I am reading the LAMMPS manual and understand that the Lennard Jones 9-3 potential in fix wall/lj93 is derived by integrating over a 3d half-lattice of Lennard-Jones 12/6 particles (see http://lammps.sandia.gov/doc/fix_wall.html).
Can someone point to me how can the 9-3 potential be derived mathematically by integration? I tried to integrate the 12-6 model with respect to dx, dy and dz but in vain.
Thank you very much.
A bit of guesswork here, but I suspect that the summation over the infinite half-lattice of LJ 12/6 particles is being approximated by the integral over the domain z > 0, and thus the integration can be carried out in spherical coordinates; the measure would then be r^2 sin \theta dr d\theta d\phi, with \theta going from 0 to \pi/2 instead of 0 to \pi. It’s easy to see then how 9-3 comes about, as the powers of u® r^2 are r^-10 and r^-4, which go to r^-9 and r^-3, respectively.
Thank you Ahmad for your suggestion.
May I know what are the limits for r and phi then? I am still unable
to obtain the 9-3 model despite following your idea.
I think Mark Stevens (CCd) knows the reference for this. I can't
recall where it is derived. Possible in Allen & Tildesley.