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# Lennard-Jones potential

Neutral atoms and molecules are subject to two distinct forces in the limit of large distance and short distance: an attractive force at long ranges (van der Waals force, or dispersion force) and a repulsive force at short ranges (the result of overlapping electron orbitals, referred to as Pauli repulsion from Pauli exclusion principle). The Lennard-Jones potential (also referred to as the L-J potential, 6-12 potential or, less commonly, 12-6 potential) is a simple mathematical model that represents this behavior. It was proposed in 1924 by John Lennard-Jones.[1]   The L-J potential is of the form

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

where $\, \epsilon$ is the depth of the potential well and $\, \sigma$ is the (finite) distance at which the interparticle potential is zero.

These parameters can be fitted to reproduce experimental data or deduced from results of accurate quantum chemistry calculations. The $\left(\frac{1}{r}\right)^{12}$ term describes repulsion and the $\left(\frac{1}{r}\right)^{6}$ term describes attraction. The force function is the negative of the gradient of the above potential:

$\mathbf{F}(r) = - \nabla V(r) = - \frac{d}{dr} V(r) \hat{\mathbf{r}} = 4 \epsilon \left( 12\,{\frac {{\sigma}^{12}}{{r}^{13}}}-6\,{\frac{{\sigma}^{6}}{{r}^{7}}} \right) \hat{\mathbf{r}}$

The L-J potential is approximate. The form of the repulsion term has no theoretical justification; the repulsion force should depend exponentially on the distance, but the repulsion term of the L-J formula is more convenient due to the ease and efficiency of computing r12 as the square of r6. The attractive long-range potential, however, is derived from dispersion interactions. The L-J potential is a relatively good approximation and due to its simplicity is often used to describe the properties of gases, and to model dispersion and overlap interactions in molecular models. It is particularly accurate for noble gas atoms and is a good approximation at long and short distances for neutral atoms and molecules. On the graph, Lennard-Jones potential for argon dimer is shown. Small deviation from the accurate empirical potential due to incorrect short range part of the repulsion term can be seen.

The lowest energy arrangement of an infinite number of atoms described by a Lennard-Jones potential is a hexagonal close-packing. On raising temperature, the lowest free energy arrangement becomes cubic close packing and then liquid. Under pressure the lowest energy structure switches between cubic and hexagonal close packing.[2]

Other more recent methods, such as the Stockmayer equation and the so-called multi equation, describe the interaction of molecules more accurately. Quantum chemistry methods, Møller-Plesset perturbation theory, coupled cluster method or full configuration interaction can give extremely accurate results, but require large computational cost.

## Alternative expressions

The Lennard-Jones potential function is also often written as

$V(r) = \epsilon \left[ \left(\frac{r_{min}}{r}\right)^{12} - 2\left(\frac{r_{min}}{r}\right)^{6} \right]$

where

$\, r_{min}$ = $\, 2^{1/6}\sigma$ is the distance at the minimum of the potential.

The simplest formulation, often used internally by simulation software, is:

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

where

$\, A = 4 \epsilon \sigma^{12}$

$\, B = 4 \epsilon \sigma^6$

$\sigma = \left( \frac{A}{B} \right)^{ \frac{1}{6} }$

and

$\epsilon = \frac{B^2}{4 A}$.

## Molecular dynamics simulation: Truncated potential

To save computational time, the Lennard-Jones (LJ) potential is often truncated at the cut-off distance of $\displaystyle r_c = 2.5 \sigma$ where

 $\displaystyle V ( r_c ) = V ( 2.5 \sigma ) = 4 \epsilon \left[ \left( \frac {\sigma} {2.5 \sigma} \right)^{12} - \left( \frac {\sigma} {2.5 \sigma} \right)^6 \right] = -0.0163 \epsilon = - \frac {1} {61.3} \epsilon$ (1)

i.e., at $\displaystyle r_c = 2.5 \sigma$, the LJ potential $\displaystyle V$ is about 1/60th of its minimum value $\displaystyle \epsilon$ (depth of potential well). Beyond $\displaystyle r_c$, the computational potential is set to zero. On the other hand, to avoid a jump discontinuity at $\displaystyle r_c$, as shown in Eq.(1), the LJ potential is shifted upward a little so that the computational potential would be zero exactly at the cut-off distance $\displaystyle r_c$.

For clarity, let $\displaystyle V_{LJ}$ denote the LJ potential as defined above, i.e.,

 $\displaystyle V_{LJ} (r) = 4 \epsilon \left[ \left( \frac {\sigma} {r} \right)^{12} - \left( \frac {\sigma} {r} \right)^6 \right]$ (2)

The computational potential $\displaystyle V_{comp}$ is defined as follows [3]

 $\displaystyle V_{comp} (r) := \begin{cases} V_{LJ} (r) - V_{LJ} (r_c) & {\rm for} \ r \le r_c \\ 0 & {\rm for} \ r > r_c \end{cases}$ (3)

It can be easily verified that $\displaystyle V_{comp} (r_c) = 0$, thus eliminating the jump discontinuity at $\displaystyle r=r_c$.