Mathematica: Lennard Jones

Define a Lennard-Jones potential function: note the underscore after the variables and parameters on the left-hand side and the :=

Plot the Lennard-Jones potential for r values ranging from 0.2 to 1.0

Manipulate allows us to vary the parameters epsilon and sigma to see their effect The parameter sigma controls the position (r –value) of the minimum The parameter epsilon controls the depth of the minimum (its y axis value). Because the scale of the graph also continuously changes, this effect is difficult to see.

Fixing the range using PlotRange I did not type an → rather I typed -> and it got converted into an arrow

Mathematica (as a symbolic manipulation program) can be used to take a derivative

We can plot and manipulate the Output of taking the derivative

Comparing plots Just confirming that when the original function had a negative slope, the derivative function is negative and when the original function has a positive slope the derivative is positive

Solving when the derivation equals 0 – finding the minimum

N[%] – N[] means get the numerical value and % refers to the previous item Note of the six solutions , the first is negative, and the last four are complex – note the i (square root of -1). So if r corresponds to the distance between two particles, then only the second solution r  1.12245 sig) makes sense

We can select out the one desired solution

Now that we know the location of the minimum (its x value) we can determine its y value by substituting the previous result into the function

In conclusion We have seen that the Lennard- Jones potential function is a minimum at r = 1.12246 σ and that the function is equal to – ε at that minimum