# Fitting parameters of Lennard-Jones potential to properties of real materials

I'm using the Lennard-Jones potential for some (very simple) protein docking simulation* and couldn't find parameters for different materials (like $\ce{C}$, $\ce{N}$, $\ce{O}$, …). Does it even make sense to differentiate between different materials when using a crude approximation like the Lennard-Jones potential?

* the potential function for each configuration of the two molecules is the sum over all pairwise energies, for which I only consider the van der Waals term by using the LJ-potential

• Your question does not make really sense within the state-of-art molecular mechanics. Please refine what you mean, with respect to en.wikipedia.org/wiki/Force_field_%28chemistry%29 – ssavec May 27 '15 at 7:38
• My potential function for each configuration of the two molecules is the sum over all pairwise energies, for which I only consider the van der Waals term by using the LJ-potential; does it make more sense now? – Peter May 27 '15 at 9:05
• Do you have any reason why not to use any existing force field? E.g. UFF? – ssavec May 27 '15 at 9:20
• Than, again, get the parameters from any existing force field, for example dasher.wustl.edu/tinker/distribution/params/tiny.prm – ssavec May 27 '15 at 9:46
• It would be much better, if you could edit your question in response to the comments, to make clear what you are trying to achieve. And also to prevent it from getting closed. – Martin - マーチン May 27 '15 at 9:51

All molecular simulation force fields are essentially "fitted" mathematical functions. The parameters are adjusted until the simulation results match some physical properties we care about. As Geoff says, Lennard-Jones force fields are not particularly realistic, in the sense that you wouldn't expect a simulation of atoms interacting solely through Lennard-Jones potentials to behave exactly like a real system. This becomes very important for things like proteins, where conformation and charge-charge interactions are very important.

However, what might not be clear from the other answers and comments is that all force fields are not 100% realistic. The trick is to intelligently choose your level of realism. It could be that a Lennard-Jones force field that ignores charge-charge interactions works perfectly well for your application. Without knowing the details, I can't tell you how suitable it would be.

I do know that for polymers all sorts of real phenomena can be modeled with very simple Lennard-Jones-like force fields. A famous example is this article by Kurt Kremer and Gary Grest:

Dynamics of entangled linear polymer melts:  A molecular‐dynamics simulation. J. Chem. Phys. 92, 5057 (1990)

Also, as Geoff mentioned, LJ potentials work pretty well for things that can be modeled by the van der Waals equation of state: noble gases, simple diatomics, and other small molecules at high temperature and low pressure.

Some coarse-grained simulations use LJ potentials for each "blob," and depending again on what you want to study, this can be very successful.

And while we are on the subject of unrealistic force fields, the hard-sphere model has been used to successfully model things like glass transitions (I don't have a good representative link for this, but you can search for "hard sphere glass transition" to find many studies).

So, in response to your actual question:

Does it even make sense to differentiate between different materials when using a crude approximation like the Lennard-Jones potential?

Maybe - it depends on what the goal of your simulation is. What physical behavior are you trying to model? If you just want to see how a string of soft, slightly sticky beads sticks to another string, then you don't need to differentiate between atoms. If you want to see what happens when some of those beads are really sticky (sort of like hydrogen bonding sites on a protein), then it would make sense to use different parameters for very sticky vs. slightly sticky beads. It probably would never make sense to worry about which beads were carbon and which were oxygen, nitrogen, etc., or to try to select parameters for them that were based on published values (unless those studies were specifically looking at the same system and properties that you are). Most LJ parameters are derived from fitting simulated phase behavior to real phase behavior - which is pretty far removed from what you are doing.

If you want to get a better idea of what real proteins would do, then the problem becomes much more complicated. Real protein behavior has a lot to do with interactions with water, which means you would need to simulate that as well. Charge-charge interactions also become much more important (and the simulations much more computationally expensive)

My advice (and I think what the other responses are saying as well) is to choose your system and goals carefully so that you don't need this level of realism. Or, if you do need it, then do it right; don't try to make a simple model work for a system that it doesn't fit.

You could, in principal, create some simulation of "Lennard-Jones-ium" based on a set of tailored parameters. There's probably some benefit from an applied mathematics perspective. However, as discussed by multiple comments, this wouldn't be at all physically realistic.

Most force fields have many atom types - not just C vs. O, but $\ce{C_\alpha}$ or $\ce{C_{O2}}$ etc., because the parameters vary considerably in terms of interatomic forces, etc. for different atom types.

Also, the bonding interactions between two carbons are very different than non-bonded interactions. In other words, if you wanted a model for a protein and docking simply from LJ parameters, you'd need separate pair-wise parameters for bonded and non-bonded interactions.

Moreover, for chemistry, you need more than just pairwise atomic interactions. Most chemical force fields include three-atom angle-bending terms, four-atom dihedral terms, and out-of-plane distortion terms (e.g., pyramidal N or P).

Finally, all of this ignores electrostatic interactions, which a Lennard-Jones model would sweep under the rug.

If you want to simulate something with LJ, I might go for a sample of gas (e.g., He, Ne, etc.) or atoms in a solid (e.g., Ag).

• As for the application, the OP is right, that most terms do vanish. For rigid docking, i.e. two fixed molecules interact, only the non-bonding terms do not cancel out. From this point of view is the approximation by L-J fine. You would not get different energies based on charge interaction, though. The described procedure will answer the question whether given molecule sterically fits in cavity in protein. Which is not that bad, after all. – ssavec May 27 '15 at 19:58