Interatomic potential

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 ${\displaystyle \textstyle V_{\mathrm {TOT} }}$ can be written as [3]

${\displaystyle V_{\mathrm {TOT} }=\sum _{i}^{N}V_{1}({\vec {r}}_{i})+\sum _{i,j}^{N}V_{2}({\vec {r}}_{i},{\vec {r}}_{j})+\sum _{i,j,k}^{N}V_{3}({\vec {r}}_{i},{\vec {r}}_{j},{\vec {r}}_{k})+\cdots }$

Here ${\displaystyle \textstyle V_{1}}$ is the one-body term, ${\displaystyle \textstyle V_{2}}$ the two-body term, ${\displaystyle \textstyle V_{3}}$ the three body term, ${\displaystyle \textstyle N}$ the number of atoms in the system, ${\displaystyle {\vec {r}}_{i}}$ the position of atom ${\displaystyle i}$, etc. ${\displaystyle i}$, ${\displaystyle j}$ and ${\displaystyle k}$ 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.[3] Alternatively, the summation of the pair term can be restricted to cases ${\displaystyle \textstyle i<j}$ and similarly for the three-body term ${\displaystyle \textstyle i<j<k}$, if the potential form is such that it is symmetric with respect to exchange of the ${\displaystyle j}$ and ${\displaystyle k}$ 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 ${\displaystyle V}$ 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 ${\displaystyle \textstyle r_{ij}=|{\vec {r}}_{i}-{\vec {r}}_{j}|}$ and angles between the bonds (vectors to neighbours) ${\displaystyle \textstyle \theta _{ijk}}$. Then, in the absence of external forces, the general form becomes

${\displaystyle V_{\mathrm {TOT} }=\sum _{i,j}^{N}V_{2}(r_{ij})+\sum _{i,j,k}^{N}V_{3}(r_{ij},r_{ik},\theta _{ijk})+\cdots }$

In the three-body term ${\displaystyle \textstyle V_{3}}$ the interatomic distance ${\displaystyle \textstyle r_{jk}}$ is not needed since the three terms ${\displaystyle \textstyle r_{ij},r_{ik},\theta _{ijk}}$ are sufficient to give the relative positions of three atoms ${\displaystyle i,j,k}$ 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 ${\displaystyle N}$ atoms. However, if the range of the interatomic potential is finite, i.e. the potentials ${\displaystyle \textstyle V(r)\equiv 0}$ above some cutoff distance ${\displaystyle \textstyle r_{\mathrm {cut} }}$, the summing can be restricted to atoms within the cutoff distance of each other. By also using a cellular method for finding the neighbours,[1] 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.

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 ${\displaystyle i}$ one should take the three-dimensional derivative (gradient) with respect to the position of atom ${\displaystyle i}$:

${\displaystyle {\vec {F}}_{i}=-\nabla _{{\vec {r}}_{i}}V_{\mathrm {TOT} }}$

For two-body potentials this gradient reduces, thanks to the symmetry with respect to ${\displaystyle ij}$ in the potential form, to straightforward differentiation with respect to the interatomic distances ${\displaystyle \textstyle r_{ij}}$. However, for many-body potentials (three-body, four-body, etc.) the differentiation becomes considerably more complex [12] [13] since the potential may not be any longer symmetric with respect to ${\displaystyle ij}$ exchange. In other words, also the energy of atoms ${\displaystyle k}$ that are not direct neighbours of ${\displaystyle i}$ can depend on the position ${\displaystyle \textstyle {\vec {r}}_{i}}$ because of angular and other many-body terms, and hence contribute to the gradient ${\displaystyle \textstyle \nabla _{{\vec {r}}_{i}}}$.

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.[14] 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.[15] 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. [16] 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.

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 written

$${\displaystyle V_{\mathrm {TOT} }=\sum _{i}^{N}E(\mathbf {q} _{i})}$$

where ${\displaystyle \mathbf {q} _{i}}$is a mathematical representation of the atomic environment surrounding the atom ${\displaystyle i}$, known as the descriptor.[51] ${\displaystyle E}$ is a machine-learning model that provides a prediction for the energy of atom ${\displaystyle i}$ 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 ${\displaystyle i}$ to its neighbours, yielding a machine-learned pair potential. However, more complex many-body descriptors are needed to produce highly accurate potentials. [51] It is also possible to use a linear combination of multiple descriptors with associated machine-learning models.[52] Potentials have been constructed using a variety of machine-learning methods, descriptors, and mappings, including neural networks,[53] Gaussian process regression,[54][55] and linear regression.[56][16]

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,[57] or to impose physical constraints on the predictions.[58]

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 .[59][60] Lennard-Jones potential can typically describe the lattice parameters, surface energies, and approximate mechanical properties.[61] 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.[62][63] 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%. [26][39][41][42][61][46] [64] [65] [66] 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.[67][68][69][70][71]). Key aspects here are the correct representation of chemical bonding, validation of structures and energies, as well as interpretability of all parameters.[47] Full transferability and interpretability is reached with the Interface force field (IFF).[46] 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.[14]

The NIST interatomic potential repository provides a collection of fitted interatomic potentials, either as fitted parameter values or numerical tables of the potential functions.[72] The OpenKIM [73] 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.

Classical interatomic potentials often exceed the accuracy of simplified quantum mechanical methods such as density functional theory at a million times lower computational cost.[47] 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.[74]

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.

• Computational chemistry
• Computational materials science
• Molecular dynamics
• Force field (chemistry)

• NIST interatomic potential repository
• NIST JARVIS-FF
• Open Knowledgebase of Interatomic Models (OpenKIM)