1. Mathematica: Lennard Jones

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

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

4. 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.

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

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

7. We can plot and manipulate the Output of taking the derivative

8. 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

9. Solving when the derivation equals 0 – finding the minimum

10. 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  sig) makes sense

11. We can select out the one desired solution

12. 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

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