Lennard-Jones potential

Example

Let $r > 0$ be the distance between two neutral atoms. The Lennard-Jones potential is the pair potential $$V(r) = 4 \varepsilon \left[ \left( \frac{\sigma}{r} \right)^{12} - \left( \frac{\sigma}{r} \right)^{6} \right],$$ where $\varepsilon > 0$ is the depth of the well and $\sigma > 0$ is the distance at which $V$ vanishes. The $r^{-12}$ term models short-range repulsion, and the $r^{-6}$ term the van der Waals attraction.

Check of the shape: $V(\sigma) = 4\varepsilon (1 - 1) = 0$. The derivative is $V'(r) = 4\varepsilon \left( -12 \sigma^{12} r^{-13} + 6 \sigma^6 r^{-7} \right)$, which vanishes when $r^6 = 2 \sigma^6$, that is at $r_m = 2^{1/6} \sigma$. There $(\sigma/r_m)^6 = 1/2$, so $V(r_m) = 4\varepsilon (\frac14 - \frac12) = -\varepsilon$, the minimum. As $r \to 0$, $V(r) \to +\infty$, and as $r \to \infty$, $V(r) \to 0$ from below.