← quantum · contents

§ Phase viii · Molecules · Chapter 03

The H₂ bond length

Two protons and two electrons. It is the simplest neutral molecule in the universe, and for the first two years of quantum mechanics nobody could explain why the protons sit precisely 0.74 ångström apart instead of flying off to infinity. The answer came from a 23-year-old postdoc at Zürich, working through the summer of 1927 on a problem his supervisor said would not yield. He wrote the first quantum calculation of a chemical bond, and the curve he drew is still the picture every chemist reaches for.

In June of 1927, Erwin Schrödinger had just left Zürich for Berlin, leaving behind a small office and a problem his successor would inherit by accident. The problem was a chemical one. Schrödinger’s wave equation, which had been published only the previous year, predicted the hydrogen atom beautifully. It said nothing yet about why two hydrogen atoms, brought close together, refused to fly apart. Classical electromagnetism had no answer. The two protons repel each other, the two electrons repel each other, and yet a vial of hydrogen gas at room temperature is, almost entirely, H₂ molecules and not free atoms. Something glued them together. What?

The man who arrived in the empty office that summer was Walter Heitler, a 23-year-old Munich graduate who had taken a Rockefeller fellowship to study under Schrödinger and discovered upon arrival that his supervisor had already left. A friend from his student days, Fritz London, was also passing through. The two of them, with no one to assign them anything, decided to spend the summer on hydrogen.

What they did was deceptively simple. They took the ground-state wavefunction of a single hydrogen atom, the 1s orbital that Schrödinger had derived a year earlier, and they wrote down a guess for the two-electron molecule. Imagine an electron labelled 1 sitting on the left proton (call it nucleus A) and an electron labelled 2 sitting on the right proton (nucleus B). The product of the two 1s orbitals would describe that configuration. But electrons are indistinguishable. Heitler and London also had to write the configuration with electron 2 on A and electron 1 on B, and they had to add the two configurations together with the right symmetry. The choice of symmetry was the whole point.

The symmetric combination, in which the spatial part is unchanged under swapping the two electron labels, requires that the spin part of the wavefunction be antisymmetric. That is the singlet state, two spins paired into total spin zero. The antisymmetric spatial combination requires the symmetric (triplet) spin state, total spin one. Heitler and London computed the energy of both. The singlet curve dropped well below the energy of two free hydrogen atoms when the protons were brought close, reaching a minimum at a separation around 0.87 ångström before climbing steeply as the protons approached zero distance. The triplet curve had no minimum at all. It was repulsive everywhere.

There, in two curves on the same plot, was the explanation of the chemical bond. Two hydrogen atoms with their electrons paired into a singlet attract each other and form a stable molecule. Two hydrogen atoms with their electrons in a triplet repel and refuse to bond. The Pauli exclusion principle, in the form of antisymmetry of the total wavefunction, was the rule that decided whether or not the protons stuck. It was not the protons themselves that bound. It was the way the electron probability piled up in the region between them when the spin state was a singlet, with the negative charge density acting as the glue.

The years immediately following 1927 brought a second framework that arrived at the same answer by a different route. Friedrich Hund in Leipzig and Robert Mulliken in Chicago, working independently, proposed that you should not start with atomic orbitals at all. You should start with molecular orbitals, single-electron states that already span both nuclei. For H₂ the two lowest molecular orbitals are formed from the symmetric and antisymmetric combinations of the two 1s atomic orbitals. The symmetric combination, which physicists came to call σ_g, has a fat lobe of electron density sitting between the two protons. The antisymmetric combination, σ_u*, has a node exactly midway between them. Put both electrons into σ_g with paired spins and you have an H₂ molecule. Leave σ_u* empty. The configuration is written (σ_g)² and it is the simplest closed-shell molecule in chemistry.

The two frameworks (Heitler-London valence-bond theory and Hund-Mulliken molecular-orbital theory) gave nearly identical predictions for H₂ near the equilibrium bond length and quite different predictions in two limits. At very large separation the valence-bond wavefunction correctly dissociates into two neutral hydrogen atoms, while the simple molecular-orbital wavefunction gives, paradoxically, a 50-50 mix of two neutral atoms and the ionic configuration H⁺ + H⁻. The ionic part costs about 13 eV and should not be there. At very short separation the molecular-orbital picture handles the strong covalent overlap more naturally than the valence-bond does. Generations of chemists argued about which was the better starting point. The answer, slowly accepted, was that both are starting points; the truth lies in adding configurations from both pictures until the wavefunction converges. The full calculation, called configuration interaction, gives essentially the exact answer.

To even ask “what is the bond length” presupposes a piece of physics that had to be put in carefully. The hydrogen molecule has four moving particles: two protons and two electrons. The full Schrödinger equation is in twelve coordinates. Solving that directly was hopeless even for H₂, and Born and Oppenheimer had pointed out in 1927 a clean way to make it tractable. The protons are roughly 1836 times heavier than the electrons. When the protons jiggle, the electrons keep up with them effortlessly, the way a swarm of bees would stay around a slowly moving hive. You can pretend the protons are clamped at a fixed separation R, solve the electronic problem, get an electronic energy E(R), and only afterwards let the protons move in this electronic energy as if it were a potential well.

The Born-Oppenheimer approximation reduces the chemistry to a one-dimensional curve. Vary R, solve the electronic Schrödinger equation at each R, plot the resulting energy. The plot has the characteristic shape every chemist learns to draw. Far apart it is flat, at the energy of two non-interacting hydrogen atoms. As R decreases the energy first dips, finds a minimum, then rises steeply as the proton-proton repulsion overwhelms the electronic attraction at very small R. The minimum sits at R_e = 0.741 ångström. The depth from the asymptote down to the minimum is the dissociation energy D_e = 4.52 electron-volts. Both numbers are reported in chemistry handbooks to four-figure accuracy and have been confirmed by experiment and theory to twelve.

The curve does more than fix the bond length. Its curvature at the minimum determines the vibrational frequency of the molecule, because near R_e the electronic energy looks like a harmonic potential, and a harmonic potential gives evenly spaced vibrational levels separated by ℏω. Fit a parabola to the bottom of the well, read off the spring constant k, and plug into ω = √(k/μ) with μ the reduced mass of the two protons. The result for H₂ is about 4400 wavenumbers, which is a vibrational period of around 7.5 femtoseconds. Spectroscopists working with infrared light see this transition at 4161 cm⁻¹, the slight discrepancy coming from the fact that the real curve is not quite a parabola (it is anharmonic, leaning slightly to the right) and that the zero-point motion is large enough to feel the anharmonicity. The Heitler-London paper did not compute the vibrational frequency, but everything you would need to compute it is implicit in their curve.

E(R) = E(R_e) + E'(R_e) (R - R_e) + (1/2) E''(R_e) (R - R_e)² + …

V(R) ≈ (1/2) k (R - R_e)², where k = E''(R_e)

ω = √(k / μ)

ν̃ = ω / (2π c)

Step back from the two curves and the calculation, and look at what the Heitler-London paper actually achieved. Before 1927 the chemical bond was a mystery covered in postulated rules: Lewis dot diagrams, valence numbers, the octet rule. Each rule worked in its narrow domain and offered no explanation of why. The Heitler-London calculation said: there is no separate physics of chemistry. There is only quantum mechanics applied to electrons and nuclei. The covalent bond is the antisymmetric combination of electron spins occupying a symmetric spatial wavefunction. The triplet is repulsive because Pauli requires the spatial part to be antisymmetric, which drives the electron density out of the bonding region. The whole edifice of organic chemistry, which would later sustain pharmaceutical research and modern biology, sits on top of this one calculation, generalized and elaborated but not, in any deep sense, fundamentally extended.

The fact that the equilibrium separation of two protons in a hydrogen molecule is 0.741 ångström, and not some other number, is therefore not an empirical fact about hydrogen but a consequence of the wave equation, the Coulomb potential, the Pauli principle, and the proton-electron mass ratio. Change the electron’s mass and you would change R_e. Change the strength of the electromagnetic force and you would change R_e. In a universe with even slightly different fundamental constants, the hydrogen bond length would shift, water would not have the geometry it has, and chemistry would not work as it does. The number 0.741 is a fingerprint of the laws of physics at small scales, and the curve it sits on is the first place quantum mechanics turned out to be quantitatively right about the way ordinary matter holds together.