Morse potential.
The harmonic oscillator is a beautiful lie — real chemical bonds break. Stretch a hydrogen-chloride molecule far enough and it dissociates into H and Cl separately. The Morse potential captures that asymmetry, and its energy levels crowd together as you climb toward the dissociation limit. Eigentone tunes a scale to those crowding levels, fit to real HCl spectroscopy.
A bond, with rules and a limit.
The curve is the actual Morse potential for HCl. Compress the bond and the energy shoots up steeply; stretch it and the curve flattens out — past about 3 Å the atoms are essentially separate. The horizontal lines are the allowed vibrational levels. They aren't evenly spaced.
Why bonds aren't springs.
Hooke's law says the restoring force on a stretched spring is proportional to the displacement: pull twice as hard, get twice the stretch. A real molecular bond behaves like that for tiny displacements around equilibrium, but it's not symmetric. Compressing the bond pushes electron clouds together, raising the energy steeply. Stretching it lets the atoms drift apart — and once the atoms are far enough, they stop interacting altogether.
The Morse potential is the simplest analytical curve that captures both behaviors: a steep wall on the compressed side, a soft asymptote on the stretched side, finite depth De. Solve the Schrödinger equation for it and the energy levels come out in closed form — and they aren't evenly spaced, the way a harmonic oscillator's are. Each step up is a little smaller than the one before.
A real molecule, sounding.
Infrared spectroscopy of HCl is a textbook experiment for first-year chemistry students. The v = 0 → 1 transition sits at about 2886 cm−1 — a vibration period of 8.65 × 1013 Hz. The Morse model predicts the higher overtones v = 0 → 2, 0 → 3 and so on with high accuracy, including the anharmonicity correction that makes them slightly closer than integer multiples.
Eigentone takes the energy of each vibrational level (in real Joules), divides by Planck's constant to get a frequency in hertz, then octave-shifts down. Climb the keyboard, climb a vibrating bond.
An exponential well.
The Morse potential as a function of bond length r is:
V(r) = De · ( 1 − e−a(r − re) )2
where De is the dissociation energy, re is the equilibrium bond length, and a sets the curvature. For HCl: De = 4.620 eV, re = 1.275 Å, a = 1.869 Å−1.
The exact bound-state energies are:
The first term is the harmonic ladder. The second — the anharmonicity — is what makes successive levels crowd together. xe is small (≈ 0.018 for HCl), but it grows quadratically and eventually closes the ladder at the dissociation limit.
Molecular vibrations sit in the infrared, around 1014 Hz. Multiply each level's wavenumber by the speed of light, divide by 239, and the anharmonic spacing arrives intact in audio.
v=0: E = 1481.5 cm⁻¹ → 80.79 Hz
v=6: E = 17231.5 cm⁻¹ → 939.67 Hz
Spectroscopic constants from Huber & Herzberg, Constants of Diatomic Molecules (1979). Same N as the Quantum Harmonic Oscillator set, so the v=0 ground state lands at the same ≈ 81 Hz in both — by design, to make the anharmonic correction audible head-to-head.
What each white key plays.
Climb the keys and the intervals tighten — the unmistakable signature of an anharmonic bond.
Above v ≈ 28 for HCl the Morse formula stops adding bound states (dE/dv ≤ 0); the levels converge on the dissociation continuum at De. Eigentone's seven white-key range stays well within the bound region.
Hear a chemical bond.
Morse Potential ships with the Pro tier. Real spectroscopy, real anharmonicity, in your DAW.