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§ Phase iii · The Hydrogen Atom · Chapter 05

The periodic table from first principles

In 1869, a Siberian chemist arranged the known elements in a grid and noticed that the chemistry repeated itself in a pattern of 2, 8, 8, 18, 18. He had no theory. Sixty years later, the Schrödinger equation, plus a single rule about how electrons share addresses, would explain every row length without fitting a single number.

The picture of Dmitri Mendeleev that survives is of a wild-bearded chemist in St. Petersburg, working at his standing desk in February of 1869, shuffling a deck of cards on which he had written the names, atomic weights, and characteristic compounds of every chemical element then known. There were sixty-three of them. He was trying to find a textbook order in which to teach them to his students. The order kept escaping him. The metals he knew sodium, potassium, lithium, rubidium, caesium clustered together by chemistry, but their atomic weights were scattered across the table like beads spilled on a floor. Iron sat near nickel. Chlorine and bromine and iodine, all corrosive yellow-green-violet halogens, marched together. There was a pattern hiding underneath. He could feel it the way a card player feels a run.

That February, Mendeleev later told his biographer, he fell asleep with the cards in his hand and dreamed the order. When he woke he wrote it out, almost without correction: a grid in which rows held elements of increasing atomic weight and columns held elements of similar chemistry. Mendeleev was bold enough to leave gaps. If the next known element in atomic weight did not fit the chemistry of its row, he skipped it and placed it where its chemistry belonged. The empty squares, he claimed, would be filled in by elements not yet discovered. He even predicted the atomic weights and chemical behaviours of three of them. Within fifteen years, gallium, scandium, and germanium had been pulled out of ores by chemists who knew exactly what to look for. The table worked.

But work and explain are different verbs. Mendeleev’s table was a description, not a derivation. The rows had lengths 2, 8, 8, 18, 18, 32, 32 and nobody knew why. The columns of similar chemistry argon, neon, krypton, xenon, the noble gases, sitting at the right margin like a series of full stops were a pattern with no mechanism. Why exactly eight elements between two noble gases in the second row, and again in the third? Why eighteen in the fourth? Why does the chemistry of the elements rhyme rather than evolve smoothly? For sixty years the periodic table was the most successful empirical chart in chemistry and the most embarrassing one in physics: an arithmetic poem with no grammar.

The grammar arrived in pieces, between 1925 and 1928. Three ideas, none of them obvious, snapped together to derive every row length in the table from scratch. The first was the Schrödinger equation, which we have already met: bound electrons in a hydrogen-like atom can sit only in discrete states labelled by three integers, the quantum numbers (n, l, m). The second was electron spin, an extra two-valued degree of freedom proposed by George Uhlenbeck and Samuel Goudsmit in 1925 that doubled every state. The third was Wolfgang Pauli’s exclusion principle, also 1925, which forbade two electrons in an atom from occupying the same complete address. Out of these three ideas drops a clean arithmetic: shell n holds 2n^2 electrons. Two, eight, eighteen, thirty-two. Mendeleev’s rows.

To see how the three ideas combine, picture a hydrogen atom with a single electron. The Schrödinger equation, solved in Phase iii Chapter 03, tells us the allowed bound states are labelled by three integers. The principal quantum number n runs 1, 2, 3, and so on, and roughly sets the size of the orbit and its binding energy. The angular momentum quantum number l runs from 0 up to n minus 1, and labels how lopsided the orbital is: l equals 0 is a sphere (the s orbital), l equals 1 is a dumbbell (p), l equals 2 is a four-lobed cloverleaf (d), l equals 3 is a more elaborate flower (f). The magnetic quantum number m runs from minus l to plus l in integer steps and labels the spatial orientation. Count the combinations. For a given n there are n values of l. For each l there are 2l plus 1 values of m. The total number of (n, l, m) addresses with the same principal quantum number is 1 plus 3 plus 5 plus … up to 2n minus 1, which by a famous schoolroom identity equals n squared. Shell n has n squared orbitals.

Now add spin. Uhlenbeck and Goudsmit, two graduate students in Leiden, suggested that every electron carries an intrinsic two-valued spin, often visualised (badly, but usefully) as a top spinning either up or down. The spin state doubles the address: (n, l, m) becomes (n, l, m, spin), with spin equal to plus one-half or minus one-half. So shell n now has 2n squared addresses, not n squared. Shell 1 has 2. Shell 2 has 8. Shell 3 has 18. Shell 4 has 32. Those are the row lengths in Mendeleev’s table, almost. The almost will turn out to be the most interesting part.

Pauli sealed the count. In December 1924 Wolfgang Pauli published the rule that no two electrons in the same atom may share the same complete set of quantum numbers. Pauli called it the Ausschliessungsprinzip, exclusion principle. The motivation was empirical: he had been staring at term diagrams for years and noticed that the helium atom never appears to occupy certain states that the symmetry of the Schrödinger equation would otherwise allow. Two electrons in the lowest shell of helium always seem to have opposite spins; never the same. Generalising, Pauli posited that electrons refuse to be exact copies of one another. If you want to drop a second electron into an atom whose lowest address is already filled, the newcomer has to take a different address. By the time you have run out of n equals 1 addresses (there are two of them), the third electron must climb to n equals 2. By the time you have run out of n equals 2 addresses (eight of them), the eleventh electron climbs to n equals 3. Build the periodic table this way, atom by atom, and you have just derived the lengths of its rows.

The periodic table is a graphic description of the periodic law, which states that the properties and atomic structures of the chemical elements are a periodic function of their atomic number. Elements are placed in the periodic table according to their electron configurations, the periodic recurrences of which explain the trends in properties across the periodic table. An electron can be thought of as inhabiting an atomic orbital, which characterizes the probability it can be found in any…

Putting the rules together gives the Aufbau principle, German for “building up”: construct each atom by adding one electron at a time, dropping each new electron into the lowest-energy unoccupied address, respecting Pauli. The first electron of hydrogen goes into the (n=1, l=0) orbital, the 1s. So does the second electron of helium, with opposite spin. The 1s shell is full at Z equals 2. Helium is a noble gas because the next available address requires climbing to n equals 2, a substantial energy gap. The third electron of lithium opens that shell, dropping into 2s. By neon, at Z equals 10, the 2s and the three 2p orbitals are all full: two plus six equals eight new electrons since helium, and we have arrived at the next noble gas. The row lengths 2 and 8 are now derivations, not observations.

So far, so simple. The puzzle starts at potassium, Z equals 19. Counting addresses, after the 3p orbitals are filled at argon (Z equals 18) the next orbital you might expect to fill is the 3d, which has five m values and ten total electron slots with spin. If the energy ordering of multi-electron atoms followed the hydrogen-like ordering (in which energy depends only on n), the next eighteen electrons would all sit in n equals 3 orbitals, and the third row of the periodic table would have length eighteen, not eight. But it does not. The third row ends at argon, with only eight elements. The fourth row opens with potassium, whose nineteenth electron sits not in 3d but in 4s, a shell with higher n. The Schrödinger equation seems to have been overruled.

The resolution is that the Schrödinger equation has not been overruled, only the hydrogen-like solution has. In hydrogen, with a single electron, the energy depends purely on n because the Coulomb potential is exactly 1 over r. The angular shape l and orientation m have no effect on the energy. As soon as you add a second electron, the inner electron screens part of the nuclear charge, and the outer electron feels a potential that is no longer pure 1 over r. The l degeneracy breaks. Orbitals with smaller l, which spend more time near the nucleus (s orbitals especially have a nonzero probability density right at r equals 0), feel more of the full nuclear charge and are pulled down in energy. Orbitals with larger l are pushed up. By the time you reach potassium, the 4s orbital sits below the 3d orbital, and the nineteenth electron sensibly takes the lower-energy address.

The empirical pattern of which orbital fills next, across the whole table, is captured by Madelung’s rule, also called the n plus l rule: orbitals fill in order of increasing n plus l, and ties broken by increasing n. Compute n plus l for the orbitals in order. 1s gives 1. 2s gives 2. 2p gives 3. 3s gives 3. The tie at 3 is broken by smaller n first, so 2p fills before 3s. Continuing: 3p gives 4. 4s gives 4. 3d gives 5. Tie at 4 broken by smaller n, so 3p fills before 4s. 4s gives 4 and 3d gives 5, so 4s fills before 3d. That is the rule that pushes potassium into the 4s orbital and leaves 3d to wait for the fourth-row transition metals. The same rule predicts the order 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d. Walk through the rule and you generate the entire periodic table in fill order. No fitting parameters. One rule.

The exceptions to Madelung’s rule are themselves instructive. Chromium, Z equals 24, should by strict counting have configuration 3d4 4s2: ten electrons past argon, with 4s filled and four added to 3d. Spectroscopy says no. Chromium’s measured ground state is 3d5 4s1, with one electron borrowed from 4s and promoted to 3d. Copper, Z equals 29, should be 3d9 4s2. It is 3d10 4s1, again with the 4s donating an electron to 3d. Silver and gold do the same thing one row down. The reason is a small extra stability that arises when a d shell is exactly half-filled (five electrons, one in each m, all spins parallel) or exactly full (ten electrons in five orbitals, paired). The exchange energy gain from maximising parallel spins, what we know as Hund’s rule, slightly outweighs the cost of promoting an electron up the energy ladder.

Hund’s rule is the third clause in the building-up algorithm, and it matters most when a subshell is only partially occupied. Friedrich Hund, working in Göttingen alongside Heisenberg and Born in 1925, observed that the lowest-energy way to populate degenerate orbitals (the three p orbitals, the five d orbitals) is to spread one electron into each spatial orbital with parallel spins before any orbital takes a second electron. Two electrons in the same spatial orbital must have opposite spins (Pauli), but they also repel each other strongly because they share a region of space. Spread them across the orbitals first, with spins aligned, and the repulsion is lowered. This is why nitrogen, Z equals 7, has all three 2p electrons in separate spatial lobes with parallel spins, and not two paired plus one unpaired. The half-filled stability of chromium and the full-shell stability of copper are the same physics writ slightly larger.

↘ Derive: shell n holds 2n^2 electrons

Start with the Schrödinger equation for an electron in a Coulomb potential. The bound states are labelled by three quantum numbers:

• principal: n = 1, 2, 3, ...
• angular: l = 0, 1, ..., n - 1
• magnetic: m = -l, -l+1, ..., +l

Count the orbitals for a given n. For each value of l there are (2l + 1) values of m. Sum over l from 0 to n - 1:

Σ (2l + 1) = (2·0 + 1) + (2·1 + 1) + ... + (2(n-1) + 1) = 1 + 3 + 5 + ... + (2n - 1)

This is a sum of the first n odd integers. A standard identity gives:

1 + 3 + 5 + ... + (2n - 1) = n²

So shell n has n² orbitals. Each orbital can hold two electrons (one spin up, one spin down), by Pauli plus the two-valued spin degree of freedom. Multiply:

(number of electrons in shell n) = 2n²

For n = 1, 2, 3, 4, 5, this gives 2, 8, 18, 32, 50.

The rows of the periodic table, in their idealised “if energy depended only on n” form, would have lengths 2, 8, 18, 32, 50. In practice the rows have lengths 2, 8, 8, 18, 18, 32, 32 because the Madelung crossings delay the filling of 3d, 4d, 4f, and 5f past their nominal shell, splitting what looks like a long shell into two periods. Add the Madelung crossings into your Aufbau order and the actual periodic table drops out exactly. No fitting parameters. Schrödinger plus Pauli plus screening.

What started as Mendeleev’s deck of cards has become a derivation. Take Schrödinger’s equation. Add Uhlenbeck and Goudsmit’s spin. Add Pauli’s exclusion. Apply Hund’s rule to maximise spin within a partly filled subshell. Track the energy ordering with Madelung’s empirical n plus l rule, itself an artefact of screening in a many-electron atom. The periodic table falls out: rows of length 2, 8, 8, 18, 18, 32, 32; noble gases at the right margin where the next available subshell sits across a large energy gap; transition metals in the d blocks of rows 4 through 7; rare earths in the f blocks of rows 6 and 7. The chemistry of a column repeats because the outermost subshell pattern repeats. Lithium, sodium, potassium, rubidium, caesium are all alkali metals because they all have a single s electron orbiting a closed noble-gas core, and that single s electron is what does the chemistry.

The periodic table is no longer a chart we have to memorise. It is a result we can derive. The arithmetic Mendeleev guessed in 1869, the empirical pattern Newlands had been laughed at for proposing in 1865, the gaps Mendeleev left for elements not yet discovered, all of it has a single underlying reason. Electrons in an atom obey the Schrödinger equation, they come in two spin states, and no two of them can share a complete quantum address. Three sentences. One chart. The whole of inorganic chemistry, reading the columns; the whole of synthesis, reading the gaps; the whole of stellar nucleosynthesis, reading the order of the rows. And we have not fitted a single number to make it work.

If you wanted a slogan for the architecture of matter, you could do worse than this: the periodic table is what Pauli’s exclusion principle looks like when you draw it. Take it away and the universe collapses (every electron in every atom would crash into the lowest 1s state, and chemistry would stop being a science of distinction). Keep it and the universe extends, atom by atom, into the rich and slightly weird grid that runs from hydrogen on the upper left to oganesson on the lower right. The grid was discovered before it was derived, the way the spectrum of hydrogen was discovered before Schrödinger and the photon was discovered before Dirac. That is the ordinary order of physics. The experiment knows. The theory catches up, sixty years later, and says: ah, of course.

§

We have used spin to derive a chart of the elements, but we have not yet shown that spin is real. The experiment that nailed it sent silver atoms through a non-uniform magnet in 1922 and watched the beam split into exactly two spots. Otto Stern was looking for something else entirely.

next chapter → Stern–Gerlach