Field

What is taught wrongly

The explanations that are confident, memorable and false — stated fairly and then tested against a calculation rather than an opinion.

methane — Td. The molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.

Hybridisation does not explain

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

$Orbitals at the 90 per cent contour. Several orbitals drawn at the same enclosed fraction and, unless stated otherwise, at the same scale — so the sizes on the page are the sizes. Each contour was solved for separately by integrating that orbital's own density. Contours drawn: 1s at 90% of its density, |ψ| = 3.94e-2; 2s at 90% of its density, |ψ| = 7.40e-3; 2pz at 90% of its density, |ψ| = 9.48e-3.$

Orbitals are not where the electron is

A many-electron atom has no exact orbitals at all. The orbital picture is a basis for an approximation — an extremely good one — and treating it as a description of reality is the source of most of the confusion in this subject.

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

The dipole is not a sum of bonds

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

Four scales, four orderings. Each column ranks the elements by one electronegativity scale, most electronegative at the top, with a line joining each element across the columns. Every crossing is a pair of elements that two scales order differently.

Electronegativity is not one quantity

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

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

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

Overlap against separation, Z = 3.25. How the overlap integral falls as two atoms are pulled apart, for several pairs of orbitals. Where a closed form exists it is drawn over the computed curve, so the integrator is checked rather than trusted.

A double bond is not two single bonds

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

4s and 3d from K to Zn. The mean radius of the 4s and 3d orbitals across the elements K to Zn, at the nuclear charge each shell actually feels. The screening is Slater's, which is fitted; the radius that follows from it is the closed form for a hydrogen-like orbital, which is not.

The aufbau order is not a property of the atom

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

Every force field the spectrum permits. 5 force fields for H₂O, each one refitted with rr:O:H,H held at the value on the axis, and every one of them reproducing all of the molecule's own frequencies. The heavy line is the error each makes on the isotopologue it was never fitted to. The spectrum cannot choose between these fields; the isotopologue can.

The force field is not in the spectrum

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

6 fitted force fields. Every valence force field fitted here, ordered by the size of its bond stretching constant, with the stretching frequencies of the molecule beside it. The two orders are not the same, which is the whole of what separates a force constant from a frequency. The last two columns say how many constants were fitted to how many observed frequencies, and a field with as many of the first as the molecule has distinct frequencies fits exactly and reports nothing.

The frequency is not the bond strength

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

methane: 2 valence bands. The measured valence photoelectron bands of methane, each labelled with the symmetry species of the orbital it comes from, and beside them the species the valence basis spans — the central atom's s and p functions and one s on each ligand, reduced in the molecule's own group. A band carrying a species the reduction does not produce would stop this figure being drawn.

What a photoelectron spectrum measures

The bands of a photoelectron spectrum are routinely read off as orbital energies. They are ionisation energies, which is a different quantity — and the identification rests on two errors of about an electronvolt each that happen to have opposite signs.

60 electrons in 60 levels. The density of states of a ring of 60, drawn with the energy up the page, and the 60 electrons filled in from the bottom. Where the filling stops is what decides whether the system has cheap excitations.

A half-filled band is not always a metal

Every band picture rests on an approximation that a whole class of materials refuses — each electron moving in an average field, never seeing another one individually. Where the repulsion is strong enough, a half-filled band describes an insulator, and no amount of care with the band fixes it.

The gap against size: uniform against δ = 0.15. The HOMO–LUMO gap of a half-filled chain plotted against the number of sites, on log axes, for a uniform chain and for one whose bonds alternate. The uniform sequence falls without limit; the alternating one settles at four times the alternation.

A band gap is not a bond energy

Silicon's gap is 1.1 electronvolts and its Si–Si bond is 2.3. Both are quoted in the same units, both describe the same material, and neither is convertible into the other — one is the cost of promoting an electron and the other is the cost of taking two atoms apart.

What orders the spectrochemical series, and what does not. Nine ligands' measured octahedral splittings, plotted against charge and against the π parameter that says whether the ligand donates or accepts π density. Charge puts 3 of 20 mixed-charge pairs in the order it predicts; the π parameter puts 35 of 35 in order.

The spectrochemical series is not electrostatics

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

Why d⁸ is square planar and nothing else is. The ligand field energy a square plane buys over a tetrahedron, for every d^n, each geometry in its own ground state. The maximum is at d⁸ (1.73 in units of eσ), it is exactly zero for an empty shell and a full one, and d⁸ is one of only two fillings where the plane is diamagnetic and the tetrahedron is not.

VSEPR does not reach a transition metal

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

Bonds the scales disagree about. For each bond, which atom each scale calls the more electronegative. Every row is a bond whose polarity would be drawn in opposite directions depending on which of these four tables in common use was consulted: Pauling, Mulliken, Allred–Rochow, Allen.

A ranking is not a difference

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

The energy is the last thing a wrong wavefunction gets wrong. Two errors against the error in the wavefunction, on log axes, for a chain of 2 at U = 4t. The energy's line has slope 2.00 and the double occupancy's has slope 1.01: the first is second order in the error and the second is first order. So the two lines diverge as the wavefunction improves, and the energy stops being evidence about anything else long before it stops improving.

A better energy is not a better answer

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

Three answers to one question. The energy to remove an electron from a half-filled four-site system, computed three ways against the repulsion: exactly, by solving a self-consistent field twice — once for the molecule and once for the ion — and by reading the highest occupied orbital energy straight off the molecule, which is Koopmans' theorem. All three agree exactly at zero repulsion. The theorem always sits above the two-calculation answer, because letting the ion relax can only lower it; the exact answer sits above both, because the molecule is more correlated than its ion. The two errors have opposite signs and do not cancel: the residue grows to 6.03.

Koopmans' theorem is exact for nothing

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

$Five sixths of the bonds, nine tenths of the binding. What a site in the outer layer of an open block keeps, measured three ways: the fraction of its bonds, the fraction the second-moment rule predicts of its binding, and the fraction the calculation gives. The last two agree and the first does not.$

A surface is not a count of broken bonds

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

Where two bands lie, as their centres are pulled apart. The σ band and the π band of a two-orbital chain, drawn as the intervals they occupy, against the difference in site energy between the two orbitals. Below a difference of 3 the two intervals overlap and the filled-band count stops deciding anything.

A full band is not an insulator

Two electrons per atom, two orbitals per atom, and the lower set exactly full: the count says insulator, and magnesium is a metal. The count is not wrong about the count. What it assumes is that the two sets of levels occupy separate ranges of energy, and whether they do is a comparison of four numbers that has nothing to do with how many electrons there are.

H³⁵Cl: the well, its states and their averages. The Morse potential built from H³⁵Cl's measured vibrational constants, with the lowest four states drawn at their computed energies and the average separation of each marked. Every average lies to the right of the minimum, because the well is not symmetric — and they move outward as the state rises.

The bond length that depends on the isotope

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

How many sites a state occupies, and whether that depends on the ring. The participation ratio of the states at the middle of the band — the number of sites a state occupies — against the width of the disorder, for rings of 50, 100, 200 sites. With no disorder the three curves are three different numbers, each two thirds of its own ring. At the right they have converged: 7.47 sites on a ring of 50 and 9.82 on a ring 4 times larger.

The third way to be an insulator

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

Two wrong numbers and a right difference

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

The bonding follows the overlap over, and turns where it turns. The stabilisation of three pairs of orbitals against how far apart they are, each scaled to its own largest, with the coupling taken from the computed overlap. Two orbitals with no radial node are stabilised less at every separation further out; the two with one turn over — and the maximum of the bonding is at exactly the separation of maximum overlap, to five decimal places, because a coupling proportional to the overlap makes the stabilisation a strictly increasing function of it.

The same overlap, a different bond

A 1s and a 2s change their overlap by two thirds between two and seven bohr, and change what they are bonded by by 0.154 per cent. The overlap turns over and the bonding turns over with it, at exactly the same separation to five decimal places — the arithmetic refused the expectation that the secular denominators would move it — and what separates one pair from another is not where the maximum is but how little of it there is.

How many times more a metal carries, and when. The ratio of the carriers in a uniform ring to those in an alternating one of the same size, at four sizes and six temperatures. Down the left-hand column the two are indistinguishable, because a ring of 42 at kT = 0.002 has a level spacing larger than the temperature and is no more a metal than the gapped one is. Along the bottom row they are indistinguishable again, because the temperature is larger than the gap. The word only means anything in the middle.

The metal a thermometer cannot find

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

A correction that stops belonging to the system it is added to. The exact ground state of a four-site Hubbard ring, the unrestricted mean field's, and the composite: the mean field plus the correlation correction computed on the symmetric molecule. At ε = 0 the two systems are the same one and the composite is exact. As the sites are made unlike, the transferred correction stops being the right one — the exact correlation energy shrinks towards nothing while the transferred number does not — and the last rows are the recipe adding a correction almost as large as the error it is meant to remove.

The correction that was computed somewhere else

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

Four unweighted rules against the answer the hardnesses give. Twelve molecules, each drawn with the exact equalised electronegativity as a filled mark and the four unweighted means of the free atoms' values as open ones. The geometric mean — Sanderson's rule — is the closest on average, at 0.0706 eV against the arithmetic mean's 0.1666, and it is the closest on only 4 of the twelve. Every rule below the arithmetic mean is constrained to sit below it, and 2 of the exact answers do not.

A mean that is low rather than right

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

A cheap number that predicts an expensive failure. Thirty-two systems. Along the bottom, how far the mean field's own symmetry breaking has moved between the reference and the target — a quantity available before any exact calculation. Up the side, how wrong the transferred correction turns out to be. They rank together at 0.902, and the open marks are the systems whose broken solution has collapsed entirely, which is where the diagnostic stops being a scale and becomes a warning.

The warning a cheap calculation gives

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

The carriers a distortion was hiding

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

The slope goes to a half, and a window fit stops short of it. The local slope of the alternation against the reduced temperature, between each neighbouring pair of points, on a ring of 40 at K = 1.6. It rises monotonically from 0.4115 to 0.5053 as the transition is approached, crossing a half at about a part in a thousand of the reduced temperature. The fitted 0.44 is the average of the left-hand end of this curve; the exponent is one half, which is what a free energy analytic in one order parameter is obliged to give.

The exponent was the window's

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

A fourth data point, and 3 negative electronegativities. Each element's electronegativity from three points on its energy curve, the cubic coefficient a fourth point adds, and what the fourth point leaves. The cubic coefficient is one sixth of a second difference of the ionisation series, so it is largest where that series has a kink — and the alkali metals, whose second electron comes out of a closed shell, are pushed to Li -7.91, Na -3.42, K -1.49 eV. The last column is where the two roots of the fixed-point equation collide, beyond which the atom has no solution at all.

Where a closed form stops being one

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

The pair's regime is not a property of the pair. How much the pair's bonding responds to its own overlap — the ratio of what it is bonded by at an overlap of 0.4 to what it is bonded by at 0.1 — with and without a third orbital coupled to both. Alone it is 11.83, which is the regime in which bonding tracks overlap. With a third orbital present it falls to 1.46, 0.92, 0.74 — and two of those are below one, meaning a fourfold increase in the overlap between the two atoms buys them less bonding rather than more. The coupling comes from the overlap by the Wolfsberg–Helmholz rule with K = 1.75, which is fitted rather than derived. Every stabilisation here inherits that; the shape of the curve against separation does not, because K is a constant.

A regime that belongs to the neighbours

Two orbitals at the same energy are bonded in proportion to their overlap and two far apart are barely bonded at all — two regimes, and the natural question is whether the regime is a property of the pair. It is not. Put a third orbital beside them and the pair's response to its own overlap falls from twelvefold to less than one: more overlap buys less bonding.

The capacity, against whether the anion exists. Each atom at its electron affinity and the capacity the cubic model gives it — the largest amount of electron the model says it can accept. A negative affinity is an anion that is not bound, which is the statement that the true capacity at the integer is zero. The three atoms with negative affinities are beryllium, magnesium and nitrogen, and they have the largest capacities in the set: two of them infinite and the third 31.5. The three smallest capacities all belong to atoms whose anions are bound.

A capacity that is largest where there is none

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

How much stronger a fundamental is than a satellite, against the repulsion. The weakest fundamental divided by the strongest satellite, for a six-site ring and chain at every filling from a third to a half, against the on-site repulsion. Below the line at two the two kinds of line cannot be told apart by their height. The half-filled systems cross it and the third-filled ones do not — not at any repulsion up to sixty-four times the hopping, where the third-filled ring is still at 8.3.

A satellite that never loses its place

The repulsion at which a satellite stops being tellable from a fundamental orders exactly with the one-electron gap across four systems. Changing the gap by the filling instead is the sharper test, and the ordering does not survive it: a six-site chain has a larger gap at half filling and a smaller boundary. Below half filling there is no boundary at all, at any repulsion up to sixty-four times the hopping.

The alternation a spring buys, three ways. The alternation against the elastic constant, on a logarithmic axis. The middle line solves (2/π)(K − E)/(1 − δ²) = K for the complete elliptic integrals; the lower one is the exponential form every account of a Peierls distortion quotes, which is its own asymptote and is 6.5 per cent low at K = 1.2; the upper one is a ring of 40, which leaves the infinite chain as the spring stiffens because a smaller alternation is a longer coherence length.

The amplitude the collapse left behind

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

Where the broken solution appears. The mean field's spin polarisation against the on-site repulsion, at half filling and no site-energy modulation, for three systems. Two of them are symmetric below a threshold and polarised above it — 1.672 for a chain of four and 2.355 for a ring of six. The third is polarised at every repulsion tested, because its half-filled shell is degenerate and the symmetric solution is unstable however small the repulsion is.

The half of the square a ring of four cannot show

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

A pair with no overlap, and a third orbital swept past it. The three levels of a trio in which the two outer orbitals have exactly no overlap with each other, as the third orbital's energy is swept. The middle line is at -13.6 at every point — the antisymmetric combination of the two, which has no partner of its own symmetry and cannot mix with anything. The other two move, so the pair is split by an orbital it has no direct contact through.

A bond order between atoms that do not interact

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

One geometry, four electron counts, four answers. The bond order between two orbitals with exactly no overlap and no resonance integral, as the third orbital's energy is swept, at every count the trio can hold. With none it is identically zero. With two it is positive and rises past one. With four it is negative and reaches -0.954. With six it is a horizontal line — the third orbital's energy stops mattering entirely.

A filled shell is not an empty statement

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

What happens to the level that was exact. The three levels of the trio as one of the two outer orbitals is raised. At zero detuning the middle one sits at -13.6 exactly — it is the antisymmetric combination, and nothing of its symmetry exists for it to mix with. The moment the two are made inequivalent that statement is gone: the level leaves linearly, and the other two barely move by comparison.

A symmetry holds or it does not

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

The second number is the first one, rearranged. The composite's error against the change in the correlation energy, at every point on both axes of the square. They lie on the diagonal because they are the same quantity: the composite is the target's mean field plus the reference's correlation energy, so its error is the reference's correlation energy minus the target's. The largest departure across 20 points is 2.2e-16, which is the arithmetic's own precision and not a measurement.

The second number is the error, rearranged

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

The capacity against the quantity it was supposed to be. The chemical capacity of 14 atoms against their second ionisation energy, with the capacity on a logarithmic axis because it spans nearly three orders of magnitude. The suspected relation is not there: N and K are 2.02 electronvolts apart in the second ionisation energy and differ by a factor of 193 in capacity, which no function of one variable can produce.

A correlation is not an account

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

Five candidates, five failures, five different places. Every point of the square, with each cheap diagnostic's failing pair joined by a line. The five tested candidates fail on five different pairs involving 10 different points — no line shares an end with another. Had they all failed on one corner the lines would have converged on it, and the honest conclusion would have been that composites are safe away from that corner.

Five failures in five different places

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

The input the question named is the worst of the six. For each candidate input, the largest capacity ratio between two atoms that are neighbours in that input. A quantity the capacity were a function of would have a small bar. The dashed line is the floor — neighbours in the capacity itself still differ by ×5.0, because fourteen atoms spread over a factor of four hundred cannot do better. The second ionisation energy is ×193, which is 39 times that floor.

The worst of the six was the one we asked about

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

No panel of this kind can find an exception

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

Six of fifteen change verdict

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

One size scale, computed for every atom at once. Each atom's valence shell at the effective charge Slater's rules give it, with the mean radius of a hydrogenic orbital of that shell and that charge, and its root-mean-square radius. Both are closed forms in the principal and angular quantum numbers and the charge, so both are on one scale for every atom by construction — which is the difficulty a tabulated radius has and this does not. Neither quantity is anywhere in the cubic, which knows three energies and no length.

A size the fit was not made from

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

Straight segments, and the curve fitted through their ends. Cl's energy against the charge it carries. The exact theory says the energy is straight between integers, with a kink at each one: the slope below the neutral atom is the electron affinity and the slope above it is the first ionisation energy, and the two are different numbers. Every quantity in this argument comes from the smooth curve fitted through those points instead — and the curve's second derivative, which is the hardness, is a property the segments do not have at all.

Four quantities go and one question stays

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

Letting disputed pairs through narrows the rule and barely moves the panel. Over 61,353 straight boundaries in the plane of difference and dispute, the fewest pairs any boundary flags for each number of disputed pairs it lets through (bars), and the number of tables that flagged count needs at the measured correlation of 0.325 (dots). Missing none, nineteen is the least — the published rule. Each miss saves one or two flags and two to five tables; at eight misses, half the disputed pairs, the rule flags 8 and still needs 46 tables.

The rule is not the lever

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

Six removal lines, each smooth, and the contrast is whichever two sit at the cut. The weight of each of the six strongest removal lines of the half-filled chain of six, followed from U = 16 upward by continuity in energy and labelled by the energy it tends to. Every line is monotone from U = 19. The contrast is the third strongest over the fourth, so it changes whenever two lines exchange those ranks: at U = 28.92 the line tending to +0.45 overtakes the one tending to −1.80 and the two weights are equal, and at U = 157 the fourth and fifth exchange. Lines at ±E, drawn in one colour, converge to one weight.

The limit of one is a parity

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

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