Lennard-Jones Potential
Atoms attract at a distance and repel up close. The 6-12 potential captures this with two terms and two parameters — enough to produce gas, liquid, and solid phases from nothing but pair interactions.
About this lab
The Lennard-Jones 6-12 potential
The Lennard-Jones potential models the interaction between a pair of neutral atoms or molecules:
V(r) = 4ε [ (σ/r)^12 - (σ/r)^6 ]
The r^(-6) term captures the van der Waals attraction — London dispersion
forces arising from correlated fluctuations in electron clouds. The r^(-12) term
models Pauli repulsion at short range: electron orbitals cannot overlap. The exponent 12 was
chosen largely for computational convenience (it is the square of 6), though it turns out to
give reasonable results for noble gases. The parameter ε sets the depth of the
energy well (the strength of attraction), and σ sets the distance at which
the potential crosses zero (the effective atomic diameter).
Molecular dynamics
This simulation integrates Newton’s equations of motion using the velocity Verlet algorithm. At each timestep, forces are computed from the gradient of the LJ potential, positions and velocities are updated, and a thermostat couples the system to a heat bath at the target temperature. The result is a trajectory through phase space that samples the canonical ensemble — the same method used in real computational chemistry to study protein folding, drug binding, and material properties.
Phase transitions
Even this minimal model produces the three phases of matter. At high temperature (T > 2.0 in reduced units), atoms fly freely — a gas. Lower the temperature and they condense into fluctuating clusters — a liquid. Cool further and the clusters freeze into a hexagonal lattice — a solid. The radial distribution function g(r) reveals these transitions: a gas shows a single broad peak, a liquid shows damped oscillations, and a solid shows sharp peaks at lattice spacings.
Why 6-12?
The r^(-6) exponent has rigorous quantum-mechanical justification: second-order
perturbation theory shows that the leading term in the London dispersion interaction between two
atoms falls off as the sixth power of distance. The r^(-12) is more empirical —
the true Pauli repulsion is better described by an exponential, as in the Buckingham potential.
But the 12th power is the square of the 6th, making force computation faster (compute
(σ/r)^6 once and square it). John Lennard-Jones proposed this form in 1924,
and it remains the default model for non-bonded interactions in molecular simulation ninety
years later.