Lennard-Jones Potential: The Physics Behind Molecular Dynamics

One formula, two competing forces

Two atoms drifting past each other feel a force that is weakly attractive at long range and violently repulsive at short range. The Lennard-Jones potential packages both into one expression with just two parameters:

V_LJ(r) = 4ε[(σ/r)¹² − (σ/r)⁶]
ε = depth of the potential well (energy)
σ = distance at which V = 0 (length)

The −(σ/r)⁶ term pulls atoms together; the (σ/r)¹² term, which wins out at short range because it grows so much faster, pushes them violently apart. Together they carve out a well with a minimum at r_eq = 2^(1/6)σ ≈ 1.122σ, where V(r_eq) = −ε — the equilibrium bond distance and bond energy for the pair.

Where the exponents come from

The attractive r⁻⁶ term has real physical grounding: it's the London dispersion force, derived by Fritz London in 1930 from quantum perturbation theory. Even non-polar atoms like argon feel it, because instantaneous fluctuations in electron density create a fleeting dipole that induces a matching dipole in a neighbouring atom; the interaction scales with the square of polarisability and decays as r⁻⁶. The repulsive term is a different story — true Pauli-exclusion repulsion from overlapping electron clouds actually falls off exponentially (the Buckingham potential uses A·e^(−Br)), and r⁻¹² has no comparable physical derivation. It's chosen for convenience: since (σ/r)¹² is just [(σ/r)⁶]², a simulation gets the repulsive term almost for free once it has computed the attractive one.

Equilibrium distance, well depth, and real atoms

The force is the negative gradient of the potential:

F(r) = −dV/dr = 24ε/r · [2(σ/r)¹² − (σ/r)⁶]

— repulsive for r < r_eq, attractive for r > r_eq, and effectively zero beyond about 3σ. Fitting ε and σ to real gases gives concrete numbers — argon: ε/k_B = 119.8 K, σ = 3.405 Å; neon: 35.6 K, 2.749 Å; krypton: 171.0 K, 3.61 Å. Reporting ε as a temperature (ε/k_B) is deliberate: it's directly comparable to thermal energy k_BT, and it explains why argon only condenses into a liquid below about 87 K — thermal agitation has to drop below the depth of the LJ well before atoms can stay bound to their neighbours.

Reduced units and the cutoff radius

Molecular dynamics codes almost never work in physical units. Instead they use reduced LJ units — length r* = r/σ, energy E* = E/ε, temperature T* = k_BT/ε, time t* = t√(ε/mσ²) — which collapses the potential to V*(r*) = 4[(1/r*)¹² − (1/r*)⁶] with no free parameters at all. Two different LJ fluids at the same T* and ρ* behave identically, a consequence of the principle of corresponding states, which is why a single simulation run in reduced units can stand in for argon, krypton, or any other noble gas just by rescaling.

Computing every pair interaction is O(N²), so real codes truncate the potential at a cutoff radius r_c (typically 2.5σ, where V(r_c) ≈ −0.016ε — already negligible against room-temperature thermal energy) and shift the whole curve down by V(r_c) so it reaches exactly zero at the cutoff without a discontinuous jump that would leak energy every timestep. Combined with Verlet neighbour lists or linked-cell lists, this turns an O(N²) force calculation into an O(N) one, which is what makes simulating thousands of Lennard-Jones particles in a browser tractable at all — and lets gas, liquid and solid phases emerge purely from these pairwise forces and a Velocity Verlet integrator, with no phase-transition rule required.

Frequently asked questions

Try it live

Everything above runs in your browser — open Lennard-Jones Molecular Dynamics and watch gas, liquid and solid phases emerge spontaneously as temperature and density change, all from the same pairwise force law. Nothing is installed, nothing is uploaded.