Molecular Geometry & Bonding
what the shape decides
The collection

Every essay — page 40

Page 40 of 42, continuing through the fields in the same order.

Orbitals Where the atoms go Bonding models What symmetry decides Beyond the octet What a spectrum settles When the molecule does not stop What the shape is for What is taught wrongly Series Named objects Orbitals Refutations Search

What is taught wrongly

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

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

The third way to be an insulator

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

7 figures 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.

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

The same overlap, a different bond

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

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

The metal a thermometer cannot find

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

5 figures 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.

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

A mean that is low rather than right

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

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

The warning a cheap calculation gives

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

7 figures The carriers a distortion was hiding. A half-filled ring of 40 — one of the 4m rings that carry exactly one pair of carriers at every temperature — allowed to distort. Cold, it alternates by 0.1232, opens a gap of 0.4927 and carries 3.6e-15 carriers rather than one pair. The alternation is undone continuously at kT = 0.1358, and the carrier count comes back as it goes.

The carriers a distortion was hiding

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

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

The exponent was the window's

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

7 figures 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.

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

A regime that belongs to the neighbours

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

5 figures 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.

6 figures