Simply false — page 5 of 5
Every claim that earns this verdict, in the form it is usually taught, with the computation that settles it. The other verdicts are indexed beside this one, and they are not interchangeable — which of them a claim earns is a statement about how it is wrong, and that is the part that generalises.
816 claims earn this verdict, 16 of them here, in the order of the essays that test them.
“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
“The icosahedron at four electrons is unaffected by re-orienting its cut shell because of its icosahedral symmetry.”
Its symmetry group has 120 operations and cannot carry an arbitrary plane of its three-fold shell onto an arbitrary other one. The reason is averaging: the twelve vertices are a spherical 5-design, so every polynomial of degree up to five has the same vertex sum however the cage is turned, and the Pipek–Mezey functional of a degree-one shell is a vertex sum of degree four.
Tested in The cage that averages like a sphere · the series on multicentre
“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 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
“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