The isovalue nobody chose

The first rule for orbital pictures is that every one states the fraction of the probability it encloses, and say what it encloses is the essay that argues for it. The rule was written against a caption — the ninety per cent surface — attached to a picture nobody had done the integral for.

There is a second half to that problem and it is the one every piece of visualisation software has. A program draws an isosurface at a value the user types. The value is in atomic units of the wavefunction, the defaults in common use are 0.05 or 0.02, and nothing anywhere in the interface reports what fraction of the density that value encloses.

It is not a constant. It is not even nearly a constant.

One number, ninety-six points of spread #

Take $\psi = 0.02$, which is a common default and roughly what a published orbital plate uses.

For hydrogen’s 1s, that surface encloses 96.2 per cent of the density. For the 2s, 51.8. For the 2p, 71.5. For the 3s, 1.3. For the 4s, 0.4.

The extremes are worth stating plainly. The 1s picture is essentially the whole orbital. The 4s picture is a surface containing four electrons in every thousand — a thin inner shell, with the outer nine hundred and ninety-six parts of the density lying outside the drawn surface entirely, invisible.

Both are drawn to the same rule and both look like orbitals. A reader comparing their sizes is comparing a nearly complete surface with a small fragment of one, and no caption anywhere says so.

The spread is not a defect of the default. Move to 0.05 and the numbers become 86.2, 3.9, 20.2, 0.5, 0.0 — different, and just as spread. Move to 0.01 and they become 98.7, 83.8, 89.2, 8.7, 0.5. There is no value that treats these orbitals alike, because they differ in how much of the density is at high amplitude and how much is spread thin.

The four measures of size in how big is an orbital span a factor of two and are all honest; the spread here is a hundredfold and is not a measure of anything, because the quantity being held fixed is the one the picture is not about.

Why they differ so much #

A wavefunction’s value at a point and the probability inside a surface are related by an integral, and the integral has a factor that does most of the work.

The density at radius $r$ contributes to the total in proportion to $r^2$, because that is how much area a shell has. So a diffuse orbital has most of its probability where the wavefunction is small: the amplitude is spread over a large volume, and a contour drawn at any appreciable value cuts inside almost all of it. A compact orbital has its probability where the amplitude is large, and the same contour encloses nearly everything.

That is where the electron is’s point applied to a surface rather than to a radius: the wavefunction is largest at the nucleus and the electron is most likely to be found a bohr out, and the same $r^2$ is responsible for both.

The number of radial nodes makes it worse rather than better. An orbital with nodes has its amplitude divided among several shells, and the outer shell — where most of the probability is — has the smallest amplitude of them, because the function has to be normalised across all of them.

The same picture, the wrong claim, across the periodic table #

Fix the orbital and vary the atom instead, and the same thing happens for a different reason.

A 2s function on a nucleus of charge $Z$ is the hydrogen 2s scaled: pulled in by a factor $Z$ in radius, raised by $Z^{3/2}$ in amplitude. Screening in a real atom softens that to an effective charge, which is what what an electron actually feels computes and what the screening model cannot see qualifies — but the direction is the same. A more highly charged nucleus holds its 2s closer and higher.

So a fixed isovalue encloses more of it. At $\psi = 0.02$, the 2s of a nucleus of charge one encloses 51.8 per cent, of charge two 90.7, of charge four 98.3 and of charge eight 99.7.

That is the case that matters for a plate of second-row atoms, which is one of the commonest pictures in a chemistry textbook. Drawn at one isovalue, lithium’s 2s and fluorine’s 2s look different sizes — correctly, they are different sizes — but the two surfaces are not the same kind of statement, and the reduction in apparent size across the row is a mixture of a real contraction and a change in what fraction is being shown. The radial distribution across the table is where the real contraction is measured without the picture in the way.

The other direction: what a stated fraction costs #

Turn the question round. Solve for the level that encloses ninety per cent of each orbital — which is what every orbital picture here does — and the levels come out at $3.9 \times 10^{-2}$, $7.4 \times 10^{-3}$, $9.5 \times 10^{-3}$, $2.6 \times 10^{-3}$, $3.0 \times 10^{-3}$, $3.6 \times 10^{-3}$ and $1.2 \times 10^{-3}$ for the seven orbitals above.

A factor of thirty-two between the largest and the smallest. And that is within one atom: bring in a nucleus of charge six and the 1s needs $5.8 \times 10^{-1}$, which is nearly five hundred times the 4s level of hydrogen.

So the two conventions are not two equally arbitrary choices. One of them fixes a number that has to vary by orders of magnitude to keep the meaning constant; the other fixes the meaning and lets the number vary. It is the difference between drawing every map at the same scale and drawing every map to fit the page.

What solving for the level does, and where it is still a choice #

Every orbital figure here solves for its level: a bisection on the enclosed probability finds the level that gives the fraction asked for, the integral is run again on that level before it is drawn, and the caption states the fraction that was actually solved for — the discipline one level is not one comparison argues for.

That removes one arbitrary choice and does not remove all of them. Ninety per cent is itself a convention. It is a defensible one — it is what the literature usually means when it says nothing — but the sensitivity above says that the same orbital drawn at fifty per cent looks meaningfully smaller, and nothing in physics prefers one to the other.

The honest form is the one used here: the number is stated on the figure. A reader who wants the fifty per cent surface knows that this is not it.

What a reader can do with a picture that does not say #

Most published orbital pictures do not state a contour, so the practical question is what can be recovered from one that does not.

The shape can be trusted. An isosurface of $\psi$ at any level has the same nodal structure, the same symmetry and the same lobe signs as the surface at any other level, because a node is where the function is zero and no choice of contour moves it. So the count of lobes, the planes through them and the sign pattern — which is nodes’s subject — survive an unstated level intact.

The size cannot. Nothing about the extent of a drawn surface is meaningful without the level, and the previous section’s numbers put a bound on how badly: two orbitals of the same atom drawn at one level can differ in enclosed fraction by ninety-six points.

The relative size across a shell can be trusted a little. Within one $n$ and one atom — a 2s beside a 2p — the amplitudes are of comparable scale and the fractions come out at 51.8 and 71.5 per cent at the common value: different, but not so different that the picture misleads about which is larger. It is across shells, and across the periodic table, that the comparison fails.

That is not a satisfying set of rules, and its unsatisfactoriness is the argument for stating the level. One number on a caption converts a picture from something a reader has to guess about into something they can check.

The one place the fixed value is right #

There is a use for a fixed isovalue and it is not a compromise: comparing the same orbital between two situations.

Draw a molecule’s orbital before and after a geometry change at one isovalue and the difference in the picture is a real difference in the wavefunction. Draw two isosurfaces of two different molecular orbitals of the same molecule at one isovalue and the relative sizes say something about relative diffuseness. What breaks is the comparison across objects whose normalisation is spread differently, which is exactly the case a plate of atomic orbitals presents.

That is a general shape and it turns up elsewhere. A ranking is not a difference is the same distinction for electronegativity scales: a quantity can be perfectly good for ordering and useless for subtracting, and which one it is depends on what stays fixed.

Why the software does it this way #

The convention is not carelessness, and understanding why it exists is part of knowing what to do about it.

Solving for a fraction costs an integral. Drawing a surface at a stated value is a marching-cubes pass over a grid; drawing one at a stated enclosed fraction is that pass wrapped in a search, with the density integrated inside the surface at every step. For a molecular orbital on a three-dimensional grid that is expensive, and it has to be redone whenever the geometry changes.

And for a molecular orbital the fraction is not obviously the right quantity anyway. An orbital delocalised over twenty atoms has its ninety per cent surface enclosing most of the molecule, which is not a useful picture; the isovalue picture, which shows where the amplitude is largest, is what a chemist actually wants to see.

So the two conventions serve different purposes, and the case this essay is about — a plate of atomic orbitals presented as a comparison — is the case where the software’s convention is applied outside the situation it suits.

The integral is affordable here because hydrogenic orbitals separate. A hydrogenic function’s isosurface meets each ray where the radial function crosses a threshold, which is a one-dimensional root-find rather than a volume scan, and the enclosed probability is a product of a radial integral and an angular sum. That is the whole reason say what it encloses’s rule was affordable enough to make a rule.

The one thing a fixed value has that a stated fraction does not #

Everything above is against the fixed isovalue, and the case is strong. It is worth putting the other side, because there is a class of picture for which the enclosed-fraction rule is not merely inconvenient but undefined — and the enclosed-fraction rule has to stop somewhere.

An enclosed fraction is a fraction of something. It requires the field being contoured to be a normalised, non-negative density, so that the integral inside a surface is a proportion of a whole. Every atomic orbital is such a field, which is why the rule works for them.

A great many useful pictures are not.

A density difference — the density of a molecule minus the densities of its separated atoms, which is the picture that actually shows what bonding did — integrates to exactly zero. There is no whole to take a fraction of, and a surface enclosing ninety per cent of it is a meaningless request. The same is true of any deformation map, any difference between two states, and any spin density in a system with equal numbers of each spin.

A field with no normalisation at all — an electrostatic potential mapped onto a surface, a gradient, a Laplacian — has units and a scale but no total.

For all of those the only available convention is a stated value, and the honest practice is the one the fixed isovalue almost achieves and never quite does: state the number, state its units, and state something that fixes the scale — the field’s maximum, or the fraction of the positive part enclosed, or the integral of the absolute value inside the surface. Any of those restores the comparability that a bare number lacks, and none of them requires the field to be a probability.

So the complaint in this essay is narrower than it first sounds, and narrowing it is the point. The fixed isovalue is not wrong as a mechanism; it is wrong as a default applied to an object that has a better convention available. An orbital is normalised and non-negative in its density, so the fraction it encloses is computable and is the thing a reader wants. A difference map is neither, so a value is all there is.

The rule this site follows should therefore be read with its scope attached: say what it encloses, where enclosing is defined; say what the value is and what it is a fraction of, where it is not.

Where the model stops #

These are hydrogenic functions. A real many-electron orbital is not a scaled hydrogen orbital, and orbitals are not where the electron is says why the object being contoured is a basis function rather than a thing. The spread computed here would be larger for real atoms, not smaller, since a self-consistent 4s is more diffuse than a hydrogenic one.

The integral is a quadrature. Enclosed fractions here come from a product Gauss rule on the radial and angular parts, validated against the closed-form 1s overlap to $6 \times 10^{-5}$ — accurate enough that the differences above are thousands of times the error.

Isovalues in the literature are not always in these units. Some programs contour the density $\psi^2$ rather than the wavefunction, which squares the number and changes nothing about the argument; some contour in units scaled to the maximum, which fixes the problem for one orbital at a time and not across a plate.

One practical figure is worth extracting from the tables. Across the seven orbitals here, the isovalue needed for ninety per cent scales roughly as the inverse cube of the orbital’s mean radius — which it must, since a normalised function spread over a volume $V$ has an amplitude of order $V^{-1/2}$ and its density of order $V^{-1}$.

That gives a rule of thumb for reading a plate whose contour is unstated: if two orbitals are drawn at one value and one of them is twice the size of the other, the larger one’s surface is enclosing a great deal less of it, by roughly the cube of the ratio in amplitude. The rule is crude and it is better than assuming the pictures are comparable, which is what a plate at a fixed isovalue invites.

What measuring the software convention adds #

The argument about contours so far is that the level has to be stated, that one orbital has four different sizes, that one level is not one comparison, and that a filled shell has no shape at all.

This one takes the convention that the rest of the world actually uses — a number typed into a program — and measures what it means. The answer is that it means something different for every orbital and for every atom: a spread of ninety-six percentage points at one common value, and a factor of thirty-two in the values needed for one common claim. Solving for a level is not fastidiousness; it is the only way the caption can be true.

The open question is the contours of molecular orbitals, where the normalisation is spread over several atoms and the question of what a surface encloses acquires a second half: enclosing how much, of what, near which nucleus.