Simply false — page 5 of 5

“A longer-ranged interaction moves the root in a definite direction relative to the one-bond zero.”

It moves it to either side. At a reach of a quarter the root is 5.192 and the one-bond zero 5.072, so the root is later by 0.121; at a reach of a half the root is 5.515 and the zero 5.870, so the root is earlier by 0.355. The sign reverses between a third and a half.

Tested in A cancellation between two separations · the series on correlation

“Adding a second-neighbour repulsion strengthens the interaction, so both crossings move earlier.”

Both move later. The root goes from 3.8955 with no reach to 5.5151 at half, and the pole from 4.1595 to 6.2212 — the second-neighbour term competes with the nearest-neighbour one for the same electrons, so a larger V is needed before the structure beyond contact stops paying for itself.

Tested in A cancellation between two separations · the series on correlation

“The pair count at two bonds rises with the repulsion, as it does with a one-bond interaction.”

Only when nothing penalises it. With a one-bond interaction the count at two bonds rises from the start — the electrons pushed off their neighbours go somewhere. With a second-neighbour term it falls at weak coupling, reaching −0.024 at V = 2 for a reach of a half, and rises again once the state begins to order.

Tested in A cancellation between two separations · the series on correlation

“Fitted to each hydrogen halide's dipole, a two-state ionic–covalent model gives one coupling per bond.”

A dipole fixes the ionic weight, and the two-state problem has two unknowns besides it: the coupling and the covalent arrangement's energy. The dipole alone leaves a one-parameter family. With the bond energy as well the inversion is exact, and the couplings it returns are 1.73, 2.81, 2.50 and 1.81 eV for HF, HCl, HBr and HI — rising and then falling with length, the shortest bond's the smallest.

Tested in Two measurements leave nothing to fit · the series on electronegativity

“The two-state model is a fit, since it has a free coupling for every bond.”

With the dipole and the bond energy both used it has no free parameter, and its covalent arrangement — which nothing in the inversion connects to any other molecule — reproduces Pauling's geometric mean of the homonuclear bond energies at 0.966, 0.985 and 1.025 of it for HCl, HBr and HI, and 1.088 for ClF. It is a model that fails once, on hydrogen fluoride, at 1.68.

Tested in Two measurements leave nothing to fit · the series on electronegativity

“Carbon monoxide's measured ωₑyₑ chooses the bent-range fit that does not dissociate, so its potential turns over.”

It does choose that fit — 0.01100 cm⁻¹ against a measured 0.01101, read from the lowest four levels of the published expansion — but only because the linear bend has no other way to raise αₑ. A range that bends and stays bounded fits all four constants with a curve that dissociates, at μ = 0.765 and λ = −0.086.

Tested in The fourth constant chose a curve, not a depth · the series on rotation

“Adding a measured constant to a fit brings the fitted potential closer to the real one.”

Held to ωₑ, ωₑxₑ, αₑ and ωₑyₑ, the dissociating curves are 5.7, 31, 25 and 26 per cent too deep for CO, HCl, DCl and HF. The plain Morse curve from ωₑ and ωₑxₑ alone is off by 2.2, 14, 14 and 3.5 per cent — closer on every one.

Tested in The fourth constant chose a curve, not a depth · the series on rotation

“A potential fitted to a molecule's spectroscopic constants is that molecule's potential.”

Hydrogen chloride and deuterium chloride share one potential. Each fitted to its own four constants, they land on two members of the same family — μ = 0.829 and 1.119 — that agree to 2.7 cm⁻¹ between 0.9 and 1.2 bond lengths and differ by 0.27 eV in depth.

Tested in The fourth constant chose a curve, not a depth · the series on rotation

“The asymptotic defects were the uncertain input.”

The measured δ₀ values differ from the four-decimal ones by at most 1.2 × 10⁻⁵ and move the turn by 0.06 of a shell. What moves it is the n-dependence of the defect: the d series' correction alone, −7.6 × 10⁻⁴ at the turn, moves it two shells, and the s series' moves it the other way.

Tested in Potassium's turn was a fine-structure question · the series on representation

“Averaging a p series over its fine structure is harmless when the fine-structure interval is small.”

At the thirty-fourth shell the np interval is 0.0199 cm⁻¹, 0.64 per cent of the s–p gap. Placing the p series at p₁/₂'s defect or p₃/₂'s instead of their 1 : 2 average puts the turn at n = 22.8 or 45.2. A sign that turns on a cancellation to a few parts in a thousand inherits every interval that size.

Tested in Potassium's turn was a fine-structure question · the series on representation

“Symmetry keeps the gap's first-order change along every zero-sum distortion at zero on both sides of the crossing at 17.90°.”

Past the crossing the highest filled level is the E pair dxz, dyz, and E ⊗ E contains B: the pair fold splits it at first order and the gap falls at 0.0258 eσ a degree at a fold of 20°, either way. Only the tilts, which carry E, stay blind.

Tested in The blindness needed a single top level · the series on electron count

“The coupling between two runs in a plane depends on the distance between them and not on the direction, up to a factor of order one in the prefactor.”

At ten sites between centres a twin in the same row splits the level by 1.65 × 10⁻², and one stacked directly above by 4.97 × 10⁻⁴ — a factor of thirty-three. Two rows apart, sliding the second run by one site changes the splitting by up to a factor of twenty-six, from 0.42 to 0.016.

Tested in The disc was a band with holes in it · the series on defect

“A run of six holds the same set of levels in a plane as in a chain.”

With the same site energies a chain's run binds six levels below its host and a plane's binds four, because the plane's host band is twice as wide. The shallowest, which the coupling is about, sits at 0.248 below the band edge in the chain and 0.239 in the plane.

Tested in The disc was a band with holes in it · the series on defect

“The profile at π/R measures a molecule's antibonding occupation.”

It measures the electrons in sine combinations. In nitrogen the largest single contribution at π/R, 0.052 per electron of the molecule, comes from the 2p σ bond. Ordered by the valence profile's share at π/R the first-row diatomics run Li₂, C₂, B₂, N₂, O₂, F₂, which is not the order of their antibonding counts or their bond orders.

Tested in The zero is a parity, not a bond · the series on orbital

“A real first-row molecule can have a zero in its momentum profile along the bond.”

The 1s core pair fills a sine combination in every one of them, so no whole profile is zero at π/R — the smallest share, carbon's, is 0.113. Only lithium's valence profile is zero, because its valence shell is one s bond, and any s–p mixing in a σ bond gives it a value proportional to the p share.

Tested in The zero is a parity, not a bond · the series on orbital

“A main-group centre loses a usable valence orbital only when its ligands lie in one plane.”

A bare ring of four to eight ligands, folded off its plane to any polar angle from 60° to 120°, matches three of the centre's orbitals and orphans n − 3, one more than the formula — the same count as the flat ring. The unmatched orbital is totally symmetric off the plane and antisymmetric on it.

Tested in Folding the ring does not give the orbital back · the series on hypervalency

“Ligands spanning all three directions in space give a main-group centre four usable orbitals.”

A folded ring spans three directions and gives three. In C₄ᵥ to C₈ᵥ the centre's s and its p along the axis are both totally symmetric, and a single symmetry-distinct set of ligands supplies exactly one totally symmetric combination for the two. An apex supplies the second and restores four at every angle, 90° included.

Tested in Folding the ring does not give the orbital back · the series on hypervalency

“The pentagonal pyramid, the one arrangement the census could not count, sits where the counting formula is expected to fail.”

Reduced in C₅ᵥ, sulfur hexafluoride as a pentagonal pyramid spans 2A₁ + E₁ + E₂ against the centre's 2A₁ + E₁: four matched, two orphaned, which is n + L − 4. The census now counts twenty-seven arrangements and the formula is right for twenty-four, the three it misses still the three flat ones.

Tested in Folding the ring does not give the orbital back · the series on hypervalency

“A splitting the solver reports as a small number is a small splitting.”

Raising phosphine's barrier from 2,020 wavenumbers to its own 12,300, the reported splitting falls cleanly for three points and then stops falling: 7.8 × 10⁻¹⁰ at 5,000, 3.9 × 10⁻¹⁰ at 6,000, zero at 7,000 and 7.8 × 10⁻¹⁰ again at 8,000. A splitting that grows with the barrier it tunnels through is not a splitting, and every one of those values is below the floor the operator's norm sets.

Tested in The same fraction at six times the barrier · the series on inversion

“The floor is a fraction of the level, since each eigenvalue is bisected to the last bit of itself.”

It is a fraction of the operator's norm. A Sturm count is a recurrence over every grid point and its rounding is set by the largest thing in the matrix, which on a 2,400-point grid is tens of hartree where the levels are thousandths of one. The two differ by four orders of magnitude — 1.7 × 10⁻⁹ against 1.3 × 10⁻¹² wavenumbers for phosphine — and the spurious values above sit between them.

Tested in The same fraction at six times the barrier · the series on inversion

“A correction sized on ammonia says nothing about a molecule whose barrier is six times larger.”

It says almost exactly the same thing. The same fraction of each molecule's own transverse zero-point difference is 4.58 per cent of ammonia's barrier and 0.56 per cent of phosphine's — ten times less — while the action it acts on is 8.86 against 60.76, seven times more. The predicted isotope ratio moves by 21.4 per cent for ammonia and 16.1 for phosphine.

Tested in The same fraction at six times the barrier · the series on inversion

“Since the overlap asymmetry moves only the spin and the coupling moves only the level, a radical with both moves each by the separate amounts.”

True at the origin, where both mixed second derivatives vanish, and false beyond it. With the A–C overlap 0.2 above B's and an end-to-end overlap of 0.1, the level sits 0.20 eV below the sum of the two shifts and the spin on A at 0.218 against the 0.236 the asymmetry alone gives. The electron also reaches the middle orbital, which carries 0.012 of it.

Tested in Two changes that do not add · the series on overlap

“The chain of four's contrast limit of 1.3125 is the fraction 21/16.”

It is (20 + 5√3 + √15)/(20 + 5√3 − √15) = 1.3124978, the ratio of 1/4 + √3/16 + √15/80 to 1/4 + √3/16 − √15/80: the √3 belongs to the four-site spin chain's singlet and the √5 to the hole's standing waves. The exact spectra extrapolated from U = 4096 and 16384 give 1.3124972, closer to the closed form than to 21/16 by a factor of nearly five.

Tested in The limit was two chains at once · the series on photoelectron

“The parity rule can only be argued, not counted, beyond the chain of six.”

At infinite repulsion the lines are a hole level times a spin state, and the spin states of each level are fixed by second-order perturbation theory on a few hundred configurations. Built that way the chains of two, six and ten have a limit of exactly one and the chains of four, eight and twelve 1.3125, 1.0458 and 1.0195. A separate exact diagonalisation of the chain of eight at U = 1,000 and 10,000 extrapolates to 1.04577.

Tested in The limit was two chains at once · the series on photoelectron

“At infinite repulsion the spin part of the problem is degenerate, so only the hole's levels matter to the intensities.”

The degeneracy is what makes the order-t²/U operator decide which spin states the lines are, however small it is. Leaving out its three-site part — the hole hopping across an occupied site — changes the chain of eight's limit from 1.0458 to 1.0746 and the chain of twelve's from 1.0195 to 1.0364. Only on the chain of four, where symmetry leaves the operator nothing to choose, are the two the same.

Tested in The limit was two chains at once · the series on photoelectron

“Each E vibration's Coriolis constant is a separate measurement of ammonia's mode shapes, so two constants test the force field twice.”

On all twenty-two fields walked, ζ₃ + ζ₄ equals I∥/2I⊥ − 1 (the moments about the axis and across it) to a part in a million: −0.22247 for NH₃ and −0.19551 for ND₃. The geometry fixes the sum, so the two constants carry one number about the field. With the translation pair (+1) and the rotation pair (1 − I∥/2I⊥) the E-species total is exactly one.

Tested in Two Coriolis constants are one number · the series on point group

“The Coriolis constants would only refine what the frequencies already fix.”

Across the band of stretch–bend constants the frequencies cannot tell apart (−1.000 to −0.634 mdyn/rad), every frequency of NH₃ and ND₃ moves by 0.28 to 0.87 per cent while ζ₄ moves from −0.807 to −0.641, by 23 per cent. ζ₄ changes by 0.453 for each mdyn/rad, so a value measured to ±0.005 leaves 0.022 of the band's 0.367.

Tested in Two Coriolis constants are one number · the series on point group

“The E′ Coriolis constant, which depends on eigenvectors rather than eigenvalues, can pick between the two fields a ¹⁰B substitution leaves.”

Under the fitted field (B–F constant 7.425 mdyn/Å) and its partner (3.966), ¹¹BF₃'s ζ₃ is 0.7910255 and 0.7910254, ¹⁰BF₃'s 0.8088492 under both, and each E′ pair's coupling to the out-of-plane wag agrees to 10⁻⁷. Around the loop ζ₃ = 2s/(1 − m(B)/M) − 1 to 3 × 10⁻⁹, with s the boron's share of the motion — and s = −∂ln λ/∂ln m(B), which two fields agreeing at every boron mass must share.

Tested in The Coriolis constant was an isotope shift · the series on normal mode

“A planar symmetric top's two E′ Coriolis constants are two measurements.”

For a planar molecule the moment about the axis is twice the moment across it, so the sum rule gives ζ₃ + ζ₄ = 0 at every point of the loop, to 10⁻¹⁵. And the couplings across the axis to the wag are 1 + ζ₃ and 1 + ζ₄. There is one Coriolis number in the E′ block.

Tested in The Coriolis constant was an isotope shift · the series on normal mode

“The pair fold from a folded square leads to a recognisable four-coordinate shape such as a butterfly or a sawhorse.”

Over the four elevations, the lowest minimum of the weighted energy is a D₂d flattened tetrahedron below a weight of 0.0762 and the square plane above it, at every weight sampled. Forty random starts per weight find nothing lower; the other minima they find are lower-symmetry near-tetrahedra and squares tipped about one axis.

Tested in The pair fold was the road to a tetrahedron · the series on electron count

“Folding a square and flattening a tetrahedron are different distortions to the d levels.”

At the same angle they give identical d levels to 10⁻¹², because lowering one trans pair instead of lifting it reverses the sign of that pair's dyz amplitude and nothing else. The 17.90° crossing of dz² with the dxz, dyz pair is at the same angle on both roads.

Tested in The pair fold was the road to a tetrahedron · the series on electron count

“The tail exponents of the Peierls chain's healing might belong to the fixed-length constraint rather than to the healing itself.”

Relaxed at zero tension, the chain's middle carries no compression (10⁻¹² against the fixed chain's −5.8 × 10⁻⁴ at K = 1.6), and the two-ended tail exponent from bond six agrees with the fixed chain's to under 0.001 at all ten stiffnesses — under 0.003 from any starting bond.

Tested in The chain that pays for nothing · the series on Peierls distortion

“The exponents' fall as the fit starts deeper is the next term of the tail's expansion, a 1/b correction, and fitting it would steady them.”

With a factor (1 + a/b) fitted alongside, the exponent's spread over fit starts 6 to 30 grows on every chain from K = 2.2: 0.041 to 0.053, 0.042 to 0.114, 0.025 to 0.146 and 0.008 to 0.118. At K = 2.8 it falls from 0.692 to 0.546 where the plain fit moved from 0.697 to 0.673.

Tested in The chain that pays for nothing · the series on Peierls distortion

“Keeping only the consecutive triples fails for a uniform chain because four fragments is too short a chain for the saving to show.”

At five, six, seven and eight fragments the consecutive assembly covers 45.6, 47.1, 48.8 and 48.1 per cent of three decades of accuracy line, against 60.0, 63.1, 65.8 and 67.9 for pairs alone. The shortfall is 5.2 points at four fragments and 19.8 at eight: it grows with length rather than closing.

Tested in Length did not rescue the consecutive triples · the series on basis

“Adding every triple to a pairwise assembly is at least as good as adding only some of them.”

On the heavy-outside chain, every triple together covers 70.8, 56.2, 59.1 and 62.3 per cent from five to eight fragments, and the consecutive triples alone cover 77.4, 80.9, 84.4 and 87.9. What every triple leaves out reaches 1.05 kilocalories a mole at five and 2.70 at eight, so the complete order is the worse assembly by up to twenty-five points.

Tested in Length did not rescue the consecutive triples · the series on basis

“The sign of the correction cannot be settled without frequencies at the planar geometry.”

It is settled by the direction of the gap. The deuterated molecule's fitted barrier is the lower one and its transverse zero point is also the lower one, so the fraction that closes the gap is positive for every band and under every construction of the mass — the transverse frequencies must stiffen as the molecule flattens, which is what an X–H bond does as its central atom goes planar.

Tested in The band that cannot supply what it changes · the series on inversion

“The bend is the obvious supplier, since a flattening pyramid changes its bends most.”

It is the worst. The degenerate bend carries a quarter of the isotope difference against the degenerate stretch's half, so it would have to stiffen by 22.25 per cent against the stretch's 10.92 — and its two components soften rather than stiffen as the molecule flattens, which turns its contribution into a debt. With the bend softening by eight per cent the stretches have to find 9.79 rather than 7.20.

Tested in The band that cannot supply what it changes · the series on inversion

“An underdetermined correction cannot fix a barrier.”

It fixes this one. The bare electronic barrier is the fitted value less the zero-point rise at the pyramid, which works out to the gap divided by one minus that band's isotope frequency ratio — and those ratios are 0.7252, 0.7422 and 0.7320, all just above 1/√2. Four choices whose required fractions span a factor of 2.04 imply barriers spanning 1.33 per cent: 1740.6, 1754.9, 1763.9 and 1750.1 wavenumbers.

Tested in The band that cannot supply what it changes · the series on inversion

“Counting which central orbitals a set of ligands can use needs the arrangement's point group and its character table.”

It needs the rank of the rows (1, x, y, z) over the ligand directions. On the 451 geometries where a table exists — the census, every folded, capped and flat ring, and every tabulated point on every path between arrangements — the rank and the table give the same count every time.

Tested in The count needed no table · the series on hypervalency

“A reduction by symmetry species gives the right count wherever the group is known.”

It gives an upper bound. Five ligands placed irregularly on a small circle of the sphere have no symmetry beyond C₁, so every function is one species and the table matches all four central orbitals. The rank is three: a combination of s and the p normal to the circle's plane reaches none of them.

Tested in The count needed no table · the series on hypervalency

“Some three-dimensional arrangement of a single ligand set might leave px and py short while matching pz.”

Two orbitals left short means a rank of two, and a rank of two means every ligand direction lies on one straight line. A line meets the sphere at no more than two points, so only a two-ligand molecule can do it — where nothing is orphaned. Every crown of four to ten alternating ligands tried reaches all four.

Tested in The count needed no table · the series on hypervalency

“The square's scores depend on where the correction is fitted less than they depended on the mean field's start.”

On the lowest mean field the polarisation's worst ratio runs from 9.73 to 213 across eight references, a factor of twenty-two; the start moved the published reference's from 48.0 to 14.7, a factor of three and a quarter.

Tested in The worst case measured the reference · the series on approximation

“A diagnostic that passes the census threshold on a square has found a square on which the transfer works.”

With the correction fitted at the band's largest miss on the conventional start, the polarisation's worst ratio is 2.05, under the threshold of three, and every one of the eighty systems' errors lies between −1.50 and −0.60.

Tested in The worst case measured the reference · the series on approximation

“The two force fields left by a ¹⁰B spectrum of boron trifluoride are a numerical coincidence of that molecule.”

They are an exact exchange of the stretching constant x and the rescaled bending constant 3z/R² at fixed stretch–bend constant, present in every planar XY₃ at any masses. The swap reproduces the partner found by polishing a traced loop to two parts in a hundred million, from the fitted field alone, without any tracing.

Tested in The partner field was a swap · the series on normal mode

“Once symmetry is broken, the first-order Coriolis constants between the split components can separate the two fields.”

Between the components of ν₃ in ¹¹B³⁵Cl₂³⁷Cl the two fields give 0.75138 and 0.75142, and between those of ν₄ 0.75127 and 0.75117. These differ by 10⁻⁴ or less and are not an independent measurement of anything frequencies reach.

Tested in The partner field was a swap · the series on normal mode

“Placing the p series at the p₁/₂ defect gives np₁/₂'s turn, near the twenty-third shell.”

That moved the level and kept the angular factors of the state that was not there. With its own factors, 1/9 to s and 2/9 to d, np₁/₂ is at 1.097 on the twenty-third shell and still at 1.017 on the hundredth. It has not turned anywhere computed.

Tested in The state that turned is not in the atom · the series on representation

“How a sweep's precision varies with temperature is a further coordinate of a design, on top of its window and spacing.”

It is the same coordinate. An even sweep with precision growing as √T and a logarithmic sweep at flat precision give 1.447 and 1.453 at 20–300 K, 1.627 and 1.632 at 40–300 K, 1.367 and 1.380 at 20–500 K, because they weight the window alike. A sweep's measure is spacing times inverse variance, and only that.

Tested in A precision is a spacing · the series on magnetism

“A realistic even sweep has the even design's constant, since its temperatures are evenly spaced.”

With a one per cent floor and constant absolute noise on the susceptibility, the constant falls from 1.696 to 1.598, 1.479 and 1.450 as the noise grows, and settles at about 1.44. A sweep's spacing does not say what its design is. Scaling every sigma, by contrast, changes nothing at all, to a part in a thousand million.

Tested in A precision is a spacing · the series on magnetism

“The splitting's unequal steps down the isotopologue series are a property of the potential's shape.”

They are the shape of a square root. Fitting the logarithm of the splitting against the square root of the reduced mass leaves a worst residual of 0.0126 for the ground line and 0.0084 for the excited one; against the number of deuteriums the same fits leave 0.100 and 0.0676 — eight times more. A tunnelling exponent is linear in √μ and √μ is not linear in a count.

Tested in The steps were a square root of a mass · the series on inversion

“The excited doublet would show a different pattern, since it sits higher in the well.”

It shows the same pattern with a smaller slope. Its sensitivity to √μ is 0.8614 of the ground line's, and the ratio of the two step patterns is 0.8323, 0.8705 and 0.8935 at the three steps — inside four per cent of that slope ratio throughout. One factor, not a change of shape.

Tested in The steps were a square root of a mass · the series on inversion

“A correction fitted to the ground doublets says nothing about the excited ones.”

It moves the excited isotope ratio from 12.442 to 10.129 against a measured 10.690 — from sixteen per cent high to five per cent low. A shift fitted entirely to the two ground lines improves the untouched excited ratio by a factor of three, which is evidence, and over-shoots it, which is a bound on what else is missing.

Tested in The steps were a square root of a mass · the series on inversion

“What is left over after the correction is more of the same missing term.”

It is not isotope-dependent. Each molecule's excited doublet, computed from a barrier fitted to that same molecule's own ground doublet, comes out 8.96 per cent too large for NH₃ and 14.99 per cent too large for ND₃ — same sign, similar size. An isotope-dependent term cannot produce that; a well of the wrong shape can, and the well here is a quartic.

Tested in The steps were a square root of a mass · the series on inversion

“The orientational average replaces cos(qR) by sin(qR)/qR, so the interference is one function with a shifted zero.”

It is three functions. The cross term multiplies the interference by the orbital's own angular density and the average of the product is not the product of the averages, so an s combination gets j₀(pR), a pσ combination j₀ − 2j₂ and a pπ combination j₀ + j₂ — with first zeros at π, 0.663π and 1.430π. The directional profile gives every orbital the same zero; the average gives each kind its own.

Tested in The pair that cancels only at zero overlap · the series on orbital

“The exact zero at π/R survives the average in weakened form.”

It does not survive at all. A profile is an integral over every momentum above q and the averaged density's zero is at one momentum, so the integral through it is not zero. Nitrogen's averaged profile at π/R differs from two free atoms' by 3.5 per cent of its peak, where its directional profile has a definite minimum there.

Tested in The pair that cancels only at zero overlap · the series on orbital

“The core pair's nearly flat contribution swamps the signature.”

The core pair contributes nothing to the signature whatever. Bonding plus antibonding of the same atomic function is twice the atom exactly, so a filled pair cancels — the 1s pair's contribution to the interference part is below 10⁻⁷ in every molecule, because its two functions overlap by 7 × 10⁻⁵. It dilutes the signature by sitting in the denominator and it does not obscure it.

Tested in The pair that cancels only at zero overlap · the series on orbital

“So the interference part is a weighted count of parity mismatches.”

The cancellation of a filled pair is exact only at zero overlap. The two combinations are normalised by 2 + 2S and 2 − 2S, so equal occupations leave a residue that grows with S — and in nitrogen the filled 2s pair, whose imbalance is zero, contributes −24.7 per cent of J(0) against the imbalanced functions' +10.9 per cent together. The largest term is the one a count says is absent, and it has the opposite sign.

Tested in The pair that cancels only at zero overlap · the series on orbital

“Whether a cut shell moves the answer is a property of the cage.”

It is a property of the cage and the criterion together. The octahedron at four electrons moves by a factor of 1.152 under Pipek–Mezey and does not move in the ninth decimal under Boys, because it is a 3-design and Boys needs cubic averages where Pipek–Mezey needs quartic ones.

Tested in The cage that averages like a sphere · the series on multicentre

“A cage good enough at averaging protects any cut shell.”

Only a shell of degree one, whose every re-orientation is a rotation of space. The icosahedron at ten electrons cuts its five-fold shell of degree two and moves by 1.207 under Pipek–Mezey, on the same 5-design that leaves its degree-one cut untouched.

Tested in The cage that averages like a sphere · the series on multicentre

“The allyl radical's central spin is known only to within the spread of McConnell's Q, −0.15 to −0.18.”

The couplings share one Q, so their ratio is the ratio of spins; with the three π spins summing to one, +4.06 against the terminal −13.93 and −14.83 G fixes the central spin at −0.164 without Q, and implies Q = −24.7 G.

Tested in Six electronvolts was a difference · the series on overlap

“A repulsion between neighbouring sites adds to the negative spin an on-site repulsion puts on the middle orbital.”

Alone, a repulsion between each end and the middle leaves the middle with no spin and one between the ends puts positive spin there. With an on-site repulsion present, the neighbour terms shrink the negative spin: at U = 6 eV from −0.162 to −0.050 as they reach 5.5 eV.

Tested in Six electronvolts was a difference · the series on overlap

“Scoring a diagnostic by a rank correlation with the size of the error removes the dependence on the reference.”

It removes the near-zero denominators and leaves the reference: the polarisation's rank correlation with the error runs from 0.47 to 0.94 across eight references. Every error is c(ref) − c(sys), so a reference shifts a straight line of errors, and the size of the error folds that line at the reference. A score free of the reference has to be against c(sys) itself, where it is 0.49.

Tested in A constant ties the polarisation · the series on approximation

“A correction interpolated between a weakly and a strongly correlated reference shrinks the largest error from the size of one correction to the size of their difference.”

Taking the published reference and the grid's weakest system, the largest error is 0.27 to 1.22 depending on the diagnostic, against the 0.35 that one reference gives. Over every pair the best result is 0.177, against the best single reference's 0.216, and every diagnostic's median pair, 0.42 to 0.64, is worse than the median single reference's 0.34.

Tested in A constant ties the polarisation · the series on approximation

“The floor is set by how different the two atoms are, so it orders by electronegativity.”

It orders by neither. B–N's floor is 1.63 per cent and C–O's 1.35, although the two have the same difference in nuclear charge and C–O is by far the more polar bond. The floor is the mismatch of two radial functions evaluated at π/R, so it depends on the bond length through the momentum at which the comparison is made, and B–N is the longer bond.

Tested in A floor no charge transfer explains · the series on orbital

“Some coefficient ratio would restore the cancellation.”

Cauchy–Schwarz on the same measure puts the square of the cross integral at or below the product of the two self integrals, so the bracket is bounded below by the squared difference of each atom's coefficient times the square root of its own self integral, and vanishes only where the two atomic momentum functions are proportional at every momentum. Two Slater functions of different exponents are nowhere proportional, so no coefficients do it — and at equal coefficients the bound is only 4.1 per cent of C–O's depth, so the part that is irreducible is nearly all of it.

Tested in A floor no charge transfer explains · the series on orbital

“The polarity that minimises the depth is a small one in the chemical direction.”

It is in the other direction. Matching the two contributions needs a coefficient ratio equal to the square root of the ratio of the two self integrals, which is 1.24 for C–O — more weight on the more diffuse atom, because the compact one has the broader momentum distribution and the larger contribution at π/R. Chemistry puts the weight on the compact atom, so the two effects add: C–O's shallowest point is at a charge imbalance of −0.24, and at the imbalance a real bond has the depth is nearly three times the floor.

Tested in A floor no charge transfer explains · the series on orbital

“Two short bonds cost twice what one does.”

Only when they are far apart. Across the ring, 144° apart, two short bonds cost 2.017 times one short bond's equatorial penalty. Side by side, 73° apart, they cost 2.335, and one axial with one equatorial, 91° apart, costs 1.147 where one would be expected. Close short bonds interact.

Tested in The ring held flat by its long bonds · the series on VSEPR

“An arrangement that leaves a central orbital unreached has weaker bonding than one that reaches all four.”

In a σ-only model with every ligand level filled, the square leaves one p orbital unreached and the seesaw reaches all four, and the square is the more stable of the two up to a coupling of 0.59. At weak coupling what costs energy is a net direction among the ligands, which the seesaw has and the square does not.

Tested in What a missing orbital costs · the series on hypervalency

“How nearly an orbital is unreached — the Gram matrix's smallest eigenvalue — measures how weak the bonding is.”

The pentagonal pyramid has a smaller eigenvalue than the trigonal prism, 0.19 against 0.25, and is more stable at every coupling. Across the eleven arrangements of four to six ligands the eigenvalue fails to order the energies; the fourth-order cost orders every group without exception.

Tested in What a missing orbital costs · the series on hypervalency

“The trio's agreement with Pariser's on-site repulsion might rest on its round overlap of 0.25, which was never allyl's.”

Carbon's own 2pπ overlap at allyl's 1.39 Å is 0.2484, and replacing both of the trio's overlaps with carbon's moves the one-parameter repulsion from 10.89 to 10.68 eV. It is the energy that carries it: replacing hydrogen's −13.6 eV with carbon's −11.4 moves it to 9.66, and both together to 9.49.

Tested in The agreement was hydrogen's energy · the series on overlap

“A shortfall common to every pair is what a strength constant is for.”

Multiplying the four-electron repulsion by 6.15 brings the six errors' mean to zero and spreads them from 1.3 points to 8.8 — magnesium oxide from −14.86 to +5.47 per cent, potassium chloride from −13.81 to −3.33. The spread grows with the strength from the first step: 3.1 points at a factor of two, 6.7 at four.

Tested in The shortfall was a length, not a strength · the series on contour

“Range and strength are two independent knobs, so the order they are set in does not matter.”

They are not independent at all. Every overlap between Slater functions depends on the exponents only through the product with the separation, so scaling every exponent by λ is exactly the unscaled model with its repulsion's strength multiplied by 1/λ, every separation then stretched by 1/λ. A range is a strength plus a length.

Tested in The shortfall was a length, not a strength · the series on contour

“A strength constant moves every equilibrium by about the same fraction.”

It moves each by the repulsion's own decay length times the logarithm of the factor, to within six per cent for all six pairs. As a fraction of the separation that is 10.5 per cent for potassium chloride and 20.3 for magnesium oxide, in the order of the decay length over the separation, 0.060 to 0.108.

Tested in The shortfall was a length, not a strength · the series on contour

“Adding the atoms at the ends of the bonds makes a repulsion model more realistic by adding more repulsion.”

Under one power law a second set of points on the same rays is the first set rescaled: the energy is multiplied by d^p for every arrangement, and the pucker's curvature times d^p comes out at exactly 1.000 at nine placements under 1/r and under 1/r¹². More of the same repulsion changes no shape. The end atoms matter only through a law of a different shape.

Tested in The fluorines' range decides the ring · the series on VSEPR

“A steeper or shorter-ranged contact repulsion always pushes crowded neighbours apart and so always favours the pucker.”

An exponential contact exp(−r/ρ) favours the pucker only for ρ below 0.346 Å. Above it the term's own pucker curvature is positive — 1.8 × 10⁻⁴ at 0.35 Å, 3.3 × 10⁻³ at 0.4 — and it holds the ring flatter at any weight, because the ring-to-axis contacts it resists with are only 18 per cent further away than the ring neighbours it relieves.

Tested in The fluorines' range decides the ring · the series on VSEPR

“For every ligand count past four there is a continuum of shapes the σ model cannot tell apart, and the trigonal bipyramid has no reason to be the only shape with its moments.”

An irregular five-ligand shape does have a two-parameter family of twins. The bipyramid has none: wherever it is the floor it is the unique minimum, rigid except for rotation, and any shape sharing its moments would be a second minimum that sixteen seeded searches never find.

Tested in Five ligands, and no ideal to aim at · the series on hypervalency

“Adding the antipodal term is a small correction to the two-bond picture.”

At the root every separation changes sign. With the two-bond reach of a half and nothing at the antipode the pair changes at one bond, two bonds and the antipode are +0.185, −0.370 and +0.527; with the 1/r tail round the ring they are −0.258, +0.588 and −0.109. The antipodal term itself is the smallest of the three.

Tested in A root that moves when nothing else does · the series on correlation

“The root of the give-back is a property of the ground state.”

Lowering the interaction by the same amount at every separation, on-site included, adds a constant to the energy and leaves every pair change unchanged to 1e-9. It moves the root from 5.920 to 5.777 when the antipode is made to cost nothing, to 5.646 when two bonds are, and removes it below the pole when one bond is. The pole, at 6.274, does not move.

Tested in A root that moves when nothing else does · the series on correlation

“Mixing p into the diffuse atom of a polar σ bond moves its momentum distribution towards the compact atom's and can reduce the floor of the profile's minimum at π/R.”

At π/R a p function on one atom and an s function on the other interfere through sin qR, which is zero there, so p on carbon adds its own density and cancels nothing. Carbon monoxide's minimum goes from 1.35 per cent of the peak to 2.17 at a tenth p and 3.63 at a quarter; the smallest depth any coefficient ratio allows rises from 1.23 to 2.17 and 3.35.

Tested in The third function fills the floor · the series on orbital

“Xenon hexafluoride's floppiness is the same kind of motion as iodine heptafluoride's: a pucker moving round a ring.”

With the fluorines' contact repulsion past its switch the lowest structure is a capped octahedron, C₃ᵥ, and the lowest route between two of them is an edge-capped structure, C₂ᵥ, with one falling direction: the lone pair passes from one face of the fluorine octahedron to the next over an edge. The barrier is 0.006 to 0.013 of the model's energy at contact shares of 3 to 5 per cent, a few parts in ten thousand of the total.

Tested in The lone pair goes through a face · the series on VSEPR

“A contact law's range decides the verdict for a seven-domain molecule, as it did for IF₇'s ring.”

For IF₇ an exponential contact longer than 0.346 Å held the ring flatter and a shorter one puckered it. For XeF₆ every range tried — 0.25, 0.345 and 0.5 Å, and fluorine's own 2p overlap — moves the lone pair into a face; the range changes only the share needed, from 0.4 per cent to 7.2.

Tested in The lone pair goes through a face · the series on VSEPR

“A repulsion between the ligands pushes a five-coordinate centre towards the square pyramid, since the pyramid spreads its basal ligands.”

A Coulomb repulsion does the opposite: it moves the σ parameter below which the square pyramid wins from 0.862 down to 0.816 at a share of five. Five points under any power below 15.05 have the bipyramid as their own least-energy arrangement.

Tested in Only a steep contact makes five square · the series on hypervalency

“A Born–Mayer contact between bulky ligands is steep enough to make a main-group square pyramid.”

An exponential contact between ligand atoms 2.2 Å from the centre switches the shape only at ranges below 0.215 Å; at Born–Mayer's 0.25 Å and above it never does. The threshold is where the exponential's local steepness at the ligands' ninety-degree separation, √2·L/ρ, is 14.5 — within four per cent of the power-law switch.

Tested in Only a steep contact makes five square · the series on hypervalency

“Because a single angular ring scatters more near the end of a chain than at its centre, giving the factor a dependence on position would close most of the residual.”

Splitting each count into angular rings next to an end ring and rings further in raises R² by 0.0006, 0.0010 and 0.0014 at seven, nine and eleven rings, and the worst residual spread falls from 2.65 to 2.63 at eleven. An angular ring under a pair does cost less next to an end — 0.69 against 0.57 at eleven rings — but few pairs sit on one.

Tested in The angular factor belongs to the chain · the series on aromaticity

“Carbon monoxide's σ bond has a momentum-profile minimum at π/R somewhere between the s-only floor of 1.35 per cent and the quarter-p value of 12.5 per cent.”

With weights from an extended Hückel calculation over both atoms' 2s and 2pσ there is no single σ bond. The lowest occupied σ orbital reads 2.5 per cent of its peak at π/R, the middle one has a minimum there at 8.4 per cent of its peak, and the highest reads 85 per cent of its peak — it has no minimum at π/R.

Tested in The lone pair sits where the floor was · the series on orbital

“The carbon lone pair's orbital, whose hybrid points away from oxygen, shows a deeper minimum at π/R than the bonding σ orbital.”

Its profile rises from the origin to a peak at 0.83 π/R and at π/R stands at 2.58 times its value at the origin. That holds at every Wolfsberg–Helmholz constant from 1.5 to 2, between 2.1 and 3.4 times.

Tested in The lone pair sits where the floor was · the series on orbital

“Carbon monoxide's polarity should make its σ profile read higher at π/R than an isoelectronic non-polar molecule's.”

Through the same calculation, carbon monoxide's six σ electrons read 39.5 per cent of their value at the origin at π/R and nitrogen's 41.6: the polar molecule sits lower, at every constant tried, and neither total has a minimum anywhere out to 2π/R.

Tested in The lone pair sits where the floor was · the series on orbital

“A bulky ligand's contact is a small steric correction on the bonding.”

At strong coupling the contact that makes the square pyramid win carries at least a seventh of the σ bond energy under every law tested, and more than all of it under the sixteenth power, because a steep law barely tells the two shapes apart near its own threshold.

Tested in Priced against the whole bond · the series on hypervalency

“The model cannot represent a stereochemically inactive lone pair, because it cannot put a lone pair nowhere.”

A weightless lone pair is the inactive limit, and the model reaches it continuously: at a weight of 0.001 the six fluorines are a regular octahedron to 0.06° and the lone pair sits over a face centre, 54.76° from three fluorines. The distortion grows from zero at 58.5° of cis-angle spread per unit weight, and the pentagonal pyramid never intervenes below a weight of one.

Tested in A lone pair weaker than a bond · the series on VSEPR

“A pair whose margin crosses one switches the localised description from canonical to mixed.”

At H₂C=F⁺ the π–lone-pair margin is 1.350 and the σ–π margin 1.179, so both pairs are endorsed and they share π. The maximum over all three orbitals is still two bent bonds sharing π at 0.496 and 0.504, with the lone pair kept 95.2% itself — the same description as at H₂C=O, where the π–n pair is refused at 0.739.

Tested in A verdict that flips and changes nothing · the series on hybrids

“When two endorsed mixings share an orbital, a pairwise localisation will depend on the order the pairs are taken in.”

Three orders run to convergence at H₂C=F⁺, one starting with the π–n pair, all reach 1.180063, the maximum over the whole rotation group. The π–n pair optimum is not a maximum of the three-orbital functional at all; a σ rotation from it climbs by 0.00087.

Tested in A verdict that flips and changes nothing · the series on hybrids

“A molecule whose occupied σ space is a single bond, such as lithium hydride, is where the polarity minimum at π/R should be readable.”

Lithium hydride's one occupied valence σ orbital, from extended Hückel over lithium 2s and 2pσ and hydrogen 1s, is 91.3 per cent hydrogen. Its momentum profile along the bond falls monotonically and reads 10.2 per cent of its peak at π/R; there is no minimum anywhere out to 2π/R.

Tested in The minimum belongs to the heavy atom's node · the series on orbital

“Hydrogen fluoride's bond, with its other σ orbital nearly pure fluorine 2s, isolates the bond's minimum.”

Its lower σ is 92.9 per cent fluorine 2s and falls monotonically; its upper σ, the bond, is 93.0 per cent fluorine 2pσ, starts near zero as a p function along its own axis does, peaks at 0.625 π/R and reads 71 per cent of its peak at π/R. Neither has a minimum.

Tested in The minimum belongs to the heavy atom's node · the series on orbital

“An angular ring has four outer carbons in two inequivalent pairs relative to its meta fusion.”

Every interior ring of a catacondensed chain has two carbons that belong to it alone and four it shares with its neighbours; only the two terminal rings have four. On an angular ring the two are adjacent and bonded to each other.

Tested in On an angular ring the carbon decides · the series on delocalisation

“On the kink's first angular ring the heteroatom sat on an outer carbon beside the long run of unshared bonds, and on the second it faced the short side.”

Both placements were ring junctions, chosen as the carbon furthest along the chain's axis. On ring five that carbon ends the ring's one short unshared bond, at the bay; on ring six it ends the run of three. The published reading had the two the other way round.

Tested in On an angular ring the carbon decides · the series on delocalisation

“In the seven-domain repulsion model, a lone pair crowds the bonds next to it, so given xenon hexafluoride's composition the model puts the longer bonds round the face the lone pair opens — as the molecule does.”

With the domains alone the long three go opposite the lone pair for every weight below 0.727, and the face is the lowest structure only below 0.657. With any contact share from half a per cent to six, they go opposite at every weight from 0.05 to 2. Of 210 cells of weight and share, the face is lowest in 94 and the long bonds go next to the lone pair in 23, and no cell is both.

Tested in The model lengthens the wrong three · the series on VSEPR

“A nodeless Slater-type 2s has no radial node and so no node in momentum.”

The transform of r·e^(−ζr) is proportional to (3ζ² − p²)/(ζ² + p²)³, which vanishes at p = √3·ζ. For lithium at ζ = 0.65 that is 1.126 bohr⁻¹, 1.080 π/R at lithium hydride's length. A node in momentum exists exactly when a function's integral and its slope at the nucleus have the same sign; the Slater 2s is positive in sum and rises from zero.

Tested in The node a nodeless orbital has · the series on orbital

“With the node above the fringe, the minimum moves onto π/R.”

With the nodeless Slater 2s the node is at 1.080 π/R, above the fringe, and the only minima any 2s–1s combination shows sit at 0.898–0.955 π/R, at hydrogen shares from 0.026 to 0.065, never at π/R and never at the node.

Tested in The node a nodeless orbital has · the series on orbital

“In the stiff limit the free-length answer reduces to the fixed-length one.”

Only for the lone pair kept among the bonds, whose crossing sits at 0.7253 at every stiffness from 5 to 1000. A pinned lone pair has no crossing at all for stiffnesses up to 50 and crosses at 0.295 at 100 and 0.673 at 1000, still 0.05 away with the bonds stretched by two tenths of a per cent.

Tested in The lone pair has no bond to stretch · the series on VSEPR

“The repulsion model accounts for xenon hexafluoride's bond-length spread and only its orientation is in question.”

At a mean stretch of two to three per cent every reading splits the near and far sets by under 0.04%, against 4.3% from electron diffraction and 6.25% from correlated calculations. The best split per unit stretch anywhere on the grid is 0.046, so the measured spread would need every bond stretched by about 94%.

Tested in The lone pair has no bond to stretch · the series on VSEPR

“A cage nearly a spherical 4-design moves under Pipek–Mezey by an amount that shrinks in proportion to its degree-four Legendre sum.”

Wrong twice. The sum has to be taken over the points the shell's eigenvectors draw and not over the vertices: an icosahedron stretched twofold has a vertex sum of 0.118 and does not move in the fifteenth decimal. And taken over the right points, the movement goes as the sum's square root, with a fitted slope of 0.499 on eleven settings within ten per cent of the design, because the sum is a sum of squares.

Tested in The average was over the eigenvectors · the series on multicentre

“The icosahedral group has nothing to do with why the icosahedron's cut shell does not move.”

It cannot carry one plane of the shell onto another, which is the reason that was tried and rightly refused. It does have no invariant harmonic of degree three or four, so the fourth moment of any point set its shell can draw is forced isotropic. That is why two kinds of bond in C60, which drag the shell off its coordinates, still move nothing.

Tested in The average was over the eigenvectors · the series on multicentre

“The diffuseness an ionic model lacks belongs to the anions, as the neutral-atom origin of their exponents suggests.”

With cations left at their own exponents and the anions scaled by 0.738 to close the shortfall, sodium fluoride and magnesium oxide, the two pairs no contraction touches, sit at +3.15 and −3.15 per cent, a spread of 6.3 points. They meet at a ratio of 1.07, cations 0.845 and anions 0.906, the other way from the suggestion.

Tested in The diffuseness was not the anions' · the series on contour