The Lennard-Jones potential (also termed the LJ potential or 12-6 potential) is an intermolecular
pair potential
In physics, a pair potential is a function that describes the potential energy of two interacting objects solely as a function of the distance between them. Examples of pair potentials include the Coulomb's law in electrodynamics, Newton's law of ...
. 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 models soft repulsive and attractive (
van der Waals) interactions. Hence, the Lennard-Jones potential describes electronically neutral atoms or molecules. It is named after
John Lennard-Jones
Sir John Edward Lennard-Jones (27 October 1894 – 1 November 1954) was a British mathematician and professor of theoretical physics at the University of Bristol, and then of theoretical science at the University of Cambridge. He was an imp ...
. The commonly used expression for the Lennard-Jones potential is
where
is the distance between two interacting particles,
is the depth of the
potential well
A potential well is the region surrounding a local minimum of potential energy. Energy captured in a potential well is unable to convert to another type of energy (kinetic energy in the case of a gravitational potential well) because it is cap ...
(usually referred to as 'dispersion energy'), and
is the distance at which the particle-particle potential energy
is zero (often referred to as 'size of the particle'). The Lennard-Jones potential has its minimum at a distance of
, where the potential energy has the value
.
The Lennard-Jones potential is a simplified model that yet describes the essential features of interactions between simple atoms and molecules: Two interacting particles repel each other at very close distance, attract each other at moderate distance, and do not interact at infinite distance, as shown in Figure 1. The Lennard-Jones potential is a pair potential, i.e. no three- or multi-body interactions are covered by the potential.
Statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
and
computer simulations
Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be dete ...
can be used to study the Lennard-Jones potential and to obtain thermophysical properties of the 'Lennard-Jones substance'. The Lennard-Jones substance is often referred to as 'Lennard-Jonesium' suggesting that it is viewed as a (fictive)
chemical element
A chemical element is a species of atoms that have a given number of protons in their nuclei, including the pure substance consisting only of that species. Unlike chemical compounds, chemical elements cannot be broken down into simpler sub ...
.
Moreover, its energy and length parameters can be adjusted to fit many different real substances. Both the Lennard-Jones potential and, accordingly, the Lennard-Jones substance are simplified yet realistic models, such as they accurately capture essential physical principles like the presence of a
critical
Critical or Critically may refer to:
*Critical, or critical but stable, medical states
**Critical, or intensive care medicine
*Critical juncture, a discontinuous change studied in the social sciences.
*Critical Software, a company specializing in ...
and a
triple point
In thermodynamics, the triple point of a substance is the temperature and pressure at which the three phases (gas, liquid, and solid) of that substance coexist in thermodynamic equilibrium.. It is that temperature and pressure at which the subli ...
,
condensation and
freezing
Freezing is a phase transition where a liquid turns into a solid when its temperature is lowered below its freezing point. In accordance with the internationally established definition, freezing means the solidification phase change of a liquid ...
. Due in part to its mathematical simplicity, the Lennard-Jones potential has been extensively used in studies on matter since the early days of computer simulation.
The Lennard-Jones potential is probably still the most frequently studied model potential.
The Lennard-Jones potential is usually the standard choice for the development of theories for
matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic partic ...
(especially soft-matter) as well as for the development and testing of computational methods and algorithms. Upon adjusting the model parameters
and
to real substance properties, the Lennard-Jones potential can be used to describe simple substance (like
noble gas
The noble gases (historically also the inert gases; sometimes referred to as aerogens) make up a class of chemical elements with similar properties; under standard conditions, they are all odorless, colorless, monatomic gases with very low chemi ...
es) with good accuracy. Furthermore, the Lennard-Jones potential is often used as a building block in
molecular models (a.k.a.
force fields) for more complex substances.
Physical background and mathematical details
The Lennard-Jones potential models the two most important and fundamental molecular interactions: The repulsive term (
term) describes the
Pauli repulsion
In chemistry and physics, the exchange interaction (with an exchange energy and exchange term) is a quantum mechanical effect that only occurs between identical particles. Despite sometimes being called an exchange force in an analogy to classical ...
at short distances of the interacting particles due to overlapping electron orbitals and the attractive term (
term) describes attraction at long ranged interactions (
London dispersion force
London dispersion forces (LDF, also known as dispersion forces, London forces, instantaneous dipole–induced dipole forces, fluctuating induced dipole bonds or loosely as van der Waals forces) are a type of intermolecular force acting between a ...
), which vanish at infinite distance between two particles. The steep repulsive interactions at short distances yield the low
compressibility
In thermodynamics and fluid mechanics, the compressibility (also known as the coefficient of compressibility or, if the temperature is held constant, the isothermal compressibility) is a measure of the instantaneous relative volume change of a f ...
of the solid and liquid phase; the attractive dispersive interactions act stabilizing for the condensed phase, especially the
vapor–liquid equilibrium
In thermodynamics and chemical engineering, the vapor–liquid equilibrium (VLE) describes the distribution of a chemical species between the vapor phase and a liquid phase.
The concentration of a vapor in contact with its liquid, especially a ...
.
The functional form of the attractive term, the exponent '6',
has a physical justification, which does not hold as rigorously for the repulsive term with the exponent '12'. The attractive dispersive interactions between simple atoms and molecules are a result of fluctuating partial charges. It has been shown by quantum-chemical calculations that this
dispersive contribution has to decay with
.
The
term is mainly used because it can be implemented computationally very efficiently as the square of
, which does not hold to the same extent for values other than '12'. Also,
approximates the
Pauli repulsion
In chemistry and physics, the exchange interaction (with an exchange energy and exchange term) is a quantum mechanical effect that only occurs between identical particles. Despite sometimes being called an exchange force in an analogy to classical ...
reasonably well. If needed, the Lennard-Jones potential can be generalized using arbitrary exponents instead of 12 and 6; the resulting model is called the
Mie potential The Mie potential is an intermolecular pair potential, i.e. it describes the interactions between particles at the atomic level.
The model is attributed to the German physicist Gustave Mie. The Mie potential is the generalized case of the Lenna ...
. The present article exclusively discusses the original (12-6) Lennard-Jones potential.
The Lennard-Jones potential exhibits a pole at
, i.e. the potential energy diverges to
, which can cause instabilities in molecular simulations, e.g. for the sampling of the chemical potential. The Lennard-Jones potential converges to
for
. Hence, from a mathematical standpoint, attractive interactions stay present for infinitely distanced particles. These dispersive 'long-range' interactions have an important influence on several properties of the Lennard-Jones substance, e.g. the pressure or heat capacity in the vicinity of the critical point and the critical point itself. The importance of the long-range interactions were noticed already in the early stages of
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
. For computer simulations, only finite numbers of particles can be used, which leads to the fact that the potential can only be evaluated up to a finite radius
, which is a so-called finite size effect. There are well-established methods to implicitly consider the thereby neglected long-range contribution for a given observable (details are given below).
It is often claimed that multiple Lennard-Jones potentials and corresponding substances exist depending on the handling of the long-range interactions. This is misleading. There is only one 'Lennard-Jones potential', which is exactly defined by Eq. (1). The Lennard-Jones potential requires the consideration and evaluation of long-range interactions up to very long (actually infinite) distances – at least so that the influence of the truncation has no influence on the
observable
In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum ph ...
of interest for the reported decimal places.
The Lennard-Jones potential implies that the particles are point masses with a mass
. Even though the parameter
is often referred to as 'size of the particle', particles interacting with the Lennard-Jones potential have no uniquely defined 'size' – opposite to the
hard sphere potential. Particles interacting with the Lennard-Jones potential rather have soft repulsive cores.
The Lennard-Jones model describes the potential intermolecular energy
between two particles based on the outlined principles. Following
Newton's mechanics, the actual force
between two interacting particles is simply obtained by negating and differentiating the Lennard-Jones potential with respect to
, i.e.
. Depending on the distance between the two particles, the net force can be either attractive or repulsive.
The Lennard-Jones potential yields a good approximation of intermolecular interactions for many applications: The macroscopic properties computed using the Lennard-Jones potential are in good agreement with experimental data for simple substances like argon on one side and the potential function
is in fair agreement with results from
quantum chemistry
Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions ...
on the other side. The Lennard-Jones potential gives a good description of molecular interactions in
fluid
In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
phases, whereas molecular interactions in solid phases are only roughly well described. This is mainly due to the fact that multi-body interactions play a significant role in solid phases, which are not comprised in the Lennard-Jones potential. Therefore, the Lennard-Jones potential is extensively used in
soft-matter physics and associated fields, whereas it is less frequently used in
solid-state physics
Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the l ...
. Due to its simplicity, the Lennard-Jones potential is often used to describe the properties of gases and simple fluids and to model dispersive and repulsive interactions in
molecular models. It is especially accurate for
noble gas
The noble gases (historically also the inert gases; sometimes referred to as aerogens) make up a class of chemical elements with similar properties; under standard conditions, they are all odorless, colorless, monatomic gases with very low chemi ...
atoms and
methane
Methane ( , ) is a chemical compound with the chemical formula (one carbon atom bonded to four hydrogen atoms). It is a group-14 hydride, the simplest alkane, and the main constituent of natural gas. The relative abundance of methane on Eart ...
. It is furthermore a good approximation for molecular interactions at long and short distances for neutral atoms and molecules. Therefore, the Lennard-Jones potential is very often used as a building block of
molecular models of complex molecules, e.g.
alkane
In organic chemistry, an alkane, or paraffin (a historical trivial name that also has other meanings), is an acyclic saturated hydrocarbon. In other words, an alkane consists of hydrogen and carbon atoms arranged in a tree structure in which ...
s or
water
Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as a ...
.
The Lennard-Jones potential can also be used to model the
adsorption
Adsorption is the adhesion of atoms, ions or molecules from a gas, liquid or dissolved solid to a surface. This process creates a film of the ''adsorbate'' on the surface of the ''adsorbent''. This process differs from absorption, in which ...
interactions at solid–fluid interfaces, i.e.
physisorption
Physisorption, also called physical adsorption, is a process in which the electronic structure of the atom or molecule is barely perturbed upon adsorption.
Overview
The fundamental interacting force of physisorption is Van der Waals force. Even ...
or
chemisorption
Chemisorption is a kind of adsorption which involves a chemical reaction between the surface and the adsorbate. New chemical bonds are generated at the adsorbent surface. Examples include macroscopic phenomena that can be very obvious, like cor ...
.
It is well accepted, that the main limitations of the Lennard-Jones potential lie in the fact the potential is a
pair potential
In physics, a pair potential is a function that describes the potential energy of two interacting objects solely as a function of the distance between them. Examples of pair potentials include the Coulomb's law in electrodynamics, Newton's law of ...
(does not cover multi-body interactions) and that the
exponent term is used for the repulsion. Results from quantum chemistry suggest that a higher exponent than 12 has to be used, i.e. a steeper potential. Furthermore, the Lennard-Jones potential has a limited flexibility, i.e. only the two model parameters
and
can be used for the fitting to describe a real substance.
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 Figure 1. Examples for other potentials are the
Morse potential
The Morse potential, named after physicist Philip M. Morse, is a convenient
interatomic interaction model for the potential energy of a diatomic molecule. It is a better approximation for the vibrational structure of the molecule than the qua ...
, the
Mie potential The Mie potential is an intermolecular pair potential, i.e. it describes the interactions between particles at the atomic level.
The model is attributed to the German physicist Gustave Mie. The Mie potential is the generalized case of the Lenna ...
,
the Buckingham potential and the Tang-Tönnies potential. Nevertheless, none of those are of such general importance as the Lennard-Jones potential.
Application of the Lennard-Jones potential
The Lennard-Jones potential is not only of fundamental importance in
computational chemistry
Computational chemistry is a branch of chemistry that uses computer simulation to assist in solving chemical problems. It uses methods of theoretical chemistry, incorporated into computer programs, to calculate the structures and properties of m ...
and
soft-matter physics, but also for the modeling of real substances. The Lennard-Jones potential is frequently used for fundamental studies on the behavior of matter and for elucidating atomistic phenomena. It is also often used for somewhat special use cases, e.g. for studying thermophysical properties of two- or four-dimensional substances (instead of the classical three spatial directions of our universe).
The Lennard-Jones potential is extensively used for molecular modeling. There are essentially two ways the Lennard-Jones potential can be used for molecular modeling: (1) A real substance atom or molecule is modeled directly by the Lennard-Jones potential, which yields very good results for
noble gas
The noble gases (historically also the inert gases; sometimes referred to as aerogens) make up a class of chemical elements with similar properties; under standard conditions, they are all odorless, colorless, monatomic gases with very low chemi ...
es and
methane
Methane ( , ) is a chemical compound with the chemical formula (one carbon atom bonded to four hydrogen atoms). It is a group-14 hydride, the simplest alkane, and the main constituent of natural gas. The relative abundance of methane on Eart ...
, i.e. dispersively interacting spherical particles. In the case of methane, the molecule is assumed to be spherically symmetric and the hydrogen atoms are fused with the carbon atom to a common unit. This simplification can in general also be applied to more complex molecules, but yields usually poor results. (2) A real substance molecule is built of multiple Lennard-Jones interactions sites, which can be connected either by rigid bonds or flexible additional potentials (and eventually also consists of other potential types, e.g. partial charges).
Molecular models (often referred to as '
force fields') for practically all molecular and ionic particles can be constructed using this scheme for example for
alkane
In organic chemistry, an alkane, or paraffin (a historical trivial name that also has other meanings), is an acyclic saturated hydrocarbon. In other words, an alkane consists of hydrogen and carbon atoms arranged in a tree structure in which ...
s.
Upon using the first outlined approach, the molecular model has only the two parameters of the Lennard-Jones potential
and
that can be used for the fitting, e.g.
and
are frequently used for
argon
Argon is a chemical element with the symbol Ar and atomic number 18. It is in group 18 of the periodic table and is a noble gas. Argon is the third-most abundant gas in Earth's atmosphere, at 0.934% (9340 ppmv). It is more than twice as abu ...
. Evidently, this approach is only a good approximation for spherical and simply dispersively interacting molecules and atoms. The direct use of the Lennard-Jones potential has the great advantage that simulation results and theories for the Lennard-Jones potential can be used directly. Hence, available results for the Lennard-Jones potential and substance can be directly scaled using the appropriate
and
(see reduced units). The Lennard-Jones potential parameters
and
can in general be fitted to any desired real substance property. In soft-matter physics, usually experimental data for the vapor–liquid phase equilibrium or the critical point are used for the parametrization; in solid-state physics, rather the compressibility, heat capacity or lattice constants are employed.
The second outlined approach of using the Lennard-Jones potential as a building block of elongated and complex molecules is far more sophisticated.
Molecular models are thereby tailor-made in a sense that simulation results are only applicable for that particular model. This development approach for molecular force fields is today mainly performed in
soft-matter physics and associated fields such as
chemical engineering
Chemical engineering is an engineering field which deals with the study of operation and design of chemical plants as well as methods of improving production. Chemical engineers develop economical commercial processes to convert raw materials int ...
, chemistry, and computational biology. A large number of
force fields are based on the Lennard-Jones potential, e.g. th
TraPPE force field the OPLS force field, and th
MolMod force fieldref name=":12"> (an overview of
molecular force fields is out of the scope of the present article). For the state-of-the-art modeling of solid-state materials, more elaborate multi-body potentials (e.g.
EAM potentials) are used.
Alternative notations of the Lennard-Jones potential
There are several different ways to formulate the Lennard-Jones potential besides Eq. (1). Alternatives are:
AB form
The AB form is frequently used in implementations of simulation software as it is computationally favorable. The Lennard-Jones potential can be written as
where,
and
. Conversely,