# How does the Lennard Jones Potential changes for interaction between particles of different sizes?

I am interested in incorporating a Lennard-Jones potential in a simulation. When the interaction only involves the same type of particle, with same characteristics, we can use reduced units, scaling the units according to the potential well. But I assume the potential well is not the same for a case with particles of different sizes. Neither the sigma parameter should remain the same in my opinion. How can I propose the units or incorporate such feature to my code?

The question can be answered theoretically, but it would be nice to have examples with real code (in Fortran, Matlab, C, or whatever language you have experience with).

• This isn't really a site for code, though. As a theoretical question, it's perfectly fine here, but you may or may not get actual code samples. – David Z Feb 17 '14 at 2:32
• Yes, I know, it was just a suggestion. Thanks for the comment anyways. – adpala Feb 18 '14 at 3:14

## 1 Answer

when you say "sizes of the particles" you need to be more specific. do you mean you have hard core repulsion? if so than each particle will have it's potential with it's radius $r_c$:

$$u(r) = 4\cdot\epsilon[(\frac{\sigma}{r})^{12}-(\frac{\sigma}{r})^{6}] \ for :r>r_c$$ $$u(r) = \infty for :r<r_c$$

you can still use reduced units, just make sure that the particles are not inside one another before you make a monte carlo or molecular dynamics move.

If you mean that some particles have a different potential that gives a different minimum to the energy, that's a whole other problem and you should be more specific and tell us what the potentials for each particle looks like.

• With respect to hard-core Lennard-Jones, the proposed function is fine for Monte Carlo, but cannot be used for Molecular Dynamics, wherein the discontinuity of the function would make you require an infinitely small time-step. – Ghersic Dec 12 '18 at 23:41