The radius that was tabulated

Measuring how much of the interaction between two atoms lies outside the surfaces a picture of them draws finds that at a three ångström contact 36.9 per cent of the overlap integral is outside both ninety per cent contours. That measurement leaves out one term, the repulsion, and a picture showing both terms would need two surfaces, at different fractions, crossing somewhere.

This essay does that computation, and it turns out to answer a different and better question on the way. The place where the two terms balance is the distance at which two closed-shell atoms come to rest, which is usually taken from a table rather than calculated.

Two terms, and only one of them is computable here #

The repulsion is computed. Two filled orbitals facing each other put four electrons into a bonding and an antibonding pair, and the antibonding one rises by more than the bonding one falls — an asymmetry that comes from the overlap in the denominator of the secular determinant. It is the reason helium is repelled rather than indifferent, and it is the whole of the short-range term here. Four electrons therefore cost energy rather than gaining it, and twice that asymmetry is what the pair costs. Summed over the whole valence shell, because a closed shell is not one orbital.

The attraction is quoted. There is no dispersion in a one-electron model and there was never going to be: the correlated fluctuation of two charge distributions is exactly what an independent-particle picture cannot contain — the same boundary a mean field cannot get out of the way draws from the other side. So London’s leading term is used, $C_6 = \tfrac{3}{4}\alpha^2 I$, with the polarisability and the ionisation energy taken from measurement. What would be dishonest is to fit something and call it computed.

The result is a model with no length in it anywhere. The only fitted constant is the Wolfsberg–Helmholz $K$, which is dimensionless and is the same 1.75 everybody uses, so it can change every energy here and no distance.

Three atoms #

Helium is the easy case and would prove little on its own. Neon and argon have four valence orbitals apiece — an s and three p — and the repulsion has to be summed over all of them.

Doing it with the σ-facing orbital alone, which is the obvious first attempt, is wrong by a great deal: argon’s contact then comes out at a quarter of the observed distance, because a single 3p pointing along the axis has an overlap that passes through zero somewhere inside the contact region — the non-monotonic behaviour of overlap with distance — and the repulsion vanishes exactly where it should be largest. Four orbitals do not have that problem, because the perpendicular pair are monotonic and carry the sum.

The wells are half what they should be, consistently #

The distances are good and the depths are not. The computed wells are 0.52, 1.55 and 5.63 meV against measured values of 0.95, 3.64 and 12.34 — too shallow by factors of 1.84, 2.35 and 2.19.

That is reported rather than tuned, and the consistency is the useful part. A model carrying London’s leading term and nothing else — no $C_8$, no $C_{10}$, no three-body term, no correlation beyond a two-atom dipole–dipole fluctuation — should underbind, and it does, by about the same factor for every atom. A well depth is a difference of two large numbers near their crossing point, so it is the quantity most sensitive to everything that was left out; a contact distance is the position of that crossing, and a position is far more robust than a depth.

This is the general reason positions are easier than energies, and it has been visible elsewhere in this collection: a bond length out of a spectrum is a good number from a poor potential for the same arithmetic reason.

Which effective charges, and it matters #

The radial functions come from Slater-type orbitals, which need one number per shell: an effective nuclear charge. There are two published ways of getting one and they are not the same kind of thing. Slater’s rules are an arithmetic recipe on a configuration — the familiar screening recipe — and Clementi and Raimondi’s numbers are a fit to Hartree–Fock atoms.

The disagreement is one number: Slater’s rules give neon’s 2p an exponent of 5.85 and the Hartree–Fock fit gives 4.4532, thirty-one per cent lower. A more diffuse 2p reaches further, the overlap at a given separation is larger, the repulsion is larger, and the contact moves out. With 5.85 it does not move out far enough and the dispersion term wins all the way in.

That is worth stating as a general caution rather than as a detail. Slater’s rules are a good enough account of screening for a periodic trend and are used all over this collection for exactly that. They are not interchangeable with a fitted exponent when the answer is a length, because a length depends on where a function’s tail is and a tail is what a screening rule fitted to energies is worst at.

The surface two atoms actually stop at #

The original question is now answerable. Half the contact distance is a radius. What fraction of the outermost orbital’s density lies inside it?

Helium 99.726 per cent, neon 99.753, argon 99.379. Three atoms of very different size, agreeing to within four tenths of a per cent on a quantity that a picture would have to choose.

So the two surfaces are not near each other. A ninety per cent contour is a statement about where an electron is; a contact surface is a statement about where the next atom stops, and the second is very much larger. The gap between them is the whole of the last three tenths of a per cent of the density, which is a region that carries almost no probability and does almost all of the intermolecular work.

Why nearly the same fraction for three different atoms #

The near-constancy is not a coincidence and it is not deep. Both terms in the balance are steep functions of the separation — the repulsion goes as the square of an overlap that decays exponentially, the attraction as an inverse sixth power — so their crossing sits at a separation where the overlap has fallen to a particular small value, roughly the same value for every atom. A fixed overlap corresponds to a fixed number of decay lengths, and a fixed number of decay lengths corresponds to a fixed enclosed fraction.

So the constancy is a statement about the shape of an exponential tail rather than about noble gases. It is also why a tabulated van der Waals radius is such a serviceable quantity: it is nearly transferable for the same reason.

None of the usual measures of size is the one a contact distance defines, and the new one is far out on the tail: half a computed contact distance falls well outside the ninety-per-cent surface of the outermost occupied orbital. A radius that is a property of two atoms meeting is not a radius any single-atom quantity was going to produce.

What the fraction is not #

It is worth separating two statements that the last figure could be read as making, because only one of them is true.

True: at the separation where two of these atoms come to rest, a sphere of half that radius encloses about 99.4 per cent of the outermost orbital’s density.

Not true: an atom “is” that big. There is no size in this calculation at all — that is the point of it. What has been computed is a distance at which a balance holds, and the fraction is a way of expressing that distance in the units this site’s pictures use. Reading it back as a radius reinstates exactly the quantity the essay set out to avoid assuming, and how big an orbital is remains four different questions with four answers, to which this adds a fifth rather than settling any of them. A momentum-space measure runs backwards from all of them, which is the sixth.

The distinction matters most where the two conventions are used in one sentence. A crystallographic argument that two atoms are “in contact” is an argument about the first quantity; a picture drawn at ninety per cent is about the second; and a reader shown the picture and told the argument has been given two things that differ by three tenths of a per cent of a density and by a great deal of space.

Where the missing half of the well is #

A well shallow by a consistent factor is a diagnosis waiting to be made rather than a defect to be reported, and most of this one is locatable without changing the calculation.

The attraction here is a London term, which is a dipole–dipole interaction and falls off as the inverse sixth power. That is the leading term of a series, not the whole of it. Two atoms polarise each other’s quadrupoles as well as their dipoles, and their octupoles after that, giving terms in $R^{-8}$ and $R^{-10}$ with coefficients of their own. Truncating at the first one is standard, and near a contact distance it is not a small approximation.

The coefficients are known accurately for helium, so the size of the omission can be read off directly. At three ångström:

The dipole–dipole term is about seventy per cent of the attraction at the distance the atoms actually sit at, so including the next two multiplies it by 1.44. That accounts for the whole of the shallowest case in this essay’s range of 1.5 to 3.5, and it runs in the right direction for the contact distance too: a stronger attraction pulls the minimum inward, and every computed contact here is too long.

The share is also a function of distance, which is the part worth carrying. At six bohr the leading term is 73 per cent of the total and at 5.4 it is 68 — so the higher terms grow faster than the leading one as the atoms approach, and truncating the series is least defensible exactly where a contact distance is decided. A dispersion expansion is a long-range expansion being used at short range, and its error is not a constant factor that could be absorbed anywhere.

What that leaves unexplained is the top of the range. A factor of 1.44 does not reach 3.5, and the remainder is likely in the other approximation the London expression makes: it replaces a sum over all excited states by a single energy, and takes that energy to be the ionisation energy. That substitution is a bound rather than an estimate, and it errs by underestimating the attraction — again in the right direction, and this time by an amount that is not calculable from tabulated scalars.

So the shallow wells are not evidence that the balance being struck here is the wrong balance. Both known approximations run the same way, and their sizes are of the order of the discrepancy: the repulsion is computed and the attraction is truncated, so the minimum sits too far out and too shallow, consistently and in three atoms at once. A defect that is consistent across a set and explicable by an omission of a known size is a better outcome than agreement would have been, because agreement would have meant two errors cancelling.

What this cannot say #

One electron, and no correlation. The repulsion here is the Pauli cost of overlapping two filled orbitals in a one-electron model. Real short-range repulsion has exchange and correlation contributions this model does not contain, and the agreement to about a tenth is better than the model deserves.

The dispersion is a quoted leading term. $C_6$ from the Slater–Kirkwood form is an approximation to a quantity that is itself the leading term of a series. Both the approximation and the truncation push the same way, which is why the wells are shallow.

Three atoms is not a series. Helium, neon and argon are the atoms whose valence shells this site’s orbital library can build. Krypton would need 4s and 4p functions and xenon 5s and 5p, and the hydrogenic radial functions get worse as the principal quantum number rises.

And the balance is spherical. Two closed-shell atoms have no orientation. Two closed-shell molecules do, and their contact distance depends on which way they are pointing — which is the reason a van der Waals radius is only ever an approximation for anything but a noble gas, and is a difficulty this calculation does not have and does not address. The nearest thing to it is what holds a solid together, which puts four kinds of interaction on one scale and treats the weakest as a single number per pair.

What the check found that nothing else did #

One result belongs to the numerics rather than to the chemistry, and it is worth recording because it is easy to misread.

An overlap computed with one quadrature rule can be checked against a different rule — more points, a wider map — and any disagreement larger than $10^{-4}$ flags a problem. Extending the contact scan inward to three bohr makes that check fire on argon’s 3s at 7.7568.

The tempting reading is that one of the stored values is wrong. It is not: at that separation the two rules genuinely disagree, by $1.1 \times 10^{-4}$, because a compact 3s function’s radial node at three bohr is beyond what a sixty-point product rule resolves. The check is right, and the obvious explanation for it is wrong.

The repair is to start the scan outside the region where the atoms are inside one another — which is physically empty as well as numerically hard — and to read a disagreement between two quadrature rules as a statement about resolution before reading it as anything else.

The input to the repulsion term is the overlap between the two closed shells, and its exponential decay is what makes the balance sit where it does: a term going as the square of something exponential meets a term going as an inverse power, and they cross once. That single crossing is the contact distance, and it exists for the same reason a Lennard-Jones minimum does without either curve being assumed.

What was checked #

Every contact within thirteen per cent, with no length fitted, and each one a balance of a positive computed term against a negative quoted one.

The wells shallow by a factor between 1.5 and 3.5, for all three, which is the consistency that makes the shallowness a property of the model rather than of one atom.

Neon out by more than a fifth under Slater’s rules, and helium unmoved, and the exponents differing by more than a quarter — three claims that together locate the failure in one number rather than in the method.

And the enclosed fraction between a half and 0.999 for every atom, which is the loose bound that would have caught a fraction of one or of zero, either of which would mean the radius or the integral had gone wrong.

Still open: the anion, and two crossing surfaces #

The natural open question is the anion, where the contour convention breaks in the other direction. A fluoride ion’s outermost density extends much further than a neutral fluorine’s and its tabulated radius is much larger, so the same calculation would say what fraction a charged closed shell stops at — and whether the near-constancy above survives a change of charge is a question with a definite answer.

The nearer question is the one sidestepped here. What has been computed is the separation at which two atoms come to rest, and the original question was about a surface: two of them, at two fractions, crossing. The fraction that corresponds to the contact is now known, but the picture of two surfaces at different fractions has not been drawn, because it would need the repulsion’s spatial distribution rather than its magnitude — and that is not available here, for the same reason there is no picture of a ring current. What is available is the fraction of the overlap that falls outside a drawn surface, which is the same question asked about the attraction alone and is where this essay started.