The aufbau order is not a property of the atom

The aufbau principle is taught as a fixed sequence. Orbitals have energies, the energies come in a fixed order, and electrons fill them from the bottom. The 4s is below the 3d, so it fills first.

Every part of that is wrong except the observation about scandium’s configuration, and the arithmetic to see it is a single line.

The arithmetic #

A hydrogen-like orbital at a screened charge has energy

$$E = -\frac{Z_{\text{eff}}^2}{2n^2} \ \text{hartree}.$$

Both quantities are computable. What an electron actually feels applies Slater’s rules to the computed configuration, and the numbers across the first transition series come out as:

At scandium the two shells feel exactly the same charge, and the 3d is still lower, because the energy goes as $1/n^2$ and 3 is smaller than 4. By zinc the 3d is lower by a hundred electron volts.

There is no element at which this estimate puts the 4s below the 3d. And yet every element in the row has its 4s filled.

Two ways of resolving it, one of which works #

The first response is to distrust the estimate. It is a crude one: hydrogenic functions, a fitted screening rule, and no electron repulsion beyond what the screening stands in for.

But the direction of the discrepancy is enormous and it never reverses, and better calculations agree. Hartree–Fock orbital energies for the neutral transition metals put the 3d below the 4s across the entire series. This is not a marginal case that a more careful treatment would tip the other way.

The second response is the right one: the question was wrong. A filling order is not a statement about orbital energies at all.

What is actually determined experimentally is the ground configuration of the whole atom — the arrangement of all its electrons with the lowest total energy. The total energy is not the sum of orbital energies. It is the sum of orbital energies minus a double-counted repulsion, plus corrections, and the differences between configurations are dominated by the repulsion terms rather than by the one-electron energies.

The 3d orbital is small. Iron’s, by the computation above, has a mean radius of 1.68 bohr against the 4s at 6.40. Putting two electrons into an orbital that compact costs a great deal of repulsion — they are forced close together — where two electrons in a diffuse 4s cost far less. So the configuration $3d^6 4s^2$ can be lower in total energy than $3d^8$ even though each individual 3d level is lower than the 4s.

The evidence that settles it #

The decisive observation is what happens on ionisation, and it is unambiguous: every transition metal loses its 4s electrons first.

Iron is $[\text{Ar}]3d^6 4s^2$. Fe²⁺ is $[\text{Ar}]3d^6$ — no 4s at all. Not $3d^4 4s^2$, which is what removing electrons from the top of a fixed order would give.

A fixed order cannot do that. If the 4s were genuinely below the 3d, the last electrons in would be 3d electrons and they would be the first out. The order in which the shells fill across the neutral series and the order in which they empty on ionisation are opposite, and no single ordering of orbital energies produces both.

What produces both is a competition between one-electron energies, which favour the 3d, and repulsion, which favours spreading electrons into the diffuse 4s. Adding an electron to a neutral atom and removing one from a cation are different competitions, and they come out differently.

A 3d function has no radial node, one peak and nothing inside it — a compact single-lobed distribution, tightly bound once the nuclear charge rises and expensive to put a second electron into. A 4s has three nodes and an innermost peak at 0.73 bohr, inside the argon core where the screening fails. Everything about the ordering follows from that one difference.

What the diagram in every textbook is #

The picture that accompanies the aufbau principle — a column of levels with 4s drawn just below 3d, and a diagonal arrow threading through a triangle of orbital labels — is doing two incompatible jobs at once, which is why it is so hard to correct.

As a mnemonic for the observed order of ground configurations, it is a good one. It reproduces the sequence for most of the periodic table and it fits on a corner of a page.

As a diagram of orbital energies, it is false, and it is false in a specific way: it draws a single fixed ordering, where the actual one-electron energies depend on which atom, which charge state, and which configuration they are computed for. Iron’s 3d energy in the neutral atom, in Fe²⁺ and in Fe³⁺ are three different numbers, and their positions relative to the 4s are not the same in all three.

The two jobs are incompatible because a mnemonic can be a list and a physical ordering cannot. Nothing would be lost by drawing the mnemonic without energy axes, and what would be gained is that nobody would infer from it that a 4s electron is easier to remove than a 3d one because it is higher up — which is true, and which the diagram gets right for the wrong reason.

How large the repulsion has to be, and whether it is #

The account above leaves a number unstated, and the argument is worth only as much as that number. The 4s wins because doubly occupying a compact 3d is expensive — and expensive has so far been a word rather than a quantity. Enough of it is available in closed form to ask whether the story is even the right size.

For a hydrogenic orbital the mean inverse radius is exact,

$$\left\langle \frac{1}{r} \right\rangle = \frac{Z_{\text{eff}}}{n^2},$$

which at iron’s screened charges gives $0.69$ hartree for the 3d and $0.23$ for the 4s — 18.9 eV against 6.4. That is the scale of the Coulomb energy between two electrons confined to those shells, and the honest correction to it is downward: for two electrons in a hydrogenic 1s the actual repulsion integral is $5Z/8$ where $\langle 1/r \rangle$ is $Z$, so about five-eighths of the naive figure. Applying that factor gives roughly 12 eV to put a second electron into the 3d against 4 eV for the 4s — a difference of about 8 eV in favour of the diffuse orbital.

Now set that beside the gap it is required to overcome. The table above puts the 3d below the 4s by 47 eV at iron, and 8 eV does not touch 47. On the arithmetic as it stands the repulsion account fails, and saying so is more useful than not having checked.

What has gone wrong is the 47 rather than the 8. Slater’s rules give a 3d electron no shielding whatever from the 4s electrons outside it — the crudest single assumption in them — so the screened charge of 6.25 is far too high and the energy, which goes as its square, is too low by a great deal. Hartree–Fock orbital energies for neutral iron put the 3d near $-17.6$ eV and the 4s near $-7.0$: a gap of about 10.6 eV, four times smaller than the hydrogenic estimate and the same size as the repulsion difference computed above.

That is the resolution, and it is a sharper one than the argument could reach without the number. Two effects of comparable size are in competition — a one-electron term of order ten electron volts favouring the 3d, a repulsion term of order eight favouring the 4s — so the ground configuration of a transition metal is settled by a near-cancellation between two large quantities rather than by either of them.

A near-cancellation explains what a fixed order cannot. It explains why chromium and copper go the other way on differences of tenths of an electron volt; why molybdenum follows chromium and tungsten does not; and why the ordering reverses on ionisation, since taking an electron away removes repulsion and leaves the one-electron term unopposed. None of that is a list of exceptions to a rule. It is what two competing terms of nearly equal size always look like.

So the sign of the earlier arithmetic survives — every level of theory keeps the 3d below the 4s in the neutral atom — and its magnitude does not. The magnitude is the part that decides which configuration wins, which is why the filling order was never readable off the energies.

The anomalies, which stop being anomalous #

Chromium is $3d^5 4s^1$ and copper is $3d^{10} 4s^1$, where the simple filling rule predicts $3d^4 4s^2$ and $3d^9 4s^2$. These are presented as exceptions to be memorised, usually with a remark about the stability of half-filled and filled shells.

The shell-stability story is at best incomplete. What the competition above says is that the balance between “another electron in the compact 3d” and “another electron in the diffuse 4s” is close — the two configurations differ by a fraction of an electron volt in the total energy — so the outcome is decided by small terms, and exchange energy between parallel-spin d electrons is one of them.

Small differences settled by small terms is not a situation with a memorable rule in it. It is a situation in which a calculation is required, and where the answer varies down a group: molybdenum is $4d^5 5s^1$ like chromium, tungsten is $5d^4 6s^2$ unlike both.

The honest summary is that the filling order is an empirical summary of ground configurations, that it is approximately reproduced by the $n + l$ rule, and that the rule has about twenty exceptions among the elements — which is a great many for something usually presented as a principle.

The shell filling in the figures here is done by the $n + l$ rule, deliberately, so that the anomalies show up as anomalies: chromium and copper come out $3d^4 4s^2$ and $3d^9 4s^2$, which is what the rule says and not what the atoms do.

What was computed, and how #

Computed: every configuration, by filling shells in $n + l$ order; every screened charge, by applying Slater’s rules to that configuration; every energy, from the hydrogenic expression; and every radius, from the closed forms checked against a forty-thousand-point quadrature.

Quoted: the ground configurations of the elements, and the fact that transition metals ionise from the 4s. Both are measurements.

The comparison is the argument, and it runs the way the essay says. The figure checks that the screened charge never falls across the series, which is a check on the rules having been applied to the right configuration; the energies then follow from a single expression that has no fitted content beyond the screening.

What is not claimed is that these energies are accurate. They are hydrogenic estimates and they are wrong by electron volts. The claim is about their ordering, which is stable — every level of theory that has been applied to this question puts the 3d below the 4s in the neutral atom — and about the fact that no ordering of orbital energies whatever can produce both the filling order and the ionisation order.

Where the model stops #

This is a one-electron argument being used to demonstrate the limits of one-electron arguments, which requires care. The demonstration works because it needs only the sign of a comparison, and the sign is the same at every level of theory. It would not support a number.

Slater’s rules are a fit and a crude one. They give the 3d and 4s of scandium the same screened charge, which better calculations do not. The conclusion survives because the discrepancy it is about is far larger than the rules’ error.

No repulsion integral is computed anywhere here. The account of why the 4s wins — that a compact 3d is expensive to doubly occupy — is carried by a scale estimate from hydrogenic closed forms rather than by a two-electron integral over the real orbitals, which is not computed here. A scale estimate is enough to establish that the two competing terms are the same size and is not enough to decide which of them wins at a given element. That is a stated gap rather than a hidden one: a minimal self-consistent-field calculation is deliberately not attempted, because a wrong one producing plausible numbers is exactly the failure this essay warns against.

And the shells are treated as sharp. Slater’s rules assign every electron to a group and every group a screening constant, which is a discretisation of something continuous. The 4s and 3d distributions overlap substantially — the radial distribution across the periodic table draws them together — and “inside” and “outside” are approximations at that scale.

Relativistic effects are absent and are not negligible for the third transition series. Gold’s colour and mercury’s liquidity are both consequences of a relativistic 6s contraction, and none of the arithmetic here would see them.

Where the fixed-order picture does work #

It would be a mistake to leave the impression that orbital energies are useless. They are extremely useful, and it is worth saying where.

Within one atom in one configuration, the ordering of one-electron energies is meaningful and predicts a great deal — which orbital an electron is removed from most easily, roughly what a photoelectron spectrum will look like, which orbitals are close enough in energy to mix. Koopmans’ theorem makes the first of those quantitative, with the approximation stated.

Across a series at fixed configuration, the trends are meaningful. The 3d energies in the table above fall steeply across the row, and that steep fall is real: it is why the late transition metals are so much harder to oxidise than the early ones.

What fails is the comparison between configurations, because that is where the repulsion terms differ and where the one-electron energies are no longer a complete accounting. The filling order is exactly such a comparison, which is why it is exactly the place the fixed order breaks.

That distinction — trends within a fixed description are reliable, comparisons between descriptions are not — is one that recurs. It is the same reason a Hückel delocalisation energy needs its reference state written down, as delocalisation sets out: a difference between two things computed the same way is meaningful, and a difference between two things computed differently is not.

The generalisation #

The failure is a standard one, in one of its sharpest forms: an approximate description was reified, and then its artefacts were treated as facts about nature.

There is no such thing as the energy of the 4s orbital of iron. There is a one-electron energy in a particular approximate scheme, which depends on the scheme; there is an ionisation energy, which is a measurable property of an atom and a cation together; and there is a filling order, which is a summary of which whole-atom configuration is lowest. Three quantities, related but not equal, and the fixed-order picture collapses them into one.

That is exactly the mistake orbitals are not where the electron is is about, applied to energies rather than to positions. An orbital is a construct in an approximation, and the properties it appears to have — a location, an energy, an ordering — are properties of the approximation until something measurable is identified.

The diagnostic that catches it is worth stating as a habit: when a rule and an observation conflict, ask whether the rule’s terms name anything measurable. Here “the energy of the 4s orbital” does not, and the conflict dissolves as soon as that is noticed.

Who found it, and when #

Bohr’s 1922 account of the periodic table introduced the building-up principle, and the German word Aufbau is his. The $n + l$ ordering rule is usually attributed to Madelung, who published it in 1936, and independently to Klechkowski in 1962; neither derived it, and it remains an empirical summary.

The observation that the 4s/3d ordering reverses on ionisation is old and its significance has been rediscovered repeatedly. The clearest modern statements are in a series of papers in the Journal of Chemical Education through the 1990s and 2000s — Vanquickenborne, Pilar, and Scerri among others — which take apart the fixed-order picture in the terms used here and note how persistently it survives in teaching.

Eric Scerri’s history of the periodic table records the deeper point: no derivation of the $n + l$ rule from quantum mechanics exists, and the periodic table’s structure, which is usually presented as a consequence of the Schrödinger equation, is only partly one. That is a more interesting situation than the textbook version and it is left out of almost all of them.

Still open: electron repulsion computed #

This essay and orbitals are not where the electron is between them mark the boundary of the one-electron picture from two directions: an orbital is not a place, and an orbital energy is not a fixed position in a sequence. What lies past that boundary needs electron repulsion computed rather than absorbed into a parameter, and that is a calculation deliberately not attempted here.