Modeling Intermolecular Interactions

Although the dispersion and exchange-repulsive interactions between molecules can be calculated by quantum mechanics, we often resort to the force field description, a simple empirical expression, to perform molecular dynamics simulations. The prevalent form used in MD programs is to use Lennard-Jones potentials in Eq. (4) to describe the interaction between two particles

$$U_{LJ}(r)=4\epsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}\right].$$ (4)

There are two parameters in the Lennard-Jones 12-6 potential: the collision diameter $\sigma$ (the distance of two particles where the energy is zero) and the potential well depth $\varepsilon$. These parameters are illustrated in Fig. 8.

Figure 8: The Lennard-Jones potential.

The Lennard-Jones equation may also be expressed in terms of the distance where the energy reaches a minimum $r_{m}$. At this separation, the first derivative of the energy with respect to the internuclear distance is zero (i.e. $\partial V/\partial r=0$), from which it can easily be calculated that $r_{m}=2^{\frac{1}{6}}\sigma$. We can rewrite the Lennard-Jones 12-6 potential function as follows

$$U_{LJ}(r)=\epsilon\left[\left(\frac{r_{m}}{r}\right)^{12}-2\left(\frac{r_{m}}{r}\right)^{6}\right].$$ (5)

Finally, Lennard-Jone's original potential can be written in the following general form:

$$U_{LJ}(r)=k\varepsilon\left[\left(\frac{\sigma}{r}\right)^{n}-\left(\frac{\sigma}{r}\right)^{m}\right]$$ (6)

$$k=\frac{n}{n-m}\left(\frac{n}{m}\right)^{\frac{m}{n-m}}.$$ (7)

The Lennard-Jones 12-6 potential is a special case for $n=12$ and $m=6$. The Lennard-Jones potential is characterized by an attractive part that varies as $r^{-6}$ and a repulsive part that varies as $r^{-12}$. These two components are depicted in Fig. 9.

Figure 9: The Lennard-Jones potential is constructed from a repulsive component $r^{-12}$ and an attractive component $r^{-6}$.

The $r^{-6}$ part comes from the dispersion energy, but there is no strong theoretical argument in favor of the repulsive $r^{-12}$ form. Indeed, quantum mechanical calculations suggest an exponential form. The twelfth power term is found to be reasonable for rare gases but is probably too hard for other systems such as methane.

We will check the effect of the repulsive part of the Lennard-Jones potential in MD simulations, in particular, in the calculation of the PMF for two solutes in water.