Simply false — page 2 of 5

“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

“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

“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

“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