Molecular Geometry & Bonding
what the shape decides
The collection

Every essay — page 17

Page 17 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

Orbitals

A one-electron wavefunction, drawn as a contour surface at a level somebody chose. Nodes, signs, and what the picture is a picture of.

A ninety per cent surface with a neighbour beside it. The contour enclosing 90 per cent of a one-electron ion's density at an effective charge of 1.6, drawn with no field as a circle and in a field of 0.05 atomic units as the closed curve. The surface moves out by 47.3 millibohr on the side the field pulls the density towards and in by the same amount on the far side — 2.84 per cent of its own radius. What the sphere encloses does not change to first order; only where the surface is does.

The surface a neighbour moves

An ion in a crystal sits in the field of the ion next to it, and that field moves its contour. The displacement has a closed form, it is checked against the polarisability it implies, and it turns out to be almost perfectly anti-correlated with the discrepancy it was proposed to explain — the pairs that need the most correction get the least.

7 figures 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.

6 figures 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.

6 figures 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.

7 figures 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.

6 figures 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.

6 figures 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.

6 figures The correction collapses and the error made assembling it grows. At a separation of 2 bohr, two quantities against the number of Gaussians a centre. The trimer's own counterpoise correction falls by a factor of 2717 from one function to six — a bigger basis has less to borrow. The fraction by which summing the pairwise corrections overshoots it rises from 5.3 per cent to 38.0. Improving the calculation makes the assembly proportionally worse.

The correction that gets harder to assemble

Summed across a trimer, pairwise counterpoise corrections come to fifteen per cent more than the trimer's own. Three Gaussians a centre is a small basis, so the natural question was whether the fraction shrinks with a better one or stays put. It does neither. The correction falls by three orders of magnitude and the fraction grows fivefold.

6 figures Both quantities, and the line they have to be read against. At the closest separation, the counterpoise correction itself and the error a pairwise assembly of it makes, against the number of Gaussians a centre. Both fall — the correction by a factor of 3631, the error by 793 — and the fraction rises by exactly the ratio of those two. The dashed line is a kilocalorie a mole. The only basis where the correction is above it and the error below it is two.

One basis size where it is worth doing

The pairwise assembly's error grows with the basis — 5.25 per cent at one Gaussian a centre, 37.98 at six — while the correction itself falls by a factor of three thousand. Which of the two should a practitioner care about? Drawing a line at a kilocalorie a mole answers it: there is exactly one basis size at which the correction is worth computing and its pairwise assembly is accurate enough to use.

6 figures 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.

6 figures 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.

6 figures 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.

6 figures