A pair of neutral atoms or molecules is subject to two distinct forces in the limit of large separation and small separation: 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.

The L-J potential is of the form

$$

V(r) = 4epsilon 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 and r is the distance between the particles.

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 Lennard-Jones potential is an approximation. 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. Its physical origin is related to the Pauli principle: when the electronic clouds surrounding the atoms start to overlap, the energy of the system increases abruptly. The exponent 12 was chosen exclusively because of ease of computation.

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.

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} - 2left(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

$$ 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 $displaystyle\; r=r\_c$.

See also

• Molecular mechanics
• Embedded atom model
• Morse potential
• Force field (chemistry)
• Lennard-Jones model on SklogWiki.

References