# The Lennard-Jones Potential

#### Prerequisite

A specific model for the atom-atom potential was constructed by John Lennard-Jones (1894-1954). It is therefore known as the Lennard-Jones potential and it is a reasonable approximation for the regions that atoms typically explore in biological systems. (Atoms don't get close enough to each other to see the details of the short ranged repulsion at biological temperatures.)

The Lennard-Jones potential includes two parts:  an attraction proportional to $1/r^6$, and a repulsion proportional to $1/r^{12}$.  We can write this as $PE = A/r^{12} - B/r^6$ , where $A$ and $B$ are constants whose values depend on the specific types of atoms.  (The positive term represents repulsion, and the negative term represents attraction.) To see what this looks like, you can try graphing it on a graphing calculator or spreadsheet, and experiment with different values of $A$ and $B$.  What you get is shown in the figure at the right.

Let's see what we can conclude from this graph.  At large $r$ the potential energy graph looks flat.  The slope is just about zero.  Thus, atoms that are far apart feel just about no force. This is a very short-ranged interaction! If you double the distance between two atoms, the potential energy associated with their attraction is divided by 64 (= 26). At small $r$ the graph climbs very steeply down as you approach the origin, indicating that there is a very strong repulsive force at close range.

The L-J potential is a phenomenological model of an interatomic potential. It's chosen to match the main features that we know about the interaction -- long range attraction and short range propulsion -- and it has two parameters that can be adjusted to fit two measured properties of the atom-atom interaction: typically the bond length and the binding energy. It is commonly used because it has a simple functional form that is easily calculated.

[Note: The dimensions of the constants $A$ and $B$ are a bit of a mess since $A$ must have units of energy times L6 and $B$ must have units of energy times L12. A nicer way of writing the LJ potential is if $A$ and $B$ are replaced by a length parameter $\sigma$ with units of length and an energy parameter $\epsilon$ with units of energy. Then the LJ potential looks like this:

$$U(r) = \epsilon \bigg[ \bigg(\frac{\sigma}{r}\bigg)^{12} - \bigg(\frac{\sigma}{r}\bigg)^6 \bigg]$$

Then the two parameters give something that can be physically interpreted - a length scale and an energy scale.

Another common model of interatomic interactions is the Morse potential. This uses exponentials rather than powers and does not have the inconvenient feature of becoming infinite at separation 0. Of course neither of these potentials represent fully the fact that atoms are made up of many particles that interact via the laws of quantum mechanics. Quantum mechanical models are sometimes used to calculate effective atom-atom interaction potentials. But for most purposes, L-J works well.

Ben Dreyfus 10/30/2011 and Joe Redish 11/15/11

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