True as far as it goes, and silent on what decides — page 3 of 3

“Triples with a gap in them matter less the wider the gap.”

Nothing more than four separations away carries more than about a per cent. But the triples with a gap carry a median of 0.42 of the three-body increment by size at four fragments, 0.82 at five and 1.13 from six on, because two one-gap shapes seen from one side carry +20.8 and −11.0 per cent and a two-gap shape carries −16.5 — the gap triples do not fall away, they take back what the consecutive ones overshoot.

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

“Spin polarisation and the one-electron leak onto the middle orbital add up.”

They have opposite signs and start at different orders. In the unequal radical at 5 eV the middle orbital's spin falls to its lowest at an end–end overlap of 0.043 and rises after, returning to zero at 0.198; at 2 eV it returns to zero at 0.148.

Tested in Spin below zero · the series on overlap

“A worst error ratio measures how badly a diagnostic fails to track the error.”

It measures that and the smallest comparable error on the grid. Seven of the eight references' worst pairs divide by an error under 0.025 — down to 0.0009 — at systems whose correlation energy happens to match the reference's; the one reference whose smallest error is 0.012 scores best.

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

“The median score is robust only to one bad system.”

It is robust to the choice of reference as well: the polarisation's median stays between 1.21 and 1.55 at all eight references, and every cheap candidate but the highest occupied level stays under two.

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

“Boron trichloride's mixed chlorine species separate the two fields by the frequencies they add.”

They add 0.06 cm⁻¹ at most, starting at the second power of the mass difference, a slope of 1.99, and far under what the harmonic model can promise. What separates the fields is the symmetry of each component: A₁ and B₂ are exchanged in every doublet of every mixed species. The frequencies were the wrong thing to compare.

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

“Once the separation of an s and a p level starts to fall in a field, it continues to fall until the next order sets in.”

For the upper m = ½ state from the thirty-fourth shell to the seventy-seventh, the local polarisability itself changes through the fine-structure crossover and passes that of ns₁/₂, at 57 V/cm on the thirty-fourth and 0.6 V/cm on the sixtieth. Those are second-order minima, reached with no hyperpolarisability at all.

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

“Evenly spaced in 1/T, a 20–300 K design has a constant of 1.36.”

That is true of a plain list of forty, and most of its distance from its limit is one point. A list counts the window's top as a whole temperature where an integral counts half, and an inverse design's top is its sparsest place. With half-weight ends it reads 1.243 at forty and 1.208 at eighty, closing on the 1.19 to 1.20 of the two other routes to the same weight.

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

“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, so that route fails. 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. The group does supply the averaging, by another route: it has no invariant harmonic of degree three or four.

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

“The polarisation tracks the size of the error well in the typical case, so it is a usable warning away from the worst.”

Its medians are what they were, but twenty-seven of the eighty-one systems have a polarisation of exactly zero with correlation energies from 0.0001 to 0.432, the entire range of the grid. No function of it can reach a largest error under 0.216, which is what one constant reaches with no reading at all.

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

“A heteronuclear bond's profile at π/R measures the orbital's polarity.”

It measures the mismatch between the two atomic contributions, which is a polarity plus a floor. At equal coefficients — no charge transfer at all — the depth is already 1.35 per cent of the peak for C–O, 1.63 for B–N and 10.2 for Li–F, and zero only when the two atoms are the same. The measure is monotone in the polarity in the chemical direction, and it has an offset rather than a zero.

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

“IF₇'s equatorial ring puckers because five domains around an equator are crowded.”

Crowded, yes — the ring's 72° neighbours make the pucker the arrangement's softest motion by a factor of a hundred under 1/r. But with IF₇'s own bond lengths the flat ring is a minimum under 1/r, 1/r² and 1/r³, and its pucker is three times stiffer than with equal bonds. Crowding makes the pucker cheap; it does not make it happen.

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

“A σ bond energy depends on the geometry of the ligands.”

Not at second order in the coupling: that term is the trace of the Gram matrix, which is the number of ligands whatever their directions. Geometry first enters at fourth order, and there only through the ligands' net direction and their departure from a tight frame.

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

“The octahedron is the σ-bonding optimum for six ligands, distinct from every other arrangement.”

It is an optimum and it is not alone. A trigonal prism with its ligands at the tetrahedral magic angle shares the octahedron's summed direction and second moment, and in a σ-only model those two moments fix every level: the two shapes have the same energy at every coupling, to 10⁻¹³.

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

“With the neighbour repulsions π-electron theory uses, the on-site repulsion the allyl radical needs is carbon's own, 11.13 eV.”

With a hopping of 2.34 eV — Pariser and Parr's β, from carbon's overlap and energy — Mataga–Nishimoto neighbours need 9.49 eV and Ohno's need 12.68. Pariser's 11.13 lies between the two standard forms rather than on either; Mataga–Nishimoto reaches it only with an orbital energy of −14.4 eV.

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

“Ignoring the overlap between allyl's two ends costs nothing, since it is small.”

The ends' own overlap of 0.032 more than halves the coupling between them that orthogonalisation creates — from 0.60 to 0.29 eV with carbon's energy — but it moves the repulsion allyl needs by only 0.12 eV, a little over one per cent.

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

“The σ-only model ranks five-ligand shapes by how close each comes to the sphere's averages, and the trigonal bipyramid comes closest.”

No five-ligand shape reaches the sphere's averages, so there is no single 'closest': the energy trades a net direction against a frame excess, and the trade is set by q, the ratio of the net-direction coefficient to the frame coefficient. Below q = 0.862 a square pyramid is the floor and the bipyramid is not.

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

“The floor of a polar bond's minimum is an artefact of a minimal basis that a more flexible basis would remove.”

A second s function on carbon lowers it — to 0.89 per cent with an exponent of 3.5 and to 0.06 per cent at 4.25 — but the best mixture is almost entirely the second function each time, and the floor vanishes only when carbon's function is oxygen's. What falls is the difference between the atoms, not a residue of the basis.

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

“A repulsion model with a lone pair as one more domain cannot represent xenon hexafluoride's distorted octahedron.”

With the domains alone it cannot: the pentagonal pyramid is the lowest structure at every lone-pair weight from 1.05 to 3. With a contact repulsion between the six fluorines it can: the capped octahedron takes over at 1.0, 2.3 and 4.4 per cent of the domain energy for weights of 1.2, 1.5 and 2 under Born–Mayer contacts, and at 0.4 to 1.8 per cent under fluorine's own 2p overlap. A stereochemically inactive lone pair remains outside it.

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

“Adding a repulsion between the ligands to the σ-only model can account for main-group square pyramids such as pentaphenylantimony's.”

Only a repulsion steeper than about the fifteenth power of the ligands' separation can. For 1/r, 1/r⁶ and 1/r¹² no weight up to a million times the bipyramid's σ cost gives a square pyramid at any σ parameter from 1 to 5; for 1/r¹⁶, 1/r²⁰ and 1/r³⁰ every one switches, at shares of 1,100, 230 and 100 at q = 2.

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

“Each angular ring between two rings of a fused chain multiplies their ring-current response by about 0.60, a property of the angular ring.”

Only on nine rings. On every chain of seven, nine and eleven rings with up to two angular rings, runs of three and both uniform chains, the factor is 0.656, 0.593 and 0.542, and the factor for an angular ring under the pair 0.681, 0.630 and 0.587. Carried from nine rings to eleven, the factor misstates a pair spanning three angular rings by a factor of 1.31.

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

“The twenty-seven nine-ring chains chosen for the first fit fixed the factors.”

Every nine-ring chain of the same kinds, thirty-five in all, gives 0.593 and 0.630 against the chosen set's 0.602 and 0.655 — close, but the factor under a pair moves by 0.025 with the choice of chains alone.

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

“A steep contact switches a main-group five-coordinate centre to a square pyramid at a share of about a hundred to two thousand times the bipyramid's σ cost.”

That is the fourth-order σ energy, which the exact energy reproduces only while the coupling is small. At β = 1 the exact shares are 1.0 to 185 and at β = 1.5 they are 0.3 to 73, because the bipyramid's exact σ advantage grows as the fourth power of β only up to about β = 0.1 and nearly linearly beyond — at β = 1.5 it is 0.65 per cent of the fourth-order value.

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

“The contact needed rises with the σ parameter q, so centres whose s and p levels are further apart need bulkier ligands.”

At fourth order the share rises 2.5- to 2.8-fold from q = 1 to 5. Measured as a fraction of the σ bond energy at β = 1.5, the contact needed is 162 to 173 per cent under 1/r¹⁶, 33 to 35 under 1/r²⁰ and 14 to 15 under 1/r³⁰ — set by the steepness and nearly independent of q.

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

“In the seven-domain repulsion model, only a contact between xenon hexafluoride's fluorines can move its lone pair off the axis into a face.”

Only for lone-pair weights above one, which is where the earlier survey looked. With the domains alone the capped octahedron is the lowest structure for every weight below 0.657 — a lone pair repelling less than a bond needs no contact to sit in a face.

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

“Between the capped octahedron and the pentagonal pyramid, the edge-capped structure is a saddle the lone pair passes over.”

With the domains alone it is a true minimum from a weight of 0.601 to 1.028, and the lowest structure of all from 0.657 to 1. The face and the edge are both minima between 0.601 and 0.821, so the change between them is a crossing of two minima rather than a slide.

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

“The crossing between oxygen and fluorine holds for any hybridisation of the heteroatom.”

It holds at every bond length tried, 1.15 to 1.334 Å, and for heteroatom s fractions from 0.15 to 0.5. At an s fraction of 0.7 the fluorine member is refused as well, at 0.941, and at 0.15 the oxygen member comes within six per cent of mixing.

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

“Where a polar σ bond's profile has a minimum, the minimum is the interference fringe at π/R.”

A bonding combination of lithium 2s and hydrogen 1s has a minimum only for hydrogen shares below 0.45, and it sits between 0.525 and 0.600 π/R — just below lithium's 2s node in momentum at 0.624 π/R, nowhere near the fringe. The same holds for beryllium and boron. Only from nitrogen on, where the node lies above π/R, does the minimum move onto the fringe.

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

“Where a heteroatom sits on a ring changes how much of the ring responds.”

On a CH carbon of an angular ring the ring's own response stays within eleven per cent of the straight ring's, and what changes is what leaves it: every other ring together receives 0.47 to 0.63 of the straight chain's, against 1.05 to 1.36 from the same ring's junctions.

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

“Lengthening the three bonds round the opened face would leave a weak lone pair's small distortion where it is.”

It narrows it, by about a third of a degree — at a weight of 0.3 the spread of the cis angles falls from 14.24° to 13.87° at the MP2 ratio of 1.972 to 1.856 — but the model does not choose it. The placement it prefers, long bonds opposite, widens the spread to 14.80°.

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

“Letting the bond lengths respond to the repulsion settles which of xenon hexafluoride's three bonds are long, where fixing two lengths by hand could not.”

It settles it only once the lone pair's distance is specified, and the model has no rule for that. At a weight of 0.5 and bonds stretched by about 8%, a lone pair kept at the mean bond's distance puts the long three opposite the face (−0.010%), one left where it was puts them round it (+0.084%), and one on a bond's spring also round it (+0.027%).

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

“A cut three-fold shell can move the answer unless it is the cage's coordinate functions on a cage of high enough design strength.”

The condition is sufficient and not necessary. The truncated icosahedron's shell is not its coordinates, its measured polynomial degree exceeds four, and it does not move; nor does a stretched icosahedron, nor a cage at a = 1.3 under a matrix tuned to keep only tetrahedral symmetry with inversion.

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

“The sharp minimum in the six pairs' spread at equal scales shows the scale is common to every ion.”

Those contractions had been fitted at equal scales. Refitted at each ratio, the spread stays between 1.14 and 1.44 points for every ratio from 0.95 to 1.1. The data exclude one charge alone and do not pick out one.

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