The tenth that is not drawn

An orbital picture drawn honestly says what fraction of the density it encloses, and the choice of fraction matters: to the size of the drawing, to comparisons between orbitals, to whether a bond is drawn as one closed surface or two.

This essay is about the part that is left out. A ninety per cent contour encloses ninety per cent of the electron and leaves a tenth of it outside, and the tenth is not drawn at all — there is no fainter shading, no dotted line, nothing on the page where it is.

The question is what that tenth is doing, and the answer is that it is doing most of the chemistry between atoms that are not bonded to each other.

Where the tenth is #

For a hydrogen 1s orbital the ninety per cent contour is a sphere of radius 2.6612 bohr, at an amplitude of 3.9366 × 10⁻². That is 1.408 ångström, which is already larger than most bonds.

The remaining tenth is spread from there outwards, and it occupies a great deal of room. The ninety-nine per cent surface has a radius of 4.203 bohr and the ninety-nine-point-nine per cent one 5.613. In volume: 79.1 cubic bohr inside the drawn surface, 310.9 inside the ninety-nine per cent one, 740.6 inside the last.

So the ninth of the electron between ninety and ninety-nine per cent occupies two hundred and thirty-two cubic bohr — nearly three times the volume of the whole drawn picture — and the thousandth between ninety-nine and ninety-nine-point-nine occupies four hundred and thirty more.

Holding very little is what makes it invisible and does not make it unimportant, because what happens out there is not weighed by how much density is present. It is weighed by an integral over the product of two atoms’ amplitudes, and the second atom is out there.

Two surfaces that stop touching #

Take two hydrogen atoms and pull them apart, drawing each at ninety per cent.

The two spheres are in contact while the separation is below twice the contour radius, which is 5.3224 bohr — 2.816 ångström. Beyond that the picture shows two objects with a gap between them.

The overlap integral behind that picture does not notice. It is 0.753 at a hydrogen molecule’s bond length, 0.0768 at 5.322 bohr where the drawn surfaces lose contact, and 0.0222 at seven bohr — falling smoothly, with nothing whatever happening at the separation where the picture changed. Overlap decides plots the whole curve.

And 5.322 bohr is not a fact about two hydrogen atoms. It is a fact about the number ninety.

An overlap of 0.077 is not a small quantity in this subject. It is larger than the overlap that produces a hydrogen bond and about the size of the overlaps that hold molecular crystals together. Two orbitals with that overlap have a splitting of several tenths of an electronvolt in any two-level treatment, and this collection’s whole account of bonding is built on the claim that overlap is the quantity that decides.

So at the separation where the picture stops showing an interaction, the interaction is well developed, and it is measurable for another few bohr after that.

How much of the integral is outside the drawing #

That comparison can be made exact rather than rhetorical.

The overlap integral is $\int \phi_A \phi_B \,\mathrm{d}V$ over the whole of space. Every point in space is either inside one of the two drawn surfaces or outside both, so the integral splits into two pieces on exactly the criterion the picture is drawn on: a point counts as drawn if either atom’s amplitude there is at or above the level its own contour is drawn at.

The two halves of the split add back to the closed form for the overlap of two 1s functions, $e^{-R}(1 + R + R^2/3)$, to two parts in a thousand, which is what makes the split believable rather than merely plausible.

The numbers: 8.3 per cent outside at a hydrogen molecule’s bond length, 11.3 at three bohr, 16.6 at four, 31.4 at the separation where the surfaces part, 36.9 at a three-ångström contact and 53.4 at seven bohr.

The trend is the finding. A fixed fraction would make this a constant correction that anybody could carry in their head. A fraction that rises without limit means the picture and the interaction diverge as the atoms separate — which is to say, the pictures are least misleading exactly where they are least needed.

The contour that would contain it #

Turn the question round: how large would the surfaces have to be for nine tenths of the overlap to be inside them?

At a hydrogen molecule’s bond length, a contour enclosing 87.8 per cent — slightly smaller than the usual one, because at that separation the two atoms are inside each other and the overlap density is concentrated between the nuclei.

At a three-ångström contact, 97.28 per cent, at a radius of 3.56 bohr.

At seven bohr, 98.76 per cent, at 4.07 bohr.

Those are not drawings anybody would make. A sphere of 3.56 bohr around each atom of a molecule overlaps every neighbour, hides the structure, and makes a page of them unreadable. The convention that draws ninety per cent is not arbitrary; it is the largest fraction whose surfaces are still separate enough to see.

The same tension appears in the choice of level rather than fraction, which has a trade-off of its own: a level chosen to make one orbital look right makes another look wrong, and there is no level that serves both. Here there is no fraction that serves both purposes for the same orbital.

So no single contour can be a picture of where the electron is and a picture of where the interaction is. That is the whole of the essay’s claim, and it is not a criticism of the convention: it is a statement that two different pictures are needed and only one of them is ever drawn.

The claim is easiest to believe by running the whole measurement again at a different fraction, since a convention that could be repaired by a better choice of number would show it here.

What a chemist draws instead #

There is an existing picture for exactly this region, and it is worth naming because it is a different object and is usually drawn on the same page.

A van der Waals surface is a sphere of the tabulated van der Waals radius — 1.20 ångström for hydrogen, 1.70 for carbon — and it is a contact radius, fitted to how close two non-bonded atoms are found in crystal structures. Hydrogen’s is 2.27 bohr, which is smaller than the ninety per cent contour of a hydrogen 1s at 2.66.

So the two pictures a chemist has for an atom’s extent — the orbital contour and the van der Waals sphere — are different sizes, are built from entirely different things, and neither of them is drawn at the size where the interactions between non-bonded atoms are computed. The tabulated radius comes from where atoms stop, which is set by the repulsion between filled shells; the contour comes from where the density is; and the overlap that produces attraction is largest well outside both.

There is a third convention available and it does not help either. Drawing several orbitals at one common level rather than at one common enclosed fraction ranks them differently — a side-by-side comparison draws 1s, 2s and 2p both ways and the two orderings disagree by a factor of several in apparent size. So three conventions are in use, they disagree with each other, and none of them is drawn at the size where the interaction between non-bonded atoms is computed.

The interactions that live out there #

It is worth naming what actually happens in the region the picture omits, because the list is most of the chemistry that is not a bond.

Hydrogen bonds. An O···H contact of about 1.9 ångström is 3.6 bohr, well outside a ninety per cent contour of either atom’s valence orbitals, and the overlap that produces the donor–acceptor interaction is computed in exactly the region this essay has been measuring.

Van der Waals contacts and dispersion. Two closed-shell atoms at their contact distance have no net charge transfer and attract anyway, and a molecule with no dipole at all still interacts through the higher moments and through the correlated motion of the electrons in each. Everything in that argument happens outside every contour anybody draws.

Non-bonded contacts inside one molecule. Two atoms on opposite sides of a ring approach each other without any bond between them, and what stops them is the same repulsion; the geometry of medium rings is decided by it.

And the approach to a reaction. A reagent’s first interaction with a molecule is at a distance where nothing is drawn, and the shapes usually offered to explain orientation — the frontier orbitals, drawn at ninety per cent — are being read outside their own surfaces.

Every item on that list involves an orbital that reaches further than a 1s. A 2p’s radial function has its maximum further out and its tail further still, so it pushes more of every interaction into the undrawn region than the case measured here does — and nothing in the argument is special to the compact case. The compact case is where the omission is smallest.

None of that is an argument for drawing bigger surfaces. It is an argument for reading a contour as what it is — a statement about where the electron is — and not as a statement about where the atom can reach, which is a different question with a different answer.

The undrawn tenth is also the least reliable tenth #

There is a second thing wrong with leaving the tail out of the picture, and it cuts the other way from everything above. The region doing the chemistry is not only the region that is not drawn; it is the region where the function being drawn is least trustworthy.

Every orbital in this collection is hydrogenic, which fixes its decay as $e^{-Zr/n}$ with whatever charge is used. That is exact for one electron and a nucleus. It is the wrong form for anything else, and the failure is specifically in the tail — because the far reaches of a many-electron atom’s density are not governed by the nuclear charge at all.

What governs them is the first ionisation energy. Far from the nucleus, the last electron is moving in the field left behind by an ion, and the rate at which its density dies away is set by how much energy it would take to remove it entirely: the wavefunction falls as $e^{-\sqrt{2I}\,r}$, with $I$ the ionisation energy in hartree. For hydrogen that is $\sqrt{2 \times 0.5} = 1$, and the hydrogenic form is recovered exactly, which is why nothing in this collection has had to notice.

For anything else the two disagree, and the direction is instructive. Helium’s ionisation energy of 24.59 electronvolts gives a decay constant of 1.344; neon’s 21.56 gives 1.259; argon’s 15.76 gives 1.077. Across a group whose nuclear charge runs from 2 to 18, the quantity that sets the tail varies by a quarter. A screening rule applied to a hydrogenic function would have produced nothing like that, because it works from the charge and this does not.

Which sharpens what the earlier sections measured. The share of the overlap lying outside the drawn surfaces was computed for hydrogen, where the tail is exact, and the arithmetic is sound. Applying the same conclusion to a heavier atom means using a tail whose decay constant is wrong by whatever the difference between $Z_{\text{eff}}/n$ and $\sqrt{2I}$ happens to be — and getting it wrong in the exponent of an exponential is not a small error at the distances this essay is about.

So the honest form of the finding has two clauses. The undrawn tenth is where the interaction between atoms lives, which is the argument here and is general. And a hydrogenic tail is the wrong shape for it in every atom but one, which is the standing limit of hydrogenic functions arriving in the one place where it does the most damage — and the repair needs no calculation, only a measured scalar that has been tabulated for every element since the 1920s.

Where the model stops #

This is one electron on each centre. A real pair of atoms has filled shells whose overlap produces repulsion, and the balance between that repulsion and the attraction the overlap above produces is what sets a contact distance. Nothing here computes that balance; what is computed is one of the two terms in it.

A surface is not a slice. Everything here is computed in three dimensions, and a printed contour is a section through the surface — which encloses a different fraction on the page than the surface does in space. The shares above are the surface’s.

The split is made on the drawn surfaces, not on physics. The criterion “inside one of the two contours” is a fact about the picture. There is no boundary in space at which anything happens, and the point is precisely that the picture implies one.

The contour is solved for the density of one orbital, not of an atom. A many-electron atom’s outermost shell sits inside the density of everything below it, and a filled shell is spherical whatever its parts look like, so the surfaces a chemist draws for a real atom are already a different object from the one computed here.

It is a 1s orbital and 1s orbitals are the compact case. A 2p reaches further, so the same computation on a pair of 2p orbitals would put an even larger share of the overlap outside the drawing.

Drawing the single orbital at ninety-nine per cent instead of ninety would make it 58 per cent larger in radius, which is the honest picture for a non-bonded argument and is also why nobody draws it: on a page with a second atom on it, that surface would already be touching it.

The stated-fraction convention, and what it does not fix #

Stating the enclosed fraction fixes the comparison between two pictures and does not fix this.

Two orbitals drawn at the same fraction are comparable: the surfaces mean the same thing about each, so relative sizes and shapes can be read off. That was the point of the convention and it holds. What it cannot do is make the surface a boundary of the atom, because the density does not stop there and the fraction chosen is a matter of taste — the same orbital’s surface at half, ninety and ninety-nine per cent differs by a factor of three in radius.

The addition worth making is one sentence rather than a new kind of picture: say what the surface leaves out where it matters. A figure about a bond, at bonding distances, is missing under a tenth of the interaction and needs no caveat. A figure about a contact, an approach or a non-bonded repulsion is missing between a third and a half, and the caption is the place to say so.

What is quoted, and what is computed #

Two numbers are quoted: the van der Waals radii of hydrogen and carbon, which are fitted to crystal structures and are measurements of a kind.

Everything else is computed here. The contour radii come from inverting the closed-form enclosed probability of a 1s orbital, which is checked against the site’s general contour solver — the one that bisects on a three-dimensional integral — at four fractions, agreeing to three parts in a thousand. The split overlaps come from a cylindrical quadrature over the whole of space, checked against the closed-form overlap. The fractions needed to contain nine tenths of the overlap come from a bisection on that split.

What the argument requires #

The split integral adds back up to the closed form at every separation, to two parts in a thousand. Without this the split is arithmetic rather than a measurement.

The share outside the drawn surfaces rises with separation, at every step. A constant share would make this a correction rather than a finding.

At the separation where the drawn surfaces stop touching, the overlap is still several per cent — 0.0768 — which is what makes the divergence between picture and interaction matter.

And the refusal: at a contour enclosing 99.99 per cent, almost none of the overlap is left out at any of these separations, below two per cent everywhere. That is what makes the rise a property of the ninety per cent convention rather than of the integral.

Still open: the repulsion #

The stated-fraction convention has now been taken as far as one orbital pair will carry it, and the honest next step is the term it leaves out: the repulsion.

Two closed shells at a contact distance repel, and the repulsion comes from the same region of space and the same integral — it is what the Pauli principle does to two overlapping filled shells, and its magnitude goes as the square of the same overlap rather than as the first power. A picture that showed the two terms together would need a surface for each, at two different fractions, and the interesting thing about that picture is that the two surfaces would cross: attraction dominates outside, repulsion inside, and the crossing is the contact distance a van der Waals radius is fitted to.

That is a feasible computation, and it would turn a tabulated radius into a computed one — the same move that turns a drawn contour into a solved one.