Lennard-Jones potential

In computational chemistry, molecular physics, and physical chemistry, the Lennard-Jones potential (also termed the LJ potential or 12-6 potential; named for John Lennard-Jones) is an intermolecular pair potential. Out of all the intermolecular potentials, the Lennard-Jones potential is probably the one that has been the most extensively studied.^[1]^[2] It is considered an archetype model for simple yet realistic intermolecular interactions. The Lennard-Jones potential is often used as a building block in molecular models (a.k.a. force fields) for more complex substances.^[3] Many studies of the idealized "Lennard-Jones substance" use the potential to understand the physical nature of matter.

History[edit]

In 1924, the year that Lennard-Jones received his PhD from Cambridge University, he published^[6]^[12] a series of landmark papers on the pair potentials that would ultimately be named for him.^[2]^[3]^[13]^[1] In these papers he adjusted the parameters of the potential then using the result in a model of gas viscosity, seeking a set of values consistent with experiment. His initial results suggested a repulsive $n=13.5$ and an attractive $m=3$ .

Before Lennard-Jones, back in 1903, Gustav Mie had worked on effective field theories; Eduard Grüneisen built on Mie work for solids, showing that $n>m$ and $m>3$ is required for solids. As a result of this work the Lennard-Jones potential is sometimes called the Mie− Grüneisen potential in solid-state physics.^[3]

In 1930, after the discovery of quantum mechanics, Fritz London showed that theory predicts the long-range attractive force should have $m=6$ . In 1931, Lennard-Jones applied the this form of the potential to describe many properties of fluids setting the stage for many subsequent studies.^[1]

The Mie potential is the generalized version of the Lennard-Jones potential, i.e. the exponents 12 and 6 are introduced as parameters $\lambda _{\mathrm {rep} }$ and $\lambda _{\mathrm {attr} }$ . Especially thermodynamic derivative properties, e.g. the compressibility and the speed of sound, are known to be very sensitive to the steepness of the repulsive part of the intermolecular potential, which can therefore be modeled more sophisticated by the Mie potential.^[17] The first explicit formulation of the Mie potential is attributed to Eduard Grüneisen.^[18]^[19] Hence, the Mie potential was actually proposed before the Lennard-Jones potential. The Mie potential is named after Gustav Mie.^[8]

Mie potential

The Buckingham potential was proposed by Richard Buckingham. The repulsive part of the Lennard-Jones potential is therein replaced by an exponential function and it incorporates an additional parameter.

Buckingham potential

The Stockmayer potential is named after W.H. Stockmayer.^[20] The Stockmayer potential is a combination of a Lennard-Jones potential superimposed by a dipole. Hence, Stockmayer particles are not spherically symmetric, but rather have an important orientational structure.

Stockmayer potential

Two center Lennard-Jones potential The two center Lennard-Jones potential consists of two identical Lennard-Jones interaction sites (same $\varepsilon$ , $\sigma$ , $m$ ) that are bonded as a rigid body. It is often abbreviated as 2CLJ. Usually, the elongation (distance between the Lennard-Jones sites) is significantly smaller than the size parameter $\sigma$ . Hence, the two interaction sites are significantly fused.

Lennard-Jones truncated & splined potential The Lennard-Jones truncated & splined potential is a rarely used yet useful potential. Similar to the more popular LJTS potential, it is sturdily truncated at a certain 'end' distance $r_{\mathrm {end} }$ and no long-range interactions are considered beyond. Opposite to the LJTS potential, which is shifted such that the potential is continuous, the Lennard-Jones truncated & splined potential is made continuous by using an arbitrary but favorable spline function.

$T_{\mathrm {c} }=1.321\pm 0.007\,\varepsilon k_{\mathrm {B} }^{-1}$

$\rho _{\mathrm {c} }=0.316\pm 0.005\,\sigma ^{-3}$

$p_{\mathrm {c} }=0.129\pm 0.005\,\varepsilon \sigma ^{-3}$

Comparison of force-field implementations

Embedded atom model

Force field (chemistry)

Molecular mechanics

and Morse/Long-range potential

Morse potential

Virial expansion

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