How two atoms feel each other: squeeze them and they shove back hard; pull them apart and a faint attraction draws them home. One curve, V(r) = 4ε((σ/r)¹² − (σ/r)⁶), holds both — a steep repulsive wall and a gentle attractive tail meeting at a single well. Down the center, data flows: the atoms go in, the engine computes the potential and the force out. The blue team builds and defends the closed form; the red team tries to break it.
source J. E. (Lennard-)Jones, On the Determination of Molecular Fields — II. From the Equation of State of a Gas, Proc. Roy. Soc. A 106 (1924) 463–477 — doi:10.1098/rspa.1924.0082. Rendered, not quoted.
The pair potential is a difference of two power laws in the reduced distance x = σ/r:
V(r) = 4ε(x¹² − x⁶). The +x¹² term is the repulsive hard core; the −x⁶ term is the attractive tail. The force follows by pure calculus: F(r) = −dV/dr = (24ε/r)(2x¹² − x⁶).
Three closed facts fall straight out, checked live below: minimum at r = 2^(1/6)σ ≈ 1.122σ, depth exactly −ε, and F = 0 there. σ itself is the zero crossing (V=0), not the well.
| region | V(r) | force F(r) |
|---|---|---|
| r < rmin | rising, >0 inside σ | + repulsive |
| r = rmin | −ε (the well) | 0 |
| r > rmin | <0, → 0⁻ | − attractive |
A steep repulsion and a gentle attraction: this is the force law two atoms obey. Hand this F(r) to a symplectic integrator — Verlet's leapfrog — and it becomes the inner loop of a whole simulation.
Each atom's neighbour list feeds this one function; the trajectory of a thousand particles is nothing but this curve, summed and stepped. The pair potential is the premise; the-molecular-dynamics is the conclusion it runs.
The blue team's live check: re-derive the well from the closed form — minimum at 2^(1/6)σ, depth −ε, force zero there — and re-run the symplectic integrator to confirm energy holds. If red swaps the exponents, this badge is where it shows.
The model needs exactly two numbers plus a distance. ε (epsilon) is the well depth — how strongly the pair binds. σ (sigma) is the size — the separation at which V crosses zero. r is the current distance between the two atoms.
σ is not the resting distance: atoms settle at the bottom of the well, at 2^(1/6)σ ≈ 1.122σ, slightly farther out than σ. Feed ε, σ and r into the panel and it returns the potential and the force.
Slide r inward past the well and the force flips from a pull to a shove — computed from −dV/dr on the spot, never looked up. The canvas plots the whole curve; below it, a bound pair oscillates and its total energy stays flat (the symplectic integrator conserves it).
What the machine produces, proven: a single well with its minimum at exactly 2^(1/6)σ, depth −ε, force zero there; a repulsive shove for r<rmin and an attractive pull for r>rmin; V→+∞ as r→0 (the hard core) and V→0⁻ as r→∞ (van der Waals dispersion). Fed to a symplectic Verlet step, total energy and momentum are conserved to tolerance.
The blue team's witness (left) confirms these numbers live; the red team (right) tries to make them wrong.
It is a two-parameter cartoon: excellent for noble-gas liquids (argon), poor for metals, hydrogen bonding, and anything covalent or strongly polar. "The" interatomic potential it is not — it is the first potential simple enough to run a whole simulation on.
"σ is the equilibrium bond distance." Cut. σ is where V=0; the minimum sits farther out, at 2^(1/6)σ ≈ 1.122σ — the engine marks both.
"The exponent 12 comes from quantum mechanics." Cut. 12 is chosen for arithmetic convenience (=6×2); real short-range repulsion is exponential, not a power law.
"Lennard-Jones fits every substance." Kept, corrected. Superb for closed-shell atoms and rare gases; a rough caricature for metals, water, and directional bonds.
The red team's move: swap the exponents — use r⁻⁶ for repulsion and r⁻¹² for attraction. Now there is no steep hard core and the well inverts to a hump. The blue team's witness (window 7) is watching.
Swap them and the point at 2^(1/6)σ becomes a maximum at +ε instead of a minimum at −ε, and V→−∞ as r→0 (atoms collapse). The witness recomputes, disagrees with the known well, and turns red. Nothing is faked; the attack is real and it is caught.