Molecules
f(r) = (1 − exp(1 − r)) · exp(1 − r)
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The Morse force
Write u = e^(1 − r) and the law is u − u², positive where the pair pulls together. It is zero at r = 1, where u = 1: that is the bond length, the distance a lone pair settles at. Inside it u > 1 and the force is negative, a repulsion that climbs steeply as the two are pressed together. Outside it the force is an attraction. It rises to a peak of 0.25 at r = 1 + ln 2 ≈ 1.69, then dies away like e^(−r), so a body three or four units off feels almost nothing.
That is exactly the derivative of the Morse potential V(r) = ½(1 − e^(1 − r))². It is a well of depth ½ with its floor at 1, steep on the near side and soft on the far side, which is how a chemical bond behaves and not at all how a spring does.
Why the drop holds
Thirty unit masses start scattered over a disc and at rest, with Swirl 0, so the swarm has no momentum at all and its centre never moves. The cloud falls together in about a second, overshoots into a knot, and rebounds. What was potential energy is now motion, and the swarm has no way to shed it. What settles is a warm droplet: its bodies rattle about a mean radius near 1.4, its surface stays ragged, and it holds indefinitely. Neighbours end up nearer than the bond length of 1, at around a third of it, because in the middle of the drop a body is pulled inward by many at once and squeezed against the one repulsion beneath it.
History
Philip M. Morse proposed this potential in 1929, in the second of two papers on diatomic molecules under the new wave mechanics. Its point was that the Schrödinger equation for it can be solved exactly. Its vibrational levels crowd together toward the top of the well and stop, so the molecule can be pulled apart, where a harmonic well gives evenly spaced levels and holds forever.
Try
- Set Swirl to 0.4: the drop is spun harder than the bonds can hold and it tears itself apart within a few seconds.
- Change the formula to (1 − exp(2 − 2r))·exp(2 − 2r): the same bond length and the same peak pull, but half the reach, so the bodies never find each other and the cloud drifts apart.
- Set Bodies to 12: a smaller drop at the same spacing, nearly all of it surface.
- Rewind and watch the first two seconds: the fall, the overshoot at about one second, the rebound.
- Top: the packing reads best from above, as a plane figure.