Swap the Electrons and the Atoms Hold — Heitler and London's Hydrogen Molecule, 1927

Our series on the old quantum theory followed physics up to the point where quantum mechanics could explain a single atom (the final part is here). This new series, "The Rise of Quantum Mechanics", follows the few years after 1927, when the brand-new theory spread quickly into chemistry, the atomic nucleus, solids and relativity, in six parts. Part 1 starts with the most everyday question of all: why do atoms stick together?

Chemists knew it. Physicists could not explain it

A water molecule is one oxygen atom joined to two hydrogen atoms. Whatever holds atoms together is called a chemical bond. In 1916 the American chemist Gilbert Lewis proposed what a bond is: two atoms each put forward electrons and share them as a pair. In a hydrogen molecule, each hydrogen atom contributes one electron, and the two atoms share the resulting pair.

The "electron pair" was enormously useful to chemists. But nobody could say why sharing a pair of electrons should hold two atoms together, or why one electron, or three, would not do.

Physics was stuck too. The best-known attempt to handle a molecule with the old quantum theory was Wolfgang Pauli's doctoral thesis of 1922. His subject was the hydrogen molecular ion: a hydrogen molecule with one electron removed, so two nuclei and a single electron, the simplest molecule imaginable. Yet when Pauli carefully worked through the Bohr–Sommerfeld rules (part 4), the answer was that the molecule would have lower energy if it split into one hydrogen atom and one bare nucleus. In other words, it could not hold together. Later experiments showed clearly that the hydrogen molecular ion is stable. Even the simplest molecule was beyond the old quantum theory.

Waking at five in the afternoon — Zurich, 1927

In 1927 two young German physicists met in Zurich, Switzerland: Walter Heitler and Fritz London. Both had come on fellowships to learn the new mechanics from Erwin Schrödinger, who had published his wave equation the year before. London had earned a doctorate in philosophy before turning to physics; Heitler had worked on solutions in physical chemistry. Their backgrounds could hardly have been more different.

In a 1963 interview, Heitler recalled the moment. He slept until late in the morning, found he could not work at all, had a quick lunch and slept again until five in the afternoon. When he woke, he had a clear picture in front of him: the waves of two hydrogen atoms combined with a plus and a minus, with the "exchange" built in. He called London at once, they worked together until late that night, and no later than the next day they had the formation of the hydrogen molecule in their hands.

It is a recollection from 36 years later, so we cannot be sure every detail is right (the story of Heisenberg on Helgoland in part 5 also spread from his own memoir). What the record does show is this: the work was presented at a regional meeting of the German Physical Society in Freiburg on 12 June, and the paper reached the journal on 30 June.

Which electron is which? No one can tell

At the heart of their idea was the "exchange" that Heisenberg had used for helium in part 5.

Suppose hydrogen atom A and hydrogen atom B come close. Common sense says electron 1 belongs to A and electron 2 belongs to B. But in quantum mechanics, electrons do not wear name tags. The two electrons are exactly alike, and there is no way to tell which is which. So the state "electron 1 at A, electron 2 at B" and the state "electron 1 at B, electron 2 at A" must both be included.

There are two ways to combine the two states: add them or subtract them. When Heitler and London did the calculation, the added version lowered its energy as the atoms approached, reaching a lowest point at a certain distance. Lower energy means the atoms are harder to pull apart: they bond. The subtracted version only rose in energy however close the atoms came, so the two atoms simply pushed each other away.

This attraction comes from the fact that it is undecided which electron sits at which atom. It has no counterpart at all in classical physics. If you calculate only the ordinary electric pull between positive and negative charges, two hydrogen atoms barely bind.

Here Pauli's exclusion principle from part 4 comes in. Electrons have a property called spin, which can point one of two ways, up or down. Applying the exclusion principle showed that the attracting combination is allowed only when the two electrons' spins point in opposite directions. If the spins point the same way, you get the repelling one.

Two electrons paired with opposite spins hold two atoms together. The "electron pair" that Lewis had drawn from chemical experience eleven years earlier finally had a physical reason. In the same paper, Heitler and London noted that the same idea explains why atoms of noble gases such as helium only push each other away and do not bond.

The explanation worked. The numbers did not

On the numbers, though, the paper was modest. The integral that expresses the exchange effect (the exchange integral) was very hard to calculate, and they did not compute it exactly; they only estimated an upper limit. Reading off their graph, the energy needed to pull the two atoms apart (the binding energy) came out at about 2.4 electron volts (the unit of energy used for atoms), with the nuclei about 0.08 nanometres apart. The actual binding energy is about 4.7 electron volts, so they had roughly half. The paper did not even compare its values with experiment.

The person who took on the integral directly was Yoshikatsu Sugiura, a Japanese physicist studying in Europe. In Göttingen, thanks to Max Born, Sugiura was able to read Heitler and London's paper before it was published. Noticing that the exchange integral had not been calculated, he solved it for each distance between the nuclei and submitted his paper on 30 August. His binding energy was about 3.1 electron volts, an agreement he considered satisfactory.

Heitler himself was unimpressed. In a letter to London in September 1927 he wrote, "It appears that there are in the world some hard working dwarfs," and went so far as to say such work "seems to me ridiculous." What mattered to Heitler was the explanation of why atoms bond, not tightening the numbers.

Still, the number-tighteners kept going. In 1928 Shou Chin Wang, a graduate student at Columbia University, allowed the strength of the nuclear charge felt by each electron to be adjusted and obtained 3.76 electron volts. Then in 1933 Hubert James and Albert Coolidge at Harvard brought in a method that had worked for the helium atom: putting the distance between the two electrons directly into the formula. They added terms one at a time and recalculated each time. At thirteen terms the binding energy was 4.70 electron volts, within 0.03 electron volts of experiment.

They began on hand-cranked desk calculators and later moved to motor-driven ones. A rumour spread that the work took three years; in fact it took about six months. James later recalled the period when the numbers came out better and better as they added more terms as "among the happiest and most exciting days of my life."

The laws are known; all that is left is to calculate — or so it seemed

In 1929 Paul Dirac opened a paper with the words: "The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble."

How much weight that "only" carried soon became clear. In a 1935 review, James wrote that in principle there appeared to be a way to treat molecules with any desired precision, but "the limits of human patience restrict the usefulness of the complete method." Two electrons in a hydrogen molecule had taken six months. With more electrons, the amount of calculation grows unmanageably fast.

What could be done by this point, and what could not

Between 1927 and 1933, quantum mechanics achieved two things for the chemical bond. The first was an explanation of why atoms join: the exchange of electrons, and a pair with opposite spins. By Heitler's own memory, he and London got that in a single night. The second was confirmation that the explanation was also right in numbers. That took six years of accumulated work, from Sugiura to James and Coolidge.

What could not be done was to calculate molecules more complicated than hydrogen to the same accuracy. That wall fell not to a new theory but to a new tool. Only after the Second World War, when electronic computers appeared, did serious calculation get going again. The hydrogen molecule remained the touchstone for testing theory against experiment. By 2019, measured and calculated values of its binding energy agreed to nine significant figures.

In 2016, the energy of the same hydrogen molecule was computed on a quantum computer built from superconducting qubits. It was a first step toward handing calculations that become hopeless for ordinary computers as electrons are added over to quantum behaviour itself. Heitler and London's hydrogen molecule is still the first problem put to a new kind of calculating machine.

Heitler wanted to know why atoms bond; Sugiura and James tightened the numbers. Understanding why something happens and being able to get the right answer are not the same thing. The quantum theory of the chemical bond became trustworthy only when it had both. Next time, quantum mechanics turns to the atomic nucleus: particles that slip through a wall they should not be able to cross — George Gamow and quantum tunnelling, 1928.