The Potential of Mean Force

It is useful to know how the free energy changes as a function of reaction coordinates, such as the distance between two atoms or the torsion angle of a bond in a molecule. Here we introduce the concept of the potential of mean force (PMF). When the system is in a solvent, the PMF incorporates solvent effects as well as the intrinsic interaction between the two particles. When the same two particles were brought together in the gas phase, the free energy would simply be the pair potential $u(r)$, which has only a single minimum. But the PMF between two particles in liquid oscillates with maximum and minimum. For a given separation $r$ between the two molecules, the PMF describes an average over all the conformations of the surrounding solvent molecules.

Various methods have been proposed for calculating potentials of mean force. The simplest representation of the PMF is to use the separation $r$ between two particles as the reaction coordinate. The PMF is related to the radial distribution function using the following expression for the Helmholtz free energy

$$A(r)=-k_{B}T\ln g(r)+{\rm const}.$$ (23)

The constant is chosen so that the most probable distribution corresponds to a free energy of zero. Unfortunately, the PMF may vary by several multiples of $k_{B}T$ over the relevant range of the distance $r$. The algorithmic relationship between the PMF and the radial distribution function means that a relatively small change in the free energy (i.e. a small multiple of $k_{B}T$) may correspond to $g(r)$ changing by an order of magnitude from its most likely value.

The standard Monte Carlo or molecular dynamics simulation methods do not adequately sample regions where the radial distribution function differs drastically from the most likely value, leading to inaccurate values for the PMF. We shall describe two calculation schemes in the following to circumvent this problem.