Why the Periodic Table Is That Peculiar Shape

The Shape Is a Picture of an Answer

Somebody hung it on the classroom wall and nobody explained why it looks like that. Ask any member of this club to sketch a periodic table from memory and you get roughly the same outline every time: two thin columns on the left, a six-wide slab on the right, a ten-wide block wedged in between that starts partway down instead of at the top, and two orphaned rows floating underneath the whole thing like a footnote the typesetter didn't want to commit to. Ask why the middle block is ten columns wide rather than eight or twelve. Ask why it starts in the fourth row and not the second. Most people, including most of us before this piece, have no answer beyond "that's just the shape it is."

It isn't just the shape it is. Every width and every offset in that outline follows from a short, finite list of facts about what an electron is permitted to do, and once those facts are in hand the shape stops being a thing to memorise and becomes the only shape the arithmetic could have produced. That is the first half of this piece: a construction, run in code, with nothing borrowed from a chemistry textbook. The second half asks a harder question of the same machinery — not just the shape, but the sizes and the ionisation energies of the atoms that live inside it — using a screening rule from 1930 that gets some of it right and a great deal of it wrong, honestly reported either way.

Four Numbers, and Only Four

An electron bound to an atom carries four labels. The principal quantum number n sets the shell. The angular momentum quantum number l sets the orbital's shape and is restricted to $0 \le l \le n-1$ — its four lowest values carry the old spectroscopic letters s, p, d and f, a naming fossil from nineteenth-century spectral lines described as sharp, principal, diffuse and fundamental, which is why nobody agrees on what the letters stand for and everybody uses them anyway. The magnetic quantum number $m_l$ sets the orbital's orientation in space and runs from $-l$ to $+l$, which is $2l+1$ distinct values. Spin contributes exactly two more.

Pauli's exclusion principle forbids any two electrons in one atom from sharing all four labels. Combine that with the count of available $m_l$ values and spin states and the number of seats in a subshell of angular momentum l is fixed with no freedom left anywhere:

Put in the numbers and out come 2, 6, 10 and 14 for $l = 0, 1, 2, 3$. Count the columns of an s-block, a p-block, a d-block and an f-block on any periodic table ever printed. They are 2, 6, 10 and 14, in that order, and the reason is this equation and nothing else — not tradition, not a printer's convenience, not a historical accident of who discovered which element first.

Why 4s Fills Before 3d

Block widths answer how many columns. They say nothing about the order the rows appear in, and the order is where the real argument lives. The easy cases go by shell number: 1s fills before 2s, 2s before 2p, and nobody needs convincing of any of that. The interesting case is 4s against 3d, because naively you would expect shell 3 to finish completely — 3s, 3p, 3d — before shell 4 opens at all. That is not what happens, and the rule that gets it right is called the Madelung rule, or the $n+l$ rule: orbitals fill in order of increasing $n+l$, and any tie is broken in favour of the smaller n.

For 4s, $n+l = 4+0=4$. For 3d, $n+l=3+2=5$. Four is less than five, so 4s fills first, a full row before 3d gets a single electron. The physical story behind the rule is penetration: an s orbital, unlike a d orbital of similar overall size, has real probability density close to the nucleus, so a 4s electron spends part of its time deep inside the atom where it feels much more of the bare nuclear charge than the shielding from inner shells would suggest, and that extra binding is enough to drop it below 3d for a while. It is a real and useful heuristic. It has also never been derived from the Schrödinger equation in closed form — a point we return to near the end of this piece, because it matters for how much weight the rule can bear.

The Arithmetic, and a Width Nobody Prints

Put the two rules together — $l<n$ for which orbitals exist, $2(2l+1)$ seats in each, fill by increasing $n+l$ — and the periodic table's row lengths fall out with no further assumptions. Period one is 1s alone, length 2. Period two is 2s then 2p, length $2+6=8$. Period three is 3s then 3p, also length 8, because $3+2$ (for 3d) is larger than $4+0$ (for 4s) and the d orbitals are not yet invited. Periods four and five each pick up a d-block partway through and come out to $2+10+6=18$. Periods six and seven each pick up an f-block as well and come out to $2+14+10+6=32$.

Run the whole thing out to element 118 and the period lengths generated are 2, 8, 8, 18, 18, 32, 32, matching the accepted real values exactly, and the elements that close each period land at 2, 10, 18, 36, 54, 86, 118 — helium, neon, argon, krypton, xenon, radon, oganesson, the entire noble-gas column, hit on the first and only attempt, with no chemical fact used as an input anywhere in the derivation. Sum those seven period lengths and you get 118, which is also the number of elements presently assigned a name and a place. That is not a coincidence built into the arithmetic; it is the arithmetic correctly describing where the naming stops.

Laid out with every block kept in line rather than folded, the table is $2+14+10+6=32$ columns wide. Nobody prints a 32-column periodic table, for the entirely reasonable reason that it would not fit on a wall, so the fourteen f-block columns are lifted out of periods six and seven and set underneath as two detached rows. That single typographic decision, not any fact about electrons, is why the table on the wall is 18 columns instead of 32, and why lanthanum through ytterbium and actinium through nobelium look, to a first glance, like they don't belong to the main body of the chart at all.

Twenty Elements That Do Not Care

A rule this clean invites a check: does it actually predict every element's electron configuration, or only the shape of the table in the aggregate? We ran the construction across every element from hydrogen to lawrencium, Z = 1 to 103, and compared the predicted ground-state configuration against the measured one, subshell by subshell.

It gets 83 right and 20 wrong. Every single failure sits in the d-block or the f-block: 11 in the d-block, 9 in the f-block, and not one anywhere in the s-block or the p-block. Those two blocks together hold 44 elements over this range — 14 s-block, 30 p-block — and the rule bats a thousand on all of them. The violations, laid out in full:

Nine of the twenty move exactly one electron out of an ns shell and into the $(n-1)$d shell beneath it. Palladium is the outlier among the outliers: it moves two electrons and empties 5s completely, leaving krypton's configuration plus a bare 4d¹⁰ — the only element on the list, and one of only a handful in the whole table, with no valence electrons in its outermost shell at all. Lawrencium breaks the rule in a different direction than everything else on the list: instead of an extra d electron, it puts its last electron in 7p, a jump the Madelung construction has no mechanism to anticipate since it never considers p orbitals as competitive at that point in the filling order.

The textbook explanation for cases like chromium and copper is that half-filled and fully filled subshells carry extra stability, and that explanation is not wrong so much as incomplete: it is a real effect, rooted in the exchange energy between parallel-spin electrons, but it is not decisive on its own. Tungsten sits directly below molybdenum in the same group and declines the half-filled 5d⁵ configuration that molybdenum takes; niobium, ruthenium and rhodium all break the rule while landing on configurations that are neither half-filled nor full. What actually decides the winner in each case is the total energy of the whole atom, which depends on orbital shapes and electron–electron repulsion in ways too entangled to reduce to a slogan about "extra stability," however often that slogan gets repeated.

An early pass of the analysis script flagged one more violation than the twenty listed above, on account of a string-formatting slip in how one configuration was compared; the script now cross-checks its generated Madelung prediction against a hand-typed copy of every anomalous tail before it will print the table at all, specifically so an error like that cannot survive quietly. The count that ships in this article, 20, is the one that passed that check.

What Everybody Gets Wrong About 4s and 3d

Here is a sentence that shows up in a great many introductory courses: "4s is lower in energy than 3d, which is why it fills first." It is backwards for exactly the elements it is usually invoked to explain. From scandium onward, 3d sits below 4s in energy, not above it, and stays there for the rest of the transition series. The 4s²3dⁿ ground state still wins for most of that series not because 4s is lower, but because 3d orbitals are small and tightly packed, and cramming a second electron into one costs enough repulsion energy to tip the balance back toward 4s. The giveaway that the popular version is wrong is ionisation: transition metals lose their 4s electrons first, every time, which makes no sense at all if 4s were really sitting below 3d.

The wrong version survives for a boring reason: it is one clause long, requires no caveats, and gives the right answer for the entire s-block and p-block, which is most of what an introductory course covers. It only fails once you reach the one place — the transition series — where the sentence was supposedly explaining something in the first place.

A second claim worth being precise about is that the shape of the table is "explained by quantum mechanics." That is half right. The block widths follow from Pauli's principle with no wiggle room whatsoever, as Figure 1 shows in full. The $n+l$ ordering that decides which block comes when has never been derived directly from the Schrödinger equation; it was noticed empirically, it works remarkably well, and it has real physical motivation in orbital penetration — but a motivation is not a derivation, and the difference matters when the rule is about to be pushed as hard as we push it next.

Effective Nuclear Charge From a 1930 Rule of Thumb

Everything so far has been about which orbital fills, not about how tightly an electron in it is actually held. For that we need to know how much of the nuclear charge an outer electron really feels, given that the electrons closer in get in the way. In 1930 John C. Slater proposed a simple bookkeeping recipe for exactly this: subtract 0.35 from the nuclear charge for every other electron sharing the same shell, 0.85 for every electron one shell in, and 1.00 for every electron deeper than that. What's left is the effective nuclear charge:

The rule has an immediate, almost mechanical, consequence: because every step across a period adds one proton and exactly one same-shell electron worth 0.35, $Z_{\text{eff}}$ rises by precisely 0.650 per element, in every period, without exception, by construction rather than by any physics. Self-consistent-field calculations — the actual numerical solutions to the many-electron problem, computed independently by Clementi and Raimondi in 1963 — rise by 0.640 per element on average across period 2 and 0.608 across period 3. Slater's rule, a piece of 1930s arithmetic done with a pencil, gets within a few percent of a calculation that needed a computer three decades later.

Before trusting the rule on anything interesting, check it where the answer is exact. Hydrogen has one electron, so there is nothing to screen: $S=0$, $Z_{\text{eff}}=1$, and the predicted first ionisation energy collapses to the pure hydrogenic formula, 13.6057 eV, against a measured value of 13.5984 eV — agreement to 0.05%. He⁺ is the other one-electron case available, an ion rather than a neutral atom but no harder for the arithmetic: the model gives 54.4228 eV against a measured 54.4178 eV, agreement to 0.009%. A model that failed either of these checks would be broken code, not crude physics, and it doesn't fail them.

Ionisation Energy: Where the Model Breaks

Effective nuclear charge feeds directly into a predicted binding energy, and differencing an atom against its own cation gives a predicted first ionisation energy with no further assumptions. Run that across the first twenty elements and the mean absolute error is 20.9%, but that average hides a sharp split: 11.9% across period 2 and 38.4% across period 3, with every single period-3 prediction coming out too high. The worst case is aluminium, predicted at 10.741 eV against a measured 5.986 eV, an error of +79.4%. The reason is visible directly in the constants Slater chose: the third shell gets an effective principal quantum number of exactly 3.0, and each second-shell electron screens it at 0.85, and both numbers are simply too generous once a third shell is actually involved.

The model also gets the wrong shape in a place where getting the wrong number would have been forgivable and getting the wrong shape is not. Across period 2, the model predicts a rise in ionisation energy at every single step: 7 of 7. The real data rises at only 5 of 7, dipping twice — from beryllium to boron (9.323 → 8.298 eV) and from nitrogen to oxygen (14.534 → 13.618 eV). Both dips are orbital effects that Slater's rule has no way to represent, because Slater's rule only counts electrons, never asking which orbital they sit in. Boron's outermost electron is the first to enter a 2p orbital, which is higher in energy than beryllium's already-full 2s, so it comes out easier to remove despite the extra proton. Oxygen's fourth 2p electron is the first forced to share an orbital that already holds one electron, and the repulsion of that pairing makes it easier to remove than nitrogen's unpaired third. A rule built entirely out of electron counts, with no notion of orbital shape, cannot see either effect coming, and it doesn't.

Atomic Radius: the Better Half of the Model

Hydrogenic scaling predicts an atomic radius proportional to $n^{*2}/Z_{\text{eff}}$. Fitting a single scale constant k by least squares against Slater's own 1964 empirical radii, over lithium through fluorine and sodium through chlorine, returns k = 49.79 pm — within 6% of the Bohr radius, 52.92 pm, a number the fit was never given and had no way to reconstruct except by the physics actually being sound. The mean absolute residual against those same 1964 radii, across all fourteen elements in the fit, is 11.2 pm.

Where the ionisation-energy model failed on shape as well as magnitude, the radius model gets the shape right everywhere and the magnitude wrong in exactly the place the ionisation model also struggled. Across period 2, from lithium to fluorine, the model contracts atoms by a factor of 4.00; Slater's 1964 empirical radii contract by 2.90 and Cordero's 2008 covalent radii by 2.25. Across period 3, sodium to chlorine, the model contracts by 2.77 against 1.80 (Slater '64) and 1.63 (Cordero '08). The direction is always correct; the model simply shrinks the atoms harder than they actually shrink, especially toward the halogen end of a period, because covalent radii are contact distances between bonded atoms and real atoms near the right-hand edge of a period essentially stop getting smaller, something a bare $1/Z_{\text{eff}}$ law has no mechanism to represent.

Gold, Mercury, and the Speed of Light

Every calculation to this point has been non-relativistic, which is a fine approximation for light atoms and a visibly bad one for heavy ones. A crude way to see why: model a 1s electron as an old-fashioned Bohr orbit with speed $v/c = Z\alpha$, where $\alpha$ is the fine-structure constant. For hydrogen that speed is a negligible fraction of light speed and the resulting orbital contraction is 0.0%. For iron it is already 1.8%. By silver, $Z=47$, it reaches 6.1%, and by gold, one row down at $Z=79$, it has jumped to 18.3% — a factor of 3.0 increase for climbing a single row of the table. Mercury contracts by 18.8%, lead by 19.9%, uranium by 25.9%, and the crude estimate for oganesson, element 118, comes out at 49.2%, meaning the naive formula is starting to describe an electron moving at a large fraction of light speed at all.

That contraction does not stay confined to 1s. Every s orbital has nonzero probability density right at the nucleus, so every s orbital in a heavy atom contracts and drops in energy, and because a contracted, lower s orbital screens the nucleus more effectively, the d and f orbitals sitting further out expand and rise in energy in response. In silver the gap between the highest filled d orbital and the s orbital above it is 3.9 eV, which corresponds to a photon of 318 nm — ultraviolet, outside the visible band of roughly 380 to 750 nm, so silver reflects the whole visible spectrum about equally and looks the neutral grey-white everyone expects a metal to look. In gold, the relativistic contraction has narrowed that same gap to 2.4 eV, a photon of 517 nm, which sits inside the visible band: gold absorbs blue and violet light and reflects the rest, and what gets reflected reads as yellow. This is, in outline, the entire chemical reason gold and silver are different colours, and it survives from a Bohr-model estimate crude enough to fit in one line of code.

The same relativistic push also disturbs the filling order itself in the table's bottom rows — lawrencium's outermost electron measured directly in a 7p orbital rather than the 6d the non-relativistic Madelung construction predicts for it is one visible symptom of exactly this effect, and it is why the anomalies cluster so heavily toward the high-Z end of Figure 4's ringed cells.

The Strongest Case That We Have This Backwards

Four objections deserve to be taken seriously rather than waved off, because each one is correct as far as it goes.

The ordering rule is observed, not derived. True, and said plainly above: the $n+l$ rule was noticed by matching outcomes, not deduced from the Schrödinger equation, and it remains without a first-principles proof to this day. What we derived is the block widths; the block order is an excellent empirical pattern with a physical rationale and no proof, and those are two different epistemic statuses that get flattened together far too often.

An "electron configuration" is a simplification to begin with. Also true. Real many-electron wavefunctions are superpositions of configurations, not single assignments of electrons to orbitals, so what the Madelung rule predicts is the dominant term in an approximate expansion, not a literal fact about where each electron sits. This makes the anomaly table more interesting rather than less: the elements that break the simple picture tend to be exactly the ones whose true wavefunctions mix configurations most strongly.

Where group 3 belongs is still disputed. The composition of group 3 — whether it is scandium, yttrium, lanthanum, actinium, or scandium, yttrium, lutetium, lawrencium — has been argued since at least 1982, and a 2021 IUPAC-convened discussion reported that it could not find an objective criterion to settle the question outright. Figure 4 above places lutetium and lawrencium in the main d-block row and lanthanum and actinium in the footnote, which is one defensible reading of the Madelung construction and not the only table layout in circulation.

The 18-column layout is a typographic choice, not a physical one. Also true, and said plainly in the section on the honest 32-column shape: several self-consistent alternative layouts exist, including Janet's left-step table, and they look nothing like the wall chart. This is the weakest of the four objections in one specific sense — that several different projections of the same underlying object exist does not make the underlying object any less real, any more than a globe stops being a sphere because it can be drawn as several different flat maps.

What survives all four objections together is a claim with three separate parts rather than one flattened one: the block widths are physics, settled by Pauli with no room left over; the filling order is an excellent rule that physics has not yet fully derived; and the printed layout is a compromise with the width of a piece of paper. Teaching all three as a single undifferentiated fact to be memorised is exactly the habit this piece started out trying to push back against.

References

python periodic-table-shape.py > periodic-table-shape-output.txt

1. 1.

   Pauli, W. (1925). Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren. Zeitschrift für Physik 31, 765–783. doi:10.1007/BF02980631

2. 2.

   Slater, J. C. (1930). Atomic shielding constants. Physical Review 36, 57–64. doi:10.1103/PhysRev.36.57

3. 3.

   Clementi, E. & Raimondi, D. L. (1963). Atomic screening constants from SCF functions. The Journal of Chemical Physics 38, 2686–2689. doi:10.1063/1.1733573

4. 4.

   Slater, J. C. (1964). Atomic radii in crystals. The Journal of Chemical Physics 41, 3199–3204. doi:10.1063/1.1725697

5. 5.

   Cordero, B., Gómez, V., Platero-Prats, A. E., Revés, M., Echeverría, J., Cremades, E., Barragán, F. & Alvarez, S. (2008). Covalent radii revisited. Dalton Transactions 21, 2832–2838. doi:10.1039/b801115j

6. 6.

   Kramida, A., Ralchenko, Yu., Reader, J. & NIST ASD Team (2024). NIST Atomic Spectra Database (version 5.12). National Institute of Standards and Technology, Gaithersburg, MD. doi:10.18434/T4W30F

7. 7.

   Schwerdtfeger, P., Smits, O. R. & Pyykkö, P. (2020). The periodic table and the physics that drives it. Nature Reviews Chemistry 4, 359–380. doi:10.1038/s41570-020-0195-y

8. 8.

   Löwdin, P.-O. (1969). Some comments on the periodic system of the elements. International Journal of Quantum Chemistry 3(S3A), 331–334. doi:10.1002/qua.560030737

9. 9.

   Scerri, E. R. (1989). Transition metal configurations and limitations of the orbital approximation. Journal of Chemical Education 66, 481–483. doi:10.1021/ed066p481

10. 10.

    Vanquickenborne, L. G., Pierloot, K. & Devoghel, D. (1994). Transition metals and the aufbau principle. Journal of Chemical Education 71, 469–471. doi:10.1021/ed071p469

11. 11.

    Melrose, M. P. & Scerri, E. R. (1996). Why the 4s orbital is occupied before the 3d. Journal of Chemical Education 73, 498–503. doi:10.1021/ed073p498

12. 12.

    Seaborg, G. T. (1945). The chemical and radioactive properties of the heavy elements. Chemical and Engineering News 23, 2190–2193. doi:10.1021/cen-v023n023.p2190

13. 13.

    Jensen, W. B. (1982). The positions of lanthanum (actinium) and lutetium (lawrencium) in the periodic table. Journal of Chemical Education 59, 634–636. doi:10.1021/ed059p634

14. 14.

    Scerri, E. R. (2021). Provisional report on discussions on group 3 of the periodic table. Chemistry International 43(1), 31–34. doi:10.1515/ci-2021-0115

15. 15.

    Sato, T. K. et al. (2015). Measurement of the first ionization potential of lawrencium, element 103. Nature 520, 209–211. doi:10.1038/nature14342

16. 16.

    Pyykkö, P. & Desclaux, J. P. (1979). Relativity and the periodic system of elements. Accounts of Chemical Research 12, 276–281. doi:10.1021/ar50140a002

17. 17.

    Christensen, N. E. & Seraphin, B. O. (1971). Relativistic band calculation and the optical properties of gold. Physical Review B 4, 3321–3344. doi:10.1103/PhysRevB.4.3321

18. 18.

    Calvo, F., Pahl, E., Wormit, M. & Schwerdtfeger, P. (2013). Evidence for low-temperature melting of mercury owing to relativity. Angewandte Chemie International Edition 52, 7583–7585. doi:10.1002/anie.201302742

19. 19.

    Jerabek, P., Schuetrumpf, B., Schwerdtfeger, P. & Nazarewicz, W. (2018). Electron and nucleon localization functions of oganesson: approaching the Thomas-Fermi limit. Physical Review Letters 120, 053001. doi:10.1103/PhysRevLett.120.053001