A slice is not the surface

The first rule of an honest orbital picture is that it should say what fraction of the probability it encloses, and that the claim should be checked rather than assumed. Every figure here obeys it: the level is found by integrating the density inside a candidate contour and bisecting until the enclosed fraction is the one the caption states, and the integral is then run again on the level that is drawn.

There is a second question underneath that one, and it has been unasked here for as long as the rule has existed. The page is flat. What a reader sees is not a surface; it is either the outline of a surface projected onto paper, or — far more often, in every textbook printed — a contour of the wavefunction in one plane through the nucleus. Those are different objects, and the fraction each of them encloses is a different number.

Two honest captions for one curve #

Take a 1s orbital and find the level whose isosurface encloses ninety per cent of the density in space. That level is $\psi = 3.937 \times 10^{-2}$, and the sphere it draws has radius 2.663 bohr. So far this is the ordinary computation.

Now ask a different question of the same level: of the density in the plane of the page, how much lies inside the curve? The answer is 96.9 per cent.

Both numbers are true. Neither is a mistake. The caption under the printed picture says ninety per cent and does not say which.

Why it cannot be otherwise #

The reason is the measure, and it is short enough to give in full. In three dimensions the probability in a shell between $r$ and $r + dr$ is $|\psi|^2 \, 4\pi r^2 \, dr$; in a plane through the nucleus, the probability in an annulus of the same thickness is $|\psi|^2 \, 2\pi r \, dr$. The volume integral weights the far tail by an extra factor of $r$.

So the ten per cent that a three-dimensional surface leaves outside it is largely a thin shell at large radius, where there is a great deal of volume and very little area. Slice through it and most of that shell is simply not in the picture. The curve looks the same; what it has left out has shrunk.

For the 1s this can be done in closed form. The in-plane fraction inside a circle of radius $r$ is

$$1 - (1 + 2Zr)e^{-2Zr},$$

which at $r = 2.6625$ bohr gives 96.921 per cent — and 2.6625 bohr is the ninety per cent radius quoted throughout. The numerical quadrature agrees to better than two parts in ten thousand, which is the check on everything that follows — the same discipline the overlap integrals get, where one case with an exact answer turns the integrator from something to be trusted into something to be tested.

The number is nearly the same for every orbital #

The striking part is not the size of the gap but its constancy.

A 1s and a 4s differ enormously in size, in the number of shells the surface has, and in how much of the density lies far out. A 3d has angular nodes and a 2s has a radial one, and a filled shell of p orbitals has no shape at all. None of that moves the answer by more than a point and a half. That is the signature of a result which comes from the measure rather than from the shape — the factor of $r$ is the same factor whatever the function is.

The one orbital that departs slightly is the 4s, at 95.4 per cent, and the direction is right: it is the most diffuse of the set, so the shell left outside is the widest, and even in two dimensions some of it survives.

The level that would make the caption true #

The other direction is the one that matters for anybody drawing. Given that the page is flat, what level should the contour be drawn at so the ninety per cent claim is true of the picture?

It is a larger level, and therefore a smaller curve. For the 1s it is $8.068 \times 10^{-2}$, a factor of 2.05 above the surface-honest one, and the circle it draws has radius 1.945 bohr rather than 2.663 — twenty-seven per cent smaller in radius, and less than half the area.

The ratio between the two levels is not a constant either: 2.05 for a 1s, 1.66 for a 2s, 1.87 for a 2pz, 1.81 for a 3dz². So there is no fixed correction to apply. Anyone wanting the caption to be true of the picture has to solve for a different level, per orbital, in the plane the picture is drawn in — which is a computation nobody does, because the ambiguity is not usually noticed.

At other fractions, and why ninety is the worst case #

Ninety per cent is a convention rather than a principle, and the gap depends on it.

The gap is largest in the middle of the range, which is exactly where the conventional choice sits. At fifty per cent the surface is well inside the density and the tail it excludes is enormous in both measures; at ninety-nine per cent almost nothing is excluded either way. Ninety per cent is chosen because it is the largest fraction at which the picture is still compact enough to be a shape, and that is the same condition as the largest fraction at which the excluded shell is still substantial.

Which of the two a reader is looking at #

There is no general answer, and that is the honest position. Three cases:

A contour plot in a plane. The overwhelming majority of printed orbitals, including almost every picture of a molecular orbital in a textbook: a set of nested curves in one plane, labelled with values or with a single chosen contour. Here the two-dimensional number is the relevant one, and it is the one never quoted.

A projected isosurface. The pictures produced by visualisation software and by three-dimensional orbital drawings generally: a surface rendered with shading, so the reader is seeing an outline rather than a slice. Here the three-dimensional number is the relevant one — and even then what the eye takes from the picture is the silhouette, which is the projection of the widest part rather than a slice through the middle.

A cross-section of a rendered surface. The hybrid case, drawn as a solid but cut open. This one is genuinely a slice, and it inherits the two-dimensional number while looking like the three-dimensional one.

The distinction is not academic when several orbitals are drawn on one plate for comparison, because the correction from surface to slice differs between them. Two orbitals whose surfaces enclose the same fraction do not have curves enclosing the same fraction in the plane, and a reader comparing sizes on the page is comparing something with no single definition behind it.

What visualisation software does #

The programs chemists actually use draw isosurfaces, and the fraction is not among the quantities they report. What is typed into them is a raw isovalue — 0.02 or 0.05 in atomic units are the defaults that have propagated through three generations of program — and what such a value encloses varies from under one per cent to over ninety-six depending on the orbital. That is the first ambiguity, and it is a much larger one than this essay’s.

This essay’s is the second, and it survives even when the first is fixed. Suppose a program were changed to accept an enclosed fraction and solve for the level. It would still have to decide whether the fraction meant the surface or the slice, and the two differ by six points at the conventional value. A program that solved for one and rendered the other would be exactly as self-consistent, and exactly as wrong, as the pictures printed now.

The reason to state which is not pedantry about a few per cent. It is that the number is the only thing tying the picture to a computation. An orbital is not a region the electron is inside; it is a function, and drawing a boundary on a function is an act of interpretation that a stated fraction makes reproducible. A fraction that names two quantities has not made it reproducible.

The measure that has no ambiguity #

Behind both numbers there is a quantity that does not depend on how the picture was made, and it is worth naming because it is the thing the other two are proxies for: the radial distribution, $r^2 |R(r)|^2$, which is the probability of finding the electron at a distance rather than at a point.

This is why the most probable radius is not where the density is largest, and it is the same distinction at one remove: a density is one thing, a density times a measure is another, and a picture always shows the first while a caption almost always describes the second.

What the check was made to refuse #

The claim that the in-plane fraction exceeds the volume fraction sounds like a theorem and is stated here as a measurement, on purpose. The argument from the volume element is sound for a monotonically decreasing function; an orbital with a radial node has density at two separate ranges of radius, and the contour cuts both, so the shell-by-shell bookkeeping is not the simple picture the argument assumes. Every orbital tested does come out above, and the check records that it was tested rather than deduced.

The refusal that gives the measure its teeth is a different one. A plane on which the orbital vanishes identically — the xy plane of a 2pz, say — contains no density at all, and asking what fraction of nothing lies inside a curve is asking for the result of dividing nothing by nothing. The routine returns zero rather than a fraction, and the check requires it to, because a figure drawn from a fraction computed there would be a picture of arithmetic noise carrying a confident caption. The plane each orbital is sliced in is therefore chosen to contain the orbital’s own maximum, and that choice is part of the figure rather than a detail of it.

Where the two readings diverge most #

Everything above is a one-centre calculation, and the effect has a natural way of growing. A molecular orbital spread over two or more nuclei has density between the nuclei and outside them, and a plane containing the internuclear axis samples the bonding region generously while a volume integral gives weight to a shell surrounding the whole molecule. The correction should therefore be larger for an extended orbital than for an atomic one, and the 4s result — the most diffuse orbital tested, and the one furthest from the others — is the first hint of it, in the opposite direction from what a casual reading would suggest.

The reason it went the other way is worth the sentence, because it shows what the correction depends on. A 4s has three radial nodes, so its outermost shell is a wide, low-amplitude region. Slicing through it does capture some of that region, because the shell is wide enough to have area as well as volume. The correction is smallest where the excluded density is spread out and largest where it is a thin skin.

The table above says the same thing across the orbitals rather than inside one of them: the two columns never agree for any of the seven, and they are furthest apart where a radial node puts a shell inside a shell — because a plane through the nucleus cuts every shell while a surface encloses only the outermost.

The third object, which is what a rendered figure actually is #

Two objects have been compared: the surface in space and the contour of the wavefunction in one plane. The opening paragraph names a third and then leaves it alone — the outline of a surface projected onto paper — and it is the one a modern rendered figure actually shows. A drawing of a shaded three-dimensional isosurface presents the reader with its silhouette, and the silhouette encloses a third fraction.

The quantity is well defined and needs no new calculation. A silhouette is the shadow of the surface, so the region it bounds on the page is the shadow of a cylinder: all the density whose distance from the viewing axis is less than the surface’s widest extent, at any depth. Integrating the density over that cylinder rather than over the sphere gives what the outline encloses.

For hydrogen’s 1s at the ninety-per-cent level — a sphere of radius 2.6612 bohr — the cylinder of the same radius holds 94.82 per cent. For the 2s at its own ninety-per-cent radius of 9.1254 bohr it holds 95.46.

So one picture supports three true captions, and they are ordered:

The ordering has a reason and it is the same one throughout. Each step discards a dimension of exclusion. The surface excludes everything outside it; the silhouette re-admits the density that lies beyond the sphere but close to the viewing axis, above and below it; and the in-plane reading discards the third dimension entirely, so all the density at any height is gone from both the numerator and the denominator at once.

Which of the three a reader is looking at is decided by how the figure was made rather than by anything in the caption. A hand-drawn textbook orbital, a contour plot and a ray-traced isosurface are three different objects, and the last two are routinely captioned with the first one’s number.

The practical form of that is a warning against the most natural reading of a rendered orbital. The outline of a shaded surface is not the surface, and taking the width of a drawn 1s as the diameter of the ninety-per-cent sphere is right, while taking the region inside that outline as holding ninety per cent of the electron is wrong by five points. Five points is not much; the fact that the picture supports three different five-point-apart answers, with nothing on the page to say which, is the finding.

There is a second reading of the same table worth having. The projected figure rises from the 1s to the 2s and the in-plane one barely moves, which says the two corrections are sensitive to different things. The in-plane reading discards a whole dimension and is therefore nearly a geometric constant; the projection re-admits a slab of density whose thickness depends on how far the orbital’s tail reaches beyond its own contour, so it grows with diffuseness. A very extended orbital would have a silhouette enclosing almost everything while its surface still enclosed nine tenths — which is the same statement, one dimension down, as the one this essay makes about a thin skin of excluded density surviving a slice.

What is left #

Nothing above is an argument against printing orbitals in a plane. A slice is the clearest way to show a nodal structure, and it is the only way to show the sign pattern of a molecular orbital without occlusion.

The argument is against the caption. Ninety per cent names a quantity, and there are two quantities. A picture that states its level — the raw value of the wavefunction — is unambiguous and useless without the integral; a picture that states an enclosed fraction is useful and ambiguous unless it says in what. The repair costs one word: ninety per cent of the density in space, or in this plane.

The rule is unchanged and one clause longer. Every orbital figure here states a fraction, the fraction is solved for rather than chosen, and the integral is run again on the level that is drawn. What the measurement above adds is that the fraction is now known to be a statement about the surface, and that the same picture supports a different and larger claim about the page — so the two are reported together wherever a figure is a slice rather than a projection.

The wider point is the one the isovalue essay makes from the other side. There, a level chosen by convention was found to enclose anything between 0.4 and 96 per cent depending on the orbital, so the number nobody computes turns out not to be nearly constant. Here, a fraction that is computed turns out to name two different things, and the two differ by about the same amount for everything — so the correction is nearly constant, and nearly constant is exactly the condition under which a discrepancy survives for a century without being noticed.