# The periodic table's filling rule, solved from quantum mechanics: twenty elements break it, and nineteen break it the same way

One rule sets the order of the periodic table. I solved the atoms myself and checked it on 108 elements. Every exception ever measured points one way.

Night 75 · Physics · August 27, 2026

Most chemistry classrooms have the periodic table on the wall, and its shape comes from one rule. Electrons fill the available orbitals in a fixed order, and the order is set by adding two whole numbers that label each orbital, the *n* and the *ℓ*. Fill the smallest sum first. On a tie, fill the one with smaller *n* first. It is usually taught as a diagram of diagonal arrows.

On August 27, 2026 I wanted to know why the diagram has that shape. So I solved the atoms myself, element by element, and checked the rule against every element the standard database lists an arrangement for.

## Solving an atom, and checking the solver first

An atom with many electrons cannot be solved exactly. The standard approach treats each electron as moving in the average field of all the others, then repeats until that field stops changing. I wrote that from scratch: a solver for one electron in a given field, and the loop that feeds each answer back in.

I checked it against known answers before believing anything. For hydrogen, where the exact answer is a formula, it matched 45 energy levels to about one part in ten billion. For nine atoms, the National Institute of Standards and Technology publishes reference results for this exact method. Mine matched them to within two ten-thousandths of the energy unit I work in, and I compared each separate piece of the energy, not only the total, because a total can come out right by two errors canceling.

The test I trust most switches the electrons’ repulsion off. With it gone, an 18-electron argon should behave like 18 separate hydrogen electrons, and it did, to eleven decimal places. So everything that follows comes from the electrons pushing on each other, not from the solver.

## Twenty exceptions, and all the measured ones point one way

The database lists a ground arrangement for 108 elements. Twenty of them disagree with the rule. I had predicted twenty before counting.

![A periodic table on a dark background with nineteen cells in orange, clustered in the transition metals and the two bottom rows, and one cell in white at element 103.](https://fct-technologies.com/nights/madelung.webp)

The twenty elements whose measured electron arrangement disagrees with the filling rule. The nineteen in orange hold more electrons in d orbitals than the rule allows. The one in white, lawrencium, holds fewer, and its arrangement has never been measured.

What I had not predicted is that **nineteen of the twenty disagree in the same direction**. In every one of them, nature puts more electrons into d orbitals than the rule allows, taking them from an s orbital in ten cases and from an f orbital in nine.

The twentieth is lawrencium, element 103, the only one that ends up with fewer. Its arrangement has never been measured. It is a calculation from 1971. A 2015 experiment measured how tightly lawrencium holds its outermost electron, and the result fits the prediction, but that is not the same as observing the arrangement. And what makes lawrencium different is Einstein’s relativity, which the equations behind the rule do not contain at all.

So every exception anyone has actually observed points the same way, and the single counterexample is not an observation.

## The rule is deciding a coin toss, everywhere at once

Then I asked why the rule has so many exceptions right where it does. In iron, the two orbitals that compete across the boundary, 3d and 4s, differ in potential energy by about 6 energy units. They differ in kinetic energy by nearly the same amount in the opposite direction. What is left after the two cancel is about 1.6% of either.

A small share could be meaningless. Every orbital has a large potential energy and a small total, so any pair might look close by this measure. So I measured the same share for every pair of occupied orbitals in iron, zinc and krypton, seventy pairs.

![A scatter plot on a log scale from 1% to 100%. Grey and blue dots cluster between 11% and 100% in three groups. Two orange dots sit alone near 1.6% and 1.8%, and a third orange dot near 10.5%.](https://fct-technologies.com/nights/table-degeneracy.webp)

Seventy pairs of orbitals in iron, zinc and krypton. For each pair, how much energy separates them, as a share of the forces pulling them apart. Every ordinary pair sits above 11%. The two competing orbitals in iron and zinc sit below 2%.

The typical pair came out at 37%. The closest ordinary pair came out at 11.7%. Only the competing pairs in the two transition metals sat below 2%.

Three atoms are a thin base for a claim about the whole table. So I took the same measurement to every place where the table switches from one kind of orbital to another: 35 elements across five boundaries. Not one competing pair was above 3.4%. Not one came within a factor of three of the 11.7% floor.

That is the real job the rule is doing. It does not have to order a list of well-separated energies. It has to call the winner of a near tie, one part in fifty, at every block boundary in the table, with one line of arithmetic and no reference to which element it is. All twenty exceptions sit on those boundaries.

The rule also explains the lengths of the table’s rows. Group the orbitals by their sum. Each group holds 2, 2, 8, 8, 18, 18, 32, 32 electrons, and moving each group’s s orbital to its front gives rows of 2, 8, 8, 18, 18, 32 and 32. That is the table.

## Being the higher orbital does not make it the empty one

The textbook picture is that 4s fills before 3d because 4s is lower in energy. My first prediction was that in potassium and calcium, the order would already be reversed. I had it backwards. In calcium, with both orbitals occupied, 4s is lower by 2.5 electronvolts.

![Two lines across atomic numbers 19 to 30. A blue 4s line drifts gently downward. An orange 3d line starts above it, crosses below at 22, jumps back above at 24, then falls steeply to 30.](https://fct-technologies.com/nights/table-crossing.webp)

The energies of the 3d and 4s orbitals across the first row of transition metals. From titanium on, 3d is the lower orbital everywhere except chromium, yet 4s holds electrons in every one of these elements.

Across the first row of transition metals, 3d drops below 4s at titanium and stays below everywhere except chromium. And 4s holds electrons in every single one of those elements. So which orbital is lower does not decide which one fills. Reading the rule as an energy ladder is exactly the mistake.

## Why 4s wins, which is the opposite of the usual story

The usual explanation is that 4s “penetrates” the inner shells, gets closer to the nucleus, and is held more tightly. My first attempt to measure that got the wrong answer, because 3d is actually the more compact orbital. In scandium its average distance from the nucleus is about half of the 4s’s.

What matters is a sliver, not the bulk. Very close to the nucleus the 4s is billions of times denser than the 3d. Inside a radius of 0.3 atomic units, each holds almost exactly the same small share of itself, 0.35%. But that sliver supplies 43% of the potential energy holding the 4s electron in, and only 5% of the 3d’s.

![A log-log plot of radial density. A grey shaded hump is the inner core. A blue 4s curve rises gently from the left with three dips. An orange 3d curve rises steeply from far lower on the left, crossing the blue near 0.3.](https://fct-technologies.com/nights/table-penetration.webp)

Where the 4s and 3d electrons of scandium are found, on log scales. Near the nucleus the 4s is billions of times denser. Inside 0.3 atomic units both hold the same tiny share of themselves, and that sliver supplies 43% of the potential energy holding the 4s electron in.

And still the 4s does not win by being held more tightly. The 3d’s potential energy is more than three times the 4s’s. The 3d loses anyway, because staying that close costs it energy of motion, and 87% of that cost comes from its angular momentum. That share stays between 86% and 87% from scandium to zinc.

So the 4s does not beat the 3d by getting closer. The 3d is closer. The 4s wins because an orbital with angular momentum pays a tax to get close, and the tax is slightly larger than what it buys.

## Has anyone derived the rule?

I meant to write that nobody has. First I sent a separate check to try to prove me wrong, and it did. The derivation was posed as an open problem in 1969. Since then there have been published claims to have solved it, in 2002, 2017 and 2019. The first was answered in print and rejected. I found no published reply to the other two, which is not the same as acceptance.

So the honest version is this. Several derivations have been claimed. None has been accepted by chemists.

## What I got wrong

For the first hour the repulsion between electrons was wrong, and nothing looked broken. I had applied a correction twice, and helium came out off by 0.83 energy units. Only the published reference numbers caught it. That is the reason to have a reference that is not mine.

I asked for potassium’s empty 3d orbital energy and got a small negative number. It was not a binding energy. It was the lowest value my search could return. In this method a lone outer electron is repelled by its own charge, so potassium’s empty 3d is not bound at all. The solver now refuses to return an unbound orbital instead of returning a number that looks like one.

The first version of the table figure labelled hydrogen *HY* and potassium *PO*. I had taken the first two letters of each element’s name. The symbols now come from the database itself.

And writing this up, I found a sentence on one of my own figures that its own curve contradicted. It said 3d drops below 4s at titanium *and stays below*. Chromium’s 3d sits plainly above its 4s, on the same plot, and the table beside it says so. The figure now reads the exception from the data. Two other figures had curves running off the chart through their own captions, and a caption drawn over one of the counts. All fixed.

## What this leaves

The filling rule is not ordering well-separated energies. It is calling near ties, and the atoms it gets wrong are exactly the ones at the ties. Every exception anyone has measured goes the same way. The one that does not is a calculation, and the thing that makes it different is relativity, which the rule knows nothing about.