# Lennard Jones Potential

The Lennard-Jones potential (after John Lennard-Jones ) describes the binding energy in physical chemistry and in atomic and molecular physics . It approximates the interaction between uncharged, not chemically bonded atoms . ${\ displaystyle V}$

Figure 1: The Lennard-Jones (12, 6) potential plotted against the particle distance . In the area of ​​a negative slope repulsive forces act, in the area of ​​a positive slope attractive forces.${\ displaystyle V}$${\ displaystyle r}$

## description

For large distances between two particles , the attractive forces predominate; They are mainly Van der Waals forces , but also permanent dipole-dipole interactions .

If the respective particles are approached, the repulsive component predominates below a certain distance (see Figure 1) and the potential energy increases rapidly. The repulsive forces come about because the electrons sometimes have to move to energetically higher orbitals when the atomic shells approach , because, according to the Pauli principle , several of them cannot occupy the same state. ${\ displaystyle r_ {m} \ approx 1 {,} 12 \ sigma}$

The attractive part of the Lennard-Jones potential is derived from the London formula ( approximation ):

${\ displaystyle V = - {\ frac {C} {r ^ {6}}}}$,

in which

• ${\ displaystyle C}$is a relatively complicated term that contains substance-specific constants such as the ionization energy for both particles under consideration, and
• ${\ displaystyle r}$ the distance between the particles.

The repulsive part is described by a similar equation:

${\ displaystyle V = {\ frac {C_ {n}} {r ^ {n}}}}$

Here is . ${\ displaystyle n> 6}$

The two parts are combined in the Lennard-Jones (n, 6) potential :

${\ displaystyle V (r) = {\ frac {C_ {n}} {r ^ {n}}} - {\ frac {C} {r ^ {6}}}}$

This is often chosen for practical reasons , because then the value only has to be squared when calculating . The result is the Lennard-Jones (12, 6) potential , which is typically written in one of the following two forms: ${\ displaystyle n = 12}$${\ displaystyle 1 / r ^ {6}}$

{\ displaystyle {\ begin {alignedat} {2} V (r) & = 4 \ varepsilon \ left [\ left ({\ frac {\ sigma} {r}} \ right) ^ {12} - \ left ({ \ frac {\ sigma} {r}} \ right) ^ {6} \ right] && = 4 \ varepsilon \ cdot \ left ({\ frac {\ sigma} {r}} \ right) ^ {6} \ cdot \ left [\ left ({\ frac {\ sigma} {r}} \ right) ^ {6} -1 \ right] \\ & = \ varepsilon \ left [\ left ({\ frac {r_ {m}} {r}} \ right) ^ {12} -2 \ left ({\ frac {r_ {m}} {r}} \ right) ^ {6} \ right] && = \ varepsilon \ cdot \ left ({\ frac {r_ {m}} {r}} \ right) ^ {6} \ cdot \ left [\ left ({\ frac {r_ {m}} {r}} \ right) ^ {6} -2 \ right ] \ end {alignedat}}}

Here is

• ${\ displaystyle \ varepsilon> 0}$the “depth” of the potential well in units of joules, which is created by the two influences.
• ${\ displaystyle \ sigma}$the particle distance at which the Lennard-Jones potential, a zero has: .${\ displaystyle V (r = \ sigma) = 0}$
• ${\ displaystyle r_ {m} = {\ sqrt [{6}] {2}} \ cdot \ sigma \ approx 1 {,} 12 \ cdot \ sigma}$the particle distance at which the Lennard-Jones potential reaches its minimum. At this distance, the forces from the attractive and repulsive part of the potential are equally large and cancel each other out, so that in this distance there is no total force between the particles.

The Lennard-Jones potential is a special case of Mie potential s

${\ displaystyle V = {\ frac {C_ {n}} {r ^ {n}}} - {\ frac {C_ {m}} {r ^ {m}}}}$

introduced by Gustav Mie in 1903 .

## Others

Another form of the Lennard-Jones potential is the Lennard-Jones (exp, 6) potential , where the repulsive term is exponential . It is a special case of the Buckingham potential :

${\ displaystyle V (r) = {\ frac {\ varepsilon} {1-6 / \ alpha}} \ cdot \ left \ langle {\ frac {6} {\ alpha}} \ cdot \ exp \ left [\ alpha \ left (1 - {\ frac {r} {\ sigma}} \ right) \ right] - \ left ({\ frac {\ sigma} {r}} \ right) ^ {6} \ right \ rangle}$

with the "steepness" as the repulsive force. ${\ displaystyle \ alpha}$

## Individual evidence

1. Mie potential (online)
2. ^ Edward A. Mason: Transport Properties of Gases Obeying a Modified Buckingham (Exp-Six) ​​Potential . In: Journal of Chemical Physics . No. 22 , 1954, pp. 169-186 , doi : 10.1063 / 1.1740026 .
3. ^ RA Buckingham: The Classical Equation of State of Gaseous Helium, Neon and Argon . In: Proceedings of the Royal Society of London . Series A, Mathematical and Physical Sciences, No. 168 , 1938, pp. 264-283 , doi : 10.1098 / rspa.1938.0173 .