The order the shells fill

The hydrogen atom’s electron is a probability distribution rather than an orbit, and one feature of it is so familiar that it is rarely noticed as strange: every state with the same principal quantum number has the same energy. The 2s and the 2p are degenerate. The 3s, 3p and 3d are degenerate.

No other atom does that. And the reason hydrogen does is a peculiarity of the inverse-square force rather than anything about quantum mechanics — which is why the degeneracy is called accidental.

What the electron actually sees #

An electron in a many-electron atom does not experience the bare nucleus. Well inside the other electrons it sees nearly the full charge $Z$; well outside them it sees $Z$ less the number of electrons inside it, which for the outermost electron of a neutral atom is about one.

The simplest potential with that shape has one parameter:

$$V(r) = -\frac{1 + (Z-1)e^{-r/d}}{r}$$

in atomic units, with $d$ the screening length. It is a caricature — one screening length for every electron, no self-consistency, no exchange — and it is enough.

Solving the radial Schrödinger equation in it, by integrating outward and bisecting on the energy until the solution has the right number of nodes, gives the levels drawn above. The solver is checked against hydrogen first: with the screening switched off it returns $-1/2n^2$ to a thousandth of a hartree, and the $\ell$-dependence disappears entirely, as it must.

Switch the screening on and the degeneracy breaks. Always in the same direction: s below p below d, at every shell and every nuclear charge.

Where the ordering comes from #

The explanation is often given as “s orbitals are closer to the nucleus”, and that is exactly backwards.

At fixed $n$, the mean radius of a hydrogenic orbital falls as the angular momentum rises. The 3d orbital is the most compact of the three; the 3s is the most spread out, with the largest tail. On any measure of where the bulk of the electron is, d is nearer in.

What decides the energy is not the bulk. It is a small piece of probability at very small radius, where the screening has not yet taken effect and the nuclear charge is still nearly $Z$.

Inside the K shell — at a radius of about $1/Z$, which is 0.053 bohr for $Z = 19$ — the 3s orbital keeps 0.88 per cent of its probability, the 3p 0.11 per cent and the 3d essentially none.

The reason is the centrifugal barrier. The radial equation contains $\ell(\ell+1)/r^2$, which is a wall that grows as the angular momentum rises and keeps the wavefunction out of the region near the nucleus: $u \propto r^{\ell+1}$ at small $r$, so the probability goes as $r^{2\ell+2}$ and a d orbital is suppressed by $r^6$ where an s orbital is suppressed by $r^2$.

So less than one per cent of an s electron spends its time where the nuclear charge is nineteen rather than one, and that is enough to move its energy by half a hartree — fifteen electron-volts — relative to a d electron that never gets there. A property decided by a small fraction of the probability is the ordinary case in quantum mechanics rather than an exception, and it is worth being alert to: the bulk of a wavefunction is very often not what a given observable is sensitive to.

The crossing #

Follow 4s and 3d down the figure and they cross. Around a screening length of 0.4 bohr for $Z = 19$, the 4s level drops below the 3d one — and that crossing is the fourth row of the periodic table.

Potassium’s nineteenth electron goes into 4s rather than 3d, so potassium is an alkali metal sitting under sodium rather than the first of a new block. Calcium adds a second 4s electron. Only at scandium does 3d begin to fill, which is why the transition metals start where they do and why the fourth row is eighteen elements long rather than eight.

The rule of thumb that summarises this — fill in order of increasing $n + \ell$, and for equal $n + \ell$ in order of increasing $n$ — gets the sequence right for most of the table. It is a mnemonic rather than a theorem: it works because penetration matters more for low $\ell$ and because the shells are close together in the middle of the table, and it fails in about twenty places.

How the levels were found #

The energies quoted here are not from a table and the method is worth a paragraph, because a naive version of it produces confident nonsense.

The radial equation is integrated outward from near the origin, and the number of nodes in the solution counts how many eigenvalues lie below the trial energy. So bisecting on the node count converges on an eigenvalue: too many nodes and the energy is too high, too few and it is too low. That is a robust method needing no starting guess.

The trap is the grid. A linear grid fine enough to reach forty bohr at an affordable cost gives the innermost lobe of a $Z = 19$ orbital — which lives at about a twentieth of a bohr — three or four points, and the node count is then decided by rounding. A first attempt here did exactly that and returned 3p below 3s, which is the opposite of the ordering the whole figure exists to explain, with nothing to indicate anything was wrong: the curves were smooth and the energies plausible.

The cure is to integrate on a logarithmic grid. With $r = e^x$ and $u = \sqrt{r}\,y$ the equation becomes

$$y'' = \left[(\ell+\tfrac12)^2 + 2r^2(V - E)\right]y$$

which has no first-derivative term, so Numerov’s method applies with a constant step in $x$ — and a constant step in $x$ is a constant fraction of the radius, which is exactly the resolution an atom needs.

The second trap is that Numerov itself fails where $h^2Q$ approaches twelve, which at the very negative energies the bisection starts from happens far out where $Q$ grows as $r^2|E|$. The recurrence then oscillates and its sign changes are counted as nodes. Stopping the integration past the outer turning point fixes it and costs nothing, since there are no nodes out there to count.

Both failures produced plausible pictures, which is the point of recording them. A numerical method whose failure mode is a smooth wrong curve needs a check that is not a look at the curve — here, that switching the screening off must return $-1/2n^2$ with no dependence on $\ell$ whatever.

Why the ordering is not the ionisation order #

The most common confusion about all this is worth addressing directly, because the rule of thumb invites it.

If 4s is below 3d, an ionised transition metal ought to lose a 3d electron first. It does not: iron ionises to $\mathrm{Fe}^{2+}$ by losing both 4s electrons and keeping six in 3d.

The resolution is that the ordering is not a fixed property of the atom. The levels move as electrons are added, because each added electron changes the screening the others feel — which in the figure above is the horizontal axis. As $Z$ rises across the row the 3d level falls faster than the 4s, because 3d is more compact and therefore more sensitive to the nuclear charge, so by the middle of the row 3d is comfortably below 4s.

So the filling order and the ionisation order are answers to different questions asked of different atoms. The filling order concerns which configuration is lowest for the neutral atom at the start of the row; the ionisation order concerns which is highest for a particular atom in the middle of it. An account that treats the level diagram as fixed gets one of them wrong and cannot say why.

The table’s shape, in three sentences #

Everything about the periodic table’s outline follows from two facts, and it is worth putting them together.

The exclusion principle allows $2(2\ell+1)$ electrons in a subshell — two for s, six for p, ten for d, fourteen for f — which fixes the widths of the blocks. Penetration orders the subshells, which fixes the sequence in which the blocks appear.

So: rows of two and six give the first two periods; the appearance of d below the next s gives the eighteen-element rows and the transition block; the appearance of f gives the thirty-two-element rows and the lanthanides. Nothing else is needed, and no part of it is an empirical rule about chemistry.

That is worth pausing on, because the periodic table was discovered chemically, sixty years before any of the above existed, and its shape was an observed regularity nobody could account for. The two facts that explain it are a counting argument and a statement about a small piece of probability at small radius. The spectrum of an element was the intermediate evidence — the link between the chemistry and the structure — and it is why spectroscopy preceded quantum mechanics rather than following it.

The ordering worked out here decides which levels are available to broaden when atoms are brought together, and therefore which bands a solid has. That is the connection worth naming: a metal’s conduction band is whichever atomic level was outermost, so the filling order of an isolated atom propagates directly into the electrical properties of the solid it makes.

What the accident was #

It remains to say why hydrogen is degenerate at all, since nothing about quantum mechanics requires it.

An ordinary central potential gives energies depending on $n$ and $\ell$ separately. The Coulomb potential does not, and the reason is an extra conserved quantity: the Laplace–Runge–Lenz vector, which for an inverse-square force is conserved and which is the same conservation that makes a Kepler orbit close rather than precess.

The classical and quantum statements are the same statement. Closed orbits and degenerate levels both follow from the inverse-square law’s extra symmetry — a symmetry group larger than the rotations, which the quantum problem inherits — and both are destroyed by any departure from it. Screening is such a departure; so is the relativistic correction that splits hydrogen’s own lines; so is a general-relativistic correction to a planetary orbit.

That is a satisfying connection and a useful diagnostic. A degeneracy that is not obviously required by a symmetry usually is required by one that has not been noticed, and looking for it is generally more productive than calling it accidental.

The alkali metals’ spectra look like hydrogen’s and are not: their s, p and d terms are separated, so what appears as one line in hydrogen appears as several. That separation is the whole accident this essay is about, written into a spectrum — and it is how the ordering was first measured, decades before there was any theory to explain it.

The centrifugal barrier that keeps a d orbital out of the core is a statement about wavelength: a state with angular momentum must have a shorter wavelength to fit inside a given radius, and shorter wavelengths cost energy. That is why the accident happens — the 4s state penetrates the core and the 3d does not, so the 4s sees a less screened nucleus and drops below it.

The measurement that reads it off #

The splitting is not an inference; it is what an alkali metal’s spectrum shows directly.

Sodium’s yellow line comes from a 3p-to-3s transition, at 2.1 electron-volts. In hydrogen the corresponding states are degenerate and there is no such transition at all. The whole visible spectrum of an alkali metal exists because the degeneracy is broken, and the size of the splitting is the size of the penetration effect.

Quantifying it is done with the quantum defect. The levels of an alkali metal fit

$$E_{n\ell} = -\frac{1}{2(n - \delta_\ell)^2}$$

with $\delta_\ell$ a number depending on $\ell$ and almost not at all on $n$ — for sodium, 1.35 for s, 0.86 for p, 0.01 for d. That the defect is nearly independent of $n$ is the statement that the penetration happens in a small region the outer electron passes through on every orbit, so the correction it picks up is a fixed phase shift rather than a fixed energy.

The d defect being 0.01 is the sharpest possible statement of the argument: a d electron in sodium is, to within one per cent of a quantum number, a hydrogen electron. It never reaches the core, so it never learns that the core is there.

An atom’s levels are states in a well whose shape every added electron alters. That is what makes the ordering a computation rather than a rule: adding an electron changes the screening, which changes the shape of the well, which changes the ordering for the next one — so the sequence has to be worked out self-consistently, and the exceptions in the transition metals are where that calculation departs from the simple rule.

The screening constant, measured off a photograph #

The number this whole essay turns on — how much of the nuclear charge an electron actually sees — was measured directly, in 1913, and the measurement rearranged the periodic table.

Knock an electron out of an atom’s innermost shell and one from the next shell falls in, emitting an X-ray. Moseley photographed those lines for some forty elements and found that the square root of the frequency was a straight line in the atomic number:

$$\sqrt{f} \propto (Z - \sigma),$$

with $\sigma$ very close to one for the innermost transition. That constant is precisely this essay’s subject: an electron falling into the K shell is screened by the one other electron still there, and by essentially nothing else, because everything else is further out. For the next shell out the constant is about seven and a half, which counts the electrons inside it.

Three consequences followed at once and none of them was about screening. The relation is a clean function of $Z$, so it gives a count — and counting settled that the ordering quantity of the periodic table is nuclear charge rather than atomic weight, which resolved the two places where weight and chemistry disagreed. It showed that between aluminium and gold exactly three elements were missing, so chemists stopped looking for others. And it fixed the number of rare earths at fourteen, which had been argued over for decades.

Moseley was killed at Gallipoli two years later, at twenty-seven. The straight line is the sharpest experimental statement of the screening picture there is, and it was obtained before anybody knew why the orbitals were arranged the way they are.

The shell that screens badly, and what it does to a whole row #

Poor screening has a consequence further down the table that is worth following, because it is the same argument with the sign of the effect reversed.

The 4f orbitals fill across the lanthanides, and they are peculiar: compact enough to sit inside the already-occupied 5s and 5p shells, so adding an f electron barely alters what a chemist can reach. That is why fourteen consecutive elements have nearly the same chemistry and are notoriously difficult to separate — the electrons being added are buried.

But an electron that is buried is also an electron that screens the nucleus badly for anything outside it. Across the row the nuclear charge climbs by fourteen while the added f electrons compensate only partly, so the effective charge seen by the outermost electrons rises steadily and the atoms shrink. The contraction accumulated over the row is only ten or fifteen picometres more than the ordinary trend would give, and the consequences are out of all proportion to it.

The element immediately after the lanthanides, hafnium, ends up almost exactly the same size as zirconium, which sits directly above it and has thirty-two fewer protons. The two are so alike chemically that they occur together in every ore and were not distinguished until 1923. Separating them matters a great deal in one industry: hafnium absorbs neutrons avidly and zirconium hardly at all, so a nuclear fuel cladding made of zirconium must have its chemically near-identical companion removed to a few parts per million.

The same contraction runs along the whole third transition row, which is why tungsten, platinum and gold are so much denser than their lighter partners: the same number of electrons, packed into an atom that a badly-screening shell two rows earlier failed to keep large.

What the picture cannot show #

The screening model is a caricature and knows it. One exponential, one length, the same for every electron, no self-consistency and no exchange. It gets the ordering right and the numbers wrong by tens of per cent, and its value is that the ordering comes out of a calculation rather than being asserted.

The screening length is treated as a free parameter and it is not. In a real atom it is fixed by the electron distribution, which is fixed by the potential, which is what the screening length is supposed to describe — so drawing energies against it, as the hero figure does, is a survey of possible atoms rather than a curve any single atom lies on. The crossing it locates is real; the value of the parameter at which it happens is a property of the model.

A real atom’s potential is not a fixed function. Each electron moves in the field of the others, which are themselves moving in its field, and the honest treatment is self-consistent: guess a potential, solve for the orbitals, rebuild the potential, repeat. That is the Hartree–Fock method, and it changes the numbers considerably while leaving the picture in this essay intact.

Exchange is absent entirely. The antisymmetry of the many-electron state produces an additional energy that depends on the spins, which is what Hund’s rules are about and which is comparable in size with some of the splittings here.

The nucleus is a point. For the innermost orbitals of a heavy element it is not, and the finite nuclear size shifts s levels measurably — an effect that is a nuisance in atomic-clock work and a measurement of nuclear radii in muonic atoms, where the orbit is small enough to be inside the nucleus altogether.

And the relativistic corrections are not included. For heavy elements they are not small: the 6s orbital of gold is contracted enough by relativity to change the element’s colour, and the ordering of levels in the sixth row cannot be got right without them.

The ladder from here #

Later rungs on this anchor: the quantum defect and its near-independence of $n$, which is a phase-shift statement in disguise; the self-consistent field and what it changes; Hund’s rules and the exchange energy that orders the terms of a configuration; the relativistic effects that decide the chemistry of the heavy elements; and the periodic table’s block structure derived rather than described, including the places where the simple ordering fails.

The neighbouring ladders are the hydrogen orbital, whose degeneracy this essay breaks, and the exclusion principle, without which the shells would not fill at all. A spectrum as a subtraction is where the splittings measured here are read off.