The surface a neighbour moves

Where is an atom’s surface? The honest answer is a negative: there is no fraction at which a contour is the boundary of an atom, because the separation two ions come to rest at is not a property of either one’s density alone. That answer points in a direction. Stop asking where the surface is, and ask how much it moves when there is another ion beside it.

That is a computable question. A field mixes an s function with a p one, the mixing is the whole of a polarisability in a one-electron picture, and the displacement of a contour follows in closed form. It has been done here, and the answer is the second negative in a row — a sharper one than the first, because it comes with a sign.

The polarisation does not account for the additivity failure. It runs against it. Across the eight measured pairs, what each one needs and what the field supplies rank at −0.9524.

The response, in closed form #

For a one-electron ion, the first-order correction to the wavefunction in a uniform field is known exactly and is a polynomial times the unperturbed function:

$$\psi = \psi_0 \left(1 - F\,u(r)\cos\theta\right), \qquad u(r) = \frac{r}{Z^2} + \frac{r^2}{2Z}$$

with $Z$ the effective charge and $F$ the field. That is the whole of the physics in this essay. Everything below is that expression, integrated or root-found.

The check on it is that it implies a polarisability, and the polarisability of a hydrogenic 1s state has a closed form of its own: $\alpha = 9/2Z^4$. Integrating the perturbed density here to get the induced dipole must reproduce it, and it does — to a part in a million at five effective charges from 1 to 5.

That is a real check rather than a restatement, because the integration and the closed form share no arithmetic: one is a Simpson rule over the radial distribution and the other is a number. A wrong polynomial would give a polarisability that was wrong by a factor, and the factor would show.

It is worth being clear about what the mixing is, because it is the one piece of the argument that is a statement about shape rather than about arithmetic. An s function has no direction in it. Multiplying it by $\cos\theta$ produces something with the angular shape of a p function, so the perturbed state is an s function with a p function added — and the amount added is the polarisability. That is why an orbital’s own picture cannot show the effect: a contour drawn from the unperturbed function is a sphere, and the deformation lives entirely in a term that a picture of the free ion has thrown away.

What moves, and what does not #

Two things follow, and they point in opposite directions.

The surface moves. Solving for the contour at a fixed amplitude, angle by angle, gives a radius that is larger on the side the field pulls the density towards and smaller on the other, by

$$\Delta r = \frac{F\,u(r_0)}{Z}$$

to first order — a prediction checked here against a root-find on the full expression at fifteen combinations of charge and field, and agreeing to within eight per cent throughout.

The enclosed fraction does not. The correction to the density is proportional to $\cos\theta$, and $\cos\theta$ integrates to nothing over a sphere. So the same sphere that enclosed ninety per cent with no field encloses ninety per cent with the field on, to first order — verified by integration to twelve decimal places at five charges.

That asymmetry is the first result and it is a useful one. The convention every orbital picture here is built on — state the fraction the picture encloses — is blind to polarisation at first order. The convention it is most often confused with, a radius measured along a direction, is not. The two have been used interchangeably for a century, and here is one place where they part company cleanly.

The bridge to a real ion, which is one number #

Everything above is a one-electron result and the ions in question are closed shells of eight or eighteen electrons. The bridge is a single stated choice: take the effective charge at which a hydrogenic function encloses ninety per cent of its density inside the same radius the real ion’s Slater shells do.

It is one parameter, it is fixed by a length rather than by the answer, and it is the whole of the correspondence. Fluoride comes out at $Z = 1.63$, sodium at 2.30, chloride at 0.95, sulphide at 0.79. Those radii are the ones computed from Slater shells, so the bridge inherits whatever they are worth and adds nothing of its own. Those numbers are not effective nuclear charges in any other sense and are not offered as such; they are the charge a one-electron ion would need to be the same size.

The implied polarisabilities are then outputs rather than inputs — 0.65 bohr³ for fluoride, 0.16 for sodium, 5.5 for chloride, 11.8 for sulphide — and their ordering is the one measured polarisabilities have, which is the only claim made for them.

What the neighbour actually does #

The field is the neighbour’s charge over the square of the measured nearest-neighbour distance. A point charge, which is the crudest thing it could be, and is stated as such: 0.052 atomic units in sodium fluoride, 0.126 in magnesium oxide, 0.028 in potassium chloride.

Each ion’s surface then moves towards its neighbour, both of them, so the two radii along the bond overlap by more than the tables say. The sum of the two displacements is a computed correction to additivity, with a definite sign and no fitting anywhere.

And the correction is useless, in a specific and informative way.

The pairs that need the most get the least, and the two that need the least are over-corrected by factors of two and of seven and a half. Ranked against each other the two columns come out at −0.9524, which is very nearly the worst a correction can do: it is not merely too small or too large, it is arranged in the wrong order.

Why it comes out backwards #

The mechanism is not mysterious once the two columns are read as what they are.

The shortfall is largest for the small hard ions, because a ninety per cent radius for a compact closed shell is a great deal smaller than the distance the ion actually keeps — most of the missing distance is repulsion between the two closed shells, which grows steeply and has nothing to do with where the ninetieth percentile of the density sits.

The polarisation is largest for the big soft ions, because it goes as the polarisability, which goes as the fourth power of the inverse effective charge. Sulphide’s is eighteen times chloride’s and seventy times fluoride’s on this model.

So the two quantities are indexed by opposite ends of the same axis. Softness makes an ion polarisable and it also makes its ninety per cent radius large, which is exactly what makes the shortfall small. Nothing about that is a coincidence to be explained away; it is the reason a single mechanism was never going to cover both ends.

This is the same shape of failure as an explanation with the wrong sign one field over, and the diagnosis is the same: a candidate mechanism that is real, computable, of a plausible size, and correlated the wrong way with the thing it was proposed for.

Where the expansion stops being an answer #

Four of the eight rows have a displacement too large to be called first order, and they are reported as such rather than quoted.

Sulphide in magnesium sulphide’s field has its ninety per cent surface displaced by 36 per cent of its own radius. A first-order calculation that returns a third of the radius has not computed a small correction; it has announced that the field is not a perturbation. Oxide in magnesium oxide is at 14 per cent and chloride in sodium chloride at 10.

That is worth keeping rather than discarding, because it says something about the physical picture rather than about the arithmetic. The soft anions in these crystals are not slightly deformed spheres. Whatever their density does in a lattice, it is not usefully described as a sphere with a small bulge, and any radius quoted for them — Shannon’s, Bondi’s, or an enclosure radius of this collection’s own — is a summary of something with a shape.

What is quoted, and what is computed #

Two things are quoted. The tabulated ionic and van der Waals radii, which are Shannon’s and Bondi’s, and the eight nearest-neighbour separations in the rock-salt structures, which are diffraction measurements.

Everything else is computed: the Slater screening for every shell of every ion, the ninety per cent radius by bisection on an integrated density, the equivalent one-electron charge by bisection on a closed form, the field from the neighbour’s charge and the measured distance, the perturbed amplitude, the surface by root-finding at a hundred and eighty angles, the induced dipole by integration, and the rank correlation.

The polarisabilities quoted above are outputs of the model, not measurements, and they are not compared with measured polarisabilities anywhere — a comparison would be a second essay and would need the many-electron response this one does not have.

What this cannot say #

One electron. The response of a closed shell is not the response of a hydrogenic function with the same size, and the difference is not a small one: the outer electrons of a real anion are held by a self-consistent field that itself relaxes, which is the whole content of a Sternheimer calculation and is absent here.

Only the outermost thing that matters. Every ion here is treated as one responding object, where a real closed shell has an inner core that barely responds and a valence shell that does most of it. The screening that separates the two is a separate calculation and is not used here, because the one-electron bridge has no room for two shells.

A point charge. A real neighbour is an extended charge distribution, and at these separations the two distributions overlap. The field an ion sits in is not uniform across it, so even the form of the perturbation is an approximation — a limitation familiar from molecules, where a molecule with no dipole still interacts through the terms a point model has no room for.

One neighbour. In a rock-salt structure each ion has six nearest neighbours in an octahedral arrangement, and by symmetry the fields they produce cancel exactly. So the uniform field used here is the field of one neighbour with the other five ignored, which is the right quantity for asking what one neighbour does and the wrong one for asking what the crystal does. A site with no field at it has no first-order polarisation at all, and the real deformation in a lattice is a quadrupolar one at second order.

That last is the sharpest limitation and it is worth stating plainly: the correction computed here is larger than the one a crystal actually applies, and it still fails to account for the shortfall. The negative result is therefore stronger than the calculation, not weaker.

Where two closed shells come to rest #

What a reader of radius tables should take from it #

An enclosure radius is not made unsafe by a neighbour. That was the worry to test, and the answer is that the convention survives — a stated fraction is a stated fraction whether or not there is an ion beside it, at first order. What does not survive is the assumption that the number can be read as a distance along a bond.

A polarisability and a radius are not two views of one quantity. They are indexed by opposite ends of the same softness axis and they enter a contact distance with different signs, so a table that ranks ions by one is not ranking them by the other. The isovalue nobody chose is the same warning at a different level: two conventions that agree on an ordering can disagree on everything else.

And the failure of additivity is still unexplained. Two candidates have now been priced and both refused: a single enclosed fraction, and polarisation. What is left is the repulsion, which is the only remaining term of the right size and the only one that grows in the direction the shortfall does.

What the calculation must satisfy #

The polarisability integrated from the perturbed density is 9/2Z⁴, to a part in a million, at five effective charges. This is the check that the closed-form response is the right one.

The surface moves by F·u(r₀)/Z, checked against a root-find on the full amplitude at fifteen combinations of charge and field, to eight per cent.

And it moves out towards the neighbour and in on the far side, which is a sign rather than a magnitude and is the thing a wrong sign in the perturbation would break.

A sphere’s enclosed fraction is unchanged by the field to first order, to twelve decimal places at five charges — the asymmetry between the two conventions.

Both surfaces move towards their neighbour in every pair, so the correction has one sign throughout.

The larger of the two ions is the one whose surface moves further, which is not the same as the anion: potassium’s eighteen-electron cation is larger than fluoride, and the claim is written about size rather than about charge so that it can catch a model that had confused the two.

The correction does not account for the additivity failure, at a rank correlation below 0.2. Measured: −0.9524.

And the refusal is a field of zero. With no neighbour there is no shift, both poles return the same radius to twelve decimal places, and the surface is the sphere it started as.

The size of an orbital has never been one number: the mean radius, the most probable radius and the ninety per cent radius disagree for every function, and each responds differently to a field. That is why the deformation here is quoted as a displacement of a stated surface rather than as a change in “the radius”.

Why an anti-correlation is the one outcome a confound cannot manufacture #

It is worth pausing on the shape of this result rather than its size, because of all the outcomes this calculation could have had, it is the only one that is safe.

A confounded set produces agreement. When size, charge and hardness all rise together, any candidate that rises with them ranks well, and a good rank is therefore weak evidence — several unrelated quantities would have produced it, including some that cannot be mechanisms at all.

Nothing about a confounded set produces systematic disagreement. A candidate that ranks backwards against the thing it was proposed to explain has failed in a direction no accident of the sample supplies: the pairs that most need a correction are receiving the least of it, and no rearrangement of which quantities happen to travel together turns that into support.

So the negative result here is more robust than a positive one of the same magnitude would have been, and it is worth recording as a refusal rather than as an absence of evidence. Polarisation is not a small contribution to the shortfall that better arithmetic might rescue. It is a contribution running the wrong way.

Still open: the octahedron, and the repulsion #

The obvious open question is the octahedron. The six neighbours’ fields cancel and their gradients do not, so a site in a rock-salt structure has no first-order polarisation and a second-order quadrupolar one — a deformation towards the six neighbours at once, which is a shape a sphere has no room for. That is computable with the same method and a different perturbation, and it would replace the overestimate above with the quantity a crystal actually applies. Whether it is larger or smaller than the shortfall is then a real question rather than a foregone one.

The nearer question is what is left when polarisation is ruled out. The shortfall is largest for the small hard pairs, it goes as something steeper than the polarisability, and a repulsion between two closed shells can be computed from the overlap of their orbitals. Fitting nothing and computing that repulsion for the eight pairs would say whether the missing distance is the repulsion’s, which is the answer the question of atomic surfaces keeps circling — the one a bond is not two atoms overlapping points at — and it would do it by the one route that stops drawing surfaces and starts pricing what happens where two of them meet.