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Interatomic potentials are mathematical functions to calculate the
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potenti ...
of a system of
atom Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, a ...
s with given positions in space.M. P. Allen and D. J. Tildesley. Computer Simulation of Liquids. Oxford University Press, Oxford, England, 1989.R. Lesar. Introduction to Computational Materials Science. Cambridge University Press, 2013. Interatomic potentials are widely used as the physical basis of molecular mechanics and
molecular dynamics Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of th ...
simulations 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 mo ...
,
computational physics Computational physics is the study and implementation of numerical analysis to solve problems in physics for which a quantitative theory already exists. Historically, computational physics was the first application of modern computers in science, ...
and
computational materials science ''Computational Materials Science'' is a monthly peer-reviewed scientific journal published by Elsevier. It was established in October 1992. The editor-in-chief is Susan Sinnott. The journal covers computational modeling and practical research fo ...
to explain and predict materials properties. Examples of quantitative properties and qualitative phenomena that are explored with interatomic potentials include lattice parameters, surface energies, interfacial energies,
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 a ...
, cohesion,
thermal expansion Thermal expansion is the tendency of matter to change its shape, area, volume, and density in response to a change in temperature, usually not including phase transitions. Temperature is a monotonic function of the average molecular kin ...
, and
elastic Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics. Elastic may also refer to: Alternative name * Rubber band, ring-shaped band of rubber used to hold objects togethe ...
and
plastic Plastics are a wide range of synthetic or semi-synthetic materials that use polymers as a main ingredient. Their plasticity makes it possible for plastics to be moulded, extruded or pressed into solid objects of various shapes. This adapta ...
material behavior, as well as
chemical reaction A chemical reaction is a process that leads to the chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the positions of electrons in the forming and breaking ...
s.N. W. Ashcroft and N. D. Mermin. Solid State Physics.Saunders College, Philadelphia, 1976.Charles Kittel.
Introduction to Solid State Physics ''Introduction to Solid State Physics'', known colloquially as ''Kittel'', is a classic condensed matter physics textbook written by American physicist Charles Kittel in 1953. The book has been highly influential and has seen widespread adoptio ...
. John Wiley & Sons, New York, third edition, 1968.


Functional form

Interatomic potentials can be written as a series expansion of functional terms that depend on the position of one, two, three, etc. atoms at a time. Then the total potential of the system \textstyle V_\mathrm can be written as :: V_\mathrm = \sum_^N V_1(\vec r_i) + \sum_^N V_2(\vec r_i,\vec r_j) + \sum_^N V_3(\vec r_i,\vec r_j,\vec r_k) + \cdots Here \textstyle V_1 is the one-body term, \textstyle V_2 the two-body term, \textstyle V_3 the three body term, \textstyle N the number of atoms in the system, \vec r_i the position of atom i, etc. i, j and k are indices that loop over atom positions. Note that in case the pair potential is given per atom pair, in the two-body term the potential should be multiplied by 1/2 as otherwise each bond is counted twice, and similarly the three-body term by 1/6. Alternatively, the summation of the pair term can be restricted to cases \textstyle i and similarly for the three-body term \textstyle i, if the potential form is such that it is symmetric with respect to exchange of the j and k indices (this may not be the case for potentials for multielemental systems). The one-body term is only meaningful if the atoms are in an external field (e.g. an electric field). In the absence of external fields, the potential V should not depend on the absolute position of atoms, but only on the relative positions. This means that the functional form can be rewritten as a function of interatomic distances \textstyle r_ = , \vec r_i-\vec r_j, and angles between the bonds (vectors to neighbours) \textstyle \theta_. Then, in the absence of external forces, the general form becomes :: V_\mathrm = \sum_^N V_2(r_) + \sum_^N V_3(r_,r_,\theta_) + \cdots In the three-body term \textstyle V_3 the interatomic distance \textstyle r_ is not needed since the three terms \textstyle r_,r_,\theta_ are sufficient to give the relative positions of three atoms i, j, k in three-dimensional space. Any terms of order higher than 2 are also called ''many-body potentials''. In some interatomic potentials the many-body interactions are embedded into the terms of a pair potential (see discussion on EAM-like and bond order potentials below). In principle the sums in the expressions run over all N atoms. However, if the range of the interatomic potential is finite, i.e. the potentials \textstyle V(r) \equiv 0 above some cutoff distance \textstyle r_\mathrm, the summing can be restricted to atoms within the cutoff distance of each other. By also using a cellular method for finding the neighbours, the MD algorithm can be an O(N) algorithm. Potentials with an infinite range can be summed up efficiently by Ewald summation and its further developments.


Force calculation

The forces acting between atoms can be obtained by differentiation of the total energy with respect to atom positions. That is, to get the force on atom i one should take the three-dimensional derivative (gradient) of the potential V_ with respect to the position of atom i: :: \vec_i = -\nabla_ V_\mathrm For two-body potentials this gradient reduces, thanks to the symmetry with respect to ij in the potential form, to straightforward differentiation with respect to the interatomic distances \textstyle r_. However, for many-body potentials (three-body, four-body, etc.) the differentiation becomes considerably more complex since the potential may not be any longer symmetric with respect to ij exchange. In other words, also the energy of atoms k that are not direct neighbours of i can depend on the position \textstyle \vec_ because of angular and other many-body terms, and hence contribute to the gradient \textstyle \nabla_.


Classes of interatomic potentials

Interatomic potentials come in many different varieties, with different physical motivations. Even for single well-known elements such as silicon, a wide variety of potentials quite different in functional form and motivation have been developed. The true interatomic interactions are
quantum mechanical Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
in nature, and there is no known way in which the true interactions described by the
Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
or Dirac equation for all electrons and nuclei could be cast into an analytical functional form. Hence all analytical interatomic potentials are by necessity
approximation An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ' ...
s. Over time interatomic potentials have largely grown more complex and more accurate, although this is not strictly true. This has included both increased descriptions of physics, as well as added parameters. Until recently, all interatomic potentials could be described as "parametric", having been developed and optimized with a fixed number of (physical) terms and parameters. New research focuses instead on non-parametric potentials which can be systematically improvable by using complex local atomic neighbor descriptors and separate mappings to predict system properties, such that the total number of terms and parameters are flexible. These non-parameteric models can be significantly more accurate, but since they are not tied to physical forms and parameters, there are many potential issues surrounding extrapolation and uncertainties.


Parametric potentials


Pair potentials

The arguably simplest widely used interatomic interaction model is the
Lennard-Jones potential The Lennard-Jones potential (also termed the LJ potential or 12-6 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 studie ...
:: V_\mathrm(r) = 4\varepsilon \left \left(\frac\right)^ - \left(\frac\right)^ \right where \textstyle \varepsilon 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 ca ...
and \textstyle \sigma is the distance at which the potential crosses zero. The attractive term proportional to \textstyle 1/r^6 in the potential comes from the scaling of van der Waals forces, while the \textstyle 1/r^ repulsive term is much more approximate (conveniently the square of the attractive term). On its own, this potential is quantitatively accurate only for noble gases and has been extensively studied in the past decades, but is also widely used for qualitative studies and in systems where dipole interactions are significant, particularly in chemistry force fields to describe intermolecular interactions - espacially in fluids. Another simple and widely used pair potential is the Morse potential, which consists simply of a sum of two exponentials. ::V_\mathrm(r) = D_e ( e^-2e^ ) Here \textstyle D_e is the equilibrium bond energy and \textstyle r_e the bond distance. The Morse potential has been applied to studies of molecular vibrations and solids , and also inspired the functional form of more accurate potentials such as the bond-order potentials. Ionic materials are often described by a sum of a short-range repulsive term, such as the Buckingham pair potential, and a long-range
Coulomb potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
giving the ionic interactions between the ions forming the material. The short-range term for ionic materials can also be of many-body character . Pair potentials have some inherent limitations, such as the inability to describe all 3 elastic constants of cubic metals or correctly describe both cohesive energy and vacancy formation energy. Therefore quantitative
molecular dynamics Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of th ...
simulations are carried out with various of many-body potentials.


= Repulsive potentials

= For very short interatomic separations, important in
radiation material science Radiation materials science is a subfield of materials science which studies the interaction of radiation with matter: a broad subject covering many forms of irradiation and of matter. Main aim of radiation material science Some of the most ...
, the interactions can be described quite accurately with screened
Coulomb potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
s which have the general form : V(r_) = \varphi(r/a) Here, \varphi(r) \to 1 when r \to 0. Z_1 and Z_2 are the charges of the interacting nuclei, and ''a'' is the so-called screening parameter. A widely used popular screening function is the "Universal ZBL" one. and more accurate ones can be obtained from all-electron quantum chemistry calculations In binary collision approximation simulations this kind of potential can be used to describe the nuclear stopping power.


Many-body potentials

The Stillinger-Weber potential is a potential that has a two-body and three-body terms of the standard form :: V_\mathrm = \sum_^N V_2(r_) + \sum_^N V_3(r_,r_,\theta_) where the three-body term describes how the potential energy changes with bond bending. It was originally developed for pure Si, but has been extended to many other elements and compounds and also formed the basis for other Si potentials. Metals are very commonly described with what can be called "EAM-like" potentials, i.e. potentials that share the same functional form as the embedded atom model. In these potentials, the total potential energy is written ::V_\mathrm = \sum_i^N F_i \left(\sum_ \rho (r_) \right) + \frac \sum_^N V_2(r_) where \textstyle F_i is a so-called embedding function (not to be confused with the force \textstyle \vec F_i) that is a function of the sum of the so-called electron density \textstyle \rho (r_) . \textstyle V_2 is a pair potential that usually is purely repulsive. In the original formulation the electron density function \textstyle \rho (r_) was obtained from true atomic electron densities, and the embedding function was motivated from density-functional theory as the energy needed to 'embed' an atom into the electron density. . However, many other potentials used for metals share the same functional form but motivate the terms differently, e.g. based on tight-binding theory or other motivations . EAM-like potentials are usually implemented as numerical tables. A collection of tables is available at the interatomic potential repository at NIS

Covalently bonded materials are often described by
bond order potential Bond order potential is a class of empirical (analytical) interatomic potentials which is used in molecular dynamics and molecular statics simulations. Examples include the Tersoff potential, the EDIP potential, the Brenner potential, the Finni ...
s, sometimes also called Tersoff-like or Brenner-like potentials. These have in general a form that resembles a pair potential: :: V_(r_) = V_\mathrm(r_) + b_ V_\mathrm(r_) where the repulsive and attractive part are simple exponential functions similar to those in the Morse potential. However, the strength is modified by the environment of the atom i via the b_term. If implemented without an explicit angular dependence, these potentials can be shown to be mathematically equivalent to some varieties of EAM-like potentials Thanks to this equivalence, the bond-order potential formalism has been implemented also for many metal-covalent mixed materials. EAM potentials have also been extended to describe covalent bonding by adding angular-dependent terms to the electron density function \rho, in what is called the modified embedded atom method (MEAM).


= Force fields

= A force field is the collection of parameters to describe the physical interactions between atoms or physical units (up to ~108) using a given energy expression. The term force field characterizes the collection of parameters for a given interatomic potential (energy function) and is often used within the
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 mo ...
community. The force field parameters make the difference between good and poor models. Force fields are used for the simulation of metals, ceramics, molecules, chemistry, and biological systems, covering the entire periodic table and multiphase materials. Today's performance is among the best for solid-state materials, molecular fluids, and for biomacromolecules, whereby biomacromolecules were the primary focus of force fields from the 1970s to the early 2000s. Force fields range from relatively simple and interpretable fixed-bond models (e.g. Interface force field, CHARMM, and COMPASS) to explicitly reactive models with many adjustable fit parameters (e.g.
ReaxFF ReaxFF (for “reactive force field”) is a bond order-based force field developed by Adri van Duin, William A. Goddard, III, and co-workers at the California Institute of Technology The California Institute of Technology (branded as Caltech ...
) and machine learning models.


Non-parametric potentials

It should first be noted that non-parametric potentials are often referred to as "machine learning" potentials. While the descriptor/mapping forms of non-parametric models are closely related to machine learning in general and their complex nature make machine learning fitting optimizations almost necessary, differentiation is important in that parametric models can also be optimized using machine learning. Current research in interatomic potentials involves using systematically improvable, non-parameteric mathematical forms and increasingly complex
machine learning Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence. Machine ...
methods. The total energy is then written V_\mathrm = \sum_i^N E(\mathbf_i)where \mathbf_iis a mathematical representation of the atomic environment surrounding the atom i, known as the descriptor. E is a machine-learning model that provides a prediction for the energy of atom i based on the descriptor output. An accurate machine-learning potential requires both a robust descriptor and a suitable machine learning framework. The simplest descriptor is the set of interatomic distances from atom i to its neighbours, yielding a machine-learned pair potential. However, more complex many-body descriptors are needed to produce highly accurate potentials. It is also possible to use a linear combination of multiple descriptors with associated machine-learning models. Potentials have been constructed using a variety of machine-learning methods, descriptors, and mappings, including
neural networks A neural network is a network or circuit of biological neurons, or, in a modern sense, an artificial neural network, composed of artificial neurons or nodes. Thus, a neural network is either a biological neural network, made up of biological ...
,
Gaussian process regression In statistics, originally in geostatistics, kriging or Kriging, also known as Gaussian process regression, is a method of interpolation based on Gaussian process governed by prior covariances. Under suitable assumptions of the prior, kriging giv ...
, and
linear regression In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is cal ...
. A non-parametric potential is most often trained to total energies, forces, and/or stresses obtained from quantum-level calculations, such as
density functional theory Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body ...
, as with most modern potentials. However, the accuracy of a machine-learning potential can be converged to be comparable with the underlying quantum calculations, unlike analytical models. Hence, they are in general more accurate than traditional analytical potentials, but they are correspondingly less able to extrapolate. Further, owing to the complexity of the machine-learning model and the descriptors, they are computationally far more expensive than their analytical counterparts. Non-parametric, machine learned potentials may also be combined with parametric, analytical potentials, for example to include known physics such as the screened Coulomb repulsion, or to impose physical constraints on the predictions.


Potential fitting

Since the interatomic potentials are approximations, they by necessity all involve parameters that need to be adjusted to some reference values. In simple potentials such as the Lennard-Jones and Morse ones, the parameters are interpretable and can be set to match e.g. the equilibrium bond length and bond strength of a dimer molecule or the
surface energy In surface science, surface free energy (also interfacial free energy or surface energy) quantifies the disruption of intermolecular bonds that occurs when a surface is created. In solid-state physics, surfaces must be intrinsically less ener ...
of a solid . Lennard-Jones potential can typically describe the lattice parameters, surface energies, and approximate mechanical properties. Many-body potentials often contain tens or even hundreds of adjustable parameters with limited interpretability and no compatibility with common interatomic potentials for bonded molecules. Such parameter sets can be fit to a larger set of experimental data, or materials properties derived from less reliable data such as from density-functional theory. For solids, a many-body potential can often describe the
lattice constant A lattice constant or lattice parameter is one of the physical dimensions and angles that determine the geometry of the unit cells in a crystal lattice, and is proportional to the distance between atoms in the crystal. A simple cubic crystal has o ...
of the equilibrium crystal structure, the cohesive energy, and linear elastic constants, as well as basic point defect properties of all the elements and stable compounds well, although deviations in surface energies often exceed 50%. Non-parameteric potentials in turn contain hundreds or even thousands of independent parameters to fit. For any but the simplest model forms, sophisticated optimization and machine learning methods are necessary for useful potentials. The aim of most potential functions and fitting is to make the potential ''transferable'', i.e. that it can describe materials properties that are clearly different from those it was fitted to (for examples of potentials explicitly aiming for this, see e.g.). Key aspects here are the correct representation of chemical bonding, validation of structures and energies, as well as interpretability of all parameters. Full transferability and interpretability is reached with the Interface force field (IFF). An example of partial transferability, a review of interatomic potentials of Si describes that Stillinger-Weber and Tersoff III potentials for Si can describe several (but not all) materials properties they were not fitted to. The NIST interatomic potential repository provides a collection of fitted interatomic potentials, either as fitted parameter values or numerical tables of the potential functions. The OpenKIM project also provides a repository of fitted potentials, along with collections of validation tests and a software framework for promoting reproducibility in molecular simulations using interatomic potentials.


Reliability of interatomic potentials

Classical interatomic potentials often exceed the accuracy of simplified quantum mechanical methods such as
density functional theory Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body ...
at a million times lower computational cost. The use of interatomic potentials is recommended for the simulation of nanomaterials, biomacromolecules, and electrolytes from atoms up to millions of atoms at the 100 nm scale and beyond. As a limitation, electron densities and quantum processes at the local scale of hundreds of atoms are not included. When of interest, higher level
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 ...
methods can be locally used. The robustness of a model at different conditions other than those used in the fitting process is often measured in terms of transferability of the potential.


See also

*
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 mo ...
*
Computational materials science ''Computational Materials Science'' is a monthly peer-reviewed scientific journal published by Elsevier. It was established in October 1992. The editor-in-chief is Susan Sinnott. The journal covers computational modeling and practical research fo ...
*
Molecular dynamics Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of th ...
* Force field (chemistry)


References

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External links


NIST interatomic potential repository



Open Knowledgebase of Interatomic Models (OpenKIM)
Condensed matter physics Computational physics Materials science Quantum mechanical potentials