# What is the difference between the differents lennard-jones parameters

I know different lennard jones potentials 12-6, 9-6, 9-3. I think the difference between them is the way they describe long-range interactions,but when I use each of them?

The 12 that describes the repulsive short range interactions between two molecules is due to Pauli exclusion principle, but you could also use 11 or 13, I mean, you just need a number of that order. While the 6 of the attractive long range interactions is fixed when you are considering interactions between instant dipole and induced dipole. If use 7 inteas of 6 you are wrong. While I have never seen the last two terms you wrote. Maybe they could work for molecules with a permanent dipole or with particular features. However remeber that LJ belongs to the class of empirical potentials, so there is not a quantum theory behind to derive it analitically.

• Actually, if one takes into account the delay effects due to the finite value of the speed of light, $1/r^7$ is the correct asymptotic behavior of the induced dipole-induced dipole interaction between tho shell closed atoms. However, this is a side remark. I'll try to focus on the original question in my answer. – GiorgioP Jun 8 at 22:05

The list of possible "m-n Lennard-Jones" potentials is not exhausted by the 12-6, 9-6, 9-3 cases.

The original (12-6) choice was intended to model interaction in closed shell systems by combining an exact result of non-relativistic perturbation theory (the attraction proportional to $$1/r^6$$), and a convenient analytical expression $$1/r^{12}$$ for the harsh epulsion at small distances. This potential has been demonstrated to be an approximate but reasonably accurate expression of the interaction among rare gas atoms, at least provided that experiments would not probe too short distances.

However, most of the interesting systems are not made by rare gases. In the case of a non-shell-closed atoms, there is no theoretical justification for the van der Waals $$1/r^6$$ attraction and it is possible to check that the repulsive part of the interaction, if fitted with an inverse power law, it may differ significantly on the original $$1/r^{12}$$. Actually, combination of theory and experiments suggests much more complicate form of the interactions. Moreover, it is evident that, but for a few systems, many-body interactions, not reducible to sum of pair-wise interactions, are required for a satisfactory modeling of the forces in condensed phases.

Nevertheless, for some applications or for analyzing general trends, simpler model interactions which still can capture the two basic ingredients of any atomic or molecular force field, i.e. short range repulsion and long range attraction, are useful and in this context it is possible to explain the blooming of different members of the m-n Lennard-Jones family.

As of today, m-n Lennard-Jones parameters can be found in the literature for a vast number of systems ranging from small molecules like N$$_2$$, O$$_2$$, CO, CO$$_2$$ to Lanthanides ions, from hydrocarbons to carbon nanotubes, from proteins to amino-acids.

Summarizing, the choice between different form of LJ potential should be guided by the existing experience on similar systems and by the level of agreement with experimental data which is required. Usually, one tries to reproduce one or more physical properties, optimizing the potential also with respect m and n.