Molecular Geometry & Bonding
what the shape decides
Orbitals

The radial distribution across the periodic table

A 4s orbital is bigger than a 3d by every measure of size except the one that decides which fills first. Penetration is a feature of a small inner peak, and the periodic table's shape depends on it.
14 min read 6 figures One electron onlyCounted, not quotedTaught confidently, and wrong

In hydrogen, 3s, 3p and 3d have exactly the same energy. In every other atom they do not, and the ordering that results is the shape of the periodic table.

The whole of the difference is one term — the repulsion between electrons — and the whole of its effect can be read off the radial distributions.

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. 1 The radial distribution of a 3s orbital. Most of the density is in the outer peak, and two small inner peaks sit much closer to the nucleus than anything a 3d orbital has. Those inner peaks are what decides the filling order.

What penetration is

An electron in a many-electron atom does not feel the full nuclear charge. The other electrons screen it, and the charge it does feel is the effective nuclear charge — larger close to the nucleus, where less of the other density lies between, and smaller far out.

So an orbital that places some of its density very close in feels a larger effective charge there and is stabilised. That is penetration, and it is a statement about the shape of the radial distribution rather than about its position.

The crucial point is that penetration and size are different properties and can point in opposite directions.

The radial function of 3pzThe 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 6.00most probable radius 12.02 a₀R(r)and 4πr²R²r / bohrmeasured off the computed function1 radial node · one electron
Fig. 2 A 3p distribution: two peaks, with the inner one much smaller and further out than 3s’s. Less penetration, so less stabilisation, so 3p lies above 3s in every atom with more than one electron.
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. 3 And 3d: a single peak with nothing inside it at all. Zero radial nodes, no inner structure, and no density close to the nucleus. That absence is the reason 3d lies highest of the three.

The pattern is exactly the radial node count. An orbital with n−l−1n - l - 1 radial nodes has n−ln - l peaks, and the extra peaks are all inside the main one. So s orbitals penetrate most, then p, then d, then f — and the node count that seems like bookkeeping turns out to be the quantity that orders the periodic table.

The case that looks wrong

The 4s-before-3d ordering is the one that reliably confuses, and stating it in terms of the four numbers this site distinguishes makes it obvious rather than mysterious.

By most probable radius, 4s is further out than 3d. By mean radius, further out. By ninety-per-cent contour, further out. By every measure of where the orbital mostly is, 4s is the larger and more diffuse orbital.

And by the one quantity that is not a measure of size — the amount of density placed very close to the nucleus — 4s wins, because it has three radial nodes and therefore three inner peaks, the innermost of which sits deep inside the argon core.

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. 4 The same effect at its smallest and clearest. A 2s orbital has one inner peak; a 2p has none. That single difference is why 2s lies below 2p in lithium and in everything after it, and why the second period fills s before p.
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. 5 The 2p distribution for comparison: one peak, nothing inside it, and nothing close to the nucleus. Nothing about its size explains its energy — it is smaller than 2s by mean radius — and the ordering is entirely a matter of what sits under the peak.

So the answer to “why does 4s fill before 3d when 4s is bigger” is that the question contains a false premise: the ordering is not decided by size, and the quantity that decides it is smallest exactly where it matters most.

The reversal, which is the interesting half

Having filled 4s first, a transition metal ionises from 4s first as well — which appears to contradict the ordering that put 4s below 3d.

It does not, and the resolution is worth having because it exposes what an orbital energy actually is.

Once electrons are in the 3d orbitals, the situation has changed. The 3d electrons screen the 4s poorly — they are further in — while 4s electrons screen each other and each other’s 3d partners more. Recomputing the orbital energies for the populated configuration puts 3d below 4s.

The ordering is therefore not a property of the atom. It is a property of the atom in a given configuration, and it changes as electrons are added. The familiar aufbau sequence is a rule for building up, and rules for building up are not statements about the finished structure.

That is the general shape of the difficulty with orbital energies in many-electron atoms, and it is a consequence of the thing the orbital approximation quietly assumes: that each electron moves in an average field. Change the population and the average field changes, so the orbital energies change with it.

What was computed, and how

Every distribution on this page is 4πr2∣R(r)∣24\pi r^2 |R(r)|^2 for a hydrogenic radial function, evaluated on a fine grid, with the peaks located by sampling rather than by a formula.

The radial functions come from the associated Laguerre recurrence, and three things are checked before any of them is drawn: the density integrates to one to a part in 10610^6, any two distinct orbitals on the same centre are orthogonal, and the radial node count is n−l−1n - l - 1 by walking the function and counting sign changes.

That last check is the one carrying this essay, because the whole argument runs through node counts. A radial polynomial with a mistaken coefficient produces a function with the right decay, the right symmetry, a completely convincing picture, and the wrong number of inner peaks — and no amount of looking at the drawing would reveal it. The node counter is the only thing in the pipeline that would notice, and the site’s gate feeds it an orbital claiming the wrong count to prove it still refuses.

The screening itself is not computed here. These are hydrogenic functions with a nuclear charge parameter, and the effective charges that would make them quantitative for a real atom are not derived anywhere on this site. What the figures establish is the shape of each distribution, which is the input to the penetration argument; the argument’s conclusion is quoted.

The surprise: a hydrogenic degeneracy is an accident

The fact that 3s, 3p and 3d are degenerate in hydrogen is usually presented as a feature of the hydrogen atom. It is better described as a coincidence.

For a general central potential, the energy depends on both nn and ll. The pure Coulomb potential −1/r-1/r is special: it has an extra conserved quantity — the Laplace–Runge–Lenz vector, the same one that keeps a Kepler orbit’s perihelion from precessing — and that extra symmetry forces the ll degeneracy.

So hydrogen’s degeneracy and the closed Kepler ellipse are the same fact. Any deviation from an exact 1/r1/r potential breaks both, and screening by other electrons is exactly such a deviation.

That reframes the whole essay’s subject. The question is not “why does screening split the degeneracy” but “why was there a degeneracy to split”, and the answer is a hidden symmetry that no many-electron atom possesses.

It also explains why the splitting goes the way it does. The screened potential is steeper than 1/r1/r close in and shallower far out, so orbitals with inner density are stabilised relative to those without — which is the penetration argument arrived at from the potential rather than from the pictures.

What it costs

The distributions cost a few hundred function evaluations each and the checks cost about as much again, which is the ratio everything on this site settles at.

The cost that is not computational is the one this essay has to be careful about, and it is a cost in scope.

The penetration argument is made with hydrogenic functions, applied to many-electron atoms, to explain an ordering that exists only in many-electron atoms. That is a borrowing, and it is worth naming rather than glossing: in hydrogen the three distributions above have exactly the same energy, so penetration decides nothing there, and the shapes are being read off a system where the effect is absent and applied to one where it is present.

The borrowing is reasonable, is universally made, and works. It is also the reason a real calculation gives 4s and 3d energies that depend on the configuration while this argument gives a fixed ordering — the fixed ordering comes from fixed functions, and the real ones are not fixed.

Where the model stops

Four limits.

One electron. Every function here is hydrogenic. Real orbitals in a many-electron atom are contracted by the nuclear charge and distorted by the other electrons, and the contraction is not uniform — it affects the inner peaks differently from the outer ones, which is exactly the region the argument turns on.

Screening is not computed. The effective nuclear charges implicit in the argument are quoted from Slater’s rules or from the literature. Nothing here derives one.

The ordering is configuration-dependent and this treatment gives a single ordering. The 4s-and-3d reversal is the standard illustration and there are others; chromium and copper are the familiar anomalies and neither is explained by anything on this page.

And the periodic table’s shape has more in it than the ordering. Relativistic effects contract the 6s orbital enough to change gold’s colour and mercury’s melting point, and no non-relativistic account reaches either. That is a large correction arriving at the bottom of the table, and this essay’s machinery has nothing to say about it.

Two more consequences of the same shapes

The penetration argument is usually deployed once, for the filling order, and then dropped. It carries two further results worth having, and both follow from the same figures.

Ionisation energies across a period. Moving from lithium to neon adds protons and adds electrons in the same shell, and electrons in one shell screen each other poorly — they are at similar radii, so on average only about half of each one lies between another and the nucleus. The effective charge therefore rises steadily across the period and the ionisation energy rises with it. The two irregularities, at boron and at oxygen, are the two places where the added electron goes somewhere different: into a p orbital where the previous ones were in s, and into an already-occupied p orbital where the previous ones were singly occupying.

Atomic size down a group. Each new period adds a shell whose distribution peaks further out, so atoms get larger down a group — but not as much larger as the shell spacing suggests, because the added nuclear charge pulls everything in. The competition between the two is why the size increase from the second to the third period is large and from the fourth to the fifth is small.

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. 6 Three s orbitals at one enclosed fraction and one scale, so the sizes on the page are the sizes. Each has one more inner peak than the last — the node count again — and each reaches substantially further out, which is the shell structure that the periodic table’s rows are.

Both results are qualitative, both are read off the shapes rather than computed, and both are the same argument as the filling order: the distribution’s inner structure decides how much charge an electron feels, and its outer extent decides how big the atom looks. Two different features of one curve, answering two different questions, which is the pattern this whole field runs on.

What a real calculation would change

It is worth naming what a proper treatment would alter, because the answer is more than a refinement.

A Hartree–Fock calculation on a many-electron atom produces orbitals that are not hydrogenic at all. They have the same node counts and the same symmetries — those are fixed — and different radial shapes: contracted overall, and contracted unevenly, with the inner peaks pulled in harder than the outer ones.

That uneven contraction acts on exactly the feature this essay’s argument depends on. A 4s orbital’s innermost peak in a real transition metal sits considerably closer in than the hydrogenic picture suggests, which makes the penetration effect larger than the figures here imply rather than smaller.

So the argument survives and its quantitative basis does not. What the hydrogenic pictures establish is that the inner peaks exist and that their number goes as n−ln - l; how deep they reach, and therefore how much stabilisation they buy, needs the calculation this site does not perform.

Who found it, and when

The radial distribution as a way of thinking about atomic structure belongs to the earliest days of quantum mechanics — the functions were solved by Schrödinger in 1926 and their shapes were being drawn within a few years.

Slater’s rules for effective nuclear charge date from 1930 and are an empirical prescription: a set of screening constants fitted to reproduce atomic energies. They remain in use, they are transparently approximate, and they are the reason a phrase like “carbon’s 2s orbital with Zeff=3.25Z_{\text{eff}} = 3.25” can be written down at all.

The aufbau principle and the n+ln + l rule for filling order were assembled through the 1920s and 1930s from spectroscopic evidence, largely by Madelung and by Klechkowski independently, and the rule remains empirical: there is no derivation of the n+ln+l ordering from first principles, only a good rationalisation through penetration.

That is worth stating plainly, because the filling order is taught with a confidence its foundations do not support. It is a pattern extracted from spectra, given a plausible account afterwards, with well-known exceptions and no proof.

Where the ladder goes next

The four different answers to “where is the electron” are in where the electron is.

The count that turns out to decide the ordering is nodes.

The standing caution about applying a one-electron picture to a many-electron atom is orbitals are not where the electron is.

And the choice of which functions to draw at all is what an orbital is.

What the whole subject rests on, then, is a curve with more than one feature. Where an orbital mostly is decides how big the atom looks; what it puts closest in decides where the orbital lies in energy; and the two can move in opposite directions, which they do for 4s against 3d. A reader who takes only one number off the distribution will get one of the two questions right and will not know which.

It is also the reason this essay belongs in the orbitals field rather than in a chapter on periodicity. Everything above is a statement about one-electron functions and their shapes; the periodic table is what happens when those shapes meet a second electron. The shapes come first and they are computable, which is why they are the part this site draws.

There is one more thing worth noticing about the argument’s status. It explains the ordering, it is universally taught, and it is a rationalisation of a pattern that was extracted from spectra rather than a derivation of one. That is a perfectly honourable position for a chemical explanation to occupy, and it is different from what a reader usually takes away.

What the pictures here cannot show. Every distribution on this page is hydrogenic and spherically averaged, and the effect it is being used to explain — the splitting of an ll degeneracy — is exactly zero in the system these functions describe. The figures show the shapes the argument reads; they do not show the energies the argument is about, and no figure here could, because computing those energies requires the electron repulsion this site does not evaluate.

Where this goes next

most probable radius 1.00 a₀R(r)and 4πr²R²r / bohrmeasured off the computed function0 radial nodes · one electron 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. 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.