Lennard-Jones Potential

Imagine two balls separated a certain distance apart. In order for these two balls to begin interacting, we add energy to push them closer together. Eventually, we will get to a certain distance in which it becomes extremely difficult to further decrease the distance between the two balls. It requires even more energy because the two balls are beginning to repel one another as they each start to invade each other’s space.

The scenario described above is similar to what happens in non-bonding atoms and molecules. We can think of these atoms and molecules not currently participating in bonding as two separate particles. When they are separated an infinite distance apart, the possibility of them coming together and forming a bond is minimal. In fact, we can say that their bonding potential energy is zero. However, as we start to decrease their distance of separation, then it becomes more likely that the two particles will interact and eventually be bound to one another. At the region where the two particles are bound, the bonding potential energy decreases and becomes a negative. The two particles now bound to each other will reach a point of equilibrium, which is specified by the separation distance at which the minimum potential energy is reached.

If we keep pushing the two bound particles together pass their point of equilibrium, repulsion will start to occur. This is due to the Pauling Exclusion Principle. The two particles are so close to each other that their electrons are being forced to occupy the same space. According to Pauling, two electrons cannot occupy the same space. Therefore, repulsion occurs as the electrons fight off each other to retain their space in their respective orbitals. Despite of the repulsive force between the two particles, their bonding potential energy rapidly increases as the distance of separation between them passes the equilibrium point. The possibility of changing the bonding of atoms and the bonding within the nuclei if we continue to push the particles even closer causes the increase in bonding potential energy. The changes in bonding potential energy are summarized by plotting the Lennard-Jones graph.

The Lennard-Jones equation is an approximation of atomic interactions of particles that do not participate in bonding. It sums up their attraction at short distances and repulsion energies. The equation takes into account all of the attractive forces, mainly dipole-dipole, dispersion, and dipole-induced-dipole interactions and subtracts it from the sum of repulsive forces.

V = 4?{(?/r)12 - (?/r)6}

Epsilon (?) is a measure of how strongly the two particles attract each other. The strength of the attractive force is represented by the depth of the well; the deeper its depth means the stronger the attraction between the two particles. Sigma (?) is the distance at which the bonding potential energy is zero. Lastly, lower cased r represents the distance between the two particles or the distance of separation.

From this graph, we can see that the attraction of the particles can be carried over longer distances than the repulsive force.