Lennard Jones Potential

Intermolecular Forces

When two molecules come close to one another, different types of interactions develop between these molecules. These intermolecular forces are between electrons and nuclei of one molecule and electrons and nuclei of another molecule. The intermolecular forces produce potential energy of interaction. The interaction between these molecules depends upon the distance between the molecules.

The Lennard-Jones Potential describes the relationship between the potential energy of interaction and the distance between molecules. At very small distances, the molecules repel each other very strongly. As these neutral molecules move far apart, at a certain distance the potential energy of interaction is negative and the molecules weakly attract each other. At very large distances, the potential energy approaches zero.

The Lennard-Jones Potential

An intermolecular potential that contains attractive long-range behavior and short-range repulsive behavior expressed simply in the equation as

$V(r)=\frac{A}{r^{12}}-\frac{B}{r^{6}}$

• A = 4εσ12
• B = 4εσ6
• r = Intermolecular Distance

So the Lennard-Jones potential equation is written as

$V\left&space;(&space;r&space;\right&space;)=4\varepsilon&space;\left&space;[&space;\left&space;(&space;\frac{\sigma&space;}{r}&space;\right&space;)^{12}-\left&space;(&space;\frac{\sigma&space;}{r}&space;\right&space;)^{6}&space;\right&space;]$

• ε = Depth of the potential well
• σ = The intermolecular distance at which V = 0

In this 6-12 potential, the first term is for repulsive forces and the second term represents attraction.

Repulsive Forces

When molecules come together the nuclear and electronic repulsions dominate the attractive forces. These repulsions increase as a decrease in intermolecular distance. The Lennard-Jones potential simplified the complex calculation regarding repulsion and represents it in terms of potential energy.

If particles are close enough when the separation between the particles is ‘d‘ then the potential is to be considered as hard-sphere potential. On that potential, it is assumed that potential energy approaches infinity as particles come within a distance ‘d’. So

V = ∞     at     r ≤ d     and     V = 0     at     r > d

The 12th power term in the Lennard-Jones potential is repulsive potential that increases steeply as a decrease in intermolecular distance.

Attractive Forces

The long-range attractive forces between neutral molecules are called van der Waals forces. The actual form of the attractive term in the Lennard-Jones potential equation is well established as compared to the repulsive term. Three contributions are included in the r-6 attraction.

If two molecules have permanent dipole-dipole moments μA and μB at temperature ‘T’, the average interaction energy results in a term of attractive r-6 form equation as$V\left&space;(&space;r&space;\right&space;)_{dd}=-\frac{2}{3k}\left&space;(&space;\frac{\mu&space;_{A}\mu&space;_{B}}{4\pi&space;\epsilon&space;_{0}}&space;\right&space;)^{2}\frac{1}{r^{6}}$

The equation for a dipole-induced dipole is given by

$V\left&space;(&space;r&space;\right&space;)_{ind}=-\frac{\alpha&space;_{B}\mu&space;_{A}^{2}+\alpha&space;_{A}\mu&space;_{B}^{2}}{\left&space;(&space;4\pi&space;\epsilon&space;_{0}&space;\right&space;)^{2}r^{6}}$

where α is the polarizability of molecules.

Similarly, for the London dispersion forces

$V\left&space;(&space;r&space;\right&space;)_{disp}=-\frac{3}{2}\left&space;(&space;\frac{E_{A}E_{B}}{E_{A}}&space;\right&space;)\left&space;(&space;\frac{\alpha&space;_{A}\alpha&space;_{B}}{\left&space;(&space;4\pi&space;\epsilon&space;_{0}&space;\right&space;)^{2}}&space;\right&space;).\frac{1}{r^{6}}$

where Eand EB are energies of the first electronic transitions of these molecules

Lennard-Jones Interaction Potential

The minimum Lennard-Jones potential occurs at a minimal distance rmin. To find the distance rmin, the differential of the Lennard-Jones equation must be zero. i.e.

$\frac{dV}{dr}=0$

So differentiating eq. 2 w.r.t ‘r’, we get

For rmin dV/dr = 0 so above equation becomes

${r_{min}}^{6}=2\sigma&space;^{6}$

$r_{min}&space;=2^{1/6}\sigma$

Therefore

$V\left&space;(&space;r&space;_{min}\right&space;)=4\varepsilon&space;\left&space;[&space;\left&space;(&space;\frac{\sigma&space;}{2^{1/6}\sigma&space;}&space;\right&space;)^{12}-\left&space;(&space;\frac{\sigma&space;}{2^{1/6}\sigma&space;}&space;\right&space;)^{6}&space;\right&space;]=4\varepsilon&space;\left&space;[&space;\frac{1}{4}-\frac{1}{2}&space;\right&space;]=-\varepsilon$

So the minimum potential at r = 21/6 is V = -ε. The potential is zero for r = σ so potential arises very steeply for the small value of r.

Lennard-JOnes Parameters

The two Lennard-jones parameters ε and σ have the following information

• ε tells about the strength of attraction between molecules
• σ is the measure of the size of molecules

Typical values of Lennard-jones parameters are tabulated below for a number of molecules

Concepts Berg

What does the Lennard-Jones potential describe?

The Lennard-Jones potential describes the potential energy of interaction (both for attraction and repulsion) between two molecules or non-bonded atoms.

What is the derivative of Lennard-Jone’s potential

The derivative of the Lennard-Jones potential equation with respect to ‘r’ is given as

$\frac{dV}{dr}=4\varepsilon&space;\left&space;[&space;-\frac{12\sigma&space;^{12}}{r^{13}}+\frac{6\sigma&space;^{6}}{r^{7}}&space;\right&space;]$

What is Epsilon in Lennard-Jone’s potential?

The epsilon ε in the Lennard-Jones potential represents the depth of the potential well. It is the measure of how strong the attraction is between the molecules.

What is intermolecular potential energy?

The intermolecular potential energy is the forces between the molecules. It depends upon the type of forces and distance between the molecules.

Which type of force is greater when the distance between the two atoms is large

At the larger distance between two particles, the forces of attraction dominate the repulsion forces.

How could potential energy be negative?

The greater attraction between molecules has negative potential energy.

Why do atoms never touch

Why should an atom be in a solid-state when calculating its van der Waals radius?

To calculate the Van der Waals radius the atoms should be considered as hard-sphered objects.

In a Lennard-Jones potential function, what happens when U(x) = 0

The minimal interaction potential occurs when U(x) is equal to zero.

Reference Books

• Physical Chemistry – Fourth edition by R. J. Silbey, R. A. Alberty, and M. J. Bawendi
• Physical Chemistry A Molecular Approach – by D. A. Mcquarrie and J. D. Simon
• Physical Chemistry – Eighth Edition by P. Atkins and J. de Paula