Molecular Geometry & Bonding
what the shape decides
Orbitals

Where the electron is

The wavefunction is largest at the nucleus, the electron is most likely to be found a bohr out, and the ninety-per-cent contour is at 2.66. Three numbers, all correct, all answering different questions.
11 min read 7 figures Counted, not quotedSay what it enclosesTaught confidently, and wrong

An orbital is a function rather than a place. Ask where the electron in a hydrogen atom is and there are at least three defensible answers, all computable, all different.

The probability density is greatest at the nucleus. The most probable radius is one bohr. The ninety-per-cent contour is at 2.661 bohr. None of these contradicts the others, and confusing them is one of the more common ways to go wrong about the shape of an atom.

The radial function of 1sThe radial part of the wavefunction, which changes sign at each node, and the radial distribution, which is the probability of finding the electron in a shell at that radius. The second vanishes at the nucleus and the first does not.most probable radius 1.00 a₀R(r)and 4πr²R²r / bohrmeasured off the computed function0 radial nodes · one electron
Fig. 1 The radial function of a 1s orbital and its radial distribution. The first is largest at the nucleus and the second peaks at one bohr, and the difference between them is entirely a matter of how much volume there is at each radius.

The density and the distribution

∣ψ∣2|\psi|^2 is a probability density: probability per unit volume. For a 1s orbital it is largest at the nucleus and falls off from there monotonically.

The radial distribution is a different quantity: the probability of finding the electron anywhere in a thin shell at radius rr. A shell of thickness drdr has volume 4πr2dr4\pi r^2 dr, so the radial distribution is

P(r)=4πr2∣R(r)∣2.P(r) = 4\pi r^2 |R(r)|^2.

The factor of r2r^2 is the whole story. Close to the nucleus the density is high and the volume is tiny; far out the volume is large and the density is negligible. The product peaks in between, and for hydrogen’s 1s orbital it peaks at exactly one bohr — the Bohr radius, which is where the name comes from and one of the few places the old model gives the right answer.

Why the density peaks at the nucleus and the electron does not

This is the point that reliably surprises people, and it has an everyday analogue.

The most likely single point at which to find a raindrop landing on a circular target may be the exact centre. The most likely distance from the centre is not zero, because there is almost no target at radius zero and a great deal of it further out. Nothing is inconsistent; the two questions differ by the area available.

The same happens in three dimensions with the volume of a shell, and the r2r^2 is the same factor. So it is entirely correct to say the electron is most likely to be found near the nucleus per unit volume, and equally correct that it is most likely to be found at about one bohr from it.

The third answer, and why it is larger

The contour that encloses ninety per cent of the density sits at 2.661 bohr, a good deal further out than the peak of the distribution.

That is not a contradiction either. The distribution peaks at one bohr and then falls away slowly; getting to ninety per cent of the total requires going well past the peak. Half the density is inside 1.337 bohr, ninety per cent inside 2.661, ninety-nine per cent inside 4.203.

Choosing a contour for 1sThe fraction of the density enclosed by a contour, against the contour's level. Picking a level is picking a point on this curve, and the usual practice of picking one that looks right is picking a point without knowing which.0%25%50%75%100%50% enclosed|ψ| = 1.48e-190% enclosed|ψ| = 3.94e-299% enclosed|ψ| = 8.44e-3contour level |ψ|fraction of the density enclosedevery level here was solved for, not chosen1s
Fig. 2 The cumulative version: how much of the density is inside a contour at each level. The peak of the distribution and the ninety-per-cent radius are different points on this curve, and quoting either as “the size of the atom” is a choice rather than a fact.

The expectation value, which is a fourth number

For completeness there is one more, and it is the one a physicist usually means.

The expectation value of the radius, ⟨r⟩\langle r \rangle, is the average distance weighted by the distribution. For hydrogen’s 1s orbital it is 1.5 bohr — larger than the most probable radius, because the distribution is skewed, falling away far more gradually on the outer side than on the inner.

So: 0 for the density maximum, 1.0 for the most probable radius, 1.5 for the mean radius, 2.661 for the ninety-per-cent contour. Four numbers, one orbital, no disagreement. Which one is “the size of a hydrogen atom” depends entirely on what the number is going to be used for.

What changes with n and l

The pattern across orbitals is worth having because it explains a good deal about periodic trends.

The radial function of 2sThe radial part of the wavefunction, which changes sign at each node, and the radial distribution, which is the probability of finding the electron in a shell at that radius. The second vanishes at the nucleus and the first does not.node 2.00most probable radius 5.24 a₀R(r)and 4πr²R²r / bohrmeasured off the computed function1 radial node · one electron
Fig. 3 The 2s radial distribution, with two peaks separated by the node. Most of the density is in the outer peak, and the small inner one is close to the nucleus — much closer than any part of a 2p.

Higher nn means further out, roughly as n2n^2. The most probable radius for hydrogen’s 2s is about 5.2 bohr against 1 for the 1s.

Higher ll within a shell means less inner density. A 2s orbital has a small peak inside its node, sitting close to the nucleus; a 2p has none. That inner peak is penetration, and it is why the s orbital of a shell lies lower in energy than the p in every atom except hydrogen: the s electron spends part of its time inside the screening of the inner shells and feels a larger effective nuclear charge.

The radial function of 2pzThe radial part of the wavefunction, which changes sign at each node, and the radial distribution, which is the probability of finding the electron in a shell at that radius. The second vanishes at the nucleus and the first does not.most probable radius 4.00 a₀R(r)and 4πr²R²r / bohrmeasured off the computed function0 radial nodes · one electron
Fig. 4 The 2p radial distribution for comparison: a single peak, no inner lobe, and nothing close to the nucleus. That absence is the reason 2p lies above 2s once there is more than one electron.

That is a real explanatory chain from a shape to an energy ordering to the structure of the periodic table, and it runs entirely through the radial distribution rather than through the contour picture.

What the contour picture cannot tell

Which brings the argument back to the drawings.

An orbital contour is a statement about a proportion of the density. It says nothing about where within that region the electron is likely to be, and in particular the region is not uniformly occupied — a 1s orbital’s ninety-per-cent sphere has far more density near its centre than near its surface.

So a contour picture answers “where is most of it” and cannot answer “where is it most likely to be”. The contour is a choice and the distribution is not, which is a good reason to show both.

The site draws both for exactly that reason, and this essay exists because the two are routinely conflated — including in the common claim that a p orbital’s lobes are “where the electron spends its time”, which is not what a contour means.

Nodes in the distribution

One consequence for reading these plots.

The radial distribution vanishes at a radial node, because the wavefunction does. It also vanishes at r=0r=0 for every orbital, because the r2r^2 factor does — including for s orbitals, whose wavefunction is maximal there.

That second zero is not a node. Nothing changes sign, and the wavefunction is perfectly finite; the distribution goes to zero only because the shell has no volume. Counting it as a node is a common slip and it makes every count come out one too high. The node counts on this site are made on R(r)R(r), not on the distribution, for exactly this reason.

Where the model stops

Two limits, both standing.

One electron. Every number here is for a hydrogenic orbital. In a many-electron atom the radial distributions are contracted by the increased nuclear charge and distorted by the other electrons, and the orbitals themselves are an approximation.

A distribution is not a trajectory, and nor is the contour a boundary. Saying the electron is most likely to be found at one bohr does not mean it travels at that radius, or travels at all. The distribution describes the outcome of a measurement, not a path between measurements.

Where the number came from

The Bohr radius is 0.529 ångström, and it is worth noticing what it is doing in a theory that abandoned Bohr’s model.

Bohr’s 1913 model put the ground-state electron in a circular orbit of exactly that radius. The model is wrong in almost every respect — it has trajectories, it has no explanation for why the orbits are stable, and it fails completely for helium — and this one number survives, as the most probable radius of the 1s distribution and as the natural unit of length in atomic physics.

That is a reasonably common pattern in physics: a superseded model leaves behind a scale that turns out to be right, because the scale was fixed by dimensional analysis rather than by the wrong dynamics.

Comparing the shells

Setting several distributions side by side makes the trends visible that individual plots do not.

The radial function of 3sThe radial part of the wavefunction, which changes sign at each node, and the radial distribution, which is the probability of finding the electron in a shell at that radius. The second vanishes at the nucleus and the first does not.node 1.90node 7.10most probable radius 13.10 a₀R(r)and 4πr²R²r / bohrmeasured off the computed function2 radial nodes · one electron
Fig. 5 The 3s distribution: three peaks, two nodes, and most of the density well out from the nucleus — but with two small inner regions that reach much closer than any 3p or 3d does.
The radial function of 3dz2The radial part of the wavefunction, which changes sign at each node, and the radial distribution, which is the probability of finding the electron in a shell at that radius. The second vanishes at the nucleus and the first does not.most probable radius 9.00 a₀R(r)and 4πr²R²r / bohrmeasured off the computed function0 radial nodes · one electron
Fig. 6 The 3d distribution for the same shell: a single peak, no inner structure at all, and nothing close to the nucleus. That difference is why the 3d orbitals lie above the 3s and 3p in a many-electron atom, and why the transition metals fill their d shell after the next s shell has begun.

Higher n moves the peak out, roughly as n². Higher l within a shell removes inner structure, because the centrifugal term keeps a high-angular-momentum electron away from the nucleus.

The second trend is the one that shapes the periodic table. Penetration is what breaks the hydrogenic degeneracy, and once broken the filling order becomes 4s before 3d — which is the reason the first transition series exists where it does, and the reason its chemistry is what it is.

Orbitals at the 90 per cent contourSeveral orbitals drawn at the same enclosed fraction and, unless stated otherwise, at the same scale — so the sizes on the page are the sizes. Each contour was solved for separately by integrating that orbital's own density.one scale across the plate1s0 radial · 0 angular2s1 radial · 0 angular3s2 radial · 0 angulareach level solved for separately90% · one electron
Fig. 7 The same trend as surfaces: 1s, 2s and 3s at one enclosed fraction and one scale. The shells nest, the outer ones are much larger, and each has one more radial node than the last.

Where the ladder goes next

The contour question is what the surface encloses, which is the site’s organising discipline.

The structural feature behind the two-peaked distributions is the node.

And the caution that applies to every number on this page is that these are one-electron functions.

What the pictures here cannot show. A radial distribution is an average over all directions, so it says nothing about shape — a 2p and a 2s of the same nn have quite different distributions and a 2px2p_x and a 2pz2p_z have identical ones. Direction is precisely what these plots integrate away.

Where this goes next

50% of the density|ψ| = 1.48e-190% of the density|ψ| = 3.94e-299% of the density|ψ| = 8.44e-31slevels solved for by integration, drawn at one scale Say what it encloses An orbital picture is a contour at a level somebody chose, and almost no source says which. Two textbooks can draw the same orbital at visibly different sizes with the same caption, and both be printed in good faith. node 1.90node 7.10most probable radius 13.10 a₀R(r)and 4πr²R²r / bohrmeasured off the computed function2 radial nodes · one electron Nodes An orbital with quantum numbers n and l has exactly n−l−1 radial nodes and l angular ones. That is a count, it is exact, and it is the fastest way to catch a drawing that is wrong. one scale across the plate1s0 radial · 0 angular2s1 radial · 0 angular2pz0 radial · 1 angulareach level solved for separately90% · one electron Orbitals are not where the electron is A many-electron atom has no exact orbitals at all. The orbital picture is a basis for an approximation — an extremely good one — and treating it as a description of reality is the source of most of the confusion in this subject.