Temperature dependence of Lennard-Jones potential

  • #1
Päällikkö
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Main Question or Discussion Point

My question is not so much about the Lennard-Jones potential, although I mentioned it in the title, but of the "force field" thinking in general.

So a lot of people are (were) interested in the phase transition temperature of the Ising model. How realistic is the model in the sense that it assumes an interaction energy of J, which is independent of temperature, between neighboring sites? If instead of magnets, we think of solutions, the very same Ising model is called the regular solution model (in the Bragg-Williams mean field approximation). Now is it true that, say, water and oil mix the way described by the model (i.e. no explicit temperature dependence)?

These are of course simple toy models, but real scientists employ Lennard-Jones potentials in their molecular dynamics codes, but never seem to give too much thought, or at least discussion in the publications, to this matter. The van der Waals forces (which the attractive part of the Lennard-Jones potential represents) for neutral particles is due to London dispersion forces, which if I recall is supposed to vary as 1/T. This is to my knowledge never accounted for in actual molecular dynamics force fields. Why?

When can I be sure that the interaction energies between two (or more) particles do not explicitly depend on temperature? When does this approximation break down? Naturally, I'd also be quite interested in how other state variables, such as pressure, might affect the microscopic interactions, and when I can ignore their effects.
 

Answers and Replies

  • #2
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Individual particles do not have a temperature. If you consider the collision between two atoms, "temperature" is not a meaningful parameter for the setup. Collision speed is relevant for the motion, but unless you have relativistic collisions, I don't think this influences the potential significantly.
 
  • #3
Päällikkö
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To my understanding it is well known that if you were to fit Lennard-Jones potentials to quantum calculations, you'd quickly go astray as the interactions depend on the environment the two particles are in. I suppose that all force fields should really then be formulated as a multibody potentials, independent of temperature. However, in classical force fields, one tends not to include potentials that are not "easily understood" (I don't recall seeing anything beyond 4-body potentials, and those are only ever included for bonded interactions), and as such this environmental impact is lost. Now that I've formulated the problem in this way, maybe I should be asking whether the situation could be rectified by allowing the Lennard-Jones coefficients to directly depend on the state variables, such as temperature? How do I know when it is safe to drop the higher terms of the multibody potential expansion?

This might sound esoteric, but it does have real relevance in coarse grained force fields, where one can see aspects of Lennard-Jones that are even more worrisome than the ones described in the previous paragraph. These coarse grained force fields are a large class of force fields, where each computational bead represents not one atom, but a collection of them. One could for example take four whole water molecules and represent them as an effective Lennard-Jones blob. This has the advantage of accelerating the computations, often by several orders of magnitude. But of course there are downsides: obviously one loses degrees of freedom and therefore dynamical processes and defining time itself become difficult problems. What one usually does retain is a consistent structure (in the sense that g(r) is reproduced correctly), especially so if iterative Boltzmann inversion was used to generate the force field of the coarse grained representation. Now what is very worrying to me is that the thermodynamics seem to be wrong: The newly created force field does not work if you change temperature (and furthermore, the A-A potential changes if a particle of type B is introduced into the mix). One can argue that this is due to the fact that one loses degrees of freedom, and therefore the representation of entropy is wrong. However, all-atom force fields are essentially coarse grained representations of the underlying world of quantum mechanics. This is why I am asking whether I can ever be assured that Lennard-Jones potential should not depend on temperature. Taking into account this entropy argument, can even the multibody representation discussed in the previous paragraph be truly temperature-independent? Is there a rule of thumb as to when I need not to worry about this aspect?
 

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