Implications of the PMF

Consider a system composed of two methane molecules freely roaming in liquid water, Figure 16 plots the distance between two methane molecules in a MD simulation. Even through there is hardly any attractive force between two methane molecules at distance larger than $4.5$ Å, they are bound by the surrounding water during the simulation time duration (tens of picoseconds). The phenomena can be roughly explained by an entropy argument (see Sec. 2.4).

We emphasize that there is always an effective potential between two methane molecules when they are immersed in a simple classical fluid. In a simple fluid, the ``attraction'' and ``repulsion'' between two solutes result from collisions with all other molecules. The peaks of radial distribution function partly come from the structure of hydration shells of solutes. The first peak represents the water in the first layer of the hydration shell. As the separation between solutes increases slightly, there is still not enough space for water to enter and the radial distribution function drops until a layer of water can separate the two solutes. The oscillations of radial distribution function become less at larger separations, and eventually the radial distribution becomes one when the separation is far enough. In addition, the water molecules around the solutes are correlated strongly, leading to a modified radial distribution.

Figure 16: Distance between two freely-roaming methane molecules in water. Two methane molecules are bound within the simulation time of 6 ps. The wiggling of the distance occurs on the time scale of mode of C-H bonds in methane.

The intermolecular interaction may be another factor to influence the PMF. In a recent work [16], reproduced in Fig. 18, quantum-mechanical simulation shows a deep contact potential minimum and an insignificant solvent-separated second potential minimum in the PMF. The PMF in Fig. 18 has both similarity and distinct difference with classic calculation results which are based on parameterized force fields and water models. First, the PMF has a characteristic contact minimum at $3.9$ Å, similar to classical studies. The insignificant second potential minimum was found near $6.7$ Å.

Figure 17: The methane-water force computed in DFT and Lennard-Jones potential. The orientation of water molecule is optimized in DFT calculations. The zero-force line is plotted for reference.

In classical force-field simulations, we use Lennard-Jones potentials which have a hard repulsive part, unlike the force between real molecules. In quantum-mechanical treatment, the interaction is softer than Lennard-Jones potentials and exhibits a more reasonable behavior interacting with other molecules. The comparison between LJ potentials and intermolecular forces calculated within the density-functional theory is shown in Fig. 17. The shape of the PMF by quantum-mechanical calculations is in conflict with classical results. We propose this conflict may be understood in terms of the methane-water interaction strength. The softer repulsive part blurs the solvent-separated minimum and contact potential minimum. The softer repulsive force gives less restriction to form methane configuration and washes out features in the PMF for distances longer than the contact separation.

Figure 18: The PMF comes from constrained FPMD[16].