Hi @ruoyan,
In addition to the (nice) resources provided by @srtee, you can also look at this other paper that was published during my PhD work. You will find more details on the different methods you can use in molecular modeling to compute the elastic constants of materials there. We also discussed the different constants you get from using NVT or NVE ensemble (i.e. adiabatic or non-adiabatic).
That said, note that the scripts you used are the previous finite deformation simulations in the NVT ensemble and not the second derivative calculation detailed in the paper you cite.
Another helpful source of information to me at that time were the 5th and 7th tomes of the Landau-Lifschitz series which make the connection between statistical physics, thermodynamics and material mechanics. The full lecture is not required, as only some (obvious) chapters are self-sufficient.