Question

Where does the factor of 4 come from in the Lennard-Jones potential?

Answer 1

An attempt to answer this question depends upon how far back we need to go in the underlying concepts.

Remember that (at least in some sense) the the L-J potential equation is empirical (It is inexact mathematical model that approximates real inter-particle interaction.)

If you are willing to accept the L-J model in the form:
`V_(LJ) = epsilon [ (r_(min)/r)^12 - 2(r_(min)/r)^6]`

then the alternate form:
`V_(LJ) = 4 epsilon [(sigma/r)^12 -(sigma/r)^6]`
(which contains the `4`, I assume you referred to)

follows arithmetically from the substitution:
`r_(min) = 2^(1/6) sigma`

As I said, I'm not sure how far back I need to go and I am assuming you are familiar with the basic parameters of this model.