# Lennard-Jones model

The **Lennard-Jones** intermolecular pair potential is a special case of the Mie potential and takes its name from Sir John Edward Lennard-Jones
^{[1]} ^{[2]}.
The Lennard-Jones model consists of two 'parts'; a steep repulsive term, and
smoother attractive term, representing the London dispersion forces ^{[3]}. Apart from being an important model in itself,
the Lennard-Jones potential frequently forms one of 'building blocks' of many force fields. It is worth mentioning that the 12-6 Lennard-Jones model is not the
most faithful representation of the potential energy surface, but rather its use is widespread due to its computational expediency.
For example, the repulsive term is maybe better described with the exp-6 potential.
One of the first computer simulations using the Lennard-Jones model was undertaken by Wood and Parker in 1957 ^{[4]} in a study of liquid argon.

## Contents

- 1 Functional form
- 2 Critical point
- 3 Triple point
- 4 Radial distribution function
- 5 Helmholtz energy function
- 6 Equation of state
- 7 Virial coefficients
- 8 Phase diagram
- 9 Zeno line
- 10 Widom line
- 11 Perturbation theory
- 12 Approximations in simulation: truncation and shifting
- 13 Cutoff Lennard-Jones potential
- 14 n-m Lennard-Jones potential
- 15 Mixtures
- 16 Related models
- 17 References

## Functional form[edit]

The Lennard-Jones potential is given by

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

or is sometimes expressed as

\[ \Phi_{12}(r) = \frac{A}{r^{12}}- \frac{B}{r^6}\]

where

- \(r := |\mathbf{r}_1 - \mathbf{r}_2|\)
- \( \Phi_{12}(r) \) is the intermolecular pair potential between two particles or
*sites* - \( \sigma \) is the value of \(r\) at which \( \Phi_{12}(r)=0\)
- \( \epsilon \) is the well depth (energy)
- \(A= 4\epsilon \sigma^{12}\), \(B= 4\epsilon \sigma^{6}\)
- Minimum value of \( \Phi_{12}(r) \) at \( r = r_{min} \);

\[ \frac{r_{min}}{\sigma} = 2^{1/6} \simeq 1.12246 ... \] In reduced units:

- Density\[ \rho^* := \rho \sigma^3 \]

where \( \rho := N/V \) (number of particles \( N \) divided by the volume \( V \))

- Temperature\[ T^* := k_B T/\epsilon \]

where \( T \) is the absolute temperature and \( k_B \) is the Boltzmann constant

The following is a plot of the Lennard-Jones model for the Rowley, Nicholson and Parsonage parameter set ^{[5]} (\(\epsilon/k_B = \) 119.8 K and \(\sigma=\) 0.3405 nm). See argon for other parameter sets.

## Critical point[edit]

The location of the critical point is
^{[6]}
\[T_c^* = 1.326 \pm 0.002\]
at a reduced density of
\[\rho_c^* = 0.316 \pm 0.002\]
The critical compressibility factor is given by ^{[7]}

\[Z_c = \frac{p_cv_c}{RT_c} = 0.281\]

Vliegenthart and Lekkerkerker
^{[8]}
^{[9]}
have suggested that the critical point is related to the second virial coefficient via the expression

\[B_2 \vert_{T=T_c}= -\pi \sigma^3\]

## Triple point[edit]

The location of the triple point as found by Mastny and de Pablo ^{[10]} is
\[T_{tp}^* = 0.694\]

\[\rho_{tp}^* = 0.84\] (liquid);

\[\rho_{tp}^* = 0.96\] (solid).

## Radial distribution function[edit]

The following plot is of a typical radial distribution function for the monatomic Lennard-Jones liquid^{[11]} (here with \(\sigma=3.73\)Å and \(\epsilon=0.294\) kcal/mol at a temperature of 111.06K):

## Helmholtz energy function[edit]

An expression for the Helmholtz energy function of the face centred cubic solid has been given by van der Hoef ^{[12]}, applicable within the density range \(0.94 \le \rho^* \le 1.20\) and the temperature range \(0.1 \le T^* \le 2.0\). For the liquid state see the work of Johnson, Zollweg and Gubbins ^{[13]}.

## Equation of state[edit]

*Main article: Lennard-Jones equation of state*

## Virial coefficients[edit]

*Main article: Lennard-Jones model: virial coefficients*

## Phase diagram[edit]

*Main article: Phase diagram of the Lennard-Jones model*

## Zeno line[edit]

It has been shown that the Lennard-Jones model has a straight Zeno line ^{[14]} on the density-temperature plane.

## Widom line[edit]

It has been shown that the Lennard-Jones model has a Widom line ^{[15]} on the pressure-temperature plane.

## Perturbation theory[edit]

The Lennard-Jones model is also used in perturbation theories, for example see: Weeks-Chandler-Andersen perturbation theory.

## Approximations in simulation: truncation and shifting[edit]

The Lennard-Jones model is often used with a cutoff radius of \(2.5 \sigma\), beyond which \( \Phi_{12}(r)\) is set to zero. Setting the well depth \( \epsilon \) to be 1 in the potential on arrives at \( \Phi_{12}(r)\simeq -0.0163\), i.e. at this distance the potential is at less than 2% of the well depth. For an analysis of the effect of this cutoff on the melting line see the work of Mastny and de Pablo ^{[10]} and of Ahmed and Sadus ^{[16]}. See Panagiotopoulos for critical parameters ^{[17]}. It has recently been suggested that a truncated and shifted force cutoff of \(1.5 \sigma\) can be used under certain conditions ^{[18]}. In order to avoid any discontinuity, a piecewise continuous version, known as the modified Lennard-Jones model, was developed.

## Cutoff Lennard-Jones potential[edit]

The cutoff Lennard-Jones potential is given by (Eq. 2 in ^{[19]}):

\[ \Phi_{12}(r) = 4 \epsilon \left\{ \left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right]+ \left[ 6\left(\frac{\sigma}{r_c} \right)^{12}- 3\left( \frac{\sigma}{r_c}\right)^6 \right] \left(\frac{r}{r_c} \right)^2 -7 \left(\frac{\sigma}{r_c} \right)^{12} + 4 \left(\frac{\sigma}{r_c} \right)^{6} \right\}\]

where \(r_c\) is the cutoff radius.

## n-m Lennard-Jones potential[edit]

It is relatively common to encounter potential functions given by:
\[ \Phi_{12}(r) = c_{n,m} \epsilon \left[ \left( \frac{ \sigma }{r } \right)^n - \left( \frac{\sigma}{r} \right)^m
\right].
\]
with \( n \) and \( m \) being positive integers and \( n > m \).
\( c_{n,m} \) is chosen such that the minimum value of \( \Phi_{12}(r) \) being \( \Phi_{min} = - \epsilon \).
Such forms are usually referred to as **n-m Lennard-Jones Potential**.
For example, the 9-3 Lennard-Jones interaction potential is often used to model the interaction between
a continuous solid wall and the atoms/molecules of a liquid.
On the '9-3 Lennard-Jones potential' page a justification of this use is presented. Another example is the n-6 Lennard-Jones potential,
where \(m\) is fixed at 6, and \(n\) is free to adopt a range of integer values.
The potentials form part of the larger class of potentials known as the Mie potential.

Examples:

- 8-6 Lennard-Jones potential
- 9-3 Lennard-Jones potential
- 9-6 Lennard-Jones potential
- 10-4-3 Lennard-Jones potential
- 200-100 Lennard-Jones potential
- n-6 Lennard-Jones potential

## Mixtures[edit]

## Related models[edit]

- Kihara potential
- Lennard-Jones model in 1-dimension (rods)
- Lennard-Jones model in 2-dimensions (disks)
- Lennard-Jones model in 4-dimensions
- Lennard-Jones sticks
- Mie potential
- Soft-core Lennard-Jones model
- Soft sphere potential
- Stockmayer potential

## References[edit]

- ↑ John Edward Lennard-Jones "On the Determination of Molecular Fields. I. From the Variation of the Viscosity of a Gas with Temperature", Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character
**106**pp. 441-462 (1924) § 8 (ii) - ↑ John Edward Lennard-Jones "On the Determination of Molecular Fields. II. From the Equation of State of a Gas", Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character
**106**pp. 463-477 (1924) Eq. 2.05 - ↑ F. London "Zur Theorie und Systematik der Molekularkräfte", Zeitschrift für Physik A Hadrons and Nuclei
**63**pp. 245-279 (1930) - ↑ W. W. Wood and F. R. Parker "Monte Carlo Equation of State of Molecules Interacting with the Lennard‐Jones Potential. I. A Supercritical Isotherm at about Twice the Critical Temperature", Journal of Chemical Physics
**27**pp. 720- (1957) - ↑ L. A. Rowley, D. Nicholson and N. G. Parsonage "Monte Carlo grand canonical ensemble calculation in a gas-liquid transition region for 12-6 Argon", Journal of Computational Physics
**17**pp. 401-414 (1975) - ↑ J. M. Caillol " Critical-point of the Lennard-Jones fluid: A finite-size scaling study", Journal of Chemical Physics
**109**pp. 4885-4893 (1998) - ↑ V. L. Kulinskii "The critical compressibility factor of fluids from the global isomorphism approach", Journal of Chemical Physics
**139**184119 (2013) - ↑ G. A. Vliegenthart and H. N. W. Lekkerkerker "Predicting the gas–liquid critical point from the second virial coefficient", Journal of Chemical Physics
**112**pp. 5364-5369 (2000) - ↑ L. A. Bulavin and V. L. Kulinskii "Generalized principle of corresponding states and the scale invariant mean-field approach", Journal of Chemical Physics '
**133**134101 (2010) - ↑
^{10.0}^{10.1}Ethan A. Mastny and Juan J. de Pablo "Melting line of the Lennard-Jones system, infinite size, and full potential", Journal of Chemical Physics**127**104504 (2007) - ↑ John G. Kirkwood, Victor A. Lewinson, and Berni J. Alder "Radial Distribution Functions and the Equation of State of Fluids Composed of Molecules Interacting According to the Lennard-Jones Potential", Journal of Chemical Physics
**20**pp. 929- (1952) - ↑ Martin A. van der Hoef "Free energy of the Lennard-Jones solid", Journal of Chemical Physics
**113**pp. 8142-8148 (2000) - ↑ J. Karl Johnson, John A. Zollweg and Keith E. Gubbins "The Lennard-Jones equation of state revisited", Molecular Physics
**78**pp. 591-618 (1993) - ↑ E. M. Apfelbaum, V. S. Vorob’ev and G. A. Martynov "Regarding the Theory of the Zeno Line", Journal of Physical Chemistry A
**112**pp. 6042-6044 (2008) - ↑ V. V. Brazhkin, Yu. D. Fomin, A. G. Lyapin, V. N. Ryzhov, and E. N. Tsiok "Widom Line for the Liquid–Gas Transition in Lennard-Jones System", Journal of Physical Chemistry B Article ASAP (2011)
- ↑ Alauddin Ahmed and Richard J. Sadus "Effect of potential truncations and shifts on the solid-liquid phase coexistence of Lennard-Jones fluids", Journal of Chemical Physics
**133**124515 (2010) - ↑ A. Z. Panagiotopoulos "Molecular simulation of phase coexistence: Finite-size effects and determination of critical parameters for two- and three-dimensional Lennard-Jones fluids", International Journal of Thermophysics
**15**pp. 1057-1072 (1994) - ↑ Søren Toxvaerd and Jeppe C. Dyre "Communication: Shifted forces in molecular dynamics", Journal of Chemical Physics
**134**081102 (2011) - ↑ Spotswood D. Stoddard and Joseph Ford "Numerical Experiments on the Stochastic Behavior of a Lennard-Jones Gas System", Physical Review A
**8**pp. 1504-1512 (1973)