Derivation of Lennard Jones 9-3 model

1. Aug 3, 2010

Casey Jay

Dear all,

I am puzzled by the derivation of the Lennard Jones 9-3 model. Many articles mentioned that the Lennard Jones 9-3 model is derived from the integration of a 3d half-lattice of Lennard-Jones 12-6 particles.

The 12-6 potential is:

$$4\epsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^6\right]$$

while the 9-3 model is:
$$\epsilon\left[\frac{2}{15}\left(\frac{\sigma}{r}\right)^{9}-\left(\frac{\sigma}{r}\right)^3\right]$$

Can anyone point me to the right direction how should I perform integration of a 3d half-lattice of Lennard-Jones 12-6 particles? Do I change $$r$$ to $$r_x, r_y, r_z$$ and integrate with respect to $$dx, dy, dz$$?