Potential of mean force from potential energy

Question

Imagine a simple model of a crystal where every atom in the crystal is the same, and the potential energy of the pairwise interaction between atoms is some radial function of distance, U = U(r). For example this could be the Lennard-Jones solid with hcp packing, U(r) = A/r^12 + B/r^6

By calculating the potential energy for different unit cell sizes, we can get the potential energy vs unit cell size curve: this is essentially a volume vs pressure curve at 0K temperature. We can in principle invert this calculation to get the pairwise potential U(r) from the volume vs pressure curve (I believe the inverse problem has a unique solution, although I can't prove that yet).

Now, suppose the same solid is simulated at non-zero temperatures. The equilibrium unit cell size at zero pressure will be larger due to thermal expansion; and also at any cell size the average potential energy will be different from the potential energy at 0K. We can calculate the pressure using the virial, and again get a volume versus pressure curve. Inverting this in the same way gives a potential of mean force E(r; T) for the pairwise interaction between particles where r is now the mean distance and E is a free energy rather than potential energy.

The question is: how can E(r; T) be calculated from U(r)? If possible, I'm interested in an analytical derivation for simple potentials like Lennard-Jones, but also in methods for getting this from a single trajectory, without running a whole lot of simulations at different unit cell sizes and temperatures.

Basically the nature of the problem is this: we know the pairwise potential energy - how do we get the probability distribution of coordinates at nonzero temperatures (in a periodic solid), and how do we get a pairwise free energy?

P.S. This is related to the "Boltzmann inversion method" used in designing coarse-grained forcefields from atomistic simulations, but is not the same: in Boltzmann inversion the goal is to match the radial density function, here the goal is to match the free energy vs volume curve.

Answer 1

Your definition for "a potential of mean force" (or PMF) is not correct. The PMF of a probe from a reference particle is the potential energy change when the probe is moved towards or further from the reference, averaged radially and Boltzmann-averaged over all resulting perturbations of other particles in the ensemble of choice.

This is not the same thing as dilating all particle positions at once away from the reference particle. As an example, consider the particle-particle PMF of an ideal gas: it does not change with density, but the pressure certainly does.

Wikipedia's article for the radial distribution function, or RDF has many useful equations, including connecting it to the PMF (eq 8) and the pressure equation of state (eq 10).

To understand the subsequent quest to derive the RDF from the pair interaction function, historically one starts with understanding the BBGKY hierarchy and subsequently various models for closing it. However, a more modern approach is to study classical density functional theory or cDFT, and ultimately the state of the art is with 3D Reference Interaction Site Model or 3D-RISM theory.