INTERACTIVE COMPANION · EXPLAINER · CHEMISTRY

The Table Builder

A live model accompanying “Why the Periodic Table Is That Peculiar Shape”

Two models sit below. The first builds the periodic table one element at a time, in the order the filling rule dictates, and flags every element whose measured ground state disagrees with it. The second takes Slater's 1930 screening rules, computes an effective nuclear charge, and plots what that predicts against what spectroscopists actually measure.

Both run the same arithmetic as analysis/periodic-table-shape.py, reimplemented in JavaScript. On their default settings they reproduce the article's headline numbers, and every one of those can be checked by hand against the equations printed beside them.

Model 1. Build the table

The rule has three parts. Orbitals exist for every pair of quantum numbers (n, l) with l smaller than n. Each holds 2(2l + 1) electrons, which is where the block widths 2, 6, 10 and 14 come from. They fill in order of increasing n + l, and where two orbitals tie, the smaller n goes first.

Drag the slider and watch the shape appear. The cell being filled is outlined in ink, the configuration under construction is printed underneath, and a ring appears on any element whose measured ground state disagrees with the rule. Switch the ordering to the naive one and the lower half of the table comes apart, which is the fastest way to see how much work n + l is doing.

Elements placed 118 of 118 Stop at 2, 10, 18, 36, 54, 86 and 118 and you land on a noble gas every time, because those are exactly the points where a p subshell closes. Stop at 20 and then step to 21 to watch the d-block open in the middle of a row that had already started. Filling order Madelung, n + l The naive order fills 3d before 4s, so element 19 comes out a transition metal instead of potassium, and from element 55 onward the noble gases land in the wrong places. Compare the period lengths in the readout against 2, 8, 8, 18, 18, 32, 32. Highlight rings on With the rings on and the slider at 118 you should count 20, all of them in the d-block or the f-block. Set the slider to 56 and there are eight so far, every one a d-block metal. Build speed 6 elements per second Press play below and let it run from hydrogen. The pauses you feel are real: the slow stretches are the f-blocks, fourteen elements wide and chemically almost identical.

current: – configuration: – period lengths so far: – anomalies flagged: –

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Block widths: 2 / 6 / 10 / 14, from 2(2l + 1)
Period lengths: 2, 8, 8, 18, 18, 32, 32
Period ends: 2, 10, 18, 36, 54, 86, 118, all noble gases
Honest table width: 32 columns; the printed one cuts 14
Madelung violations: 20 of the first 103 elements, 19.4%
Violations in s or p: 0 out of 62

Model 2. What effective nuclear charge predicts, and what it misses

An outer electron does not feel the full nuclear charge, because the electrons between it and the nucleus get in the way. Slater's 1930 recipe puts a number on that: subtract 0.35 for every other electron in the same shell, 0.85 for every electron in the shell below, and 1.00 for anything deeper. What is left is the effective nuclear charge:

$$Z_{\text{eff}} \;=\; Z - S$$

Feed that into a hydrogen-like energy expression and you can predict a first ionisation energy by differencing an atom against its own cation, or an atomic radius from n*²/Z_eff with a single fitted scale constant. Both predictions are plotted below against published measurements, with the gap drawn as a vertical bar so you can see where the model is failing rather than only how much.

Property ionisation energy Ionisation energy is where the model hurts. Radius is where it does surprisingly well. Effective nuclear charge is the input to both, so switching to it shows you the error before it gets amplified. Series period 3 Across a period the nuclear charge climbs while the shell stays the same, so atoms shrink and get harder to ionise. Down a group the shell number climbs, so atoms grow and get easier. The model gets both directions right and only one of the magnitudes. Same-shell screening 0.35 Slater's own value is 0.35, and that is the default. Raising it makes electrons shield each other more, which flattens the whole period. Try 0.45 on period 3 and watch the error fall, then remember that a fitted constant is no longer a prediction. Inner-shell screening 0.85 0.85 is Slater's value for an electron one shell in. It is too generous for the third shell, which is most of why period 3 comes out 38.4% too high. Push it to 0.95 and period 3 improves while period 2 gets worse, which is the whole problem with one constant for every shell.

points plotted: – mean absolute error: – worst element: – direction: –

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Three numbers you can check by hand

Default settings on both models are meant to land on the article's figures. Here are three of them, worked out on paper, so you can tell whether the page is lying.

One: the Slater step is exactly 0.650 per element. Go from nitrogen to oxygen. Nitrogen is 1s² 2s² 2p³, so a 2p electron is screened by six others in its own shell at 0.35 each, plus two 1s electrons at 0.85 each: S = 6(0.35) + 2(0.85) = 2.10 + 1.70 = 3.80, giving Z_eff = 7 − 3.80 = 3.90. Oxygen adds one proton and one same-shell electron: S = 7(0.35) + 1.70 = 4.15, so Z_eff = 8 − 4.15 = 4.55. The difference is 1 − 0.35 = 0.65, and it will be 0.65 for every step in every period, which is exactly why the prediction line in Model 2 is perfectly straight and the measurements are not.

Two: hydrogen must come out at one rydberg. With a single electron there is nothing to screen, so S = 0 and Z_eff = 1, and Slater's effective principal quantum number for the first shell is 1. The energy is then 13.6057 × (1/1)² = 13.6057 eV, which is the exact hydrogenic answer. Measurement says 13.5984 eV. Select group 1 in Model 2 and read the leftmost point. A model that failed this check would be broken arithmetic rather than crude physics.

Three: aluminium is off by 79%. Set the property to ionisation energy and the series to period 3. The readout names aluminium as the worst element, at +79.4%: predicted 10.741 eV against a measured 5.986 eV. The reason is visible in the equation. Slater gives the third shell an effective principal quantum number of exactly 3.0 and lets the eight second-shell electrons screen at only 0.85 apiece, and both choices leave aluminium's outermost 3p electron far more tightly bound in the model than it is in reality.

What to try, in order

Set Model 1 to 20 elements and step it forward one at a time to 21. The table has been growing tidily along a row and then abruptly starts a new block in the middle. That is the n + l inversion happening in front of you, and it is the origin of every transition metal.

Then switch the filling order to naive and watch the period lengths in the readout stop matching the real table. Element 19 becomes a transition metal, the period ends drift to 2, 10, 18, 36, 68, 100 and 118, and every noble gas from xenon down lands in the wrong column. The naive rule is the one most people would guess, and it goes wrong at element 19.

In Model 2, choose ionisation energy across period 2 and look for the two places where the measured line dips while the prediction keeps climbing. Boron is the first, because its outermost electron is the first into a 2p orbital and is held less tightly than beryllium's 2s. Oxygen is the second, because its fourth 2p electron is the first forced to pair inside an orbital that already holds one. Slater's rules have no way to see either, since they never ask which orbital the electron occupies.

Finally, drag the inner-shell screening from 0.85 to 0.95 with period 3 selected. The mean error collapses. Then switch to period 2 without touching the slider and watch it get worse. One constant cannot serve both shells, which is the honest summary of what a 1930 rule of thumb can and cannot do.