Interatomic potential

Interatomic potentials are mathematical functions to calculate the potential energy of a system of atoms with given positions in space. Interatomic potentials are widely used as the physical basis of molecular mechanics and molecular dynamics simulations in computational chemistry, computational physics and computational materials science 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, cohesion, thermal expansion, and elastic and plastic material behavior, as well as chemical reactions.
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 formula_1 can
be written as 
Here formula_3 is the one-body term, formula_4 the two-body term, formula_5 the
three body term, formula_6 the number of atoms in the system,
formula_7 the position of atom formula_8, etc. formula_8, formula_10 and formula_11 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 formula_12
and similarly for the three-body term formula_13, if
the potential form is such that it is symmetric with respect to exchange
of the formula_10 and formula_11 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 formula_16 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 formula_17
and angles between the bonds
(vectors to neighbours) formula_18.
Then, in the absence of external forces, the general
form becomes
In the three-body term formula_5 the
interatomic distance formula_21 is not needed
since the three terms formula_22
are sufficient to give the relative positions of three atoms formula_23 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 formula_24 atoms.
However, if the range of the interatomic potential is finite,
i.e. the potentials formula_25 above
some cutoff distance formula_26,
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 formula_8 one should take the three-dimensional
derivative (gradient) with respect to the position of atom formula_8:
For two-body potentials this gradient reduces, thanks to the
symmetry with respect to formula_30 in the potential form, to straightforward
differentiation with respect to the interatomic distances
formula_31. 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 formula_30 exchange.
In other words, also the energy
of atoms formula_11 that are not direct neighbours of formula_8 can depend on the position formula_35
because of angular and other many-body terms, and hence contribute to the gradient
formula_36.
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 in nature, and there is no known
way in which the true interactions described by
the Schrödinger equation or Dirac equation for
all electrons and nuclei could be cast into an analytical
functional form. Hence all analytical interatomic
potentials are by necessity approximations.
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 
where formula_38 is the depth of the potential well
and formula_39 is the distance at which the potential crosses zero.
The attractive term proportional to formula_40in the potential comes from the scaling of van der Waals forces, while the formula_41 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, 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.
Another simple and widely used pair potential is the
Morse potential, which consists simply of a sum of two exponentials.
Here formula_43 is the equilibrium bond energy and
formula_44 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
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 simulations 
are carried out with various of many-body potentials.
Repulsive potentials.
For very short interatomic separations, important in radiation material science,
the interactions can be described quite accurately with screened Coulomb potentials which have the general form
Here, formula_46 when formula_47. formula_48 and formula_49 are the charges of the interacting nuclei, and "formula_50" 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
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
where formula_53 is a so-called embedding function
(not to be confused with the force formula_54) that is a function of the sum of the so-called electron density
formula_55. formula_56
is a pair potential that usually is purely repulsive. In the original
formulation the electron
density function formula_55 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 NIST 
Covalently bonded materials are often described by 
bond order potentials, sometimes also called
Tersoff-like or Brenner-like potentials.
These have in general a form that resembles a pair potential:
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 formula_8 via the formula_60term. 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 formula_61, 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 community. The force field makes 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 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) 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 methods. The total energy is then writtenformula_62where formula_63is a mathematical representation of the atomic environment surrounding the atom formula_8, known as the descriptor. formula_65 is a machine-learning model that provides a prediction for the energy of atom formula_8 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 formula_8 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, Gaussian process regression, and linear regression.
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, 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 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 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 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 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.