How big is an orbital

Ask how big a 1s orbital is and there are four defensible answers, all of them computable from the same function in a few lines, and the largest is two and a half times the smallest.

That is not a defect in the question. It is what happens when a quantity with no boundary is asked for a boundary.

The four answers #

The most probable radius is where the radial distribution $r^2R(r)^2$ peaks — the single distance at which an electron is most likely to be found. For a 1s orbital it is exactly one bohr, which is the Bohr radius and the reason that number has a name.

The mean radius is the expectation value $\langle r\rangle$, the average distance weighted by probability. For 1s it is 1.5 bohr, half again as large, because the distribution has a long tail on the outside and nothing at all on the inside of zero.

The root-mean-square radius is $\sqrt{\langle r^2\rangle}$, which is 1.73 bohr. It is larger still, for the same reason a root-mean-square is always at least a mean: squaring weights the tail more heavily. It is the measure that appears in most physical formulae, because energies depend on $r^{-1}$ and $r^{-2}$ rather than on $r$.

The ninety-per-cent radius is the sphere holding nine tenths of the density, and it is 2.66 bohr — nearly three times the most probable radius. It is the largest of the four because it is not an average at all: it is a boundary chosen to leave a stated remainder outside.

Why they cannot be reconciled #

There is a temptation to declare one of them the real size and the others approximations to it. That does not survive contact with what each is used for.

An X-ray crystallographer measuring electron density wants the distribution, not a radius. A spectroscopist computing a transition moment needs $\langle r\rangle$, because the dipole operator is linear in $r$. A diamagnetic susceptibility depends on $\langle r^2\rangle$ and on nothing else. And an illustrator drawing an orbital needs a surface, which means choosing an enclosed fraction and accepting whatever radius comes out.

Four questions, four answers, and each is exactly right for its own question. What is not defensible is quoting one of them as the size with no statement of which — which is what almost every published orbital picture does, and the reason say what it encloses is the first rule this site follows.

What the spread is made of #

The four measures separate because the radial distribution is skewed, and the skew is not the same for every orbital.

For a 1s orbital the distribution rises from zero, peaks at one bohr and decays exponentially, with a tail that is long relative to the peak. That long tail is what drags the mean out to 1.5 and the ninety-per-cent radius out to 2.66, and it is why the ratio across the four measures is largest for the tightest orbital on the list. The distribution itself, and the three quite different questions it answers, is the subject of where the electron is.

For a 4s orbital the picture is nearly the opposite. Its outermost peak is at 24.62 bohr, its mean is 24.00 — smaller than the most probable radius, which happens when the distribution is skewed inwards — and the ninety-per-cent sphere is 33.62. The four span a factor of 1.37 rather than 2.66. The bigger the orbital, the more the outer lobe dominates and the closer the four answers come together.

The fifth number, which is not a size #

All four measures answer the question how far out does the density reach. None of them answers the question chemistry actually cares about, which is how far in.

Look at the inner peaks. Every s orbital has one at almost exactly the same place:

• 2s has peaks at 0.77 and 5.24 bohr;
• 3s at 0.74, 4.18 and 13.07;
• 4s at 0.73, 4.00, 10.65 and 24.62.

The innermost peak barely moves. A 4s electron in hydrogen spends most of its time twenty-four bohr out and has a small but real amplitude at three quarters of a bohr — right in among the core electrons, where the screening fails and the full nuclear charge is exposed. That is penetration, it is what binds the 4s below the 3d, and not one of the four measures of size records it, because it is a feature of the shape rather than of the extent.

What was computed, and how #

Every number in this essay comes from integrating a hydrogenic radial function on a forty-thousand-point grid out to 120 bohr, and two of the four measures have closed forms that make the integration checkable rather than merely plausible:

$$\langle r\rangle = \frac{3n^2 - l(l+1)}{2Z}, \qquad \langle r^2\rangle = \frac{n^2\left[5n^2 + 1 - 3l(l+1)\right]}{2Z^2}.$$

Both are checked against the closed forms, to two parts in a thousand for the mean and two in a hundred for the mean square. A quadrature that agreed for 1s and drifted for 4s is exactly the failure that would otherwise go unnoticed — the numbers stay plausible and the trend across the table quietly bends.

The most probable radius is found by scanning the distribution for its maximum, and the ninety-per-cent radius by accumulating the distribution until nine tenths of it is behind. Neither has a general closed form, and for 1s both can be checked against results that do: the peak is at $a_0/Z$ exactly, and the ninety-per-cent sphere solves $1 - (1 + 2r + 2r^2)e^{-2r} = 0.9$, which orbitalcheck verifies to five parts in a thousand at four different fractions.

The claim the figure exists to make is checked too: for every orbital drawn, the four measures must differ by more than a third. A version of this figure whose four dots landed on top of each other would be making the opposite argument, and it would fail rather than mislead.

Where the model stops #

Everything here is hydrogen-like and one-electron. The four measures are properties of a function, not of an atom, and a real many-electron atom has no exact radial function to take them of.

Two consequences are worth stating.

A “sphere holding ninety per cent” is not a boundary. A wavefunction that decays exponentially has no boundary at all, and every number in this essay is a way of summarising an infinite tail. An atom does not stop anywhere, and the various tabulated atomic radii — covalent, ionic, van der Waals, metallic — are operational definitions from measurements on collections of atoms rather than measurements of an atom.

A sphere and a contour are different objects except for an s orbital. The four measures in this essay are all radial: they integrate over angles and report a distance. A contour is a surface of constant $|\psi|$, which for an s orbital is a sphere — and the two agree exactly, as they must, so 1s comes out at 2.66 bohr by either route. For a 2p orbital the ninety-per-cent contour is two lobes and reaches further along the axis than the ninety-per-cent sphere does, while enclosing nothing at all in the nodal plane. Quoting a radial measure and drawing a contour under the same caption is not wrong, but it is two different statements, and every orbital, by what it encloses lists the contours rather than the spheres for that reason.

The measure that is missing, and why it has to be #

There is a fifth thing a reader might reasonably mean by “size”, and it is the one orbital pictures actually draw: the surface enclosing a stated fraction. It is not in the list above because it is not a property of the orbital at all — it is a property of the orbital and a choice, and the choice is free.

That is worth being blunt about. The other four measures are determined: given the function, the mean radius is what it is. The contour radius is determined only once somebody has decided what fraction to enclose, and there is no principle anywhere that fixes the fraction. Ninety per cent is a convention, and a weak one: say what it encloses traces published pictures at levels from around eighty to around ninety-nine per cent under identical captions.

So the fifth measure is the one a reader sees most often and the only one that is partly a decision. The honest response is not to pretend otherwise but to state the level under every picture and hold it fixed — which is why the pictures here can be compared with each other and most published ones cannot be compared with anything.

What this does to a comparison #

The practical consequence shows up whenever two orbitals are put side by side, which is most of what an orbital picture is for.

Comparing a 1s at ninety per cent with a 2p at ninety per cent is a fair comparison: the same question has been asked of both. Comparing a 1s drawn at ninety with a 2p drawn at eighty is not, and the difference is large — for a 1s orbital the two radii are 2.66 and 2.14 bohr, a fifth of the size, which is easily enough to reverse the apparent ordering of two similar orbitals.

That is the reason overlap decides computes its overlaps by integration rather than by eye. A picture of two orbitals touching or not touching is a picture of two chosen contours touching or not touching, and the integral is indifferent to both choices: it uses the whole function, tail and all. When the pictures and the integral disagree, the integral is right and the contour was the wrong instrument.

The generalisation #

The habit worth taking away is asking what question a single number is answering before quoting it.

The same trap is set everywhere in this subject and it has been sprung several times already. Electronegativity is not one quantity is the same failure with four scales instead of four radii. The dipole is not a sum of bonds is the same failure applied to a vector, and hybrids are a basis is the same failure applied to an orbital rather than to a number. In each case a family of related quantities has been collapsed to one word, and the word is used as though the collapse had been justified.

The test in each case is the same and it is cheap: compute two of the candidates and see whether they agree to the precision the argument needs. Here they do not agree even to a factor of two.

A note on what the four measures are not #

None of the four is an orbital radius in the sense a table of atomic radii means, and the difference is larger than it looks.

A tabulated covalent radius is half of a measured bond length, assigned to one atom by a convention about how to split it. A van der Waals radius comes from how closely two non-bonded atoms of the same element approach in a crystal. An ionic radius comes from the same kind of measurement with a further convention about where the boundary between a cation and an anion falls, and the standard tables disagree with each other by tenths of an ångström because that convention is a choice.

All three are properties of atoms in company. Everything in this essay is a property of a solitary hydrogenic function. The two families of number are not interchangeable and are not even measuring the same kind of thing — carbon’s tabulated covalent radius of 0.76 ångström is 1.44 bohr, which is smaller than the mean radius of a hydrogenic 2p at carbon’s screened charge and larger than its most probable radius, and neither comparison means anything.

That is the ordinary situation in this subject rather than an unusual one: a word with a clear everyday meaning turns out to name several different measured quantities, and the sentence containing it is only as precise as the statement of which. The rule this site follows is to name the measurement every time, which is why the figures above are labelled with four dots rather than one.

Which of the four an instrument ever returns #

There is a way of choosing between the measures that does not involve preference, and it disposes of most of them: ask which one appears in an expression for something that can be measured.

The root-mean-square radius does. An atom’s diamagnetic susceptibility is proportional to the sum of $\langle r^2 \rangle$ over its electrons, and susceptibility is measured routinely — so $\sqrt{\langle r^2 \rangle}$, which is 1.732 bohr for hydrogen’s 1s, is the one entry in the table that a laboratory instrument is sensitive to.

The other three do not. The mean radius, the most probable radius and the ninety-per-cent radius are not the natural variable of any standard expression in atomic physics. They are quantities computed from the wavefunction for the purpose of describing it, and no measurement is proportional to them.

That is already an uncomfortable result for a table of four sizes. The uncomfortable part gets worse when the quantities that do govern spectra are listed, because they are all moments of the wrong sign.

A spin–orbit coupling constant goes as $\langle r^{-3} \rangle$ — which is why an s electron has no spin–orbit coupling at all, since that integral diverges for $l = 0$ and the angular factor vanishes to compensate. A hyperfine coupling for a non-s electron goes as $\langle r^{-3} \rangle$ too. The potential energy, and through the virial theorem the total energy, goes as $\langle r^{-1} \rangle$, which for hydrogen’s 1s is exactly 1 bohr⁻¹ — the fifth number this essay names as not being a size.

Every one of those is weighted towards small $r$, and every one of the four sizes is weighted towards large $r$. A ninety-per-cent contour is a statement about where the outer tenth stops; an inverse cube is a statement about the innermost part of the density, where the contour has nothing to say.

So the picture and the measurement look at opposite ends of the same function. That is not a contradiction — an orbital is one object and both ends of it are real — but it does settle what an orbital picture is for. It shows where the electron is likely to be found, which is the question a chemist asks about shape and packing. It shows nothing about the region that decides how tightly the electron is bound, how its levels split, or how it couples to a nucleus, because all three of those live inside the first bohr and the contour is drawn at two and a half.

Who found it, and when #

The expectation values quoted above are in Schrödinger’s own papers of 1926 and were standard within a year of them; the closed forms follow from a recursion Kramers gave in 1926 and are in every quantum mechanics textbook as an exercise.

The most probable radius has a longer pedigree than the wavefunction does — it is Bohr’s 1913 orbit radius, which the wave picture recovers as the peak of a distribution rather than as an orbit. That the two agree exactly for 1s is a coincidence of the hydrogen problem and does not survive to any other orbital: the 2s peak is at 5.24 bohr where Bohr’s second orbit is at 4.

The choice of ninety per cent for drawing orbitals appears to have no single origin and no argument behind it. It is a convention, it is not always adhered to, and whether a given published picture obeys it is usually not checkable, because the level is not stated. That is the gap every orbital, by what it encloses exists to close.

Still open: two orbitals brought together #

This essay and what an electron actually feels between them say what an orbital’s size is and what it is not. What neither has touched is what happens when two of them are brought together, which is where the whole of bonding starts — and where the arithmetic stops being about one atom at all.