Lennard-Jones potential
In computational chemistry, molecular physics, and physical chemistry, the Lennard-Jones potential 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. 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 for more complex substances. Many studies of the idealized "Lennard-Jones substance" use the potential to understand the physical nature of matter.
Overview
The Lennard-Jones potential is a simple model that still manages to describe the essential features of interactions between simple atoms and molecules: two interacting particles repel each other at very close distances, attract each other at moderate distances, and eventually stop interacting at infinite distance, as shown in the figure. The Lennard-Jones potential is a pair potential, i.e. no three- or multi-body interactions are covered by the potential.The general Lennard-Jones potential combines a repulsive potential,, with an attractive potential,, using empirically determined coefficients and :
In his 1931 review Lennard-Jones suggested using
to match the London dispersion force and based on experimental data. Setting and gives the widely used Lennard-Jones 12-6 potential:
where is the distance between two interacting particles, is the depth of the potential well, and is the distance at which the particle-particle potential energy is zero. The Lennard-Jones 12-6 potential has its minimum at a distance of where the potential energy has the value
The Lennard-Jones potential is usually the standard choice for the development of theories for matter as well as for the development and testing of computational methods and algorithms.
Numerous intermolecular potentials have been proposed in the past for the modeling of simple soft repulsive and attractive interactions between spherically symmetric particles, i.e. the general shape shown in the Figure. Examples for other potentials are the Morse potential, the Mie potential, the Buckingham potential and the Tang-Tönnies potential. While some of these may be more suited to modelling real fluids, the simplicity of the Lennard-Jones potential, as well as its often surprising ability to accurately capture real fluid behavior, has historically made it the pair-potential of greatest general importance.
History
In 1924, the year that Lennard-Jones received his PhD from Cambridge University, he published a series of landmark papers on the pair potentials that would ultimately be named for him. 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 and an attractive.Before Lennard-Jones, back in 1903, Gustav Mie had worked on effective field theories; Eduard Grüneisen built on Mie work for solids, showing that and 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.
In 1930, after the discovery of quantum mechanics, Fritz London showed that theory predicts the long-range attractive force should have. In 1931, Lennard-Jones applied this form of the potential to describe many properties of fluids setting the stage for many subsequent studies.
Dimensionless (reduced units)
Dimensionless reduced units can be defined based on the Lennard-Jones potential parameters, which is convenient for molecular simulations. From a numerical point of view, the advantages of this unit system include computing values which are closer to unity, using simplified equations and being able to easily scale the results. This reduced units system requires the specification of the size parameter and the energy parameter of the Lennard-Jones potential and the mass of the particle. All physical properties can be converted straightforwardly taking the respective dimension into account, see table. The reduced units are often abbreviated and indicated by an asterisk.In general, reduced units can also be built up on other molecular interaction potentials that consist of a length parameter and an energy parameter.
Long-range interactions
The Lennard-Jones potential, cf. Eq. and Figure on the top, has an infinite range. Only under its consideration, the 'true' and 'full' Lennard-Jones potential is examined. For the evaluation of an observable of an ensemble of particles interacting by the Lennard-Jones potential using molecular simulations, the interactions can only be evaluated explicitly up to a certain distance – simply due to the fact that the number of particles will always be finite. The maximum distance applied in a simulation is usually referred to as 'cut-off' radius . To obtain thermophysical properties of the 'true' and 'full' Lennard-Jones potential, the contribution of the potential beyond the cut-off radius has to be accounted for.Different correction schemes have been developed to account for the influence of the long-range interactions in simulations and to sustain a sufficiently good approximation of the 'full' potential. They are based on simplifying assumptions regarding the structure of the fluid. For simple cases, such as in studies of the equilibrium of homogeneous fluids, simple correction terms yield excellent results. In other cases, such as in studies of inhomogeneous systems with different phases, accounting for the long-range interactions is more tedious. These corrections are usually referred to as 'long-range corrections'. For most properties, simple analytical expressions are known and well established. For a given observable, the 'corrected' simulation result is then simply computed from the actually sampled value and the long-range correction value, e.g. for the internal energy. The hypothetical true value of the observable of the Lennard-Jones potential at truly infinite cut-off distance can in general only be estimated.
Furthermore, the quality of the long-range correction scheme depends on the cut-off radius. The assumptions made with the correction schemes are usually not justified at short cut-off radii. This is illustrated in the example shown in Figure on the right. The long-range correction scheme is said to be converged, if the remaining error of the correction scheme is sufficiently small at a given cut-off distance, cf. Figure.
Extensions and modifications
The Lennard-Jones potential – as an archetype for intermolecular potentials – has been used numerous times as starting point for the development of more elaborate or more generalized intermolecular potentials. Various extensions and modifications of the Lennard-Jones potential have been proposed in the literature; a more extensive list is given in the 'interatomic potential' article. The following list refers only to several example potentials that are directly related to the Lennard-Jones potential and are of both historic importance and still relevant for present research.- Mie potential The Mie potential is the generalized version of the Lennard-Jones potential, i.e. the exponents 12 and 6 are introduced as parameters and. 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. The first explicit formulation of the Mie potential is attributed to Eduard Grüneisen. Hence, the Mie potential was actually proposed before the Lennard-Jones potential. The Mie potential is named after Gustav Mie.
- Buckingham 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.
- Stockmayer potential The Stockmayer potential is named after W.H. Stockmayer. 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.
- Two center Lennard-Jones potential The two center Lennard-Jones potential consists of two identical Lennard-Jones interaction sites that are bonded as a rigid body. It is often abbreviated as 2CLJ. Usually, the elongation is significantly smaller than the size parameter. 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 and no long-range interactions are considered beyond. While the LJTS potential is shifted such that the potential is continuous but the force is discontinuous, the Lennard-Jones truncated & splined potential is made continuous by using a spline function that ensures a continuous force.
Lennard-Jones truncated & shifted (LJTS) potential
with
Hence, the LJTS potential is truncated at and shifted by the corresponding energy value. The latter is applied to avoid a discontinuity jump of the potential at. For the LJTS potential, no long-range interactions beyond are required – neither explicitly nor implicitly. The most frequently used version of the Lennard-Jones truncated & shifted potential is the one with. Nevertheless, different values have been used in the literature. Each LJTS potential with a given truncation radius has to be considered as a potential and accordingly a substance of its own.
The LJTS potential is computationally significantly cheaper than the 'full' Lennard-Jones potential, but still covers the essential physical features of matter. Therefore, the LJTS potential is used for the testing of new algorithms, simulation methods, and new physical theories.
Interestingly, for homogeneous systems, the intermolecular forces that are calculated from the LJ and the LJTS potential at a given distance are the same, whereas the potential energy and the pressure are affected by the shifting. Also, the properties of the LJTS substance may furthermore be affected by the chosen simulation algorithm, i.e. MD or MC sampling.
For the LJTS potential with, the potential energy shift is approximately 1/60 of the dispersion energy at the potential well:. The Figure on the right shows the comparison of the vapor–liquid equilibrium of the 'full' Lennard-Jones potential and the 'Lennard-Jones truncated & shifted' potential. The 'full' Lennard-Jones potential results prevail a significantly higher critical temperature and pressure compared to the LJTS potential results, but the critical density is very similar. The vapor pressure and the enthalpy of vaporization are influenced more strongly by the long-range interactions than the saturated densities. This is due to the fact that the potential is manipulated mainly energetically by the truncation and shifting.