ is.

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

“The protection is unconditional.”

It ends at eπ = eσ/4 exactly, where d(xy) rises past d(z²) and becomes the highest filled orbital. Beyond it the gap is 3eσ − 4eπ and falls at four times the π strength, which is the octahedron's expression.

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

“A molecule's totally symmetric species appears once per orbit of internal coordinates, less one for each redundancy.”

Computed for fifteen molecules it is right for seven. Sulfur hexafluoride has three orbits and six redundancies, so the formula predicts −3 totally symmetric vibrations against an answer of 1; benzene predicts −4 against 2; each ferrocene predicts −11 against 4.

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

“A redundancy removes one totally symmetric vibration.”

Only a totally symmetric redundancy does. Sulfur hexafluoride's six redundancies span 2A₁g ⊕ 2Eg and only two are totally symmetric; benzene's nine span 2A₁g ⊕ E₂g ⊕ B₁u ⊕ 2E₁u, again two; each ferrocene's sixteen contain exactly two.

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

“The capacity a cubic implies is a measure of how much charge an atom will accept.”

The three atoms that will accept none — beryllium, magnesium and nitrogen, whose anions are unbound — are among the four largest capacities, two of them infinite and one at 31.5. The three smallest, lithium at 0.073, sodium at 0.122 and potassium at 0.164, all bind their anions.

Tested in A capacity that is largest where there is none · the series on electronegativity

“The capacity is simply noise, so the failure says nothing.”

Among the fourteen atoms with a finite capacity the ordering agrees with the electron affinity at a rank correlation of 0.433. It is measuring something across the cases that cannot test it, and inverting on the three that can.

Tested in A capacity that is largest where there is none · the series on electronegativity

“A negative cubic coefficient marks the atoms with no capacity.”

Three atoms have one: beryllium, magnesium and phosphorus. Two of them have no bound anion and the third binds its own at 0.746 eV, so the sign does not separate them either.

Tested in A capacity that is largest where there is none · the series on electronegativity

“The capacity is a statement about the anion side of the curve.”

It runs against the second ionisation energy at −0.503 — the energy to remove a second electron, which for lithium is a core ionisation of 75.6 eV and gives it the smallest capacity in the set. The cubic coefficient is a second difference, and the second difference is dominated by the cation side.

Tested in A capacity that is largest where there is none · the series on electronegativity

“A bonding orbital is one that accumulates electron density between the nuclei.”

Below 5.03 bohr the combination this model puts lower has a nodal plane through the midpoint, so two electrons in it put exactly zero density there against the 1.0 the two free atoms put. The model calls the pair bonded by 0.165 hartree at two bohr.

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

“Something changes about the orbitals at the separation where their overlap vanishes.”

Nothing does. Each 2p function is what it was, the density at the midpoint of the plus combination is zero at every separation and always was, and the crossing at 5.0265 bohr is a property of an integral. What changes is which of the two combinations the model fills.

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

“An overlap that comes out at 10⁻¹³ is a symmetry-forbidden one.”

This one is not. The head-on 2p overlap passes through zero by cancellation at 5.0265 bohr, where the computing rule gives −1.1 × 10⁻¹³ and the checking rule gives 6.4 × 10⁻⁷. A symmetry zero is arithmetic noise under both rules; an accidental zero is a small number under both and can fall below any threshold by luck.

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

“The tables' disagreement about bond direction is a property of the elements involved.”

Of thirty-one bonds, the six disputed all have a difference under 0.0507 of a table's range and the twenty-five agreed all have one over 0.0584. Nothing lies between. Sorting by that one number separates the two groups completely, without reference to which elements a bond joins.

Tested in A dispute is a small difference · the series on dipole

“A disputed bond is one the tables are unusually far apart on.”

The spread across the four tables does not separate the groups at all. C–S is disputed and has the smallest spread of any bond here, 0.0318; O–H is agreed and has one of the largest, 0.3625.

Tested in A dispute is a small difference · the series on dipole

“The rate at which a depolarisation ratio leaves three quarters is one number for a molecule.”

It is one number per coordinate. Methane's three non-symmetric species give 4.54, 0.152 and 0.627 — a factor of thirty — and across three molecules the coefficients span a factor of sixty-four.

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

“The ratio is most sensitive to the coordinates a molecule is soft along.”

In ammonia the stiff pair at 3567 cm⁻¹ is five times more sensitive than the soft pair at 1680, and in boron trifluoride the stiffer pair is more sensitive too. Only methane has its softest coordinate most sensitive, and its second-softest is its least.

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

“Each component of a degenerate set has its own sensitivity.”

Methane's three T₂ components come back at 6.80, 3.42 and 3.40, and which three vectors those are is the eigensolver's choice rather than the molecule's. Only the mean of a set, 4.54, is a property of the species.

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

“Adding up pairwise counterpoise corrections gives a three-fragment system's correction.”

The sum exceeds it by 30.4 per cent at 1.6 bohr and 15.7 at two, and reaches agreement only past five, where the correction itself is a fortieth of what it was. A second basis does not help a fragment as much as the first did, so counting the two separately counts their overlap twice.

Tested in The assembly that counts one share twice · the series on basis

“Whatever the assembly gets wrong, it gets wrong in one direction.”

At two bohr the heavy central fragment is over-corrected by 16.98 per cent and each light outer one is under-corrected by 2.54. The two have opposite signs and do not cancel, and the total follows the heavy one because its correction is thirty times a light one's.

Tested in The assembly that counts one share twice · the series on basis

“The error is a property of the closest pair, so it can be estimated from that pair alone.”

Holding the first separation at two bohr and moving only the second, the overshoot falls from 15.69 per cent to 0.45 — the near pair has not moved and five sixths of the error has gone.

Tested in The assembly that counts one share twice · the series on basis

“The energy denominator is what makes an e_π larger for iodide than for fluoride.”

The denominator over-predicts the trend at every metal level a π donor permits. Its smallest possible value for the iodide-to-fluoride ratio is 1.667, with the metal orbital at the vacuum level; the fitted ratio is 1.429, and the prediction rises without limit as the metal level falls.

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

“The gaps can be computed even crudely from Slater orbitals.”

The Slater estimate puts fluorine's 2p at 92.0 eV below vacuum against a measured 17.42, and the factor by which it is wrong runs from 3.01 for iodine to 5.28 for fluorine. It is not a consistent scaling, so it cannot supply even the ratios.

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

“The overlap must grow down the halide group, since the orbitals get larger.”

Dividing the fitted ratios by the denominator's gives what the overlap squared must supply: 0.857 for iodide, 0.872 for bromide, 0.851 for chloride, against 1 for fluoride. It has to be smaller for every heavier halide, not larger.

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

“A variational estimate of a longer-ranged interaction is a small overestimate of its cost.”

The energy the on-site-only state gives in the extended Hamiltonian exceeds the true ground state by 0.025, 0.117 and 0.962 at V = 1, 2 and 4. Doubling the neighbour term multiplies the shortfall by 4.6 and then by 8.2, so it is not second order over the range where it matters.

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

“The enhancement of nearest-neighbour pairs is a robust feature of a correlated ring.”

It is 1.52 times the uncorrelated value with the on-site term alone and 1.07 once a neighbour repulsion of U/2 is in the Hamiltonian. The structure the give-back was computed from is very nearly gone at the interaction it was priced with.

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

“The chain length at which the levels form a band is a property of the disorder.”

It is a different number for every run length in the same chain: 985 sites for runs of four, 2,906 for five, 8,675 for six, 26,300 for seven and 81,500 for eight, at one composition. A single chain is a band in some of its gap states and a set of isolated levels in others.

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

“Two runs stop interacting at a stated distance.”

The splitting falls exponentially with a decay length of 2.003 sites and never reaches zero: 3.36 × 10⁻² at one site of separation, 2.67 × 10⁻⁴ at ten. What decides whether they are one system or two is the comparison against the level spacing, which depends on the chain.

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

“The field at which a broken symmetry's linear behaviour returns is one field.”

In the n = 3 m = 0 shell there are two coupled pairs, s with p and p with d, at gaps of 5.16 × 10⁻⁴ and 9.97 × 10⁻⁵ and dipoles of 7.35 and 5.20. Their crossovers are 3.51 × 10⁻⁵ and 9.60 × 10⁻⁶ atomic units, a factor of 3.66 apart.

Tested in Two events where there was one · the series on representation

“The transition from quadratic to linear is a crossing.”

The local exponent of the shell's excess width against the field runs 1.997, 1.905, 1.726, 1.522, 1.283, 1.122, 1.050, 1.020 — more than a decade of field between the two ends, and it is passing through 1.5 between the two crossovers rather than at either.

Tested in Two events where there was one · the series on representation

“A higher shell behaves like a lower one at a correspondingly higher field.”

Both of the n = 3 crossovers are below the n = 2 one: 9.60 × 10⁻⁶ and 3.51 × 10⁻⁵ against 2.92 × 10⁻⁴. A higher shell has smaller gaps and larger dipoles, and both differences push the same way.

Tested in Two events where there was one · the series on representation

“Every pair of levels in a shell has a crossover.”

Only the pairs the field couples. The s and d of the n = 3 shell have a dipole of exactly zero between them — a field changes the angular momentum by one — so there are two coupled pairs and three levels, and a rule counting neighbouring levels would report three crossovers in the n = 2 shell where there is one.

Tested in Two events where there was one · the series on representation

“A satellite becomes indistinguishable from a fundamental at a large enough repulsion, whatever the filling.”

Below half filling it does not, at any repulsion. On eight lattices the contrast flattens at 2.4999, 2.6180, 2.7986, 2.9051, 3.9999, 5.8284, 6.8541 and 7.4641 — every one above the factor of two the intensity test needs, and the largest repulsion tried was sixteen thousand times the hopping.

Tested in A contrast with a closed form · the series on photoelectron

“A flattening curve's limit can only be estimated.”

The approach is first order in the reciprocal repulsion: the departure from the limit multiplied by the repulsion is one number to better than a per cent from U = 256 to U = 16,384. Two points a factor of four apart therefore give the limit to six figures.

Tested in A contrast with a closed form · the series on photoelectron

“The limits are lattice-specific numbers with no structure to them.”

For a ring the limit is (1 + 2cos(π/n))². Read off rings of four and six, it predicts 4.000000 for a ring of three and 6.854102 for a ring of five against measured limits of 3.999996 and 6.854100.

Tested in A contrast with a closed form · the series on photoelectron

“The same expression should then hold for a chain.”

It does not. A chain of four's limit is 2.618033 against the ring expression's 5.828427, and a chain of six's is 2.905130 against 7.464102. The chains' limits rise with size as the rings' do and no expression has been found for them.

Tested in A contrast with a closed form · the series on photoelectron

“The repulsion at which satellites stop being distinguishable is set by the one-electron gap.”

Only when the gap is changed by the geometry. Changed by the filling it fails: a six-site chain has a gap of 0.890 at half filling with a boundary at U = 2, and a gap of 0.802 at two-thirds filling with a boundary at U = 4 — a smaller gap with a larger boundary.

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

“A large enough repulsion always makes satellites indistinguishable from fundamentals.”

Not below half filling. A third-filled ring of six has a contrast of 8.3 at a repulsion sixty-four times the hopping, and a third-filled chain 3.2 — both above the factor of two the test needs, and both still falling too slowly to reach it.

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

“A response that decays between rings decays in the distance between them.”

In a chain of six hexagons whose fusions are mixed, rings 1–3 and 3–5 are both 3.4641 units apart and respond at 2.3312 × 10⁻³ and 1.5247 × 10⁻³. Rings 2–4 and 4–6 are both 3.0000 units apart and respond at 1.9744 × 10⁻³ and 2.6580 × 10⁻³. The largest response is at the shorter distance and the smallest at the longer one.

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

“Anthracene and phenanthrene are two drawings of one π system.”

Both are fourteen carbons and sixteen bonds, and their Hückel π energies differ by more than 0.05β because the two fusion bonds of the middle ring lie opposite each other in one and one edge apart in the other. They are not isomorphic graphs.

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

“The comparison could be made temperature-free by choosing the right activation-energy difference.”

The ceiling's own apparent activation energy is r·g/(1 + 2x), which depends on temperature through x: it falls from 5.719 kJ/mol at 253 K to 4.787 at 373 K, a drift of 19 per cent. The best constant difference, found by minimising the margin's spread, is 5.271 kJ/mol and still leaves the margin moving by 1.8 per cent across the range.

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

“A band's shape departs from its uncoupled value at second order in the gap because the departure is second order in perturbation theory.”

The departure is second order in the coupling at every pairing tested — 1.971 to 2.044 — and its gap exponent runs from −2.011 to −0.985 across the same four pairings. The two orders are set by different things and only the first is perturbation theory.

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

“A count protected by symmetry stays protected under a small distortion.”

The square plane's sixteen-electron gap is 2eσ at every π strength, so its derivative with respect to eπ is exactly zero. At a two-degree fold it is −0.0097, at five −0.0603 and at ten −0.2340 — growing as the square of the bend, from exactly nothing.

Tested in The gap that only a tetrahedron closes · the series on electron count

“If the protection goes at the first degree, the count goes with it.”

The gap itself is 99.98 per cent of 2eσ at a five-degree fold, 99.73 at ten and 98.65 at fifteen — more than two fifths of the way to a tetrahedron. Size and exactness are different quantities and they fail at different rates.

Tested in The gap that only a tetrahedron closes · the series on electron count

“The gap closes somewhere along the path, at a distortion that can be located.”

It is positive at every angle short of the tetrahedron and exactly zero there, where the upper three levels are the degenerate t₂ set. There is no interior closing to locate.

Tested in The gap that only a tetrahedron closes · the series on electron count

“A gap that closes at a symmetric point closes gently, as a distortion squared.”

It closes at first order: the gap against the angle left to the tetrahedron has a slope of 1.013 across a factor of eighty. Half a degree of flattening from a tetrahedron opens a gap proportional to the half degree, where half a degree from the plane changes it in the fifth decimal.

Tested in The gap that only a tetrahedron closes · the series on electron count

“Formaldehyde's localised double bond is σ and π rather than two bent components.”

That answer came from a σ overlap of 0.00718, which is a hydrogenic 2s radial node cancelling the integral. With nodeless Slater functions at the same effective charges the overlap is 0.69752 and the inequality gives 0.3937 against a boundary of 1, so the description is bent.

Tested in The node that decided a picture · the series on hybrids

“The difference between hydrogenic and Slater radial functions is a refinement.”

It is a factor of 97 in the one integral that has a node in it and exactly nothing — 1 × 10⁻¹⁶ — in the one that does not. A hydrogenic 2p and a Slater 2p are the same function, so the π overlap is an exact control on what changed.

Tested in The node that decided a picture · the series on hybrids

“The no-interior-maximum theorem needed the integral.”

None of it did. That the functional is a quadratic with no linear term, that its maximum is at forty-five degrees or at the ends, and that the boundary is |Δ| < 2|d| are all closed-form results. Only which side of the boundary a molecule is on needed an integral, and it needed a good one.

Tested in The node that decided a picture · the series on hybrids

“The shortfall between a sum of radii and a measured separation is a property of the pair.”

These separations are additive to 0.0098 ångström. The best possible assignment of radii reproduces all six computable ones to within 0.0140, where the radii read off the ions' own densities miss by 0.1828 — nineteen times as far.

Tested in The residue that is two numbers · the series on contour

“How well an additive model can do has to be found by fitting.”

It is a subtraction. Each block of separations is a complete two-by-two, whose alternating sum d(AX) − d(AY) − d(BX) + d(BY) every additive model gives exactly zero — so the floor is a quarter of that combination, which is −0.03100 for the alkali halides and −0.05600 for the alkaline-earth chalcogenides.

Tested in The residue that is two numbers · the series on contour

“A universal table of radii gives up a great deal by being universal.”

The tabulated radii miss these separations by 0.0237 ångström against a floor of 0.0098 — a factor of 2.4, on a set of eight compounds they were not fitted to individually. Whatever is wrong with the enclosure radii is not that they are a compromise.

Tested in The residue that is two numbers · the series on contour

“Two sets of radii can be compared by comparing the radii.”

Adding a tenth of an ångström to every cation and taking it from every anion changes every radius in a block and no predicted separation and no residual, to the last digit. Only residuals against measured distances are comparable.

Tested in The residue that is two numbers · the series on contour

“A gap in the basin populations is a stopping rule with a stated confidence.”

It is one of three shapes. A twelve-vertex cage at twelve electrons has a step of 5.15 in the middle of its population; a nine-vertex cage at eight has no step above 2.00 anywhere among sixty-three descriptions; a twelve-vertex cage at twenty has a step of 79.7 at the first rank. Only the first has two groups.

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

“The multiplicity of localised descriptions is a property of the cage.”

At the filling this collection's cage builder uses, cages of six, nine, ten and eleven vertices have exactly one description each and only the twelve-vertex one has more. Change the electron count and nine vertices gives sixty-three, which is more than four times the twelve-vertex cage's.

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

“A search of four thousand starts has converged.”

On the twelve-vertex cage at twelve electrons it has, with a miss probability of 0.0 per cent. On the nine-vertex cage at eight electrons the same four thousand starts leave a 90.0 per cent chance that at least one description was never reached.

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

“The energy denominator accounts for the spectrochemical series once the metal level is chosen properly.”

On the donor side it over-predicts at every admissible metal level — 1.667 at the vacuum against a fitted 1.4286, rising to 18.4 at the bound. On the acceptor side it under-predicts at every one — 1.507 at the vacuum against a fitted 1.8000, falling to 1.066. One parameter, two requirements, opposite directions.

Tested in A denominator that fails both ways · the series on ligand field

“An acceptor's π* level cannot be brought into the arithmetic because it is a molecular orbital rather than an atomic one.”

It is measured directly. Electron transmission spectroscopy captures a slow electron in the π* and reads the energy: 1.50 eV for carbon monoxide, 2.30 for dinitrogen, and 2.26 for hydrogen cyanide, which stands in for cyanide's.

Tested in A denominator that fails both ways · the series on ligand field

“Since the denominator over-predicts for donors, the overlap must be roughly constant across the series.”

It must shrink down the halides to cancel part of an over-prediction and grow from cyanide to carbon monoxide to make up an under-prediction. Two statements about an overlap, in opposite directions, neither of which comes from computing one.

Tested in A denominator that fails both ways · the series on ligand field

“One number separates disputed bonds from agreed ones with nothing in between.”

On thirty-one bonds it does. On all one hundred and fifty-three pairs the four tables cover, the largest disputed difference is O–Br at 0.1628 and the smallest agreed one is Na–K at 0.0398, and twenty-seven agreed pairs lie below the largest disputed one. The gap was a property of which thirty-one bonds were chosen.

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

“Measuring the difference against how much the four tables disagree would restore the separation, since a dispute needs the difference to be small compared with the disagreement.”

The steepest disputed pair sits at 0.5950 of the way to the diagonal and the shallowest agreed one at 0.3330, so the two groups overlap on that measure too. What is exact is only the arithmetic bound: four numbers straddling zero cannot have a mean above their range, so every disputed pair must fall below the diagonal, and they all do.

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

“The disagreement is about one awkward element.”

Fifteen of the eighteen elements appear in at least one disputed pair; only fluorine, phosphorus and potassium never do. Hydrogen, which the interhalogen series was designed to exclude, appears in three of sixteen — fewer than carbon, sulphur, iodine or beryllium.

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

“The arrangements that break the counting formula are the expensive ones, which is why the formula is reliable in practice.”

The three failures sit 4.20, 6.29 and 9.80 per cent above their own repulsion minimum. The arrangements the formula gets right run from exactly zero to 26.47 per cent, so the most expensive arrangement in the census is one it survives.

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

“An arrangement decides whether the formula holds for it.”

A square plane of four ligands breaks it on methane and holds on xenon tetrafluoride, at an identical 4.197 per cent excess. Two lone pairs take the orbital perpendicular to the plane, and the orphan then has no spare orbital to sit beside.

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

“A centre cannot have an unused valence orbital and an unmatched ligand combination at the same time.”

It has both in exactly three arrangements, and they are exactly the three where the formula fails. A planar arrangement of four or more ligands cannot reach the p orbital perpendicular to its plane, so that orbital is spare while a combination goes unmatched.

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

“An amplitude that a scaling collapse divides out is a number that has to be measured.”

All five of the amplitudes the collapse divided by are reproduced to better than one part in three million by a sum of n/2 square roots — 0.23894451 against 0.23894449, 0.12316658 against 0.12316659, and so on — with no matrix diagonalised anywhere.

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

“A more detailed version of a working expression fits better.”

Split by principal axis, the coefficient the expression needs spans a factor of 2.33 among the positive cases — 0.4944 to 1.1499 — against the 1.36 the molecule-averaged version spans. Three times as many parameters, twice the spread.

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

“Zero-point motion makes a molecule's moments of inertia larger.”

It makes eleven of these twelve principal moments larger and one smaller. Water's smallest moment falls by 0.46 per cent, so its coefficient is −0.1499 and no positive constant can describe it.

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

“The expression fails per axis because the moment enters it wrongly.”

Plotted against their own moments the twelve coefficients lie on no curve: two axes of boron trifluoride whose moments differ by exactly two need 0.5863 and 0.8274, while boron trifluoride's smaller axis and ammonia's larger one differ in moment by a factor of eighteen and need 0.5863 and 1.0112.

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

“The assembly's overshoot is a small-basis artefact that a better basis removes.”

From one Gaussian a centre to six, at a separation of two bohr, the overshoot goes 5.25, 7.10, 15.69, 23.16, 32.27, 37.98 per cent. It grows at all four separations tested, by factors of 4.6, 7.2, 9.1 and 13.0.

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

“Six Gaussians a centre is enough to see where the overshoot is heading.”

Plotted against the reciprocal of the basis size, where a limit would show as a flattening, none of the four curves flattens: every one is still rising at six functions and rising faster in those coordinates than in the basis size itself.

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

“A structural response that stops falling has stopped.”

On a chain of twelve fused rings the response flattens at 4.48 × 10⁻⁵ after five rings when each molecule's couplings are measured from its own mean bond order, and falls to 7.89 × 10⁻⁹ at the twelfth ring when both are measured from the same one. The molecule is identical in the two runs.

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

“A floor in a self-consistent calculation is the convergence tolerance.”

The floor is 2.2 × 10⁻⁵, 4.5 × 10⁻⁵ and 7.0 × 10⁻⁵ at feedback strengths of 0.2, 0.4 and 0.6 — proportional to a model parameter — while the fixed point's own residual is 7.3 × 10⁻¹⁴ at all three. It scales with the physics and not with the arithmetic.

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

“A reach set by a gap goes as the reciprocal of the gap.”

Across a factor of five in the gap the structural reach goes as the gap to the power −0.688 and the magnetic reach to −0.680, both fitted over profiles spanning three to ten decades. Neither is −1, and the two agree with each other far better than either agrees with the argument.

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

“Starting a local search from the arrangement with the most unlike bonds finds the best one.”

It does on sixteen sites at both contrasts and on thirty-six at a contrast of one. On thirty-six at a contrast of four it arrives at 2.22569848 with 28 unlike bonds and is beaten by a random start reaching 2.23046805 with 24 — fewer unlike bonds, more binding.

Tested in Twelve basins where there were two · the series on cohesion

“An exhaustive search is available for any net worth studying.”

Twelve raised sites among thirty-six is 1,251,677,700 arrangements. One steepest ascent on the same net used 1,484 diagonalisations. The exhaustive answer is not expensive there; it does not exist.

Tested in Twelve basins where there were two · the series on cohesion

“A shell with three levels in one subshell and two in another has four crossovers.”

It has three. The m = 0 half has two coupled pairs and the m = ±1 half has one, because it is a two-level ladder — and the m = ±1 subshell occurs twice over, at m = +1 and at m = −1, which gives one field rather than two.

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

“The events of one subshell fall together, so a shell's crossovers separate by subshell.”

They interleave. The three fields are 9.5965 × 10⁻⁶ (m = 0), 1.1081 × 10⁻⁵ (m = ±1) and 3.5087 × 10⁻⁵ (m = 0), so the m = ±1 event sits between the m = 0 pair's two.

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

“Every pair of levels in a subshell can be coupled by a field.”

The s and the d of the m = 0 subshell have a dipole of exactly zero between them — a field changes the angular momentum by one — and no matrix element joins the m = 0 subshell to the m = ±1 one at all, because a field along z leaves m alone.

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

“A defect band's width grows without limit as the chain grows.”

The largest coupling any chain can produce is the splitting between two runs one site apart, which is 0.0877β for runs of four and 0.0158β for runs of eight. No chain, however long, contains a closer pair, so the width has a ceiling that no length can raise.

Tested in A band that is a hundred and seventy decades of nothing · the series on defect

“As runs get longer their levels crowd together, so their bands must eventually merge.”

The spacing between the levels of runs of L and L + 1 and the sum of their widest half widths both fall as the cube of L + 1. Their ratio has a minimum of 1.6187 at L = 5 and rises at both ends, so two defect bands never touch.

Tested in A band that is a hundred and seventy decades of nothing · the series on defect

“A defect band can broaden until it overlaps the host band and closes the gap.”

At a barrier of four the highest defect level of a run of twelve sits 0.0581β below the barrier band and its band is at most 0.0026β across — a margin of 22.7 of its own half widths, which is larger than the 8.7 a run of four has, not smaller.

Tested in A band that is a hundred and seventy decades of nothing · the series on defect

“A localisation criterion that has no interior maximum for two orbitals has none for three.”

The Boys functional over the whole rotation group of a carbonyl's σ, π and oxygen lone pair peaks at 1.106642 Ų against 1.079600 for the best two-orbital mixing and 0.966555 for the canonical set — an interior point 0.80 radians from any relabelling.

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

“What the third orbital adds is the third orbital's own localisation.”

The lone pair is barely moved: the maximum takes 8.44 degrees of it into the two bonds, and what improves is the bonds. Removing the lone pair from the space costs 2.50 per cent of the functional, which is the amount no pair of orbitals can find.

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

“The rabbit-ear description of a carbonyl's lone pairs is the same phenomenon as a bent bond.”

It is the two-orbital criterion applied to two orbitals, and it wins outright: the two lone pairs' centroids are 0.31652 Å apart and twice the dipole between them is 0.54826, a ratio of 1.732128. A double bond's mixing is a competition; two equivalent lone pairs' is not.

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

“A description a search reaches rarely is a worse description.”

On a nine-vertex cage at eight electrons the fifty descriptions span 2.20 × 10⁻⁵ of the functional's value, and on a twelve-vertex cage at twenty electrons the five span 1.68 × 10⁻⁶. The rare ones are not worse; they are the same answer reached from different starts.

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

“Where the descriptions do differ, the one a search usually returns is the best.”

On the twelve-vertex cage at twelve electrons the descriptions differ by up to 3.9 per cent, and the most-reached one is 0.34 per cent below the best. Two rarer descriptions beat it on the functional the search is maximising.

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

“A four-site ring is a fair miniature of a correlated system for questions about symmetry breaking.”

Its half-filled shell is degenerate to 3 × 10⁻¹⁷, so its symmetric solution is unstable at every repulsion however small and its broken solution survives down to 0.25 with a polarisation of 0.516. A chain of four breaks only above 1.67 and a ring of six only above 2.36.

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

“The two ways of moving away from a reference are symmetric, so one measurement covers both.”

The broken solution exists BELOW a critical site-energy modulation and ABOVE a critical repulsion. On the site-energy axis the failure comes from turning the modulation up; on the repulsion axis it comes from turning the repulsion down, and the worst case there is 1,538 per cent worse than the calculation it was meant to correct.

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

“A composite correction is at worst no better than the cheap calculation it corrects.”

On a chain of four with a correction built at a repulsion of six, transferring it to a repulsion of a half gives an error of 0.425 where the mean field alone was wrong by 0.026 — sixteen times worse, and the practice's own assumption forbids it.

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

“A sixteen-electron complex has a gap, and adding a ligand is a separate question from whether it does.”

The gap is (2 − kf)eσ with k axial ligands at a fraction f of the equatorial strength, exactly, at every point. It is 1eσ at a full square pyramid and exactly zero at an octahedron — so the gap and the coordination are one quantity, not two.

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

“Distorting a square plane costs the gap its exactness, so adding a ligand will too.”

The derivative of the gap with respect to eπ is zero at every point of both axial approaches, to 10⁻⁹. Bending all four ligands loses it at the first degree. The fourfold axis is what protects it, and an approach along that axis keeps it.

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

“As the gap closes the complex becomes less stable, which is why it stops at sixteen.”

The stabilisation of the sixteen filled electrons rises the whole way — 2.0, 3.0, 4.0, 5.0, 6.0 eσ as two axial donors come in from nothing to full strength — while the gap falls from 2 to 0. The two move in opposite directions along the same path.

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

“Comparing a molecule's moment of inertia with its deuterated form's compares the same quantity twice.”

The principal axes are eigenvectors of a tensor built from the masses, so changing one mass turns them. One deuterium turns water's frame by 21.122°, and the daughter's moment about the parent's own smallest axis is 0.87431 u Ų against the 0.72911 it has about its own — a difference of 19.9 per cent, none of which is physics.

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

“An isotopic substitution always changes a rotational constant, which is why the method works.”

¹¹BF₃ → ¹⁰BF₃ changes every rigid moment by exactly zero, to machine precision, because the boron is at the centre of mass. The substitution reports no coordinate at all, which is Kraitchman's known blind spot arriving as an identity rather than as a caution.

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

“A substitution that leaves the rigid moment alone leaves the spectrum alone.”

It does not. ¹⁰BF₃'s rotational constants differ from ¹¹BF₃'s by 310 kHz on two axes and 58 kHz on the third, entirely from zero-point motion; and CH₃D's constant about its own threefold axis differs from methane's by 4,038 MHz on a moment that is identical to twelve figures.

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

“The scatter within a separation class is an end effect, so it collapses against how far the pair sits from the nearer end.”

The end pair's response relative to the deepest pair at the same separation is 0.6579 in a straight chain of nine and 1.1610 in a zigzag of nine, at four fusions' separation. One variable cannot take two signs.

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

“An end correction measured on one molecule can be carried to the next.”

At two fusions' separation the ratio runs 0.820, 0.762, 0.789, 0.839 for straight chains of five, seven, nine and eleven rings, and 1.512, 1.184, 1.277, 1.244 for zigzags. Neither converges and the parity of the ring count moves it as much as the length does.

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

“The decay length of a fused system's response is a property of the system.”

It is a property of which pairs are fitted. On a straight chain of nine it is 1.9005 fusions over all pairs, 2.0690 over the interior ones and 1.718 over the end pairs alone — a spread of a fifth, with no measurement between them to prefer.

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

“Two orbitals with no overlap and no resonance integral are not bonded to each other, so their bond order is zero.”

It runs from +0.031 to −0.954 as a third orbital is swept past them, and to −0.999899 when that orbital is put 80 eV below. A bond order is computed from the occupied orbitals, and the third orbital decides which combinations are occupied.

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

“The three regimes a third orbital creates survive the pair's own interaction vanishing.”

They collapse to one point. What the pair is bonded by, on the difference-of-two-calculations definition, is exactly zero at every third-orbital energy — there is no interaction to remove, so the quantity the regimes were measured in has no variation left.

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

“If the pair contributes nothing, the system is not bound.”

The trio is bound at every third-orbital energy tested, most strongly by 5.33 eV. Every electronvolt of it is A–C and B–C, and none of it is A–B.

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

“A chain has to be angular along its length to change how the response scatters.”

One turned fusion is enough. A straight chain of nine rings scatters within a separation class by a factor of 1.520 at worst; a chain turned at its first fusion, which has one angular ring, scatters by 3.052, and the chains turned at an interior fusion, which have two adjacent angular rings, by 4.065 to 4.600.

Tested in One integer, and everything it changes · the series on aromaticity

“The worst place to turn a fusion is the middle, where it divides the molecule most evenly.”

The worst is at the third fusion of eight, at 4.600, and the centre positions give 4.065. The trend along the chain is 3.052, 4.304, 4.600, 4.065 and then its mirror — though the first member has one angular ring and the rest two, so the trend mixes a count with a position.

Tested in One integer, and everything it changes · the series on aromaticity

“Two pairs at the same separation and the same depth in one molecule respond alike.”

In a chain bent at its second fusion, the pair touching the left end and the pair touching the right differ by a factor of 2.58 at two fusions' separation — 0.331 against 0.855 of the deepest pair's response. Both are at depth zero.

Tested in One integer, and everything it changes · the series on aromaticity

“The scatter is worst at the largest separation, where the response is smallest and a fit is most fragile.”

Only for the straight chain, which is the one case that rises monotonically. The chain bent at its second fusion peaks at 4.304 at four fusions but the one bent at its third peaks at 4.600 at three, in the middle of the range where a fit carries the most points.

Tested in One integer, and everything it changes · the series on aromaticity

“The dimension of a net decides how its coupled bands behave.”

A chain and a cubic structure are one- and three-dimensional and both give a gap exponent of −2; a chain reaching to its second neighbours and a triangular net are one- and two-dimensional and both give −1. Each group contains both, so no dimension sorts them.

Tested in Seven points that looked like a switch · the series on Bands in a solid

“The coordination decides it.”

Coordination six occurs in both groups — a cubic structure at −2.0277 and a triangular net at −1.1054 — and coordination four occurs in both, a square net at −2.0118 and a chain of reach two at −1.0644.

Tested in Seven points that looked like a switch · the series on Bands in a solid

“A census whose points fall on two lines has found a switch.”

Turning the third moment on continuously slides the exponent from −2.0118 to −1.0835, and it is half-way at a skewness of 0.0300. The smallest non-zero skewness any of the seven nets has is 0.7500, twenty-five times larger. The two lines are where the nets sit, not where the behaviour changes.

Tested in Seven points that looked like a switch · the series on Bands in a solid

“The size of an effect and the power it follows change together.”

Along the same family the quotient falls by a factor of 27.6, smoothly and monotonically, with no feature at the skewness where the exponent turns over. Watching the amplitude alone would never locate the crossover.

Tested in Seven points that looked like a switch · the series on Bands in a solid

“Distorting a square-planar complex closes the gap above eight electrons.”

Folding two of the four ligands to the same side raises it, from 2.000000 to 2.093826 eσ at twenty degrees. It falls only past twenty-five, and reaches 1eσ at forty-five. The four-ligand path and the axial approach both close it monotonically; this one does not.

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

“A gap that is getting larger is a gap that is getting safer.”

The derivative with respect to eπ is −0.00490 at two degrees, −0.13698 at ten and −0.63921 at twenty — where the gap is at its largest. Every point at which the gap exceeds the plane's is a point at which its exactness is already gone.

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

“Any tilt of two ligands out of the plane is a distortion of it.”

Tilting them so that they stay trans to each other changes no level at any angle, to nine figures — because two perpendicular linear pairs always span a plane, so the result is the same square plane in a different orientation. The levels are 0, 0, 0, 1, 3 at zero degrees and at ninety.

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

“A decay measured from the end of a molecule might be the end's rather than the molecule's.”

Moved to the fifth ring of twelve, the response decays at 0.687 rings in one direction and 0.707 in the other, against 0.719 measured from the end. The three agree to within three per cent, so one number describes the reach of every case.

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

“Where the heteroatom sits does not matter, then.”

The peak response is 3.763 × 10⁻² on the end ring and 1.942 × 10⁻² on every interior one — a factor of 1.94 — and it is shared between two rings rather than one. The end-ring measurements report a decay that is the molecule's and an amplitude that is a choice of position.

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

“The two directions differ because one has more molecule beyond it.”

The step ratios run 4.00, 4.86, 3.78 on the short side and 3.96, 4.80, 4.08, 3.47 on the long one. They track each other until the short side runs out of rings, and the departure is the boundary rather than the amount of molecule.

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

“A larger block would settle whether the non-additivity has a systematic sign.”

A model with no fitted length in it supplies twenty-one blocks, nine negative and twelve positive, with a root-mean-square of 0.1118 Å. The measured residues are 0.031 and 0.056 Å. The model's scatter is twice the larger measurement, so its split in sign is its own error sorted, not a verdict.

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

“A model that reproduces the measured separations can be trusted on their differences.”

It reproduces six measured separations to a mean of 0.242 Å and a worst of 0.563 Å. The residue is a fourfold alternating difference of four such numbers, and the largest measured residue is 0.056 Å — four times below the model's error on any one term. On the one block it can check, it returns +0.115 Å against a measured −0.031.

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

“The survey covers every block the four cations and four anions make.”

The overlap rule refuses two of the sixteen separations — magnesium and calcium against sulfide, a compact p against a diffuse one at three bohr — and says so rather than returning a number. Those two appear in fifteen of the thirty-six blocks, including the alkaline-earth chalcogenide block, which is one of the two the measurements supply.

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

“A harmonic zero-point correction to a moment of inertia is the leading term, and anharmonicity is a refinement on it.”

A rotational constant averages 1/r², whose expansion carries +2⟨Δr⟩/rₑ from the anharmonicity and −3⟨Δr²⟩/rₑ² from the harmonic spread. In all four molecules the anharmonic term is the larger, by 1.94 to 2.58 times, and it is exactly zero in any symmetric well.

Tested in The term a harmonic field cannot produce · the series on rotation

“The harmonic and anharmonic corrections to a moment both make it larger.”

They have opposite signs. For hydrogen chloride the anharmonic term is +2.4008 per cent and the harmonic term −1.1335, and the exact answer, +1.3289, is a difference between them rather than a sum.

Tested in The term a harmonic field cannot produce · the series on rotation

“A more anharmonic well is one in which the anharmonic term matters more.”

The ratio runs the other way: carbon monoxide, the most nearly harmonic of the four at ωₑxₑ/ωₑ = 0.61 per cent, has the largest ratio at 2.575, and hydrogen fluoride at 2.17 per cent has the smallest at 1.941. Both terms grow with the anharmonicity and the mean square grows faster.

Tested in The term a harmonic field cannot produce · the series on rotation

“The moment of inertia of a vibrating diatomic is a single number.”

One ground-state wavefunction gives two. μ⟨r⟩² sits 2.415 per cent above the equilibrium moment for hydrogen chloride and μ/⟨1/r²⟩ sits 1.329 per cent above it — nearly a factor of two apart, and the second is the one a spectrum reports.

Tested in The term a harmonic field cannot produce · the series on rotation

“A molecule sitting slightly off a square plane might have either count.”

It has the tetrahedron's. At 89.99° a ligand sits 1.7 × 10⁻⁴ of a bond length from where the plane would put it — two orders below the 0.06 tolerance the symmetry finder uses and far below anything diffraction resolves — and the count there is zero, as it is at 54.74°.

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

“The count is defined at every geometry, whether or not it is useful.”

Within about two degrees of the tetrahedron the symmetry finder refuses: the tetrahedral operations still map the molecule onto itself inside its tolerance but cannot be refined to the precision it demands, so no group is returned and no count exists. Four of the twenty geometries computed have no count at all.

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

“A filled shell's bond orders vanish, because the sum over occupied orbitals is a sum over a complete set.”

The sum over a complete set in a non-orthogonal basis is the inverse overlap matrix, not the identity. Between two orbitals whose overlap is exactly zero the filled-shell bond order comes out at 0.142857 — one seventh — by diagonalisation and by inverting a three-by-three matrix, two routes that share nothing.

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

“The bond order between two orbitals says something about those two orbitals.”

On one geometry with one Hamiltonian it is 0.00000 at zero electrons, +0.36940 at two, −0.63060 at four and +0.14286 at six. Nothing about the pair changes between those rows.

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

“A bond order between non-interacting orbitals is a small artefact.”

At four electrons with the third orbital eighty electron volts below the pair it is −0.999899, which is a full antibond to four figures; at two electrons with the third orbital at zero it is 1.03059, which is more than a full bond.

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

“The contrast limit at half filling is a measurement of how distinguishable satellites become.”

On a ring of six it is exactly 1.000000 from U = 64 upward, and at every one of those repulsions the third and fourth strongest lines agree in weight to 6 × 10⁻¹⁶ and in energy to 8 × 10⁻¹⁵. The contrast is a line divided by its own degenerate partner.

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

“The coincidences are an artefact of deciding when two computed fields count as one.”

Two degrees either side of each coincidence the count is back to six, so each is a point rather than a stretch. A tolerance of one, which calls every crossover the same as every other, reports one event at every angle — that is what a threshold artefact looks like and it is not what is measured.

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

“Propagating uncertainties would remove one of the three discordant pairs.”

The three need errors of 51.6, 43.0 and 16.0 per cent of their set's measured range to be reversed — 59, 49 and 18 kJ/mol on strain energies spanning 115. They are the three most robust refutations in the collection.

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

“A claim's fragility can only be assessed once the uncertainties are known.”

The error that would destroy a claim is computable from the claim alone: a difference Δ between two measurements is reversed at about Δ/√2 on each. Expressed as a fraction of the set's own range it is comparable across four predictors in four different units, with nothing quoted.

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

“The π overlap must shrink from fluoride to chloride, because the energy denominator over-predicts the trend and something has to cancel it.”

Computed at the measured bond lengths with Slater radial functions, chloride's π overlap squared is 2.919 × 10⁻² and fluoride's is 8.317 × 10⁻³ — 3.5 times larger, not smaller. The two factors push the same way and their product over-predicts a fitted ratio of 1.14 by about five.

Tested in The overlap the model is not proportional to · the series on ligand field

“The angular overlap model's eπ/eσ is the ratio of the squared overlaps, as its derivation says.”

The computed ratio is 0.3052 for fluoride and 0.9456 for chloride; the fitted values are 0.14 and 0.16. The computation over-predicts by 2.2 and 5.9, and the two columns are not even monotone together.

Tested in The overlap the model is not proportional to · the series on ligand field

“A ligand with no π interaction is one with no π overlap.”

Ammonia's fitted eπ is exactly zero and its computed Sπ²/Sσ² is 0.3546 — larger than fluoride's. The zero records that nitrogen has no p orbital to spare, which a bare p function placed on the donor atom cannot know.

Tested in The overlap the model is not proportional to · the series on ligand field

“Whatever eπ is, a computed overlap is the right kind of quantity for it.”

The fitted value runs +0.14 for fluoride, 0 for ammonia and −0.10 for cyanide. A ratio of two squared overlaps is positive by construction, so no computation of that kind can produce the column at all.

Tested in The overlap the model is not proportional to · the series on ligand field

“A band's own third moment decides how it responds to being coupled to another band.”

Two triangular bands, whose own third moment is 7.296, give an exponent of −1.1054 coupled along their own bonds and −2.1040 coupled along a matching. The bands are identical in both rows and the answers are a whole power apart.

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

“Two bands with no odd cycles between them keep the constant.”

Two square bands coupled along a triangular set of bonds have an intra-band third moment of exactly zero and give −1.0732, with a quotient spread of 128 per cent. The triangles are entirely in the perturbation.

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

“A weaker coupling is a weaker version of the same perturbation.”

A matching has one coupling bond per site where a square net has four, and the difference is not a factor. A matching has no site joined to two others across the gap, so it closes no triangle at all, and its gap-crossing third moment is 4 × 10⁻¹³ rather than a quarter of anything.

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

“Two couplings — the closest and the typical — describe the distribution between them.”

The closest is the extreme: 0.125 per cent of runs of six are coupled at least that strongly. The typical is the mean of a geometric distribution, at 250 sites against a median of 177, so 95.8 per cent of pairs are coupled more strongly than the coupling quoted as typical.

Tested in A count rather than an average · the series on defect

“How much of a defect band is a resonant pair is a property of the disorder.”

It is a property of the chain length, and it rises logarithmically. Runs of six go from 1.26 per cent of runs in a resonant pair at a thousand sites to 16.1 per cent at a million million — about 1.7 percentage points per decade, slowing slowly as the share grows.

Tested in A count rather than an average · the series on defect

“The share cannot be predicted without running the chains.”

It rises by ρξ ln 10 per decade, the run density times the coupling's decay length. For runs of four to eight that predicts 5.172, 3.093, 1.802, 1.027 and 0.575 points against measured 4.518, 2.878, 1.735, 1.007 and 0.569.

Tested in A count rather than an average · the series on defect

“Longer defects couple over shorter distances, which is why they are more isolated.”

They couple over longer distances: the decay length rises from 1.4376 sites for runs of four to 2.5563 for runs of eight, a slope of 0.2801 sites per site. What makes them isolated is the density, which halves with every extra site, and the density wins.

Tested in A count rather than an average · the series on defect

“The measure therefore survives everywhere.”

It fails at the extreme. The nine-vertex cage at eight electrons finds 44 descriptions — three times the next case — and their spread is 2.2 × 10⁻⁵, which is one answer. By count it is the most ambiguous system in the family; by spread it is not ambiguous at all.

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

“The two readings rank the family similarly.”

The case at the top of the count ranking has the third-smallest spread of any case with more than one description. The case at the top of the spread ranking, an eleven-vertex cage at twenty electrons, finds two descriptions and does not appear in the count ranking's top three at all.

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

“A large spread needs many descriptions to be spread over.”

Ten of the thirteen cases with more than one description find exactly two, and those two include the largest spread in the family at 5.3 × 10⁻². Two descriptions that differ by five per cent of the functional are more ambiguity than forty-four that differ by two thousandths of a per cent.

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

“One of the two-state estimates agreeing with the exact crossing to one per cent shows the picture works near that angle.”

At a general tilt the six estimates span a factor of four to twenty-one while the exact crossing moves by a factor of 1.34 across the entire ninety degrees. Scattering six numbers over that range guarantees one lands near the target. At 60° the nearest is 1.5 per cent away and at 20° the nearest is 37 per cent away, with no angle in between behaving differently.

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

“The depolarisation ratio's departure from three quarters measures the distortion.”

Adding any amount of the totally symmetric breathing distortion to any other changes the departure by at most 0.35 per cent, and breathing alone gives exactly zero at every size tried. The ratio reads the distortion's component outside the totally symmetric species, not the distortion.

Tested in The distortion the ratio cannot see · the series on spectrum

“Distortions the ratio can see separately combine into one it can see.”

Each of the three single-bond stretches of boron trifluoride moves the ratio by 1.07 × 10⁻⁵. Their sum moves it by zero — not a small number, exactly zero, at a displacement of 0.02 and at five times that.

Tested in The distortion the ratio cannot see · the series on spectrum

“A blind direction of a second-order probe is a small effect that a larger distortion would expose.”

The breathing distortion gives exactly zero at displacements from 0.005 to 0.1, a factor of twenty. A small effect grows as the square of the displacement, as the visible distortions do; this one does not grow because it is an identity of the point group.

Tested in The distortion the ratio cannot see · the series on spectrum

“The criterion tests whether the two lone pairs are better described as mixed rabbit ears or as a σ and a π lone pair.”

It equals 1/√(p fraction), which exceeds one for every hybrid with any s character at all. Over nine hybrids from 90 per cent s to 10 per cent s it returns 3.162, 2.236, 2.000, 1.732, 1.581, 1.414, 1.291, 1.155 and 1.054 — mixed wins nine times out of nine. It has no input that could make it say otherwise.

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

“A criterion with a closed form can be checked at one molecule.”

At 50 per cent s character the correct form 1/√(p) and the wrong form 1/√(s) are both √2, and the integrals return 1.414234. A single carbonyl at that hybridisation would confirm both. The sweep over nine hybrids is what separates them, and away from that one point the wrong form misses by at least 8.9 per cent.

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

“A geometry with no orphan count is one the symmetry finder could not resolve.”

Two different failures are being run together. In two bands, 0.070° to about 6° and about 54° to 59.930°, the finder refuses — its own check that every operation maps the molecule onto itself to better than 10⁻³ Å fires. Across the eleven interior angles sampled it does not refuse: it succeeds, reports order 6, and produces no count. The geometry is exact and the arithmetic is absent.

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

“A level placed exactly by a symmetry is at least approximately placed when the symmetry is nearly there.”

Detuning one of the two equivalent orbitals by 0.05 eV moves the level by 0.025106 eV — 0.502120 of the detuning. There is no regime in which most of the exactness survives; the ratio approaches one half as the detuning falls, not zero.

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

“Breaking the symmetry changes the level and leaves everything else alone.”

The two bond orders that were equal by symmetry separate at once, to 0.525795 and 0.518996 at a detuning of 0.05, and the pair's own antibond deepens from −0.630602 to −0.632329. Every quantity moves first order.

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

“The free combination is ρ·χ_TIP raised to some power, and the power is a property of the model.”

The power is +0.412 for an open chain, +0.225, +0.278 and +0.285 for rings of six, eight and ten, and −0.857, −0.463 and −0.287 for rings of five, seven and nine. The sign is the ring's parity.

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

“Being a ring rather than a chain is what changes the answer.”

Rings of six, eight and ten give positive exponents, like the open chain. Only the odd ones — where the antiferromagnetic couplings cannot all be satisfied — go negative, so it is the frustration and not the boundary condition.

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

“The exponent is a fixed number worth quoting for a frustrated ring.”

It decays with size: 0.857 at five spins, 0.463 at seven, 0.287 at nine. An odd ring's frustration is a finite-size property and the exponent follows it down, so a value quoted without the ring size is not a value.

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

“The metal's contraction could account for the discrepancy between the computed and fitted π/σ ratios.”

Across the entire range Slater's rules allow chromium — 4.60 at the metal to 6.00 at chromium(VI) — the computed ratio changes by a factor of 2.07 to 2.18 depending on the ligand. The discrepancy runs from 2.2-fold at fluoride to 5.9-fold at chloride. The window is smaller than the gap for three of five ligands and irrelevant to the other two.

Tested in A contraction that cannot reach three of them · the series on ligand field

“A different effective charge per ligand would work, since a real 3d contracts as the ligand field strengthens.”

Water needs 9.49 and fluoride 7.20 — two different charges, neither inside 4.60 to 6.00, and 2.30 apart. Chromium has 24 protons and eighteen core electrons; an effective charge of 9.49 on a 3d electron would need nine of the eighteen to stop screening. Chloride needs more than 16, which is where the overlap rule stops agreeing with its own verifier, so the model cannot even evaluate its own answer there.

Tested in A contraction that cannot reach three of them · the series on ligand field

“The discrepancy is a matter of size that a better radial function could close.”

Ammonia's fitted e_π is exactly 0.00 and cyanide's is −0.10. The model's ratio is S_π² over S_σ², which is strictly positive for every radial function, every bond length and every basis. Two of the five ligands are outside the range of values the model can produce at all.

Tested in A contraction that cannot reach three of them · the series on ligand field

“Contracting the metal could be tuned to trade one ligand against another.”

Every ligand's computed ratio falls with contraction — by 2.17, 2.18, 2.16, 2.07 and 2.09 for chloride, fluoride, water, ammonia and cyanide. They move together and by nearly the same factor, so the ordering across the series is fixed and one charge cannot be chosen to improve one ligand at another's expense.

Tested in A contraction that cannot reach three of them · the series on ligand field

“The hardest composition to optimise is the one with the most arrangements.”

On thirty-six sites at a contrast of one the half-filled composition has 9.08 × 10⁹ arrangements, one local optimum and a hundred per cent hit rate; nine raised sites has 9.4 × 10⁷ arrangements, six local optima and a hit rate of ten per cent. The larger space is the easier problem.

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

“A search over arrangements gets steadily harder as the composition rises towards half.”

It is not monotone in either direction. On sixteen sites at a contrast of one the hit rate runs 100, 100, 100, 33.5, 21.0, 67.0, 100, 100 per cent from one raised site to eight.

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

“Difficulty is a property of the problem, so the two contrasts should agree about which composition is hardest.”

On thirty-six sites the hardest composition is nine raised at a contrast of one and twelve at a contrast of four, and the second has twelve distinct local optima against the first's six. Changing the contrast moves the hard composition by a third of the way across the range.

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

“The nineteen tables were a principled set and D3 was excluded for a reason.”

C3v is also of order six and was present; D2, D4, D5d and D6h were present. D3 needs three classes and three irreducible representations, its class sizes sum to 6, its squared dimensions sum to 6, and every pair of rows and columns is orthogonal to a part in 10⁹. It was absent because nothing had asked for it.

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

“Reading the contrast at a large repulsion gives its limit.”

The chain's half-filled contrast is 1.0316 at U = 32 and 1.0617 at U = 128 — it moves away from one over a factor of four in the repulsion before turning back. A reading at U = 32 is nearer the limit than a reading at U = 128, and neither is the limit.

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

“The guard refuses an extrapolation when the number is wrong.”

It refuses both of the ring's degenerate fillings and permits all three of the chain's. The half-filled ring, which it refuses, gives 1.000000; the half-filled chain, which it permits, gives 0.999999. The guard is a statement about the evidence, not about the answer.

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

“The product of the polarisation and the correlation-energy change collapses the scatter better than either alone.”

The change alone scores 1.157, which is the floor this statistic can reach. Multiplied by the polarisation it scores 1.910 — the multiplication moves it off the floor rather than towards it.

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

“Recomputing every price under perfect correlation gives the other extreme, so the true value lies between two numbers.”

At perfect correlation between measurements of equal precision the difference of the two has standard error zero, so no measurement error overturns any claim and the price is unbounded. The bracket runs from the independent price to infinity, which is not a bracket.

Tested in The other end of the bracket is not a number · the series on models

“Correlated errors would turn each price from a point into a bracket around the ordering.”

The multiplying factor is √2/√(1 + r² − 2ρr), which depends on the correlation and the precision ratio and on nothing about the claim. Every one of the five priced claims is multiplied by the same number, so the ordering by fragility is identical at correlations of 0, 0.25, 0.5, 0.75, 0.9, 0.99 and 1.

Tested in The other end of the bracket is not a number · the series on models

“The most fragile claim might not be the most fragile once errors are correlated.”

Overtaking would require a factor below one, and every factor is at least one — the smallest is exactly 1 at zero correlation and equal precision. Raising the correlation on one claim alone moves it further from the next, never towards it. The required correlation is reported as non-existent rather than as a value outside [0, 1].

Tested in The other end of the bracket is not a number · the series on models

“The one avoided crossing's field varies by a third across the tilt.”

In the complete shell it does not vary at all. Thirteen directions spread over the sphere give the same field to eleven decimal places, because the zero-field Hamiltonian depends only on l and is therefore invariant under every rotation, while a uniform field's coupling rotates with the field. The whole spectrum is independent of direction, exactly.

Tested in The variation was the basis · the series on representation

“The second crossing at ninety degrees is a feature of the field lying along x.”

It is the truncation's. The five-function basis retains two d functions and a field along x leaves them degenerate, so a gap closes to nothing. The whole shell's field lifts the entire d quintet at every direction, and the nine-level count is one at ninety degrees as it is everywhere else.

Tested in The variation was the basis · the series on representation

“The chemical capacity is essentially a statement about the second ionisation energy.”

Fitted directly, the second ionisation energy accounts for 2.9 per cent of the capacity's variation across fourteen atoms. The best of five two-parameter forms accounts for 43.8, and needs a logarithm the proposal did not include.

Tested in A correlation is not an account · the series on electronegativity

“The capacity's relation to the second ionisation energy is monotone enough to be a rule.”

Nitrogen and potassium differ by 2.02 eV in the second ionisation energy, less than almost any other pair, and by a factor of 193 in capacity — 31.54 against 0.164. No function of one variable gives two nearly equal inputs two answers two hundredfold apart.

Tested in A correlation is not an account · the series on electronegativity

“The direction dependence appears when the molecule is pushed hard.”

The departure divided by the square of the amplitude varies by 2.0 per cent across a factor of eight in amplitude, and the anisotropy itself moves only from 1.989 to 1.915. Both extremes are second order and so is the ratio between them.

Tested in One number was a direction too · the series on spectrum

“The two extremes are an arbitrary pair of directions on the circle.”

The minimum is at one bond stretched against another and the maximum at two bonds against the third — the two kinds of distortion the point group distinguishes within the plane. Every direction related to the minimum by the three-fold rotation and the mirror planes reads the same to two parts in a million.

Tested in One number was a direction too · the series on spectrum

“The verdicts on the gem-dimethyl effect depend on a rotor count nobody had settled.”

Nine calculations — three measured accelerations against three conventions — and the verdict column never changes. The five-membered ring's 250-fold and 11,000-fold accelerations exceed the ceiling under every convention; the six-membered ring's 10-fold falls under it under every convention. The unsettled choice is not load-bearing for any of those verdicts.

Tested in The count that was never written down · the series on strain

“The two conventions differ by an offset, so one is the other shifted.”

They order the ring sizes oppositely. n − 2 gives the five-ring 3 rotors and the six-ring 4; the decided verdict used 4 for the five-ring and 2 for the six. One rises with ring size and the other falls, so no constant reconciles them and at most one can be a count of anything.

Tested in The count that was never written down · the series on strain

“The tail exponent settles near two thirds as the gap closes.”

The four cases that produce the flattening have between 3.8 and 6.2 coherence lengths of visible tail. Raising the floor from 10⁻⁸ to 10⁻⁵ moves their fitted powers from 0.21 to 0.58, from −8.2 to 0.30, from −37.8 to −6.0 and from −196 to −36.9. A number that moves by tens when a threshold moves is not a measurement.

Tested in The exponent was the floor · the series on Peierls distortion

“The drift in the fitted power comes from fitting over a fixed window of bonds.”

Refitting each case over a window scaled to its own coherence length gives 0.47, 0.51, 0.54, 0.57, 0.61, 0.67, 0.68 — the same drift. The window's position in bonds is not what decides it.

Tested in The exponent was the floor · the series on Peierls distortion

“The drift is a subleading term the two-parameter fit is absorbing.”

Fitting rate = 1/λ + p/b + c/b² makes it worse, not better: p then runs 0.42 to 0.75 and c changes sign midway through the range.

Tested in The exponent was the floor · the series on Peierls distortion

“A chain of 320 sites is too short to hold the tail.”

The centre of the chain is bulk to better than 10⁻⁸ at every stiffness here: the excess forty bonds before the centre is 8 × 10⁻⁵ at K = 3.1 and the centre itself is −3 × 10⁻⁸. The limit is the arithmetic's resolution, not the chain's length, and a longer chain would not move it.

Tested in The exponent was the floor · the series on Peierls distortion

“The two-state picture is a limit this problem is far from, and closing the l degeneracy would bring the two counts together.”

Swept from a defect of 0.02 to 0.0002 the exact count is one throughout and the estimated count is eight throughout, with the estimates spanning a factor of 7.08 at both ends. Nothing approaches anything.

Tested in Consistently wrong is not a limit · the series on representation

“The nearest estimate agreeing to nine per cent shows the two-state picture is roughly right.”

It is the nearest of eight scattered over a factor of seven, and its ratio to the exact field converges to 0.9067 rather than to one. An estimate that settles at a fixed wrong value is a different quantity computed accurately, not the right quantity computed approximately.

Tested in Consistently wrong is not a limit · the series on representation

“The convergence of these ratios is a numerical artefact of the sweep's smallest defect.”

They move over the sweep and then stop moving: the crossing's fraction of the gap runs 0.04097, 0.04048, 0.04024, 0.04009, 0.04004, 0.04001, 0.04000 as the defect falls by a factor of a hundred. A quantity constant by construction would not have moved at the top of the range.

Tested in Consistently wrong is not a limit · the series on representation

“The exponent's sign is negative because the ring is frustrated.”

Open chains of five, seven and nine spins are not frustrated — a chain is bipartite and every antiferromagnetic bond can be satisfied — and their exponents are −0.792, −0.440 and −0.272. Three unfrustrated systems carry the sign the account reserves for frustrated ones.

Tested in It was the count, not the frustration · the series on magnetism

“Removing the frustration would change the sign.”

Weakening a ring of five's closing bond from full strength to absent removes the frustration continuously and leaves the count alone. The exponent moves by 0.065 across the whole path — from −0.857 to −0.792 — and crosses nothing. At zero strength the system reproduces the open chain of five to a part in 10⁹ in every singular value.

Tested in It was the count, not the frustration · the series on magnetism

“A bend in a fused chain is a boundary of the same kind as an end, so the reach measured on a straight chain is a property of straight chains.”

Putting one kink — two adjacent angular fusions — in the middle of a chain of twelve changes the response on the heteroatom's own ring by 0.12, 0.09 and 0.52 per cent when the heteroatom is three rings before, three rings after or on the end, and by 18.9 per cent only when it sits on the bent ring itself. An end halves it.

Tested in A bend is not an end · the series on delocalisation

“The bend changes how far the response reaches.”

The fitted decay lengths on the bent chain and the straight one agree to within 10.7 per cent everywhere, and the straight chain's own two directions already differ by up to 7 per cent. The bend moves the range by less than the direction of measurement does.

Tested in A bend is not an end · the series on delocalisation

“Which side of the bend the heteroatom sits on matters.”

Three rings before the bend the ratio is 0.9991 and three rings after it is 0.9859; one ring before, 0.9948, and one ring after, 0.9685. The effect does not change sign or size across the bend, which is what a boundary would do.

Tested in A bend is not an end · the series on delocalisation

“A bend is therefore structurally irrelevant to a fused system's response.”

A ring-current response changes by a factor of four with a single bend. The same structural change costs this response a fifth of its amplitude on one ring. Two responses, one structural change, and a difference of twenty in sensitivity.

Tested in A bend is not an end · the series on delocalisation

“The cheap diagnostics all fail because of one pathological corner of the square.”

The five testable candidates fail on five distinct pairs involving ten distinct systems, and no system appears in more than one failure. There is no small set whose removal leaves the diagnostics working.

Tested in Five failures in five different places · the series on approximation

“The 3.41 per cent residual is what the collapse is worth.”

Re-run on five cases matched at n·δ∞ = 9.6 — rings of 40, 56, 78, 108 and 150 — the worst spread is 1.62 per cent, a factor of 2.10 tighter. The residual was finite size.

Tested in Five rings that were five different sizes · the series on metal

“The square plane is the natural place to look for the blind direction.”

The gap's gradient there is zero in both directions — 4.9 × 10⁻⁸ against 2.9 × 10⁻³ ten degrees out — so every direction is flat to first order and none is distinguished. A null direction computed there is undefined, and the calculation reports that rather than an angle.

Tested in The direction the gap cannot see · the series on electron count

“The blind direction is a fixed coordinate of the four-ligand problem.”

It is at 45° to the two fold axes only on the symmetric line. At ten degrees and zero it is at 90°; at fifteen and five it is at 59.52°; at twenty and ten it is at −72.76°. The direction is a property of where the molecule sits.

Tested in The direction the gap cannot see · the series on electron count

“A molecule distorting along the blind direction keeps its gap.”

It keeps it to first order only. Three degrees along the direction from a ten-degree symmetric fold moves the gap by 0.55 per cent, and from fifteen degrees by 1.04 — upward in every case, so the symmetric line is a valley floor in the gap and a molecule leaving it pays quadratically.

Tested in The direction the gap cannot see · the series on electron count

“The symmetry argument that predicted an isotropic reading was wrong.”

It was right about each band and applied to the wrong observable. The sum over bands is the quantity the argument governs, and it is constant on the circle to four parts in ten thousand — which is confirmation rather than consistency.

Tested in The sum was flat all along · the series on spectrum

“Following one band instead of the maximum removes the anisotropy.”

Ordering the four bands by size at each direction and following the k-th gives curves that vary by a factor of two or more, because an ordering crosses between bands where they cross. An ordering is not a band, and the distinction is the whole mechanism.

Tested in The sum was flat all along · the series on spectrum

“The residual in the sum's isotropy is the fourth-order term in the distortion.”

A fourth-order origin makes the relative residual scale as the square of the amplitude. Measured across a sixteen-fold range, the residual scales as amplitude to the power 1.248, and a single power describes it to within 7 per cent over that whole range. The predicted exponent of 2 is 0.75 away from the measurement, which is five times the scatter.

Tested in The suspect that did not fit · the series on spectrum

“The residual is the solver's own precision showing through.”

A precision limit does not move with the amplitude. This one falls by a factor of 32 as the amplitude falls by a factor of 16, from 3.31 × 10⁻³ to 1.05 × 10⁻⁴. It is a term in the expansion, whatever order it is.

Tested in The suspect that did not fit · the series on spectrum

“A ring with a twofold axis through two of its bonds would test whether an orbit of torsions can be reversed.”

A twofold axis through a bond's midpoint carries the torsion about that bond onto itself end for end, and being proper it cannot change a dihedral's handedness. Benzene's ring torsion is fixed by four of its group's twenty-four operations, and the two that reverse it are the mirror bisecting the bond and the plane of the molecule.

Tested in Five coordinates for six vibrations · the series on point group

“Torsions complete any coordinate set that stretches and bends leave short.”

They complete ethene, benzene and hydrogen peroxide. Ferrocene's iron is 2.064 ångström from every carbon, beyond the 1.85 at which bonds are listed, so it has no coordinates at all: torsions take the rank from 44 to 48 of 57, and only bonds to the iron reach 57.

Tested in Five coordinates for six vibrations · the series on point group

“Completing a coordinate set brings the old orbit formula closer to the answer.”

It moves it further away. Benzene goes from −4 to −22 and each ferrocene from −11 to −66 against true counts of 2 and 4, and ethene, which the formula had right, comes out wrong.

Tested in Five coordinates for six vibrations · the series on point group

“The near-end cut is a second window with a second exponent hiding in it, so the quantity carries two arbitrary choices.”

From six bonds outward the exponent moves by between 0.5 and 9.6 per cent of itself, against a range of 0.41 to 0.66 across the stiffnesses — a sixty per cent effect. The quantity carries one arbitrary choice, and sweeping it is what establishes that.

Tested in The rule of thumb was on the flat part · the series on Peierls distortion

“Since the choice does not matter, the first few bonds could have been kept.”

Fitting from two bonds gives 0.403, 0.429, 0.452, 0.478, 0.499, 0.515, 0.526, 0.535, 0.536 and 0.534 — every one lower than its value at six, and compressed into a range of 0.133 against 0.247. The rule of thumb is avoiding a real systematic error, not a hypothetical one.

Tested in The rule of thumb was on the flat part · the series on Peierls distortion

“Every stiffness can be fitted from the same range of starts.”

A fit needs at least six local rates and the soft chains run out of visible tail early: K = 1.1 admits no start beyond fifteen bonds while K = 1.4 and above reach thirty. Comparing across stiffnesses at a fixed large start would be comparing fits with different numbers of points behind them.

Tested in The rule of thumb was on the flat part · the series on Peierls distortion

“The three windows in this fit are all plateaus, like the two that have been swept.”

The third is not. Sweeping the read fraction from a tenth to a half moves the fitted exponent by 13.3 per cent at K = 3.1, 9.9 at 2.8 and 5.3 at 2.5. The two windows already swept move their exponents by under ten per cent across their whole range and mostly by far less; this one moves more than either, on the cases quoted at the upper end of the range.

Tested in The window that was not a plateau · the series on Peierls distortion

“Opening the window fixes the affected cases.”

It fixes two of five. A profile that ends on its own runs 13.26 of its own decay lengths, the same to within 3 per cent across every case where the quarter rule was not binding. At half the chain K = 2.5, 2.8 and 3.1 have run 9.2, 6.5 and 5.2 lengths, so they are still cut. Their exponents are not mis-measured; they are unmeasured, and no fraction of a 320-site chain measures them.

Tested in The window that was not a plateau · the series on Peierls distortion

“The exponents move in whichever direction the extra points happen to push them.”

Every one of the five affected exponents falls. The published upper end of the range, 0.6586 at K = 2.8, becomes 0.5932. A window that truncates a decay whose local rate is still falling systematically overestimates the exponent, so the direction is forced and the published range's top was the window rather than the chain.

Tested in The window that was not a plateau · the series on Peierls distortion

“The second ionisation energy is one candidate among several that the capacity might have been a function of.”

It is the worst of the six by a factor of twenty-four over the best, and by thirty-nine times the floor that fourteen atoms allow. Its rank correlation with the capacity is −0.5033, so the relation a rising straight line was fitted to is a falling one.

Tested in The worst of the six was the one we asked about · the series on electronegativity

“A pair test can find the quantity the capacity really is.”

The capacity comes from a cubic fitted through the dication, the cation, the atom and the anion, so every input the model can offer is either one of that cubic's three coefficients or one of the three energies it was fitted through. The three coefficients rank ahead of the three data with no interleaving, which is the test behaving rather than a discovery.

Tested in The worst of the six was the one we asked about · the series on electronegativity

“Frustration decides the sign of the free direction's exponent.”

Four frustrated clusters with even spin counts — a tetrahedron, a square with one diagonal, an octahedron and a triangular prism — give +0.2084, +0.2136, +0.1924 and +0.2048. The frustration account predicts a negative exponent for every one of them, and none is negative.

Tested in The frustrated cluster with an even count · the series on magnetism

“A design of chains and rings alone could have settled this.”

Every one of its eleven cases had frustration and odd parity aligned or both absent, because an odd ring is frustrated, an even ring is not, and a chain never is. A frustrated system with an even count needs a graph that is neither a chain nor a ring, and its solver could not build one.

Tested in The frustrated cluster with an even count · the series on magnetism

“The sign follows the ground multiplicity, which is the mechanism the parity rule appeals to.”

A tetrahedron's ground state is two singlets at one energy, so its degeneracy is two where every other even cluster's is one — and its exponent, +0.2084, sits with theirs rather than with the odd clusters'. What the sign follows is the ground state's total spin: zero for every even cluster here and one half for every odd one.

Tested in The frustrated cluster with an even count · the series on magnetism

“So the exponent is a function of the spin count and nothing else.”

Topology moves its magnitude by up to 38 per cent at a fixed count. A trigonal bipyramid of five gives −1.1809 against a ring of five's −0.8573, and the four four-spin clusters span 0.2084 to 0.2871. What no topology moves is the sign.

Tested in The frustrated cluster with an even count · the series on magnetism

“The crossing's field converges to four hundredths of the zero-field s–p gap.”

With the defect removed the problem has no λ in it, and its minimum sits at 0.0399865255. The swept sequence passes 0.04000 between the two smallest defects and is still falling; a line through those two reaches 0.0399869 at zero.

Tested in The crossing nothing couples · the series on representation

“The limit is most likely a ratio of the shell's own angular integrals, like the four coincidence angles.”

It is the field at which two eigenvalues have equal slopes — one the lowest root of a cubic, the other 11/10 − √(1/100 + 81f²/4) — with the squared dipoles 54, 27 and 81/4 as coefficients. That is an algebraic number, and not a ratio of two integrals.

Tested in The crossing nothing couples · the series on representation

“The whole shell has one avoided crossing, at the minimum of its s–p gap.”

The two levels at that minimum are the lowest m = 0 state, 92.5 per cent 3s, and the lowest |m| = 1 state, 74.3 per cent 3pₓ. The field commutes with the angular momentum about its own axis, the z matrix has no element between those blocks, and the minimum is where their induced dipoles are equal, −3.93340 bohr each. Nothing avoids anything.

Tested in The crossing nothing couples · the series on representation

“A Morse curve built from a molecule's measured ωₑ and ωₑxₑ reproduces its vibration–rotation constant.”

Averaged exactly over its solved states it gives αₑ of 0.27747, 0.10240, 0.01674 and 0.68164 cm⁻¹ for HCl, DCl, CO and HF, against measured 0.3072, 0.1133, 0.0175 and 0.798 — ratios of 0.903, 0.904, 0.957 and 0.854.

Tested in The cubic a Morse curve guesses · the series on rotation

“The shortfall is a numerical error in the average.”

The average agrees with Pekeris's closed form for the same curve to 0.04 per cent or better in all four, and six hundred grid points instead of four hundred move HCl's value by under half a per cent. The formula and the solver describe the same curve; the curve is what misses.

Tested in The cubic a Morse curve guesses · the series on rotation

“The mismatch could belong to either molecule's data rather than to the potential.”

HCl and DCl share one Born–Oppenheimer potential, and their measured constants imply the same cubic Dunham coefficient, −2.3646 and −2.3644. Their Morse curves imply the same one too, −2.2330 and −2.2334, and miss by the same fraction. Two masses, one miss.

Tested in The cubic a Morse curve guesses · the series on rotation

“That the two models agree on the symmetric line is a coincidence worth measuring.”

It cannot be otherwise. Exchanging the two trans pairs maps the arrangement to itself, and the repulsion is a function of the arrangement alone — so it inherits the same parity and the same vanishing first derivative. The two null directions agree to the last representable bit at every amplitude, and no computation could have made them differ.

Tested in A blindness that is inherited · the series on electron count

“So a blind direction found in one model can be expected to be blind in another.”

Only when a symmetry produces it. Off the symmetric line the gap's null direction and the repulsion's have alignments of 0.3425, 0.0242 and 0.8026 — the second is all but perpendicular — and there is no geometry off the line where they agree.

Tested in A blindness that is inherited · the series on electron count

“The square plane is the natural place to compare the two models.”

Neither has a gradient there. Both fold coordinates change sign under reflection through the plane and both models are even in each of them, so every direction is null in both and the comparison has no content. The comparison is declined rather than reported as an angle.

Tested in A blindness that is inherited · the series on electron count

“The 1.62 per cent left after matching may be the even-site rounding rather than a departure from scaling.”

At a target of five the rounding leaves 6.48 per cent un-matched and the collapse residual is 2.51 — the collapse is tighter than the mismatch it is supposed to be limited by. The rounding bounds the residual loosely at one end and not at all at the other, and it cannot be what is being measured.

Tested in Three points, and they all go down · the series on metal

“The collapse can be run at any target.”

Below about one, two stiffnesses round onto the same ring — at a target of 0.5 the sizes are 2, 4, 4, 6, 8 — and two identical rings agree with each other perfectly. A target that produced them would report a tighter collapse for a reason that has nothing to do with scaling, and the right response is to report the collision rather than collapse the two into one case.

Tested in Three points, and they all go down · the series on metal

“Six misclassified pairs is the floor for a rule of this kind, because the difference is what decides a sign.”

Three is reachable, using the same data. Adding the inter-table dispute as a second variable and taking the best straight boundary in the plane brings the count from 6 to 3, and the exact one-dimensional threshold is used at each slope so the comparison is between two optima rather than between an optimum and a guess.

Tested in Two numbers caught what one could not · the series on dipole

“The right two-variable rule separates them completely.”

No straight line in the plane does. Of 401 slopes swept from −2 to 2, none puts every disputed pair below the line and every agreed pair above it. The best possible is 3 wrong of 153, so the exceptions are not an artefact of projecting onto the wrong axis.

Tested in Two numbers caught what one could not · the series on dipole

“Enough tables would eventually show whether the rule has a real exception.”

Independent tables would need ten. At the measured correlation the requirement is seventy-three, a bootstrap over the pairs puts the correlation anywhere from 0.11 to 0.55, and at a correlation of a half not even four hundred tables bring the chance of a spurious agreement below five per cent.

Tested in No panel of this kind can find an exception · the series on dipole

“Four tables are four independent opinions.”

Over the nineteen pairs with the smallest differences, Allred–Rochow and Allen correlate at 0.92, Pauling correlates with both at about 0.7, and Mulliken runs slightly against both. The participation ratio of that correlation matrix counts 2.02 effective tables.

Tested in No panel of this kind can find an exception · the series on dipole

“The eight two-state estimates scatter around the shell's avoided crossing.”

They are projections. With the field along its own axis the same formula gives three estimates, along x three different ones, and at the tilt eight, none equal to an axial one — while the exact spectrum is the same in all three directions. The eight described the axes the functions were written along.

Tested in Two levels cannot make a minimum · the series on representation

“The nearest estimate, two per cent from the coupled minimum, was estimating that minimum.”

Moving the d level moves the estimate up in proportion and the minimum down: 0.0327 at a small offset, 0.0196 at the shell's own fifth, nothing past a quarter. The two curves cross at an offset of 0.2013, and at an offset of 0.001 they differ by a factor of 340.

Tested in Two levels cannot make a minimum · the series on representation

“A two-state estimate locates where its pair of levels comes closest.”

Two coupled levels separate as √(Δ² + 4f²d²), which never falls. The p₁–d₁ pair and the s–p₀ pair on its own have no minimum at any field, and the one minimum between coupled levels needs its block's third level.

Tested in Two levels cannot make a minimum · the series on representation

“The two-parameter well is simply too crude, and any error is spread across everything it computes.”

It is not spread. The zero-point energy comes out at 586.81 wavenumbers and the harmonic frequency at the minimum at 1226.6, both within a few per cent of what a fitted potential gives; four states sit below the barrier in both. Only the splittings are wrong, and they are wrong by 70 and 91 per cent.

Tested in A barrier is not what a splitting measures · the series on inversion

“The difference between the two wells is where their barriers are tallest.”

At the ground state's own energy the two barriers are 0.2591 and 0.2607 ångström wide at half height — a difference of six parts in a thousand. Their actions are 5.6872 and 6.1229, and e raised to that difference is 1.546, which is nine tenths of the factor of 1.70 in the splitting. The area differs where the width does not.

Tested in A barrier is not what a splitting measures · the series on inversion

“The accuracy line is a convention, so moving it rescales the verdicts without changing their shape.”

It changes which size is the answer and whether there is one. The window in which the published answer holds runs from 0.9926 to 1.7224 times a kilocalorie a mole; below 0.9926 no basis size is usable at every separation. A convention seven tenths of a per cent stricter erases the finding.

Tested in The line was holding the answer up · the series on basis

“The four answers are a scatter, so the sweep is reading noise in a marginal comparison.”

They are a staircase. The usable size falls 4 → 3 → 2 → 1 in order, no size repeats, and each stretch occupies a comparable width on a logarithmic axis — roughly a fourfold loosening of the line per basis size. That is the structure a threshold cutting across quantities that fall geometrically with basis size produces.

Tested in The line was holding the answer up · the series on basis

“The far end of the fitting window is the one that matters, because it runs into the noise floor.”

Inside the ends each profile actually reaches, the exponent moves by 0.00, 0.12, 0.23, 0.72, 0.87, 0.50, 1.24, 2.89, 3.52 and 3.02 per cent of itself. The worst is 5.3 per cent of the published value, against the near end's 9.6. Both windows are plateaus and the near one is slightly the worse.

Tested in The other window was a plateau too · the series on Peierls distortion

“The sign of the exponent is a property of the spin count.”

A star of four spins and the complete bipartite graph on two and four are even and unfrustrated, and their unequal sublattices give each a ground spin of one. Their exponents are −6.841 and −3.812, where the parity rule predicts positive. Across sixteen clusters there is no coupling from 2 to 1000 cm⁻¹ at which the parity of the count predicts every sign.

Tested in The sign rule holds between two poles · the series on magnetism

“The exponent that has been followed is the free direction's.”

It is the third singular direction's, fixed to between 1.0 and 16.1 per cent. The free direction is the fourth, fixed to between 11.5 and 121 per cent. The two lie in one plane and their exponents are negative reciprocals to within three per cent, so every sign reported for the third is the opposite sign for the free one.

Tested in The sign rule holds between two poles · the series on magnetism

“The capacity's ordering across the atoms is a property of the quantity rather than of the fit.”

A fifth point changes the verdict on six of fifteen atoms. Carbon, nitrogen, fluorine, silicon and chlorine go from a finite capacity to an unbounded one and beryllium goes the other way, so a third of the set is reclassified by one more measurement in a fit that was already exact through its four.

Tested in Six of fifteen change verdict · the series on electronegativity

“Whatever a fifth point does, it will do it evenly across the set.”

The five it unbounds are exactly the five largest finite capacities the cubic reports — nitrogen at 31.5, chlorine at 6.28, silicon at 4.72, fluorine at 3.98 and carbon at 3.20. The seven that stay bounded keep their order exactly, and five of those seven still move by more than a fifth.

Tested in Six of fifteen change verdict · the series on electronegativity

“Beryllium is one of the three atoms with the largest capacities, two of which are unbounded.”

Under five points beryllium's capacity is 0.0122 of an electron, the smallest in the whole set. It crosses from one end of the ordering to the other on the addition of a single measured energy.

Tested in Six of fifteen change verdict · the series on electronegativity

“A larger molecule has a larger blind spot in its force field, since it has more constants and the same kind of redundancy.”

Methane has ten flat directions in fifty-five constants and boron trifluoride seven in twenty-eight — eighteen per cent against twenty-five. One redundancy each, and the smaller molecule loses the larger share, because a three-angle centre spreads its redundancy over a smaller field.

Tested in The second molecule with a blind spot · the series on normal mode

“The number of unmeasurable constants is the number of redundancies.”

Each molecule has exactly one redundant coordinate and the flat spaces are ten-dimensional and seven-dimensional. A redundancy is one linear relation among coordinates; what it costs in constant space is a relation among every product of coordinates that involves it, which is many.

Tested in The second molecule with a blind spot · the series on normal mode

“Every coordinate of a planar molecule is caught by its angle-sum redundancy.”

The out-of-plane coordinate is not. Its own constant is fixed on its own and so are all three of its couplings to the stretches, because the redundancy is the statement that three angles at a planar centre sum to a fixed value and a wag is not one of those angles.

Tested in The second molecule with a blind spot · the series on normal mode

“The shell's own offset of one fifth is a property of the n = 3 shell too.”

It is one fifth at every shell. As the quantum defect goes to zero it tends to λ/(l + ½), so the three levels' shifts are in the ratio 2 : ⅔ : ⅖ whatever n is — an s–p gap of 4λ/3 and a p–d gap of 4λ/15 — and their ratio carries no n at all.

Tested in The quarter, generalised · the series on representation

“A shell with a smaller threshold would have its minimum only marginally.”

The second shell's threshold is exactly zero and not small. Its m = 0 levels are s and p and stop there, so there is no third level for the push that makes a minimum, and the numerator of the ratio is absent rather than tiny. No quantum defect produces a minimum there.

Tested in The quarter, generalised · the series on representation

“The third shell's minimum is a special case, found because that shell was the one available.”

It is the first case. The m = 0 levels of a shell number n and a minimum needs three, so n = 3 is the smallest shell that can have one — and every shell above it has one with more room, from twenty per cent of the threshold at n = 3 to nearly half at n = 10.

Tested in The quarter, generalised · the series on representation

“Two published inversion barriers that differ by ten per cent are close enough to be treated as the same number.”

The splitting goes as the barrier to the power −3.56, so ten per cent on the barrier is thirty-six on the splitting. Between 2020 and 2262 wavenumbers — the published value and the one a fit to both measured splittings returns — the computed ground splitting runs from 1.3508 to 0.7935, a factor of 1.70.

Tested in The exponent that runs both ways · the series on inversion

“That sensitivity makes the barrier a badly determined quantity.”

It makes it a well determined one. The same exponent divides in the other direction: a splitting known only to a factor of two fixes the barrier to nineteen per cent, and ammonia's splitting is known to five figures. The exponent is a problem for prediction and a gift for inference, and the two are usually cited the wrong way round.

Tested in The exponent that runs both ways · the series on inversion

“The seventy per cent over-prediction of ammonia's splitting is an error in the reduced mass.”

The same potential solved for ND₃, whose only difference is a reduced mass 1.70 times larger, over-predicts by 1.61 against ammonia's 1.70. An error that survives a seventy per cent change of mass and moves by five per cent is not the mass's.

Tested in The exponent that runs both ways · the series on inversion

“The splitting is a power law in the barrier, with exponent −3.56.”

The local slope is −2.5325 at 1212 wavenumbers, −3.5628 at 2020, −4.5692 at 3030 and −6.1504 at 5050. It more than doubles across the swept range, and a single straight line through the sweep leaves a worst residual of 0.90 in the logarithm — a factor of 2.5, which is larger than the whole seventy per cent discrepancy in ammonia's splitting.

Tested in The exponent that runs both ways · the series on inversion

“The barrier, the reduced mass and the pyramid height are three independent sensitivities.”

Two rescalings of the equation fix two of them. Multiplying the mass by λ and dividing the coupling by λ divides every eigenvalue and leaves every eigenfunction, so the mass exponent is exactly one below the barrier's; substituting x = αy leaves the barrier invariant and forces the geometry exponent to be exactly twice the mass's. Checked at four barriers, the two hold to six decimal places.

Tested in The exponent that runs both ways · the series on inversion

“So the ordering by fragility is not worth quoting at all.”

Its top is safe for a reason no numbers can disturb. The two most fragile claims both depend on the Hückel eigenvalue, so any correlation in that predictor multiplies both prices by the same factor and their ratio stays 2.933 at every value. No per-predictor structure reorders them, whatever it is.

Tested in The ranking moved and the headline did not · the series on models

“The claims that can be reordered are reordered only by implausibly strong correlation.”

The easiest swap needs ρ = 0.154 in one predictor with the other independent. That is a weak correlation and an entirely ordinary one between two quantities measured in the same literature with the same instruments. It is the middle of the ranking that a per-predictor structure disturbs, at a cost nobody would call implausible.

Tested in The ranking moved and the headline did not · the series on models

“Displacing the collection's most fragile claim needs a correlation of about the same size.”

It needs ρ = 0.954 — near-perfect correlation — because the nearest challenger from a different predictor is nearly five times less fragile. A factor of six separates the two thresholds, so the ordering's top and its middle have quite different standing and quoting them together conceals that.

Tested in The ranking moved and the headline did not · the series on models

“Giving each donor the effective charge its own formal charge implies would move the computed π/σ ratios towards the fitted parameters, since a real ligand is an ion and the sweeps below used neutral atoms.”

Three of the five ratios move and all three move outwards. Chloride goes from 0.9456 to 1.4827 against a fitted 0.16, fluoride from 0.3052 to 0.3373 against 0.14, and cyanide from 0.5013 to 0.6562 against −0.10. Water's and ammonia's donors are formally neutral, so they do not move at all. Not one of the five moves nearer.

Tested in The correction that moves three of them backwards · the series on ligand field

“A donor's own contraction is a wider lever than the metal's, so somewhere inside it the fitted values must be reachable.”

It is wider — 2.45 units of effective charge against the metal's 1.40 — and every window still lies entirely above its target. The smallest π/σ ratio chloride's whole chemistry can produce is 0.2674 against a fitted 0.16, fluoride's is 0.2146 against 0.14, and water's is 0.2297 against 0.10: short by 1.67, 1.53 and 2.30 at the best each donor can do.

Tested in The correction that moves three of them backwards · the series on ligand field

“The donor sweep is the metal sweep again with the other end moved, so it can say nothing new.”

The metal's window moved the five ratios by 2.07, 2.09, 2.16, 2.17 and 2.18 — one factor, so no charge could reorder them. The donor's moves them by 1.57, 2.05, 3.15, 5.54 and 9.51, a six-fold difference between ligands. This correction reorders the series where the other could not, and reorders it into a different wrong order.

Tested in The correction that moves three of them backwards · the series on ligand field

“A charge outside the window is still a charge, so the search returning one means the repair exists in principle.”

A metal's effective charge can be pushed past Slater's window by supposing a core electron screens less. A donor's cannot be pushed past its own bare nucleus, which is where its window ends — 7.00 for chlorine, 7.30 for fluorine, 6.30 for oxygen. Chloride's answer of 9.28 would require 2.28 of chlorine's ten core electrons to stop screening altogether.

Tested in The correction that moves three of them backwards · the series on ligand field

“Half filling is the easiest composition to search because on a bipartite net the two-colouring is the unique arrangement with every bond unlike.”

A triangular net has no two-colouring and at the smaller contrast its half-filled composition has one basin reached by all two hundred starts — as uniquely easy as the square net's. And the arrangement it reaches makes thirty of forty-eight bonds unlike, against a counting ceiling of thirty-two, so it is not even the arrangement a count would pick.

Tested in A net with no two-colouring · the series on cohesion

“Removing the two-colouring should remove the recovery at half filling.”

All four sweeps have the same shape — easy at the sparse end, hardest between a quarter and a third full, easy again at half filling. Both nets at both contrasts have the half-filled composition no harder than their own hardest point, so the finding the explanation was for holds on a net the reasoning does not cover.

Tested in A net with no two-colouring · the series on cohesion

“On a frustrated net the arrangement with the most unlike bonds is the one the energy picks.”

At half filling and the larger contrast two arrangements attain the counting ceiling of thirty-two exactly and differ in binding by 1.8 × 10⁻², and a third arrangement making only thirty unlike bonds sits between them. So the count neither identifies the optimum nor orders the arrangements.

Tested in A net with no two-colouring · the series on cohesion

“Refitting a redundant force field with two isotopologues rather than one would reduce the spread the projection leaves.”

It produces a runaway the single fit does not have. With methane's bend–bend constant restored, a fit to CH₄ alone converges to a C–H constant of 5.4248 and a bend–bend of −0.0891; a fit to CH₄ and CD₄ together walks to −97596.4 and −97596.9 in two constants, equal to four figures and opposite in effect.

Tested in Adding data made it worse · the series on normal mode

“More data constrains a fit, so a second isotopologue must at least not hurt.”

A redundancy is a combination of force constants that displaces no atom, so it is invisible at every mass. Moving methane's field ninety times its own norm along that direction — putting a constant at 289 mdyn per ångström — changes no frequency of CH₄ or of CD₄ by more than 1.67 × 10⁻⁶, and the two molecules give the same number.

Tested in Adding data made it worse · the series on normal mode

“The residual left by that move is a small real curvature the extra data could grip.”

It is round-off. Making the move ten times smaller makes the residual ten times smaller, not a hundred times, on both isotopologues — which is how a difference of large numbers behaves and is not how a quadratic does.

Tested in Adding data made it worse · the series on normal mode

“A bond list needs one distance cutoff and a clause for the awkward cases.”

No cutoff works at all. The longest bond in this collection is a platinum–chlorine bond at 2.3200 ångström and the shortest pair that is not a bond is water's two hydrogens at 1.5144 — so the two populations overlap by a factor of 1.53 and every cutoff is wrong about something.

Tested in No length separates them · the series on point group

“A rule from covalent radii would need its own special cases, since a metal is not a halogen.”

It needs none. Every bond drawn here has a length at most 1.1174 times the sum of its two covalent radii and every pair that is not a bond has one at least 1.2112 times it, so every tolerance from 1.1174 up to — but not including — 1.2112 classifies all twenty-three molecules correctly, including hydrogen peroxide and without the clause.

Tested in No length separates them · the series on point group

“The window is narrow enough that the choice of radii would decide the answer.”

The window is a factor of 1.0839 wide and the two edges are set by molecules at opposite ends of the collection — hydrogen peroxide's O–O bond above and bromochlorofluoromethane's chlorine–bromine contact below. Neither involves a transition metal, so the one genuinely ambiguous radius in the table decides neither edge.

Tested in No length separates them · the series on point group

“Phosphine does not invert because phosphorus is heavier than nitrogen.”

Substituting phosphine's reduced mass into ammonia's well and changing nothing else takes the splitting from 1.3508 to 0.8366 wavenumbers — a factor of 1.61. Its pyramid height alone costs 9,641 and its barrier alone 93,992. The mass is the smallest of the three contributions by four orders of magnitude, and it is the only one the usual account names.

Tested in It was never the mass · the series on inversion

“A heavy enough central atom would suppress the inversion by mass alone.”

It cannot, at any mass. The umbrella coordinate's reduced mass is 3mₗmₓ/(3mₗ + mₓ), with mₗ a ligand's mass and mₓ the central atom's, which is bounded above by three times the ligand mass — 3.0235 unified mass units for three hydrogens, against ammonia's 2.4866. An infinitely heavy centre lowers the splitting by a factor of 2.55, and the fall being explained is 5 × 10¹⁵.

Tested in It was never the mass · the series on inversion

“The three contributions multiply, so the whole is the product of the parts.”

In the action they would add. They do not: the three one-at-a-time costs are 0.4304, 8.5039 and 12.3896, summing to 21.32, against 36.24 for all three at once. The three reinforce each other by 70 per cent, because each of them moves where the turning points are.

Tested in It was never the mass · the series on inversion

“The ionic model's error on a separation has no identifiable pattern.”

It has a clean one. Counting how many of the pair's two ions have a third-row outermost p shell sorts the six errors into three groups that do not overlap: −13.6 and −14.9 per cent at zero, +0.6 to +6.7 at one, +17.9 at two. The narrowest gap between adjacent groups is 11.2 percentage points.

Tested in The error was the row, not the charge · the series on contour

“The charge is the likely culprit, since the model treats a double charge by squaring a Coulomb term.”

The charge separates nothing. The doubly-charged pairs run from −14.9 to +6.7 per cent, spanning 31.4 of the 32.7 points the whole set covers, and interleave with the singly-charged pairs throughout. Mg²⁺O²⁻ sits with Na⁺F⁻ and Ca²⁺O²⁻ with K⁺F⁻, in each case because the shells match and despite the charges differing.

Tested in The error was the row, not the charge · the series on contour

“Six points cannot distinguish two explanations.”

These six can, because the two labels cross: both charges appear at more than one shell count, so each label varies while the other is held. That is checked before either is tested — a set of pairs in which charge and shell moved together would make the comparison vacuous however clean the result looked.

Tested in The error was the row, not the charge · the series on contour

“So the model's error is a fixed offset that could be absorbed into the radii.”

It changes sign. The model puts two compact second-row ions about fourteen per cent too close together and two diffuse third-row ions eighteen per cent too far apart. No shift of any radius produces an error that reverses with the shell, because a radius shift is additive and this is not.

Tested in The error was the row, not the charge · the series on contour

“An acceptor ligand's π parameter comes out negative because its empty π* lies closer to the metal level than its filled π does.”

It does not lie closer. At a metal level of −4.75 eV, cyanide's filled π is 8.86 eV below and its π* only 7.01 eV above — but carbon monoxide's π is 12.16 eV below against 6.25 eV above, and dinitrogen's 12.23 against 7.05. The gaps favour the donor channel in all three, so if the two overlaps were equal every one of them would come out a π donor.

Tested in The channel that points at the metal · the series on ligand field

“The far atom of a diatomic ligand cancels part of the π* overlap, which is what separates the two channels.”

It contributes between 4.7 and 9.6 per cent of the near atom's overlap and it does cut the π* and add to the π. What decides the comparison is the near-atom coefficient: 0.394 in cyanide's filled π against 0.971 in its π*, a factor of 6.06 in the square. The far lobe modifies that by a tenth.

Tested in The channel that points at the metal · the series on ligand field

“A two-channel model lifts the refusal that kept ammonia out of the single-channel one.”

It lifts the arithmetic half — a difference of two positive terms can be zero. It does not reach ammonia, whose nitrogen has no π level to donate from and no π* to accept into, its other two p functions being in N–H bonds. The model computes a donor term of 1.39 × 10⁻² on a p orbital the molecule has already spent, and there is nothing to subtract from it.

Tested in The channel that points at the metal · the series on ligand field

“The chemical capacity might turn out to be a function of atomic size, since the quantities it was tested against were all inside the fit that produced it.”

The worst adjacent pair in the mean valence radius — lithium and iodine, neighbours in size — differs in capacity by a factor of 78.7, against a control floor of 4.95. That is sixteen times the floor, and the second-worst pairs are comparable, so the failure is not one outlier.

Tested in A size the fit was not made from · the series on electronegativity

“The verdict depends on which definition of size is used.”

The mean radius and the root-mean-square radius order a few of the atoms differently and give the same worst pair and the same ratio of 78.7. Their median adjacent ratios are 6.92 and 6.20. A single definition would have left that untested; two agreeing is what makes the answer about size.

Tested in A size the fit was not made from · the series on electronegativity

“Size and capacity are unrelated, since the pair test rejects size.”

Their rank correlation is −0.433, which is a respectable association — both quantities track the periodic table. The pair test asks something stronger and the correlation does not survive it, which is the same lesson a rank correlation of 0.857 taught three essays earlier when a control that could not be a mechanism matched it.

Tested in A size the fit was not made from · the series on electronegativity

“Once the redundancy is projected out, what four fits still disagree about is a search or rounding residual.”

Every one of the four fits reproduces all six frequencies, and a fit with a convergence criterion a million times tighter and eight restarts moves the stretch–stretch difference by seven ten-thousandths. The four sit within a few thousandths of a traced curve of exact fits; their 0.115 spread in the stretch constant is a distance along that curve.

Tested in The residual was a loop · the series on normal mode

“A large quantum defect gives a shell more room below its threshold, not less.”

That followed from the screening model's defects keeping the ratio 2 : ⅔ : ⅖ as they grow. Measured defects do not: the ratio of the p–d difference to the s–p difference, which each shell's offset approaches, is 0.128 for lithium and 1.70, 3.06, 2.68 and 2.29 for the heavier alkalis, against the model's one fifth.

Tested in Four alkalis the model cannot hold · the series on representation

“Where the gap fails to name the winner it is a small numerical disagreement between nearly equal arrangements.”

The frustrated net's two contenders, held fixed, exchange binding at a contrast of 3.790 and exchange gap at 4.849. Over that whole interval the one that binds better has the narrower gap, by up to 0.14 at a contrast of 4.

Tested in The gap follows the winner late · the series on cohesion

“The contrast at which a second basin appears at half filling is where the count of unlike bonds stops picking the winner.”

The count failed on the square net above 1.5272. A second half-filled basin appears on the square net at 1.237 and on the triangular net at 2.714, the first reached by five starts in two hundred. Neither is within a quarter of the count's threshold.

Tested in The gap follows the winner late · the series on cohesion

“The reduced mass of an umbrella coordinate is a defined quantity.”

Three constructions are defensible and none is derivable from a measurement. The apex against a rigid ligand plane gives 2.4866 unified mass units, the moving atom's own mass gives 14.0031, and holding the bonds at their measured length gives a function of position running from 2.4866 to 2.9874. The splittings they produce are 1.35075, 5.5 × 10⁻⁶ and 0.94420 wavenumbers.

Tested in The mass nobody chose · the series on inversion

“A position-dependent mass can be replaced by a suitable constant.”

It sits nowhere near the average. The two constants at the ends of its range give 1.35075 and 0.56237 wavenumbers; the varying mass gives 0.94420, which is 1.68 times the heavy constant and 1.43 times off the light one. A tunnelling integral is taken across the middle of the barrier, which is where this mass is lightest, so the answer is set by where the mass is rather than by its average.

Tested in The mass nobody chose · the series on inversion

“The usual construction is safe, since the honest one is a small correction.”

It changes the splitting by 43 per cent and moves it towards the measurement — from 1.7023 times the measured value to 1.1899. A correction that improves the agreement by that much is not a correction that can be neglected; it says the standard choice is the one in error.

Tested in The mass nobody chose · the series on inversion

“The choice does not affect a barrier fitted to the measurement, since the fit absorbs it.”

The barrier the same well infers from the same measured splitting is 2330 wavenumbers under the usual construction and 2117 under the bond-conserving one — a spread of 10 per cent, comparable to the whole disagreement between published values. And the third construction cannot reach the measurement at any barrier between 200 and 40,000, which is reported as a refusal.

Tested in The mass nobody chose · the series on inversion

“The mixing criterion is a tautology on the lone pairs and was discriminating on the double bond.”

The margins run the other way. The lone pairs clear the boundary by 1.732 and the carbonyl's σ–π pair by 2.455 — and ethene's σ–π by an infinite factor, since symmetry puts both centroids at the bond midpoint and the separation is 2.5 × 10⁻¹⁹. The case offered as the counterexample is the least marginal of the three.

Tested in The criterion that has never said no · the series on hybrids

“The σ–π analysis of the double bond found the criterion refusing to produce bent bonds.”

That analysis contains no refusal. Its two verdicts are that the σ–π and bent descriptions are one occupied space in two bases, unchanged to 10⁻¹⁵ at every mixing angle, and that the bent pair are sp⁵ hybrids at 50.768° — a quantitative picture rather than a rejected one. The refusal is attributed rather than found.

Tested in The criterion that has never said no · the series on hybrids

“So the criterion is vacuous and measures nothing.”

It refuses two of the five pairs available. The carbonyl's σ–n pair scores 0.111 and its π–n pair 0.739, both from the same three orbitals and the same dipole matrices as the σ–π pair that scores 2.455. The instrument discriminates; what was one-sided is the set of cases put to it.

Tested in The criterion that has never said no · the series on hybrids

“Sweeping the polarisation would find the boundary the double bond sits near.”

Its margin has a floor of 2.412 across s fractions from 0.05 to 0.99, and it rises to 39 at the top of that range. No physical polarisation of a carbonyl brings the pair within a factor of two of the boundary, so there is no boundary inside the range where molecules live.

Tested in The criterion that has never said no · the series on hybrids

“A fourth fragment narrows the window in which a pairwise counterpoise assembly is usable, because it leaves more terms out.”

Asked of three pairs of arrangements that differ by one added centre, the covered share of three decades of accuracy line goes from 44 to 58 per cent with every centre alike, from 27 to 54 with the heavy centres inside, and from 67 down to 60 with the heavy centres outside. Two widen and one narrows.

Tested in The overshoot was one arrangement · the series on basis

“Whether a fragment is over- or under-corrected follows from where it sits and what it is.”

An end fragment of the uniform chain, at two Gaussians a centre and 1.6 bohr, is under-corrected by 43 per cent with three centres and over-corrected by 18 per cent with four. Same position, same charge, opposite sign.

Tested in The overshoot was one arrangement · the series on basis

“Writing electronegativity theory on the exact piecewise-linear energy would repair the difficulties the fitted curves have.”

It removes them by removing the quantities. The hardness is a second derivative of straight segments, the capacity a third; neither exists. The chemical potential is a pair of one-sided slopes rather than a number, so there is nothing for equalisation to set equal. Four quantities go and one question stays.

Tested in Four quantities go and one question stays · the series on electronegativity

“The piecewise model at least predicts sensible charge transfer, since it has the right shape.”

On isolated atoms it predicts none, anywhere. Moving one whole electron costs the donor's ionisation energy less the acceptor's affinity, and the cheapest such pair among these seventeen atoms — potassium to chlorine — costs 0.729 electronvolts. That becomes favourable only at a separation of 19.75 ångström, longer than any bond.

Tested in Four quantities go and one question stays · the series on electronegativity

“Adding the ion pair's own attraction gives the model back a quantitative prediction.”

It gives it an integer one. At the measured bond lengths the attraction runs from 5.10 to 15.71 electronvolts and exceeds the transfer cost for five of the eight bonds with measured dipoles, so the prediction swings from no charge to a whole electron with nothing in between. Every measured dipole implies a fraction, from 0.058 to 0.842.

Tested in Four quantities go and one question stays · the series on electronegativity

“So the piecewise model is simply worse than the quadratic one.”

Rounded to the nearest integer its prediction agrees with six of the eight measured charges, with no parameters at all, and it gets the alkali halides and the heavy hydrogen halides on the right sides. It classifies better than a reader would expect and quantifies nothing, which is a different failure from being wrong.

Tested in Four quantities go and one question stays · the series on electronegativity

“A potential carrying the cubic and quartic coefficients the measured αₑ and ωₑxₑ imply will reproduce those measurements.”

Solved on four thousand points and read from its four lowest levels, the quartic with HCl's measured coefficients gives αₑ = 0.3215 against a measured 0.3072 and ωₑxₑ = 50.55 against 52.82. In all four molecules its αₑ lands above the measurement, where the Morse curve's landed below it.

Tested in Two coefficients are not a potential · the series on rotation

“Where two potentials share their cubic and quartic, the difference in αₑ must come from the mean displacement.”

The quartic's mean displacement in HCl's ground state is only one per cent larger than the Morse curve's, and its first difference B₀ − B₁ is smaller, 0.2506 against 0.2802. Its αₑ is larger because its γₑ is +0.0355 against the Morse curve's −0.0014: the truncation shows up in the constant that was supposed to be a small correction.

Tested in Two coefficients are not a potential · the series on rotation

“The identity between orbits and totally symmetric species was established on the molecules that happened to have bond lists, and might not hold for the ones that did not.”

Rebuilt on the radius rule, xenon tetrafluoride, the hexafluoridocobaltate ion and both tetrachloride ions join the census. The number of totally symmetric vibrations equals the symmetric orbits less the totally symmetric redundancies for all four, as for the fifteen before them.

Tested in The census a bond rule was hiding · the series on point group

“A bond list that gets every molecule's bonds right gives every molecule a complete set of internal coordinates.”

Xenon tetrafluoride and the tetrachloridoplatinate ion have the right four bonds each and coordinates spanning seven of their nine vibrations. The two they miss are exactly the out-of-plane A₂u and B₂u species, which stretches and in-plane bends cannot describe at a planar four-coordinate centre.

Tested in The census a bond rule was hiding · the series on point group

“The tolerance chosen in the middle of the radius rule's window is safe against the choice of radius table.”

A second single-bond tabulation gives a window from 1.171 to 1.262, bounded by the same two pairs as the first table's 1.117 to 1.211. The factor in use, 1.163, lies below the second window, where hydrogen peroxide's O–O bond is lost; a factor from 1.171 to 1.211 works under both.

Tested in The census a bond rule was hiding · the series on point group

“A compound whose purity is not known cannot know which side of its own sign pole it sits on.”

From half a per cent monomer to sixteen, the couplings at which the monomer fraction and the temperature-independent term are uncorrelated move by under 1.6 per cent for the two stars of spin three halves and two, under 3.1 for the two clusters of spin one, and under 11 for the three doublets. Where a sample sits relative to them is decided by its coupling, not its purity.

Tested in Purity renames the poles · the series on magnetism

“The share of runs in a resonant pair grows by a fixed amount per decade of chain length and does not saturate.”

Each decade adds −ln(1 − ρ) ξ ln 10 times (1 − ρ)^s, the share not yet resonant. For runs of six at x = 0.5 the rise is 1.74 points a decade over 10³–10⁶, 1.57 over 10⁸–10¹² and 1.20 over 10²⁰–10³⁰, and the share reaches 57.7 per cent by 10⁵⁰.

Tested in The share that was read as a line · the series on defect

“Half of the runs of six are in resonant pairs at a chain of about 10³¹ sites.”

That is where a straight line through 10³ and 10¹² reaches a half. The exact share reaches it where s = ln 2 / −ln(1 − ρ), which at x = 0.5 is a chain of 10^40.7. At every concentration from 0.2 to 0.9 the straight line puts the half-way length too early.

Tested in The share that was read as a line · the series on defect

“A plausible range in the gauche energy is a smaller lever than the factor of eleven the rotor conventions spanned.”

It is larger where it matters. The conventions moved the ceiling by 11.0-fold and moved no verdict. The gauche energy's reported range of 2.5 to 4.5 kilojoules a mole moves it by 8.7-fold and moves two of the nine verdicts, because the ceilings it moves are the ones sitting nearest a measurement.

Tested in The lever that was supposed to be smaller · the series on strain

“The five-membered refutation is safe, since a factor of eleven in the ceiling did not reach it.”

At 4.422 kilojoules a mole the ceiling under the convention behind the decided verdict reaches 250 and the 250-fold acceleration stops being refuted. That is inside the range the gauche energy is reported over. Only the 11,000-fold case is refuted at every energy under every convention.

Tested in The lever that was supposed to be smaller · the series on strain

“Whatever moves here is the spread between published determinations rather than the uncertainty in any one of them.”

The six-membered verdict changes at 3.630, and the standard compilation value is 3.8 ± 0.4. That is 0.42 standard deviations from the centre — well inside a single determination's own error bar — while a band of ±0.02 moves nothing, so the sweep is measuring instability rather than manufacturing it.

Tested in The lever that was supposed to be smaller · the series on strain

“The gap in the spread distribution might be the localisation criterion's coarseness rather than a property of the cages.”

Boys localisation, which scores descriptions by orbital centroids in space rather than by atomic populations, leaves a gap of 3.2 × 10⁷ on the same forty-eight pairs — seven decades, from numerical zero to a real spread with nothing between. Pipek–Mezey's is a factor of 210. Two independent criteria both bimodal is a property of the family.

Tested in A second criterion left a gap too · the series on multicentre

“So the two criteria are measuring the same thing and either would do.”

They classify six of the forty-eight pairs differently, and the disagreements go both ways — two are degenerate under Pipek–Mezey only and four under Boys only. A criterion that was simply the other with more resolution would disagree in one direction.

Tested in A second criterion left a gap too · the series on multicentre

“The family's most ambiguous cage is a fact about that cage.”

The eleven-vertex cage at twenty electrons has the largest Pipek–Mezey spread in the family, 5.3 × 10⁻², and under Boys its descriptions are identical to machine precision. The extreme case of one criterion sits on the floor of the other.

Tested in A second criterion left a gap too · the series on multicentre

“A higher-order assembly is a cheaper route to the correction a full calculation would give.”

Under a model that charges one cubic-cost diagonalisation per ghost set, the three-body assembly of four fragments costs 1.64 times the full calculation. It becomes the cheaper of the two only at eleven fragments. The pairwise assembly costs 0.375 times.

Tested in A repair that costs more than the whole · the series on basis

“The trio's level at the free-atom energy is exact because the two outer orbitals are equivalent.”

Raising the A–C overlap from 0.25 to 0.45 while B's stays at 0.25 leaves no symmetry relating A and B, and the level stays at −13.6 eV to 2.7 × 10⁻¹⁴ eV at every step and at seven energies of the third orbital from −24 to +2 eV.

Tested in A level no symmetry was protecting · the series on overlap

“Any change that makes A and B inequivalent destroys the exact level.”

Changes that make A and B inequivalent to C do nothing. A site-energy difference moves it at 0.5 per electronvolt and a coupling between A and B at 10.20 eV per unit of overlap, and a fourth orbital that sees the pair in a different ratio from C moves it by 0.025 eV — while one that sees it in the same ratio, 1.4 to one, leaves it exact with no symmetry anywhere.

Tested in A level no symmetry was protecting · the series on overlap

“A quantity that responds to the change the level ignores must be tracking the same physics.”

The filled-shell A–B bond order moves at 0.653 per unit of A–C overlap and at nothing per electronvolt of site energy — the exact mirror of the level. It is twice an entry of the inverse overlap matrix and knows the metric, not the energies.

Tested in A level no symmetry was protecting · the series on overlap

“A residual that scales with a non-integer exponent has a non-analytic origin.”

The reversal average and difference split it exactly. The even half's anisotropy scales as the amplitude to 2.005, with a single power describing it to 0.6 per cent; the odd half scales as the amplitude to 0.981 below 0.04. The combined 1.248 lies between them, and the two halves cross at an amplitude of 0.084.

Tested in Two integers made one exponent · the series on spectrum

“The absolute value in |0.75 − ρ| could make the reading non-analytic wherever the ratio crosses three quarters.”

A depolarisation ratio 3γ²/(45ᾱ² + 4γ²) cannot exceed three quarters for any Raman tensor, so the quantity inside the absolute value never changes sign. Across every distortion swept, forward and reversed, the least departure is −1.1 × 10⁻¹⁶, which is rounding.

Tested in Two integers made one exponent · the series on spectrum

“The flatness of the sum over depolarised bands is what an experiment that cannot resolve the bands would measure.”

An unresolved degenerate pair is measured with its intensities added in each polarisation before the ratio is taken. That observable's anisotropy over the plane is 2.9 × 10⁻³ at an amplitude of 0.01, against 1.05 × 10⁻⁴ for the sum — 28 times larger — and it scales as the first power of the amplitude.

Tested in Two integers made one exponent · the series on spectrum

“The upper coupling at which a susceptibility fit's monomer fraction and temperature-independent term decouple scales as the reciprocal of the ground spin plus a half.”

True to a spread of 1.127 across the seven clusters it was found on. Of five clusters built afterwards, only the star of seven lands inside the predicted range: the star of eight is 8.5 per cent above it, K₂,₅ 12.0 per cent below, K₂,₆ 21.6 per cent below and K₃,₅ 0.7 per cent below. Across all twelve the spread is 1.559.

Tested in Five more clusters break the band · the series on magnetism

“The upper separation is where a cluster's susceptibility has become the Curie law of its ground multiplet across the measuring window.”

At the seven separations, χT at 300 K is 2.39, 2.01 and 2.37 times the ground Curie value for the doublets, 1.08 and 1.33 for the spin-one clusters, and 0.81 and 0.63 for the stars of spin three halves and two. A fixed tolerance would give one number.

Tested in Five more clusters break the band · the series on magnetism

“The energy of the first excited multiplet sets where the fit's nuisance parameters separate.”

At the twelve upper separations the first excitation runs from 40 cm⁻¹ on the star of eight to 345 cm⁻¹ on K₂,₃. The star of five and K₂,₃ share a band product of 114.7 and 114.8 cm⁻¹ with gaps of 57 and 345 cm⁻¹.

Tested in Five more clusters break the band · the series on magnetism

“A rule that flags fewer disputed-looking pairs at the same miss rate would bring the panel of tables needed within reach.”

No straight boundary in the plane that misses no disputed pair flags fewer than the published nineteen, over 61,353 boundaries searched. Letting disputed pairs through saves one or two flags each: at eight misses, half the disputed pairs, the rule flags eight and still needs 46 tables at the measured correlation, against 73.

Tested in The rule is not the lever · the series on dipole

“With a narrow enough rule, four tables could establish an exception.”

Four independent tables agree on a sign one time in eight, above the one in twenty a significant exception needs, so a rule flagging a single pair needs six tables even at zero correlation. At the measured correlation of 0.325 it needs fourteen, and at the top of the bootstrap range, 0.547, it needs fifty-eight.

Tested in The rule is not the lever · the series on dipole

“The number of tables needed is set mainly by how many pairs the rule flags.”

At the measured correlation, every rule from one flagged pair to all 153 spans 14 to 215 tables. At nineteen flagged pairs, the correlation's bootstrap range from 0.110 to 0.547 spans 16 to 1,903 — nine times as many tables.

Tested in The rule is not the lever · the series on dipole

“Sweeping the temperature and sweeping the gauche energy are two independent checks on the same verdict.”

The ceiling is a function of g/RT alone, so a change in one is reproduced exactly by a change in the other. Evaluated at five matched pairs the two routes agree bitwise — a relative difference of exactly zero, not a small number — and the temperature range swept spans 1.2248 to 1.8054 in g/RT while the gauche range spans 1.0085 to 1.8153, which contains it.

Tested in Two sweeps and one lever · the series on strain

“So the temperature sweep was wasted work.”

Its subject was not the ceiling. It put an activation-enthalpy term on the measured rate ratio, and that term is a function of temperature and not of g/RT — at 5.308 kilojoules a mole the ratio moves 22 per cent between 298 and 340 kelvin while the ceiling's dependence is already accounted for. The half of that calculation that was new is the half the identity does not cover.

Tested in Two sweeps and one lever · the series on strain

“The temperature sweep and the gauche-energy sweep found two different places where the six-membered verdict gives way.”

They found one. The temperature sweep's 312.1 kelvin at 3.8 kilojoules a mole is u = 1.4644; the gauche sweep's 3.630 kilojoules a mole at 298.15 kelvin is u = 1.4643. Two sweeps, two units, four figures of agreement, and one crossing reported twice.

Tested in Two sweeps and one lever · the series on strain

“The identity is an approximation valid where the ceiling is large.”

It is exact and it is algebraic. g and T appear in ((1+2x)/2x)ʳ only inside x = exp(−g/RT), and nowhere else, so the ceiling is a function of one variable at every rotor count, every energy and every temperature. The agreement is to the last representable bit rather than to a tolerance.

Tested in Two sweeps and one lever · the series on strain

“The cages whose classification depends on the criterion are the ones with a large enough symmetry group.”

The automorphism group separates nothing. The icosahedron's graph has 120 automorphisms — the most in the family — and gives three disagreements out of twelve fillings and nine agreements. The octahedron's has 48 and gives none out of six. The eleven-vertex cage's has 4, the fewest, and gives one out of eleven.

Tested in The cage is on both sides · the series on multicentre

“A disagreement is a property of a cage.”

It is a property of a cage and a filling. The icosahedron disagrees at fourteen, sixteen and eighteen electrons and agrees at the other nine fillings, with the same graph and the same symmetry throughout. Any explanation appealing to the cage alone predicts twelve disagreements or none.

Tested in The cage is on both sides · the series on multicentre

“A criterion-dependent case is one the two criteria half-agree about.”

In all six, one criterion's relative spread is numerically zero — 4 × 10⁻¹⁴ or exactly nothing — while the other's runs from 7 × 10⁻⁴ to 5 × 10⁻². There is no intermediate case in the family where both see a spread and rank it differently.

Tested in The cage is on both sides · the series on multicentre

“The ionic model's error runs with how diffuse the pair's softer ion is.”

An oxide's 2p exponent is 1.925 and a chloride's 3p exponent is 1.917, and the four pairs whose softer ion is one of them have errors from −14.9 to +17.9 per cent. The rank correlation of the error with the softer ion's exponent is −0.36, matched or beaten by 384 of the 720 orderings of six points.

Tested in The sum of the exponents, not the softer ion · the series on contour

“A continuous measure that orders the three middle pairs correctly has thereby shown it carries the pattern.”

Three points have six orders, so an uninformative measure lands on the right one a sixth of the time. Of seven measures tried, two order them — the exponent sum and the ratio of the mean radii — and the second fails the test on all six pairs at p = 0.175.

Tested in The sum of the exponents, not the softer ion · the series on contour

“The collection's fragility ranking is an ordering of its claims, with the most fragile at the top.”

It is a partial order. Five claims fall into chains of two, two and one — one chain per predictor — and only within a chain is the relative order independent of anybody's error correlations. Thirty of the hundred and twenty orderings respect the chains and ninety do not, so the ranking makes thirty times fewer statements than a list of five would.

Tested in Three chains and ninety orderings ruled out · the series on models

“The rigid part of the ranking is the top of it.”

The rigid parts are the chains, and one chain occupies ranks two and four with a claim from a different predictor at rank three. That intruder can be lifted above rank two at a correlation of 0.736 or dropped below rank four at 0.154, so a chain constrains its own members and nothing else — the rigid region is not an interval of the ranking.

Tested in Three chains and ninety orderings ruled out · the series on models

“The top pair being locked is a fact about how much more fragile the most fragile claim is.”

It is a fact about which chain both claims are in. They are the Hückel eigenvalue's discordance and its slope floor, they share the predictor and therefore the factor, and their ratio of 2.93 is untouched by any correlation at all. The headline would survive any numbers whatever, which is a stronger claim than a comfortable margin.

Tested in Three chains and ninety orderings ruled out · the series on models

“The chain's contrast approaching one from U = 32, moving away to U = 128 and returning is the shape of its approach to the limit.”

Every one of the six strongest lines is monotone in weight from U = 19. The turns in the contrast are exchanges of rank: an exact tie at the cut at U = 28.916, the third-ranked line changing identity at U = 17.91 and 32.40, and the strongest satellite changing identity at U = 156.6, where the contrast peaks at 1.0632.

Tested in The limit of one is a parity · the series on photoelectron

“The orphan count is what a symmetry matching produces, so a transition metal has one too.”

It has none, on fourteen of fifteen answerable arrangements. Nine valence orbitals against six ligand combinations means every combination finds a partner; the leftover is three spare metal orbitals at an octahedron, four at a trigonal bipyramid, five at a tetrahedron. Spare minus orphan is nine minus the ligand count at every arrangement, which is arithmetic rather than a finding.

Tested in The leftover changes sides · the series on hypervalency

“So a metal centre can always match whatever a ligand set spans.”

One arrangement in the census refuses. A planar hexagon's six sigma combinations span a B1u component in D6h, and no s, p or d function of a centre transforms as B1u there. It orphans one combination, and reaching it would need an f orbital.

Tested in The leftover changes sides · the series on hypervalency

“The count is constant along the Bailar twist because the twist keeps one point group.”

It passes through three — D3h at the prism, D3 in the interior, Oh at the octahedron — and the spare set is A1' + E', then A1 + E, then T2g. Three decompositions with no species and no dimension pattern in common, and the count is three in all of them, because nine minus six does not depend on what anything is called.

Tested in The leftover changes sides · the series on hypervalency

“Stiffening the repulsion about a common separation tests whether the overlap's range is what sorts the model's errors by shell.”

It cannot fail. Steepened about five bohr, the model's repulsion moves the second-row pairs out by 0.50 and 0.35 bohr and potassium chloride in by 0.33, as predicted. One exponential of one range for every pair, which carries no shell information at all, moves them out by 0.58 and 0.46 and in by 0.34. Under both, the shift falls steadily with each pair's own separation.

Tested in One contraction for two conditions · the series on contour

“The order of the three pairs with one third-row ion is part of the same range defect.”

Contracting the third-row shells moves all three and leaves their order — sodium chloride, potassium fluoride, calcium oxide — unchanged at every factor from 1.0 to 1.5, with their own spread falling only from 6.1 points to 5.2.

Tested in One contraction for two conditions · the series on contour

“The improvement a bond-conserving mass makes to ammonia's splitting could be an artefact of choosing the BenDaniel–Duke ordering.”

The five orderings in use — BenDaniel–Duke, Gora–Williams, Zhu–Kroemer, Li–Kuhn and Mustafa–Mazharimousavi — give 0.94036 to 0.94477 wavenumbers, a spread of 0.47 per cent. The mass construction moves the splitting by 43.1 per cent, ninety-two times as much.

Tested in An ordering worth half a per cent · the series on inversion

“Each ordering has to be solved separately to know what it does.”

Every ordering is BenDaniel–Duke plus s·μ′²/μ³ + c·μ″/μ², so first-order theory needs two integrals over one doublet. On fifteen orderings with exponents between −1 and 0 it reproduces the exact change to within 0.005 per cent of the splitting.

Tested in An ordering worth half a per cent · the series on inversion

“The slope floor's denominator can be priced the way its numerator is, once somebody does the arithmetic.”

It needs three prices, because the three runs are three kinds of object. The spin-only moment's run is a difference of two square roots of integers and no defensible variation moves it at all. The Hückel run is a graph eigenvalue difference, exact given the connectivity and moved only by changing the form of the model. The angle-strain run is computed through a force constant and a reference angle, both conventions.

Tested in A denominator needs three currencies · the series on models

“A denominator that carries no experimental error is a denominator that carries no uncertainty.”

Two of the three floors are cheaper to move through their computed run than through their measured rise. The angle strain's floor halves on an 11.6 per cent change in its run against a 33.8 per cent measurement error, a ratio of 2.91; the Hückel floor's ratio is 1.91. Only the spin-only floor is priced entirely by its measurement, and it is the one whose run cannot move.

Tested in A denominator needs three currencies · the series on models

“The largest slope floor in the collection rests on a run that is one per cent of that predictor's range.”

One per cent is what 0.0308 is as a fraction of the measured range of 3.10 electronvolts. As a fraction of the predictor's own range of 0.5858 it is 5.26 per cent, five times larger. Both numbers are true statements about different things, and the earlier report quoted one and named the other.

Tested in A denominator needs three currencies · the series on models

“An unsweepable denominator should be reported as an infinitely expensive one.”

An infinite price and an impossible one are different claims. √(n(n+2)) on a count of unpaired electrons has no parameter, no basis, no convention and no model form in it, so the spin-only run is not expensive to move — there is nothing there to move. It is reported as having no model price rather than a very large one.

Tested in A denominator needs three currencies · the series on models

“A geometry with no answer is a geometry whose symmetry the finder could not establish.”

Three unrelated things produce no answer. The finder failing its tolerance accounts for 25 of the 27; a continuous group, which is declined deliberately rather than missed, accounts for the excluded path; and a finite group with no character table accounts for 2. Only the first is about how nearly symmetric a geometry is.

Tested in The gap found on purpose · the series on hypervalency

“Those cases were already distinguished, since the Bailar-twist analysis made exactly that distinction.”

It made it on one path and not in the census. The census reported an infinite group whenever the symmetry search produced no name, which covers both a linear arrangement and a group of order ten — so the pentagonal pyramid was reported as having an infinite group. It has C5v, of order 10.

Tested in The gap found on purpose · the series on hypervalency

“The remaining gap is in a path's interior, like the first one.”

Both occurrences are at t = 1.000, which is the pentagonal-pyramidal endpoint of two different paths. The D3 gap was a whole stratum of interior geometries; this one is a single arrangement that two paths happen to end at, and it is in the census in its own right.

Tested in The gap found on purpose · the series on hypervalency

“The bond-conserving mass is confirmed by reproducing ND₃'s ground splitting at the published barrier.”

It lands on 0.05308 against a measured 0.05310, within 0.04 per cent, in the same well where it puts NH₃ at 0.94420 against 0.79350 — nineteen per cent high. An agreement on one isotopologue that the other refutes is a coincidence of where one number fell.

Tested in Deuterium cannot tell the masses apart · the series on inversion

“The better well should favour the better mass on the isotope test.”

In the well whose shape is fitted to both of NH₃'s lines, the bond-conserving mass widens the gap between the two isotopologues' barriers from 3.42 to 4.25 per cent and moves the predicted ratio from 17.63 to 18.49, away from the measurement.

Tested in Deuterium cannot tell the masses apart · the series on inversion

“The middle group's order is a separate question from the range of the third-row shells.”

It is the same question asked one ion at a time. Sodium chloride, potassium fluoride and calcium oxide are ordered as the contractions their third-row ions need, 1.302, 1.387 and 1.441, and a contraction of a whole shell could not reach that order only because it gives all three ions one factor.

Tested in Three contractions for one shell · the series on contour

“The two cations share a shell, so one factor should serve both.”

Carried to calcium oxide, potassium's factor leaves its error at −12.21 per cent; carried to potassium chloride, calcium's leaves it at −15.38. Both land outside the band the two second-row pairs span, −13.56 to −14.86 per cent.

Tested in Three contractions for one shell · the series on contour

“Three factors fitted to three pairs is a tuning and cannot test anything.”

Potassium chloride contains two of the three ions and was fitted on nothing. At potassium's and chloride's factors its error is −13.81 per cent, inside the second-row band and 0.40 points from its middle. The potassium factor it would need on its own, 1.401, agrees with potassium fluoride's to one per cent.

Tested in Three contractions for one shell · the series on contour

“The slope floor is a bound on every model of the form measurement = f(predictor), established without fitting.”

It is a bound on every model of that form with the predictor written on one particular scale. Under thirteen strictly increasing reparameterisations the floor moves by a factor of 19.1 on the spin-only moment, 10.2 on the Hückel eigenvalue and 11,746 on the angle strain. Every reparameterised predictor makes the same predictions about order and a different claim about derivatives.

Tested in A floor on models written in one scale · the series on models

“The tie and discordance instruments are approximately scale-free, like most robust statistics.”

They are exactly scale-free. Across all thirteen maps on all three predictors the count of exact ties is identical, the count of discordant pairs is identical, and the share of the variation the ties leave unexplained agrees to the last bit. Both quantities are functions of the order of the predicted values alone, and a strictly increasing map is a bijection on order.

Tested in A floor on models written in one scale · the series on models

“The power family is an artificial stress test, so the movement it produces says nothing about a real modelling choice.”

The overlap family is not artificial. Hückel theory sets every overlap integral to zero; restoring a uniform overlap s gives eigenvalues x/(1 + sx), and every member of the family for s between 0 and 0.3 is a defensible calculation. Across it the discordance and both exact ties are unchanged and the slope floor moves from 28.5 to 36.4, a rise of 27 per cent.

Tested in A floor on models written in one scale · the series on models

“A floor that moves is still a floor, so the reported value is a safe lower bound.”

The reported value is the smallest of the thirteen on the Hückel eigenvalue and nowhere near the smallest on the other two. The angle strain's reported 3.38 sits between 0.005 and 55.9 across the family, and on two of the three predictors the steepest pair changes identity — so the quoted fragility of the floor moves as well as its value.

Tested in A floor on models written in one scale · the series on models

“The three icosahedral disagreements at fourteen, sixteen and eighteen electrons are one degeneracy being filled rather than three separate coincidences.”

They are one calculation, which is stronger and has a different cause. All three hand the localisation the same nine orbitals, because the search selects by non-zero occupation and Hund's rule fills a five-fold shell singly before pairing. Their spreads are identical because they are the same arithmetic run three times, not because a degeneracy is being filled.

Tested in Six disagreements and three calculations · the series on multicentre

“A partly occupied degenerate shell is what makes the two criteria disagree.”

Twenty-one of the forty-eight fillings leave a shell partly occupied and three of those disagree; three of the six disagreeing rows have a closed shell. The condition neither implies the disagreement nor follows from it, so the explanation is refused in both directions.

Tested in Six disagreements and three calculations · the series on multicentre

“The family survey covers forty-eight cage-and-filling pairs.”

It covers forty-eight rows and thirty-three questions. Fifteen rows hand the localisation an orbital set another row has already handed it: four inputs on the six-vertex cage rather than six, six on the nine-vertex rather than nine, seven of ten, eight of eleven and eight of twelve.

Tested in Six disagreements and three calculations · the series on multicentre

“Counting the repeats is only a correction to a denominator.”

It changes the finding the earlier survey rested on. Four Boys-degenerate cases against two Pipek–Mezey ones, read as a trend with cage size, is two against one. The direction survives and the evidence is a third of what it was, which is the difference between a trend and two cases.

Tested in Six disagreements and three calculations · the series on multicentre

“The momentum picture has fewer nodes than the position one, since a transform smooths a function out.”

It has exactly as many. The position radial function carries a Laguerre polynomial of degree n − l − 1 and the momentum one a Gegenbauer polynomial of the same degree, because the substitution carrying one to the other is a rational map of degree one in the squared momentum. Checked on nine orbitals from 2s to 5d, and against the numerical transform on the 3s.

Tested in The nodes in the other variable · the series on orbital

“A position node at radius r corresponds to a momentum node near 1/r.”

Within one orbital the products of paired nodes are not equal under either ordering. The 4s gives 1.130, 1.653 and 1.607 pairing innermost with outermost and a different set pairing innermost with innermost; the 5s spreads from 1.144 to 2.082. A rule would collapse one of those sets to a single value on every orbital and neither does on any.

Tested in The nodes in the other variable · the series on orbital

“Nothing exact relates the two node sets.”

The product of all the position nodes times the product of all the momentum nodes is (n+l)! / ((2l+1)! · 2^(n−l−1)), to twelve digits on every orbital tested — and it contains no nuclear charge. Each position node contracts as 1/Z and each momentum node expands as Z, and across Z from 1 to 10 the product does not move.

Tested in The nodes in the other variable · the series on orbital

“A Compton experiment would show an orbital's momentum node as a zero in the measured profile.”

A Compton profile is an integral of a squared wavefunction from the measured momentum upwards, so it is strictly positive. Its derivative is minus the momentum density times the momentum, which vanishes exactly at the node — so the node appears as a horizontal tangent. On the 2s that tangent sits at q = 0.5 with the profile still at 24 per cent of its peak.

Tested in The nodes in the other variable · the series on orbital

“With mixed ligands the bond-conserving reduced mass is no longer a single function of one coordinate.”

It is. For the symmetric umbrella motion the three ligands slide outward together along fixed bonds, and their radial kinetic energy sums to a single function of the apex height. What the asymmetry adds is one extra term, not an extra variable: the horizontal centre of mass moves, and subtracting its motion takes half the sum of the squared mass differences, divided by the total mass, off the radial coefficient.

Tested in The two that are not on the line · the series on inversion

“A partly deuterated molecule's splitting is an interpolation between the two pure ones.”

Not even under this model. The steps in the logarithm are 1.175, 0.960 and 0.809, so the series is concave in the substitution count and both middle molecules sit off the line joining the ends. The reason is arithmetic rather than chemistry — a splitting falls with the square root of the mass and the mass grows linearly with substitutions — and the departure from a square-root law is what a measurement would test.

Tested in The two that are not on the line · the series on inversion

“The asymmetric term is a refinement too small to matter.”

It changes the predicted splitting by 0.507 per cent for NH₂D and 0.423 for NHD₂. The isotope test could not choose between two whole constructions of the mass because they differed by less than either missed the measurement by; this term is of that order, and it exists only for the molecules that test did not have.

Tested in The two that are not on the line · the series on inversion

“If a change leaves the radical's singly occupied level where it was, it leaves the unpaired electron where it was.”

Raising the A–C overlap from 0.25 to 0.45 leaves the level at −13.6 eV to rounding and moves the spin on A from 0.500 to 0.236 and on B to 0.764. The split is S(B–C)² : S(A–C)² exactly, at seven energies of C from −24 to +2 eV and at three Wolfsberg–Helmholz constants.

Tested in The spin the count does not hold · the series on overlap

“A localisation is invariant to how the occupied orbitals are chosen, since any unitary mixing of them leaves the density alone.”

It is invariant to mixings of the occupied set with itself, which is what the basin search exploits. It is not invariant to a rotation inside a degenerate shell the occupied set only partly takes: that changes which combinations are occupied. On the icosahedron at ten electrons the best Pipek–Mezey functional moves from 2.5171 to 2.0852 across four re-orientations, a factor of 1.207.

Tested in The basis a diagonaliser happened to return · the series on multicentre

“The number of distinct localised descriptions is a property of a cage and a filling.”

On the icosahedron at twelve electrons it is 10, 8, 14, 11 and 13 from five orientations of one shell. Nothing about the cage or the filling differs between them. The count moves on three of the six affected inputs.

Tested in The basis a diagonaliser happened to return · the series on multicentre

“The degeneracy classification the family survey is built on is a fact about each cage.”

On three of the six affected inputs it is not. The six-vertex cage at four electrons gives Pipek–Mezey spreads of 0, 5.5 × 10⁻², 4.9 × 10⁻², 0 and 0 against a threshold of 3.1 × 10⁻⁴; the ten-vertex cage at sixteen flips under Boys instead; the icosahedron at twenty flips under Pipek–Mezey. Each is labelled both ways depending on the orientation an eigenvalue routine returned.

Tested in The basis a diagonaliser happened to return · the series on multicentre

“The effect is really the search's numerical noise under a different name.”

An input whose degenerate shells are wholly occupied or wholly empty is put through the same rotation and does not move at all — 0.1666666667 against 0.1666666667, identical to ten decimal places. The rotation is then a mixing of occupied orbitals with occupied orbitals, which is the invariance the method rests on, and the search reproduces it exactly.

Tested in The basis a diagonaliser happened to return · the series on multicentre

“A bonding orbital is extended along the bond, so its momentum distribution is extended along the bond too.”

The two anisotropies have opposite sign. In position ⟨z²⟩/⟨x²⟩ = 2.056 and in momentum ⟨p∥²⟩/⟨p⊥²⟩ = 0.600, on the same orbital at the same separation. The interference factor cos²(p∥R/2) cuts the momentum density off at π/R along the bond and does not touch it across, so the same orbital is flattened in the direction it is stretched in.

Tested in Oblate in the picture nobody draws · the series on orbital

“The interference fringes wash out of any measurement, since a Compton profile integrates over two momentum components.”

They wash out in one direction and survive intact in the other. The cosine depends only on the component along the bond, so measuring along the bond leaves it untouched and the profile is the atomic one times cos²(qR/2), with exact zeros at odd multiples of π/R. Measuring across the bond integrates over the cosine's own variable and the fringes become a Bessel weight with no zeros in it.

Tested in Oblate in the picture nobody draws · the series on orbital

“Reading a bond length off a momentum measurement would need a model of the electron density.”

The first zero of the profile along the bond is at π/R exactly, whatever the atomic function is. Across separations from 1.4 to 4.0 bohr the exponent the energy chooses contracts by a quarter and the overlap falls from 0.72 to 0.19, and π/q₁ returns the separation to twelve digits at every point.

Tested in Oblate in the picture nobody draws · the series on orbital

“The momentum oblateness is an artefact of writing the orbital as a sum of two atomic functions.”

Separate the centres and it goes. At two bohr the anisotropy is 0.600; at forty, with the overlap down to 2 × 10⁻¹⁵, it is 1.0000 — the fringes crowd together faster than the atomic density falls, their average becomes one half, and the half cancels between the two directions. A formula that kept a molecular signature at forty bohr would be reporting itself.

Tested in Oblate in the picture nobody draws · the series on orbital

“The ordering of the kinetic operator for a position-dependent mass is a choice that has to be made by hand.”

Not for a mass that comes from a constraint. The reduction is a one-dimensional Riemannian manifold with metric μ(x), which has a distinguished Laplacian; carried to the flat measure by ψ ↦ μ^(¼)ψ it is a von Roos ordering with square coefficient −7/32 and curvature coefficient 1/8. Fitted numerically over forty-one positions those come out to six decimals, with a residual of 2 × 10⁻⁸.

Tested in The ordering a manifold picks · the series on inversion

“The covariant operator is BenDaniel–Duke, since that is the ordering with no extra potential.”

It is Mustafa–Mazharimousavi. The fitted coefficients give α + γ = −1/2 and αγ = 1/16, whose roots are a double root at −1/4. BenDaniel–Duke is at (0, 0), a quarter of the von Roos plane away, and adds no potential at all — which is what makes it the one to reach for and not the one a manifold picks.

Tested in The ordering a manifold picks · the series on inversion

“Resolving the ordering removes the last unsettled choice in the one-dimensional model.”

It removes the last one inside the von Roos family. A constraint imposed as the limit of a stiff potential leaves the zero-point energy of the five degrees of freedom it froze, and that term is not in the family: it depends on the masses, where the two basis functions of the family depend only on the mass profile the constraint already fixes.

Tested in The ordering a manifold picks · the series on inversion

“A family of chains that differ in which single fusion is turned varies the position of one bend and nothing else.”

A fusion's entry is a direction, so turning one turns the chain and the next fusion turns it back. Turned at the first or last fusion that leaves one angular ring; turned at any of the six between, two adjacent ones. Built from those angular rings, the chains reproduce the family's spreads — 3.052 for the ends, 4.065 to 4.600 for the rest — to four figures.

Tested in The scatter counts angular rings · the series on aromaticity

“Whether two angular rings turn the same way or opposite ways — whether the chain's overall envelope bends — decides the scatter.”

At seventeen arrangements of two or three angular rings, turning them all one way instead of alternately changes the worst spread by at most 0.81 per cent, while the envelope goes from straight to bent by up to 180°.

Tested in The scatter counts angular rings · the series on aromaticity

“The more angular rings a chain has, the more its response scatters.”

Two angular rings at positions two and six scatter by 2.449, below every single angular ring (3.029 to 3.660). Two at one and three scatter by 5.244. A chain with all seven interior rings angular scatters by 1.367, less than the straight chain's 1.520.

Tested in The scatter counts angular rings · the series on aromaticity

“Changing the sign between the two atomic functions only moves where the momentum fringes fall.”

It changes the orbital's shape in momentum from oblate to prolate. The bonding combination's ratio of second moments along and across the bond is 0.600; the antibonding combination's, at the same separation and exponent, is 1.731. The antibonding orbital is prolate in position too, at 4.335, so the opposite senses in the two pictures belong to the bonding orbital alone.

Tested in The zero belongs to one determinant · the series on orbital

“Filling both the bonding and the antibonding orbital cancels the interference, leaving two atoms.”

It inverts it. One electron in each gives a profile along the bond equal to the atom's times (1 − S cos qR)/(1 − S²): a dip to 0.683 of the atom at zero momentum and a rise to 1.867 at π/R, with contrast equal to the overlap. The dip costs 0.156 hartree of kinetic energy per electron at this separation, which is closed-shell repulsion.

Tested in The zero belongs to one determinant · the series on orbital

“A single flag for an angular fusion in a chain of fused rings makes one angular ring.”

The flags are fusion directions, not turns. A single one among zeros turns the chain and the next zero turns it back, so rings five and six are both fused across meta bonds. The bend measured along the chain was a kink of two angular rings, and the heteroatom on the first of them reads 0.811 of the straight chain while on the second it reads 0.969.

Tested in An angular ring rescales what lies beyond it · the series on delocalisation

“A heteroatom's response passes through angular fusion unchanged, because the fitted decay length does not move.”

The decay length moves by at most eleven per cent, from 0.719 rings to 0.638 in the fully angular chain. The responses on the rings beyond angular fusions move by factors: the third ring keeps 0.453 of its straight-chain response and the rings beyond the fifth between 0.127 and 0.313, alternating. A length is a slope, and angular fusion changes the level.

Tested in An angular ring rescales what lies beyond it · the series on delocalisation

“The fifth an angular ring costs belongs to its two shared bonds.”

With the heteroatom on the kink's first angular ring, the summed response over that ring's two shared bonds rises from 0.0352 to 0.0434 and over its four outer bonds falls from 0.0815 to 0.0513. The loss is in the bonds that are not shared.

Tested in An angular ring rescales what lies beyond it · the series on delocalisation

“Changing one bond's length changes where it sits and leaves the arrangement alone.”

At a ratio of 1.5 neither seven-coordinate placement survives a nudge: both relax to a capped octahedron, C₃ᵥ, with the long bond as the cap and its three nearest neighbours at 69°.

Tested in The long bond goes to the crowded site · the series on VSEPR

“The profile at π/R measures a molecule's antibonding occupation.”

It measures the electrons in sine combinations. In nitrogen the largest single contribution at π/R, 0.052 per electron of the molecule, comes from the 2p σ bond. Ordered by the valence profile's share at π/R the first-row diatomics run Li₂, C₂, B₂, N₂, O₂, F₂, which is not the order of their antibonding counts or their bond orders.

Tested in The zero is a parity, not a bond · the series on orbital

“A real first-row molecule can have a zero in its momentum profile along the bond.”

The 1s core pair fills a sine combination in every one of them, so no whole profile is zero at π/R — the smallest share, carbon's, is 0.113. Only lithium's valence profile is zero, because its valence shell is one s bond, and any s–p mixing in a σ bond gives it a value proportional to the p share.

Tested in The zero is a parity, not a bond · the series on orbital

“A main-group centre loses a usable valence orbital only when its ligands lie in one plane.”

A bare ring of four to eight ligands, folded off its plane to any polar angle from 60° to 120°, matches three of the centre's orbitals and orphans n − 3, one more than the formula — the same count as the flat ring. The unmatched orbital is totally symmetric off the plane and antisymmetric on it.

Tested in Folding the ring does not give the orbital back · the series on hypervalency

“Ligands spanning all three directions in space give a main-group centre four usable orbitals.”

A folded ring spans three directions and gives three. In C₄ᵥ to C₈ᵥ the centre's s and its p along the axis are both totally symmetric, and a single symmetry-distinct set of ligands supplies exactly one totally symmetric combination for the two. An apex supplies the second and restores four at every angle, 90° included.

Tested in Folding the ring does not give the orbital back · the series on hypervalency

“The pentagonal pyramid, the one arrangement the census could not count, sits where the counting formula is expected to fail.”

Reduced in C₅ᵥ, sulfur hexafluoride as a pentagonal pyramid spans 2A₁ + E₁ + E₂ against the centre's 2A₁ + E₁: four matched, two orphaned, which is n + L − 4. The census now counts twenty-seven arrangements and the formula is right for twenty-four, the three it misses still the three flat ones.

Tested in Folding the ring does not give the orbital back · the series on hypervalency

“Keeping only the consecutive triples fails for a uniform chain because four fragments is too short a chain for the saving to show.”

At five, six, seven and eight fragments the consecutive assembly covers 45.6, 47.1, 48.8 and 48.1 per cent of three decades of accuracy line, against 60.0, 63.1, 65.8 and 67.9 for pairs alone. The shortfall is 5.2 points at four fragments and 19.8 at eight: it grows with length rather than closing.

Tested in Length did not rescue the consecutive triples · the series on basis

“Adding every triple to a pairwise assembly is at least as good as adding only some of them.”

On the heavy-outside chain, every triple together covers 70.8, 56.2, 59.1 and 62.3 per cent from five to eight fragments, and the consecutive triples alone cover 77.4, 80.9, 84.4 and 87.9. What every triple leaves out reaches 1.05 kilocalories a mole at five and 2.70 at eight, so the complete order is the worse assembly by up to twenty-five points.

Tested in Length did not rescue the consecutive triples · the series on basis