Molecular Geometry & Bonding
what the shape decides
Concept

Approximation — where it appears

A model adopted knowing what it leaves out, which is most of what this subject is made of. What distinguishes a useful one is not its accuracy but that the direction and size of its error can be stated.

Named by 95 essays across 9 fields — each of them below, with the objects they name alongside it.

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

A band with no structure in it

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

solids · Bands in a solid How long a chain has to be before its ends stop mattering. The difference in energy per site between a ring and a chain of the same length, against that length. It falls as one over the length, which is what it means for the difference to be an end effect, and the size at which it drops below a thousandth of a β is printed.

Where a molecule stops being one

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

solids · Bands in a solid 1s, 2s, 2pz, 3s, 3dz2, 4s at one level and at one fraction. The orbitals 1s, 2s, 2pz, 3s, 3dz2, 4s, each with the contour level that encloses 90 per cent of its own density, and beside it what each one encloses when they are instead all drawn at 1s's level. The first column is a set of pictures that can be compared; the second is what a plate drawn at a single contour value actually shows. 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; 3s at 90% of its density, |ψ| = 2.64e-3; 3dz2 at 90% of its density, |ψ| = 3.60e-3; 4s at 90% of its density, |ψ| = 1.24e-3.

One level is not one comparison

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

orbitals · Contour 2s and 2p from B to Ne. The mean radius of the 2s and 2p orbitals across the elements B to Ne, 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.

What the screening model cannot see

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

orbitals · Orbital 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.

wrong · Metal 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.

wrong · Bands in a solid Two orbitals, 2 electrons, S = 0 and S = 0.2. Two interacting orbitals with 2 electrons in them, drawn twice: once with the overlap set to zero and once with it kept at 0.2. Dropping the overlap makes the two shifts equal, which is the picture usually taught; keeping it makes the upper level rise by more than the lower falls, which is why four electrons in two orbitals is a repulsion.

Overlap is not interaction

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

applied · Overlap The exponent the molecule chooses, and what it buys. The 1s exponent that minimises the energy of a one-electron diatomic, against the separation of the nuclei, with the binding curves at that exponent and at the free atom's. Held at ζ = 1 the bond comes out at 2.49 bohr and binds 0.0648 hartree; with the exponent free it comes out at 2.00 bohr at ζ = 1.238 and binds 0.0865. The exact answer for this molecule is 2.00 bohr and 0.1026.

The atom does not bring its own orbital

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

orbitals · Orbital 7 molecules, three moments each. The principal moments of inertia of water, sulfur dioxide, formaldehyde, boron trifluoride, ammonia, methane, hydrogen peroxide, in u Ų, with the inertial defect and the asymmetry parameter beside them. The defect vanishes exactly for a planar structure and does not for any other, so three numbers computed from the coordinates decide planarity with no model anywhere in the argument. Every classification is checked against the one the molecule's point group forces.

The moment that is the sum of the other two

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

spectra · Rotation What holds matter together, per pair. Four kinds of interaction between two units of matter, each computed from the model named beside it, on a logarithmic energy scale. The range from top to bottom is a factor of several hundred, which is the number behind why a molecular solid melts hundreds of degrees below a covalent one.

What holds a solid together

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

solids · Cohesion What holds matter together, per pair. Four kinds of interaction between two units of matter, each computed from the model named beside it, on a logarithmic energy scale. The range from top to bottom is a factor of several hundred, which is the number behind why a molecular solid melts hundreds of degrees below a covalent one.

The lattice sum that depends on the order of adding

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

solids · Cohesion The group of benzene, against how much error is forgiven. Four distortions of benzene, none larger than six hundredths of an ångström, and the point group the search reports for each at nine tolerances. The exact structure is D6h at every one of them, so the staircases below belong to the distortions rather than to the search. The cell marked in warning colour is a step at which the order of the group named at the tighter tolerance does not divide the order of the one named at the looser: C6h of order 12, so the sequence is not a chain of subgroups.

The tolerance is a decision

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

symmetry · Point group The wrong shape, fitted as well as it can be. The exact hydrogen 1s orbital and the best sums of one, two, three and six Gaussians, each with its exponents optimised for the energy. Three of them already reproduce the exact function to 99.94 per cent by overlap, which is why the method works at all — and the two places it goes wrong, at the nucleus and far out, are exactly where the other faces of this figure look.

A Gaussian is the wrong shape

Sixty years of molecular calculation are built on functions that get the two ends of an orbital wrong. A Gaussian has no cusp at the nucleus and dies too fast far away, and no number of them fixes either — while three of them already reproduce hydrogen's 1s to better than 99.9 per cent by overlap, and that is why the method works.

orbitals · Basis Two humps and a dip, which is not what a trend looks like. The measured enthalpy of hydration of the first transition series, in kilojoules per mole, against a straight line fitted through it. The measurements do not fall on the line: they rise and dip at manganese, rise and dip again at zinc. Both dips are at configurations with no ligand field stabilisation — d⁵ high spin and d¹⁰ — and the line alone accounts for only 73 per cent of the variation.

The double hump and what removes it

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

applied · Ligand field The bond sum, the measurement, and what is left over. For each pyramid: the vector sum of three bond moments estimated from the electronegativity difference and the measured geometry, the measured dipole, and the difference between them. Positive is towards the lone pair. Ammonia's bonds point that way and nitrogen trifluoride's point the other, which is why the trifluoride's far more polar bonds give it a dipole six times smaller. The leftover is 0.58 D for ammonia and at least 1.48 D for the trifluoride, so it is not one lone pair's property.

The lone pair is not the missing term

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

shape · Dipole The cost of a ratio decided somewhere else. Four bases containing exactly the same six primitives, differing only in how many of the linear coefficients the calculation may choose. The horizontal axis is the effective nuclear charge, which is this one-electron problem's only knob for a different environment; the contraction was fitted at one. At 1.238 — the exponent H₂⁺ chooses when a bond forms — the fully contracted basis is 0.0283 hartree above what the same six functions could give, and one freed coefficient removes most of it.

A contraction is a decision made once

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

orbitals · Basis 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.

wrong · Approximation The same dimer, one, twice and three times over. Independent Hubbard dimers with nothing between them, solved exactly and solved in a space with the configurations that make more than one of them ionic thrown away. The exact energy is exactly additive; the truncated one is exact for a single dimer, because there is nothing there to throw away, and falls behind by 0.193 for two and 0.485 for three. The error per dimer grows, which is what makes a method size-inconsistent rather than merely approximate.

A method that is not additive

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

bonding · Correlation A microwave constant predicted from an infrared one. For four diatomics: the rotational constant and the stretching frequency, both measured, and the centrifugal distortion constant predicted from them by 4B³/ω² — then the constant a microwave spectroscopist fits to the line positions. The prediction and the fit agree within a few per cent across three orders of magnitude in the quantity, and nothing connects the two measurements except the assumption that the bond stretching under rotation is the same bond that vibrates.

The rotor that stretches

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

spectra · Rotation 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.

wrong · Photoelectron The Compton profile, exact and fitted. The momentum density integrated over the two perpendicular directions, for the exact 1s and for three fitted bases. The exact curve is 8/3π(1 + q²)³ in closed form; the fitted ones are sums of Gaussians and are cusped differently at the origin, which is the position-space cusp showing up as a shape in momentum.

The measurement a basis was not fitted to

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

orbitals · Basis 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.

wrong · Approximation One of these is guaranteed to improve, and it is not the one anybody measures. The relative error in the energy and in four properties of the fitted function, against the number of Gaussians. The energy falls at every step, because that is what the variational principle promises. The mean radius is exact for the single-function basis and 332 thousand times worse for the two-function one, and the density at the nucleus is still 6.2 per cent wrong where the energy is wrong by 0.011 per cent.

The property that gets worse

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

orbitals · Basis Nearly all of the error cancels, and the answer gets worse. For each repulsion: the error a spin-paired mean field makes in the total energy of one four-site system and of two two-site ones with the same number of electrons, and the error left in the difference between them. The cancellation improves from 83 to 97 per cent along the axis. The residue as a share of the quantity being computed goes the other way, from 1 to 423 per cent, because the reaction energy shrinks faster than what survives.

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.

wrong · Approximation Two errors, opposite signs, four orders of magnitude apart. A finite basis makes H₂⁺'s binding too large by letting each atom borrow the other's functions, and too small by describing the molecule incompletely. Both are computed here against the exact binding of 0.102634 hartree. The second is thousands of times the first at every basis size, and it is the first that counterpoise removes — so the corrected number is further from the true one than the uncorrected at every row of this table.

The basis the other atom lent

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

orbitals · Basis 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.

wrong · Approximation The 6-ring at 111.5°: the alternating form and what a search finds. The alternating ring — every atom displaced above or below the plane in turn — with its dihedral angles and its two strain terms, beside the lowest-torsion member a search constrained only by bond angles and closure returns. Both satisfy every geometric constraint; only one of them is staggered.

The explanation with the wrong sign

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

shape · Strain A control that ranked better than the mechanism. Rank correlations against the additivity shortfall, over eight ion pairs. The overlap of the two closed shells ranks at 0.8571 — but the cation's formal charge, which cannot be a mechanism, ranks at 0.9524, so the set is confounded: its eight pairs split four and four by charge and everything else rises with it. Held fixed within a charge group the overlap still ranks at 0.80 — and so does the softness, at -1.00. Four pairs cannot separate two candidates.

A control that outranked the mechanism

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

orbitals · Contour The shared bond's response falls and reverses, at every coupling. How far the relaxation moves the most interior cross bond of an acene, against the number of rings, at three couplings. Every series falls monotonically and every one changes sign — at λ = 0.2 between 3 and 4 rings, at λ = 0.4 between 3 and 4 rings, at λ = 0.6 between 4 and 5 rings. Where it crosses is a property of the coupling; that it crosses is not. The natural reading — that the effect grows with the number of rings feeding a bond — is refused by every one of these curves.

An anomaly that is not the first of a series

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

bonding · Delocalisation How far the rotor count would have to be wrong. The ceiling against the number of rotations a closure freezes, with the two measured accelerations drawn across it. The five-membered closure freezes three and its ceiling is 36.46; the ceiling does not reach the measured 250 until 5 rotors, so the count would have to be wrong by 2 on a ring that has three rotations to freeze. The six-membered closure freezes four at a ceiling of 120.88, and stays above its measured 10 down to 2 — so the refusal is airtight and the sufficiency is comfortable.

An estimate that can be wrong by two

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

shape · Strain How much of a curve each extra parameter has left to work with. The singular values of the design matrix for a susceptibility curve, for two, three and four parameters fitted to the same data, on a logarithmic scale. With four they run 12.411, 2.026, 0.149, 0.025 — a span of 500 — so one per cent data fix the first two to under 3 per cent and the last to 37. Each value is what is left of the measurement after the directions above it have taken their share, so a short bar is not a hard parameter but an absent one.

How many parameters a curve is worth

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

applied · Magnetism 10 of 55 force constants that no spectrum can see. The map from methane's 55 independent force constants to its Cartesian Hessian, as a spectrum: 45 directions the frequencies respond to and 10 they do not, out of 55. The null block is exactly the size the redundancy count predicts — 10 for 1 redundancy on 10 coordinates — and the two are computed by different routes, one a rank and one a closed form. A force field quoted to four figures is quoted along 45 directions that were measured and 10 that were chosen.

Ten directions no frequency can see

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

symmetry · Normal mode The two halves change places. What fraction of the counterpoise correction belongs to the lighter of two unlike atoms, against their separation. At a bonding distance it is 1.26 per cent — essentially the whole correction is the heavier atom's — and by 9.0 bohr it is 67. The two change places at 6.29 bohr. A symmetric pair's share is exactly a half everywhere, which is what makes one frozen number a complete description there and nowhere else.

A correction that is two functions

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

orbitals · Basis The two rows of a table, and the curve between them. The rotamer ceiling of a closure freezing 4 rotations, against how hard each of a stated number of them is to turn. The top curve is the free-rotor row and the right-hand end of the bottom curve is the absent-rotor row; everything else is what a real hindered tether does. The measured acceleration of 10-fold is drawn across it. It falls below that line when 3 of the 4 are hindered by 4.09 kJ/mol, or when 4 of the 4 are hindered by 2.70 kJ/mol.

The curve between two rows

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

shape · Strain Two curves that cross, and two that do not. Every pair of parameters that reproduces one measured number, for three numbers, with the true system marked at Δ = 12 and t⊥ = 0.06. The lower band's shape is a function of the ratio of the two, so its curve is a straight line through the origin; the excess gap is a function of the coupling squared over the separation, so its curve bends. The two cross at one point. The upper band's shape draws a line almost on top of the first, because it is a function of the same ratio — a second measurement lying along the first fixes nothing the first had not already fixed.

Two ways of being second order

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

solids · Bands in a solid What each of these predictors cannot see. Four predictors built elsewhere in this collection, each audited by its ties: the share of the variation in what was measured that the predictor demonstrably cannot account for, because two systems it assigns the same number to were measured to differ by that much. the spin-only moment leaves 23 per cent; the VSEPR angle leaves 92 per cent; the highest occupied Hückel eigenvalue leaves 41 per cent. The control has no ties at all and the instrument returns nothing for it, which is what it must do. No fitting anywhere: a tie is a claim a model of that form cannot escape.

Two systems a model cannot tell apart

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

bonding · Models Eight wells, and where each one puts its pair. The total energy of each pair against separation, with the measured distance marked on every curve. The wells are deep and their minima are in the right region — tenths of an ångström from the measurements — which is what makes the comparison worth making. What they are not is closer to the measurements than the sum of two tabulated radii, and that is the result.

A size a confound cannot supply

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

orbitals · Contour A functional with no interior maximum. The Boys functional against the mixing angle for a carbonyl, evaluated directly at a hundred and eighty-one angles. It is a quadratic in cos²θ − ½ with no linear term, so it is symmetric about forty-five degrees and its maximum is at the middle or at the ends and nowhere else. Here the coefficient is positive, so the best is at 0° — the canonical σ and π. There is no angle to search for, however unsymmetrical the molecule is.

The angle that does not have to be searched for

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

bonding · Hybrids The four things a susceptibility curve measures, in order. The four directions in the parameter space, best fixed first, each written as the product of powers it is. The Jacobian is logarithmic, so a direction is a set of exponents and a combination is a product — which is why the answer can be printed. The best-determined is g · J^-0.33, fixed to 0.08 per cent by a curve measured to one per cent; the worst is tip · rho^-0.40, fixed to 40. Neither is one parameter's own axis.

The product a curve measures

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

applied · Magnetism 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.

wrong · Electronegativity One expression, four molecules, a factor of eleven. The computed zero-point correction to each molecule's moment of inertia against the expression A²·Σ(1/ν)/I, which has no fitted quantity in it. The dashed line is the mean dimensionless coefficient, 0.6343; the four points lie within 18 per cent of it, on corrections that span a factor of 11.4. The expression is evaluated from a moment of inertia and a list of wavenumbers, which is what a spectroscopist has before doing anything.

An expression for what was a warning

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

spectra · Rotation The two sides of a verdict, with error bars. The computed ceiling for the six-membered closure — 10.9945 — with the band a gauche energy of 3.8 ± 0.4 kJ/mol puts on it, against the measured tenfold rate ratio with an assumed 20 per cent uncertainty. The two bands overlap over most of their length, and the nine per cent margin the verdict was decided by sits inside both of them.

A verdict inside its own error bar

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

shape · Strain One half is a curve and the other is not. The two halves of a counterpoise correction for an unequal pair, against separation, on a logarithmic axis. The heavier centre's falls smoothly over two decades; the lighter centre's scatters over more than one decade between neighbouring points. It is not a rough function — it is a difference of two energies of order a hartree whose difference is a millionth, and the solver does not have seven figures to spare.

The half that cannot be computed

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

orbitals · Basis Two instruments, and they do not agree. Each of this collection's four predictors under both tests. The exact-tie instrument reports the share of the variation a predictor demonstrably cannot account for, and it ranks VSEPR worst and has nothing at all to say about the ring-strain control. The near-tie instrument reports how much steeper any model must be somewhere than the set is overall, and it ranks the control worst and cannot speak about VSEPR, whose predictor is a label rather than a number. Neither instrument is the general one, and a predictor that passes one has not been audited.

The pair that is not a tie

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

bonding · Models The count is a curve, and the curve flattens. How many distinct localised descriptions of a twelve-vertex cage had been found after each batch, out of 4,000 random starts. The last new one appears at start 124; the remaining 3,876 add nothing. The dashed curve is what the basin sizes measured here predict — the chance of having hit each description at least once — and it is a closed form rather than a fit to the points.

Where the count stops being an effort

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

beyond · Multicentre 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.

wrong · Electronegativity The filled orbital at 3 bohr, with a node in the middle of the bond. The combination the two-level model puts lower at 3 bohr, drawn as contours of the wavefunction with the two signs in the two colours. The overlap here is 0.4825, so the interaction is of the sign that fills the plus combination. There is a nodal plane through the midpoint: the pair the model calls bonded has no density at all between its nuclei.

A bond with nothing in the middle

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

bonding · Overlap The trimer's correction, and the sum of its pairs. The counterpoise correction of a three-fragment system computed directly — each fragment's energy alone less its energy in the whole trimer's basis — against the sum of the three pairwise corrections, on a logarithmic axis. The sum is the larger everywhere the difference is above the solver's noise: 30.4 per cent at 1.6 bohr and nothing by six.

The assembly that counts one share twice

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

orbitals · Basis What the gap alone would predict, and what was fitted. Each halide's π scale as a multiple of fluoride's: the value fitted to the spectrochemical series, against what the energy denominator alone gives with the metal orbital at the vacuum level — which is the weakest the denominator effect can be. It over-predicts at every ligand, and moving the metal level down makes it worse.

The gap that would have to be smaller

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

applied · Ligand field Two events, not one — and both below the n = 2 shell's. The field at which each coupled pair's shift stops being quadratic and starts being linear, for the n = 3 shell and for the n = 2 shell. The n = 3 shell has two, a factor of 3.66 apart, and both are far below n = 2's single one — so the linear effect returns is two events in a shell with a d, and it happens at a field thirty times weaker than in a shell without one.

Two events where there was one

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

symmetry · Representation The measurement is not a horizontal line. The rotamer ceiling against temperature, with the measured tenfold drawn four ways: flat, and as a rate ratio whose two activation energies differ by 3, 5.31 and 8 kJ/mol. Flat, it is crossed at 312.1 K, which is the flat-rotor answer. At 5.31 kJ/mol the two curves are parallel and never meet. At 8 they meet on the other side, and it is cooling rather than heating that refutes the account.

One number decides which way it breaks

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

shape · Strain The inequality, decided twice. The quantity the localisation criterion compares — the centroid separation over twice the off-axis dipole — computed with hydrogenic radial functions and with Slater ones at the same effective charges. Below one the localised description is a pair of bent components; above it, σ and π. The hydrogenic answer is 1.8090 and the Slater answer is 0.3937, and they are on opposite sides.

The node that decided a picture

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

bonding · Hybrids Four ways of predicting the same six separations. How far each model's predicted separation is from the measured one, pair by pair. The best additive model misses by 0.0098 ångström on average, the tabulated radii by 0.0237, radii read off the ions' own densities by 0.1828, and the balance of a Madelung attraction against a computed repulsion by 0.2421.

The residue that is two numbers

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

orbitals · Contour Three basin populations from one search. The share of 4000 random starts landing on each localised description, in rank order, on a logarithmic axis, for three cages and fillings. a twelve-vertex cage at twelve electrons gives 14 descriptions and two groups with a gap; a nine-vertex cage at eight electrons gives 63 descriptions and a smooth tail; a twelve-vertex cage at twenty electrons gives 15 descriptions and one basin and dust. Only the first is the twelve-vertex shape, and the rule it suggests is a rule about that shape.

Three shapes from one search

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

beyond · Multicentre The acceptors' π* levels, as measured. For each π acceptor: the energy at which a slow electron is temporarily captured, which is the π level above the vacuum, and the π scale this collection's series carries for it. Carbon monoxide's and dinitrogen's resonances are measured on the molecule itself; cyanide's cannot be, because an electron cannot be attached to an anion, so hydrogen cyanide's stands in for it — the same π with a proton where the metal would be.

A denominator that fails both ways

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

applied · Ligand field The repulsion against the count, and they do not sort together. Every arrangement of the census by how many ligand σ combinations are left without a central partner and by how far its ligand repulsion sits above the best arrangement of that many points. The three that break the formula are marked. They are not the expensive ones: they sit at 4.2, 6.3, 9.8 per cent while the arrangements the formula gets right run to 26.5.

Expensive is not the same as unadopted

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

beyond · Hypervalency The coefficient each principal axis needs. The dimensionless coefficient the molecule-averaged expression requires, evaluated on each principal axis separately rather than on the mean of the three. Across the twelve axes the positive ones span a factor of 2.33, against the 1.36 the molecule-averaged version spans — and one of them is negative, which no positive constant can be.

The axis that goes the other way

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

spectra · Rotation Three cages, three answers, and none of them reassuring. For each cage: how many descriptions the search finds, what share the commonest takes, how far the whole set spreads in the functional, whether the commonest is the best, and the verdict. A search that always agrees with itself is agreeing about a choice that does not matter; a search whose descriptions genuinely differ does not return the best one.

Fifty descriptions of one molecule

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

beyond · Multicentre 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.

wrong · Approximation What a fifth ligand does to the gap above eight electrons. The gap between the fourth and fifth d levels as one axial σ donor is brought in, and as two are. It closes exactly linearly — 2eσ less one eσ for each unit of axial σ strength — and a full octahedron has none of it left. The rule of sixteen has a gap to be about only while the axial positions are empty, and how much of it survives is a number rather than a yes or no.

The ligand the rule was waiting for

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

applied · Electron count The frame H₂O → HDO turns. H₂O → HDO: the parent's principal axes and the daughter's, drawn on the same nuclei. Two of the three turn by 21.12° and the third does not move at all, because it is the normal to a plane no substitution can tilt. A per-axis comparison between these two species is comparing moments about lines this far apart, which is a rotation of the frame rather than a correction to a number.

Two moments about two different lines

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

spectra · Rotation The end pair against the deepest pair, at each separation. For each separation, the response of the pair that touches an end divided by the response of the pair at the same separation sitting deepest in the molecule. A straight chain is below one at every separation and a zigzag is above one at every separation, so the end effect has opposite signs on the two shapes. The third chain — two straight arms meeting at one angular ring — is above two at three separations and below one at the fourth, which is a third behaviour and not an intermediate one.

An end effect with two signs

Neither of two separations accounts for the scatter in a fused ring system's response, and the natural guess is the end: pairs with more molecule outboard should behave differently from pairs at an edge. They do. In a straight chain an end pair responds a third less than an interior one, in a zigzag a quarter more, and in a chain of two straight arms meeting at one angular ring the anomaly is in the middle.

symmetry · Aromaticity One turned fusion, moved along the chain. Chains of 9 rings differing in one integer: which fusion's direction is turned. Turned at the first or last fusion, that leaves one angular ring beside an end; anywhere between, it leaves two adjacent angular rings whose turns cancel. The widest ratio between two pairs at the same separation, against which fusion is turned. A straight chain gives 1.520 and every bent one gives more — from 3.052 to 4.600. The two ends of the curve are the one-ring members; every interior point is a two-ring step.

One integer, and everything it changes

Eight molecules with the same rings, the same carbons and the same graph distance between every pair, differing in which fusion's direction is turned — one angular ring when the turned fusion is at an end, two adjacent ones anywhere else. The scatter within a separation class runs from three to four and a half times, against a straight chain's one and a half — and the two ends of one molecule disagree by up to a factor of four.

symmetry · Aromaticity Folding two ligands makes the gap bigger before it makes it smaller. The gap above eight electrons as two of the four ligands fold to the same side. It rises first, to 2.0938eσ at 20°, before falling. The four-ligand path only ever closes it, so the direction the gap moves is not a property of bending — it is a property of which ligands bend.

The distortion that opens the gap

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

applied · Electron count A heteroatom on ring 5, and the profile in both directions. The response of each ring to a heteroatom placed on ring 5 of 12, on a logarithmic scale. Two rings share the peak, because an interior ring's outermost carbon belongs to two rings at once, and the profile falls away at the same rate on both sides — which is the answer wanted: the reach is the molecule's and not the end's. Every point is a mean over the six bonds of its ring.

The reach is the molecule's

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

bonding · Delocalisation The residue is four times below the model's own error. The quantity whose sign is wanted, beside the accuracy of the numbers it is a difference of. The measured residues are 0.031 and 0.056 ångström; the model gets a single separation right to 0.242 on average. A fourfold alternating difference of quantities known that badly cannot resolve something that small, and that arithmetic was available before any of this was computed.

The residue is below its own noise

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

orbitals · Contour The two terms in a vibrationally averaged rotational constant. A rotational constant averages 1/r², not r², so the moment it reports carries +2⟨Δr⟩/rₑ from the anharmonicity and −3⟨Δr²⟩/rₑ² from the harmonic spread. The two have opposite signs in every molecule here, and the anharmonic one — which is exactly zero in any symmetric well and therefore absent from every harmonic force field — is larger by a factor of 1.94 to 2.58.

The term a harmonic field cannot produce

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

spectra · Rotation One steps, the other slides — and the step is at the far end. The repulsion energy and the orphan count along the path from a tetrahedron to a square plane. The energy rises smoothly and monotonically, lowest at the tetrahedron and highest at the plane. The count is zero everywhere — including at 89.99° — and becomes one only at 90° exactly. The step is not near the energy's minimum; it is at its maximum, and it is at a single point.

A count that changes at one point

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

beyond · Hypervalency The two overlaps, squared, at the measured bond lengths. For each chromium(III) donor: the σ overlap squared, a metal 3d(z²) against the donor's p(z), and the π overlap squared, a 3d(xz) against its p(x). Everything is computed — the radial functions from Slater's rules, the separation from the measured bond length, the integral by quadrature. Chloride's π overlap is 3.5 times fluoride's, which is the opposite of what the overlap argument required of it.

The overlap the model is not proportional to

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

applied · Ligand field The count, for five run lengths. The fraction of runs whose nearest neighbour of the same length is coupled more strongly than a threshold, against how many decades below the closest possible coupling that threshold sits. Every curve is a straight line over this range, because the fraction is small and the geometric tail is linear in the separation there. The slope is what the next figure is about: it is the whole content of the distribution, and it is a product of two numbers.

A count rather than an average

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

solids · Defect How many descriptions, against how much they differ. Every cage-and-filling pair, by the number of distinct descriptions its localisation finds and by how far apart they are in the functional. If the count measured ambiguity the points would rise from left to right. The case with the most descriptions — 44 of them — has a spread of two parts in a hundred thousand, and sits at the bottom right.

Counting was right except where it mattered

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

beyond · Multicentre 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.

wrong · Overlap The exponent's sign is the ring's parity. The power the temperature-independent term carries in the free combination, for an open chain and for rings of five to ten spins. Every odd ring is negative and every even ring and the chain is positive — so the free product is ρ·χ_TIP raised to a power whose SIGN changes, which is a different combination rather than a shifted one. Nothing here is a near miss: the closest pair on either side of zero are +0.23 and −0.29.

The sign a frustrated ring changes

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

applied · Magnetism Two charges, and three ligands no charge reaches. The metal effective charge each ligand would need on its own for the model's ratio to equal the fitted one. The band is the range Slater's rules allow chromium. Two ligands have an answer, both far outside it and 2.30 apart from each other. Chloride needs more than the overlap rule can be trusted to compute. Ammonia's fitted parameter is exactly zero and cyanide's is negative, and a quotient of squares is neither.

A contraction that cannot reach three of them

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

applied · Ligand field How often a random start reaches the best of them. The share of random starts that reach the best arrangement found, against the fraction of sites raised, for two nets at two contrasts. Every curve dips in the middle of its left half and recovers: the hard compositions are between a quarter and a third, and the half-filled one — the rightmost point of each curve — is among the easiest. The hardest points are 5 of 16, 6 of 16, 9 of 36, 12 of 36.

The composition that is hard is not the full one

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

solids · Cohesion The five-membered ring would need more rotors than its chain has bonds. For each measured acceleration, the least rotor count whose ceiling reaches it, against the most rotors any convention gives that ring and against the number of bonds the chain has at all. The five-membered cases need five and eight rotors from a chain with four bonds, so they are refused by supply rather than by a margin — a refutation that needs no convention to be chosen, because every convention is below the requirement.

The count that was never written down

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

shape · Strain The sign follows the count, on a ring and on a chain alike. The free combination's exponent against the number of spins, for open chains and for rings. Every even count is positive and every odd count is negative, whichever topology it is — and an open chain has no frustration at all. The usual comparison sets frustrated odd rings against an unfrustrated chain of eight, which varies the frustration and the parity together.

It was the count, not the frustration

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

applied · Magnetism One kink, and the heteroatom's own ring notices only at the kink. The response on the heteroatom's own ring, on a straight chain of twelve and on one with a kink — two adjacent angular fusions — in the middle, as the heteroatom is moved along. The two agree to within one and a half per cent except on the kink's two rings: on the first the bent chain reads 18.9 per cent lower and on the second 3.1 per cent lower. An end halves the amplitude.

A bend is not an end

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

bonding · Delocalisation One fold opens it, two close it, and the antisymmetric one does neither. The gap along three directions out of the square plane: folding one pair, folding both equally, and — from a symmetric point ten degrees out — folding one pair further while unfolding the other by as much. The first rises, the second falls, and the third leaves at zero slope. That third direction is the one the question asked for.

The direction the gap cannot see

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

applied · Electron count Reversing the distortion changes the answer, which settles the order. How much the sum changes when the distortion is reversed — the same direction taken backwards — at each angle in the plane. Along a basis direction it changes by two parts in ten million, which is nothing. Halfway between them it changes by 7.36e-4, which is the same size as the isotropy residual itself at 5.24e-4. A response with only even powers of the amplitude cannot do that, so the leading anisotropic term is the cubic one and not the quartic.

The suspect that did not fit

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

spectra · Spectrum The exponent against where the fit is allowed to start. For each stiffness, the fitted tail exponent as the near end of the fitting window is moved outward from two bonds to thirty. The standard start is six, by a rule of thumb — three coherence lengths — and the question was whether that choice is doing any work. Below six the exponent rises steeply; from six outward it is nearly flat. The rule of thumb sits on a plateau.

The rule of thumb was on the flat part

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

solids · Peierls distortion Half the series stops on its own; the other half stops where it is told. How far from the chain's end the local decay rate can still be read, against how much of the chain the reading is allowed to cover. The dashed line is the cap itself. A curve that flattens below it has ended on the noise floor and the window never mattered; a curve that tracks the cap is being cut, and has more to say. The soft chains do the first and the stiff chains do the second.

The window that was not a plateau

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

solids · Peierls distortion The gap is flat along it everywhere, and the repulsion is flat only where a symmetry says so. At each distorted geometry, how far the sixteen-electron gap and the ligand–ligand repulsion move along the direction the gap is blind to, over 5 degrees. On the symmetric line the repulsion moves by about a tenth of a per cent and downward, and its own null direction is the same one; off it the repulsion climbs by up to 1.30 per cent and its null direction is elsewhere. A blindness a symmetry produces is inherited by every function of the arrangement; one a gradient search produces is inherited by nothing.

A blindness that is inherited

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

applied · Electron count Three targets, and the residual keeps falling. The worst spread across the five scaled curves, at three values of the matched product n·δ∞. It falls from 2.51 per cent at 5 to 1.62 at 9.6, monotonically. The unmatched cases sit at 3.41 per cent throughout, because they are the same five rings whatever target is being aimed at — which is what makes the comparison a comparison.

Three points, and they all go down

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

solids · Metal Two quantities, and the best straight line between the two kinds of pair. Every pair the four electronegativity tables all cover, by how much the two elements differ and by how much the tables disagree about them. Filled points are pairs whose sign the tables dispute. The line is the boundary that misclassifies fewest — 3 of 153, against 6 for the best rule using the difference alone. It slopes upward, which is the mechanism: more dispute buys a larger difference and still leaves the sign in doubt.

Two numbers caught what one could not

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

shape · Dipole Which basis size is usable, against where the line is drawn. The basis size usable at every separation — big enough that the three-body correction matters, small enough that the pairwise assembly reproduces it — against the accuracy line, over two decades. It is not one answer. Four different sizes are the answer over this range, and for much of it there is no answer at all. The standard kilocalorie a mole gives basis 2, and it sits 0.7 per cent above the edge where that answer stops.

The line was holding the answer up

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

orbitals · Basis The far end stops mattering, abruptly. The fitted tail exponent as the far end of the window is opened from twenty bonds to a hundred and twenty. Each curve is flat past a stiffness-dependent point and exactly flat past it — because beyond a profile's own reach there are no more local rates to add, so a larger window is the same fit. The standard sixty bonds is inside the flat part for every case.

The other window was a plateau too

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

solids · Peierls distortion The ranking is fragile and the headline is not. The correlation required to disturb two different things. Reordering the easiest neighbouring pair needs ρ = 0.154, which is weak enough to expect. Displacing the collection's most fragile claim outright needs ρ = 0.954, which is near-perfect correlation. So a per-predictor structure scrambles the middle of the ordering and leaves the top of it alone.

The ranking moved and the headline did not

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

bonding · Models The error runs with the row of the periodic table. The model's relative error on each measured separation, against how many of the two ions have a third-row outermost shell. The three groups do not overlap and they run in order: two second-row ions and the model is about fourteen per cent short, one of each and it is within seven per cent long, two third-row ions and it is eighteen per cent long. The narrowest gap between groups is 11.2 percentage points.

The error was the row, not the charge

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

orbitals · Contour Five pairs, one boundary, and every case anybody asked about on one side. The mixing criterion's margin for every pair of orbitals available from the carbonyl and from ethene. Above one the mixed description wins. Three pairs were put to the criterion and all three clear the line; two more come from the same three orbitals and the same dipole matrices, and both fail it. The instrument discriminates — the selection did not.

The criterion that has never said no

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

bonding · Hybrids The resonant share is not a straight line; it bends over and saturates. The share of runs of six in a resonant pair against the logarithm of the chain length, from a thousand sites to 10⁵⁰, at five concentrations of low sites. Dashed lines carry the slope from 10³ to 10¹² straight on. At x = 0.5 the share is 16.1 per cent at 10¹² and 57.7 per cent at 10⁵⁰, where the straight line would be at 78.6. At x = 0.75, the concentration with the most runs of six, it passes 76 per cent by 10³⁰.

The share that was read as a line

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

solids · Defect The ceiling moves by 8.7-fold across the gauche energy's own reported range. The rotamer ceiling against butane's gauche energy, for the rotor counts each convention assigns to a five-membered and a six-membered closure, with the three measured accelerations drawn as horizontal lines. Across the reported range the ceiling moves by up to 8.70-fold, and two of the nine verdicts cross a measurement during the sweep. The natural prediction is that this input would be the smaller lever of the two; it is the larger.

The lever that was supposed to be smaller

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

shape · Strain Both criteria leave a gap, and Boys leaves a chasm. Every non-zero relative spread under each criterion, on one logarithmic axis, with the largest empty stretch shaded. Pipek–Mezey's runs a factor of 206; Boys's runs 2.8e+7 — seven decades, from numerical zero to a real spread with nothing between. So the bimodality belongs to the cages rather than to the functional, and under the second criterion the threshold matters even less.

A second criterion left a gap too

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

beyond · Multicentre The usable stretches, assembled from pairs and from triples. For each four-centre arrangement, the stretches of accuracy line with a usable basis size when each fragment's correction is assembled from pairs, and when the three-body increments are added to it, labelled with the sizes usable there. The covered share rises from 58% to 67% (1-1-1-1), 54% to 75% (1-2-2-1), 60% to 80% (2-1-1-2), and stretches where more than one basis size is usable appear where there were none.

A repair that costs more than the whole

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

orbitals · Basis Six pairs off the diagonal, and every one of them on an axis. Each cage-and-filling pair's Pipek–Mezey spread against its Boys spread, both logarithmic, with the two thresholds drawn. Agreements sit in the two opposite corners: spreads that are zero in both, or large in both. The six disagreements do not sit between them — they sit on the axes, with one coordinate at the floor. A criterion-dependent cage is not one the two criteria half-agree about; it is one where the difference between its descriptions is invisible to one of them entirely.

The cage is on both sides

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

beyond · Multicentre

Named alongside it

The objects these essays reach for when they reach for this one.

Model limitConventionReference stateClosed formDegeneracyConvergenceUnderdeterminationLeast-squaresOverlap integralEigenvalueBasisExact diagonalisation

All concepts