Interatomic potential explained

Interatomic potentials are mathematical functions to calculate the potential energy of a system of atoms with given positions in space.^{[1] [2] [3] [4]} 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.^{[5] [6] [7] [8] [9] [10] [11]}

Functional form

Interatomic potentials can be written as a series expansion offunctional terms that depend on the position of one, two, three, etc.atoms at a time. Then the total potential of the system

style V canbe written as V = \sum ^{N} _{i =1} V _{1(\vec} r _{i)} + \sum ^{N} _{i,j =1} V _{2(\vec} r _{i,\vec} r _{j)} + \sum ^{N} _{i,j,k =1} V _{3(\vec} r _{i,\vec} r _{j,\vec} r _{k)} + …

Here

style V _{1} is the one-body term, style V _{2} the two-body term, style V _{3} thethree body term, style N the number of atoms in the system,\vec r _{i} the position of atom i , etc. i , j and k are indicesthat loop over atom positions.

Note that in case the pair potential is given per atom pair, in the two-bodyterm the potential should be multiplied by 1/2 as otherwise each bond is countedtwice, and similarly the three-body term by 1/6. Alternatively,the summation of the pair term can be restricted to cases

style i<j and similarly for the three-body term style i<j<k , ifthe potential form is such that it is symmetric with respect to exchangeof the j and k indices (this may not be the case for potentialsfor multielemental systems).

The one-body term is only meaningful if the atoms are in an externalfield (e.g. an electric field). In the absence of external fields,the potential

V should not depend on the absolute position ofatoms, but only on the relative positions. This meansthat the functional form can be rewritten as a functionof interatomic distances style r _{ij} = |\vec r _{i-\vec} r _{j|} and angles between the bonds(vectors to neighbours) style \theta _{ijk} .Then, in the absence of external forces, the generalform becomesV _{TOT} = \sum ^{N} _{i,j} V _{2(r} _{ij} ) + \sum ^{N} _{i,j,k} V _{3(r} _{ij} ,r _{ik} ,\theta _{ijk} ) + …

In the three-body term

style V _{3} theinteratomic distance style r _{jk} is not neededsince the three terms style r _{ij} ,r _{ik} ,\theta _{ijk} are sufficient to give the relative positions of three atoms i, j, k in three-dimensional space. Any terms of order higher than2 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 onEAM-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 style V(r) \equiv 0 abovesome cutoff distance style r _{cut} ,the summing can be restricted to atoms within the cutoffdistance of each other. By also using a cellular methodfor finding the neighbours,^[1] the MD algorithm can bean O(N) algorithm. Potentials with an infiniterange can be summed up efficiently by Ewald summationand its further developments.

Force calculation

The forces acting between atoms can be obtained by differentiation ofthe total energy with respect to atom positions. That is,to get the force on atom

i one should take the three-dimensionalderivative (gradient) of the potential V _{tot} with respect to the position of atom i :\vec{F} _{i} = - \nabla _{\vec} _{i}} V _{TOT} For two-body potentials this gradient reduces, thanks to thesymmetry with respect to ij in the potential form, to straightforwarddifferentiation with respect to the interatomic distancesstyle r _{ij} . However, for many-bodypotentials (three-body, four-body, etc.) the differentiationbecomes considerably more complex ^{[12] [13]} since the potential may not be any longer symmetric with respect to ij exchange.In other words, also the energyof atoms k that are not direct neighbours of i can depend on the position style \vec{r} _{i} because of angular and other many-body terms, and hence contribute to the gradientstyle \nabla _{\vec{r} _{k} }.

Classes of interatomic potentials

Interatomic potentials come in many different varieties, withdifferent 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 interactionsare quantum mechanical in nature, and there is no knownway in which the true interactions described bythe Schrödinger equation or Dirac equation forall electrons and nuclei could be cast into an analyticalfunctional form. Hence all analytical interatomicpotentials 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-parametric 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 ^{[17] [18] [19]}

V _{LJ(r)} = 4 \varepsilon \left[ \left( (\sigma)/(r) \right) ^{12} - \left( (\sigma)/(r) \right) ^{6} \right] where style \varepsilon is the depth of the potential welland style \sigma is the distance at which the potential crosses zero.The attractive term proportional to style 1 /r ^{6} in the potential comes from the scaling of van der Waals forces, while the style 1 /r ^{12} repulsive term is much more approximate (conveniently the square of the attractive term).^[6] On its own, this potential is quantitatively accurate only for noble gases and has been extensively studied in the past decades,^[20] 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 - especially in fluids.^[21]

Another simple and widely used pair potential is theMorse potential, which consists simply of a sum of two exponentials.

V _{M(r)} = D _{e} ( e ^{-2 a(r - r _{e)}} -2 e ^{- a(r - r _{e)}} ) Here style D _{e} is the equilibrium bond energy andstyle r _{e} the bond distance. The Morsepotential has been applied to studies of molecular vibrations and solids, ^[22] 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 theBuckingham pair potential, and a long-range Coulomb potentialgiving the ionic interactions between the ions forming the material. The short-rangeterm for ionic materials can also be of many-body character.^[23]

Pair potentials have some inherent limitations, such as the inabilityto describe all 3 elastic constants ofcubic metals or correctly describe both cohesive energy and vacancy formation energy.^[7] 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

V(r _{ij} ) = { 1 \over 4 \pi \varepsilon _{0}} {Z _{1Z} _{2} e ^{2} \over r _{ij} } \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.^[24] and more accurate ones can be obtained from all-electron quantum chemistry calculations^[25] In binary collision approximation simulations this kind of potential can be usedto describe the nuclear stopping power.

Many-body potentials

The Stillinger-Weber potential^[26] is a potential that has a two-body and three-body terms of the standard form

V _{TOT} = \sum ^{N} _{i,j} V _{2(r} _{ij} ) + \sum ^{N} _{i,j,k} V _{3(r} _{ij} ,r _{ik} ,\theta _{ijk} ) 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 otherelements and compounds^{[27] [28]} and also formed the basis for other Si potentials.^{[29] [30]}

Metals are very commonly described with what can be called"EAM-like" potentials, i.e. potentials that sharethe same functional form as the embedded atom model.In these potentials, the total potential energy is written

V _{TOT} = \sum ^{N} _{i} F _{i} \left(\sum _{j} \rho (r _{ij} ) \right) + (1)/(2) \sum ^{N} _{i, j} V _{2(r} _{ij} )

where

style F _{i} is a so-called embedding function(not to be confused with the force style \vec F _{i} ) that is a function of the sum of the so-called electron densitystyle \rho (r _{ij} ) . style V _{2} is a pair potential that usually is purely repulsive. In the originalformulation ^{[31] [32]} the electrondensity function style \rho (r _{ij} ) was obtainedfrom true atomic electron densities, and the embedding functionwas motivated from density-functional theory as the energy neededto 'embed' an atom into the electron density. .^[33] However, many other potentials used for metals share the same functionalform but motivate the terms differently, e.g. basedon tight-binding theory^{[34] [35] [36]} or other motivations^{[37] [38]} .^[39]

EAM-like potentials are usually implemented as numerical tables.A collection of tables is available at the interatomicpotential repository at NIST http://www.ctcms.nist.gov/potentials/

Covalently bonded materials are often described by bond order potentials, sometimes also calledTersoff-like or Brenner-like potentials.^{[40] [41]}

These have in general a form that resembles a pair potential:

V _{ij} (r _{ij} ) = V _{repulsive(r} _{ij} ) + b _{ijk} V _{attractive(r} _{ij} )

where the repulsive and attractive part are simple exponentialfunctions similar to those in the Morse potential.However, the strength is modified by the environment of the atom

i via the b _{ijk} term. If implemented withoutan explicit angular dependence, these potentialscan be shown to be mathematically equivalent to some varieties of EAM-like potentials^{[42] [43]} Thanks to this equivalence, the bond-order potential formalism has been implemented also for many metal-covalent mixed materials.^{[43] [44] [45] [46]}

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).^{[47] [48] [49]}

Force fields

See main article: Force field (chemistry). 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.^[50] 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,^{[51] [52]} molecular fluids,^[21] and for biomacromolecules,^[53] 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,^[54] 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-parametric mathematical forms and increasingly complex machine learning methods. The total energy is then written$$V\_\backslash mathrm\; =\; \backslash sum\_i^N\; E(\backslash mathbf\_i)$$where

q _{i} is a mathematical representation of the atomic environment surrounding the atom i , known as the descriptor.^[55] 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.^[56] Potentials have been constructed using a variety of machine-learning methods, descriptors, and mappings, including neural networks,^[57] Gaussian process regression,^{[58] [59]} and linear regression.^[60]

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,^[61] or to impose physical constraints on the predictions.^[62]

Potential fitting

Since the interatomic potentials are approximations, they by necessity all involveparameters that need to be adjusted to some reference values. In simplepotentials 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 strengthof a dimer molecule or the surface energy of a solid.^{[63] [64]} Lennard-Jones potential can typically describe the lattice parameters, surface energies, and approximate mechanical properties.^[65] Many-bodypotentials 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 materialsproperties derived from less reliable data such as from density-functional theory.^{[66] [67]} For solids, a many-body potentialcan 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%.^{[30] [43] [45] [68] [69] [70]} Non-parametric 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 potentialtransferable, i.e. that it can describe materials properties that are clearlydifferent from those it was fitted to (for examples of potentials explicitly aiming for this,see e.g.^{[71] [72] [73] [74] [75]}). 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 potentialsof 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 numericaltables of the potential functions.^[76] The OpenKIM ^[77] 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.

Machine-learned interatomic potentials

See main article: Machine learning potential. Since the 1990s, machine learning programs have been employed to construct interatomic potentials, mapping atomic structures to their potential energies. These are generally referred to as 'machine learning potentials' (MLPs)^[78] or as 'machine-learned interatomic potentials' (MLIPs).^[79] Such machine learning potentials help fill the gap between highly accurate but computationally intensive simulations like density functional theory and computationally lighter, but much less precise, empirical potentials. Early neural networks showed promise, but their inability to systematically account for interatomic energy interactions limited their applications to smaller, low-dimensional systems, keeping them largely within the confines of academia. However, with continuous advancements in artificial intelligence technology, machine learning methods have become significantly more accurate, positioning machine learning as a significant player in potential fitting.^{[80] [81] [82]}

Modern neural networks have revolutionized the construction of highly accurate and computationally light potentials by integrating theoretical understanding of materials science into their architectures and preprocessing. Almost all are local, accounting for all interactions between an atom and its neighbor up to some cutoff radius. These neural networks usually intake atomic coordinates and output potential energies. Atomic coordinates are sometimes transformed with atom-centered symmetry functions or pair symmetry functions before being fed into neural networks. Encoding symmetry has been pivotal in enhancing machine learning potentials by drastically constraining the neural networks' search space.^{[80] [83]}

Conversely, message-passing neural networks (MPNNs), a form of graph neural networks, learn their own descriptors and symmetry encodings. They treat molecules as three-dimensional graphs and iteratively update each atom's feature vectors as information about neighboring atoms is processed through message functions and convolutions. These feature vectors are then used to directly predict the final potentials. In 2017, the first-ever MPNN model, a deep tensor neural network, was used to calculate the properties of small organic molecules. Advancements in this technology led to the development of Matlantis in 2022, which commercially applies machine learning potentials for new materials discovery.^[84] Matlantis, which can simulate 72 elements, handle up to 20,000 atoms at a time, and execute calculations up to 20 million times faster than density functional theory with almost indistinguishable accuracy, showcases the power of machine learning potentials in the age of artificial intelligence.^{[80] [85] [86]}

Another class of machine-learned interatomic potential is the Gaussian approximation potential (GAP),^{[87] [88] [89]} which combines compact descriptors of local atomic environments^[90] with Gaussian process regression^[91] to machine learn the potential energy surface of a given system. To date, the GAP framework has been used to successfully develop a number of MLIPs for various systems, including for elemental systems such as Carbon^[92] Silicon,^[93] and Tungsten,^[94] as well as for multicomponent systems such as Ge₂Sb₂Te₅^[95] and austenitic stainless steel, Fe₇Cr₂Ni.^[96]

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.^[97]

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 materials science
• Molecular dynamics
• Force field (chemistry)

External links

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

Notes and References

1. M. P. Allen and D. J. Tildesley. Computer Simulation of Liquids. Oxford University Press, Oxford, England, 1989.
2. Daan Frenkel and Berend Smit. Understanding molecular simulation: from algorithms to applications. Academic Press, San Diego, second edition, 2002.
3. R. Lesar. Introduction to Computational Materials Science. Cambridge University Press, 2013.
4. Brenner. D.W.. The Art and Science of an Analytic Potential. Physica Status Solidi B. 217. 1. 2000. 23–40. 0370-1972. 10.1002/(SICI)1521-3951(200001)217:1<23::AID-PSSB23>3.0.CO;2-N. 2000PSSBR.217...23B.
5. N. W. Ashcroft and N. D. Mermin. Solid State Physics.Saunders College, Philadelphia, 1976.
6. Charles Kittel. Introduction to Solid State Physics. John Wiley & Sons, New York, third edition, 1968.
7. Daw. Murray S.. Foiles. Stephen M.. Baskes. Michael I.. The embedded-atom method: a review of theory and applications. Materials Science Reports. 9. 7–8. 1993. 251–310. 0920-2307. 10.1016/0920-2307(93)90001-U. free.
8. Tersoff J . New empirical approach for the structure and energy of covalent systems . Physical Review B . 37 . 12 . 6991–7000 . April 1988 . 9943969 . 10.1103/physrevb.37.6991 . 1988PhRvB..37.6991T .
9. FINNIS . M . Bond-order potentials through the ages . Progress in Materials Science . 52 . 2–3 . 2007 . 0079-6425 . 10.1016/j.pmatsci.2006.10.003 . 133–153.
10. Sinnott . Susan B. . Susan Sinnott. Brenner . Donald W. . Three decades of many-body potentials in materials research . MRS Bulletin . 37 . 5 . 2012 . 0883-7694 . 10.1557/mrs.2012.88 . 469–473. free . 2012MRSBu..37..469S .
11. Fischer . Johann . Wendland . Martin . October 2023 . On the history of key empirical intermolecular potentials . Fluid Phase Equilibria . en . 573 . 113876 . 10.1016/j.fluid.2023.113876. 2023FlPEq.57313876F . free .
12. Beardmore . Keith M. . Grønbech-Jensen . Niels . Direct simulation of ion-beam-induced stressing and amorphization of silicon . Physical Review B . 60 . 18 . 1 October 1999 . 0163-1829 . 10.1103/physrevb.60.12610 . 12610–12616. cond-mat/9901319v2. 1999PhRvB..6012610B . 15494648 .
13. Albe . Karsten . Nord . J. . Nordlund . K. . Dynamic charge-transfer bond-order potential for gallium nitride . Philosophical Magazine . 89 . 34–36 . 2009 . 1478-6435 . 10.1080/14786430903313708 . 3477–3497. 2009PMag...89.3477A . 56072359 .
14. Balamane H, Halicioglu T, Tiller WA . Comparative study of silicon empirical interatomic potentials . Physical Review B . 46 . 4 . 2250–2279 . July 1992 . 10003901 . 10.1103/physrevb.46.2250 . 1992PhRvB..46.2250B .
15. Plimpton SJ, Thompson AP . Computational aspects of many-body potentials . MRS Bull. . 37 . 513–521 . 2012. 5 . 10.1557/mrs.2012.96 . 2012MRSBu..37..513P . 138567968 .
16. Shapeev . Alexander V. . 28970251. 2016-09-13 . Moment Tensor Potentials: A Class of Systematically Improvable Interatomic Potentials. Multiscale Modeling & Simulation . en . 14 . 3 . 1153–1173 . 10.1137/15M1054183 . 1540-3459 . 1512.06054.
17. Lennard-Jones . J. E. . 1924 . On the Determination of Molecular Fields . Proc. R. Soc. Lond. A . 106 . 738 . 463–477 . 10.1098/rspa.1924.0082 . 1924RSPSA.106..463J . free . .
18. Lenhard . Johannes . Stephan . Simon . Hasse . Hans . June 2024 . On the History of the Lennard-Jones Potential . Annalen der Physik . en . 536 . 6 . 10.1002/andp.202400115 . 0003-3804.
19. Fischer . Johann . Wendland . Martin . October 2023 . On the history of key empirical intermolecular potentials . Fluid Phase Equilibria . en . 573 . 113876 . 10.1016/j.fluid.2023.113876. 2023FlPEq.57313876F . free .
20. Stephan . Simon . Thol . Monika . Vrabec . Jadran . Hasse . Hans . 2019-10-28 . Thermophysical Properties of the Lennard-Jones Fluid: Database and Data Assessment . Journal of Chemical Information and Modeling . en . 59 . 10 . 4248–4265 . 10.1021/acs.jcim.9b00620 . 31609113 . 204545481 . 1549-9596.
21. Stephan . Simon . Horsch . Martin T. . Vrabec . Jadran . Hasse . Hans . 2019-07-03 . MolMod – an open access database of force fields for molecular simulations of fluids . Molecular Simulation . en . 45 . 10 . 806–814 . 10.1080/08927022.2019.1601191 . 119199372 . 0892-7022. 1904.05206 .
22. Girifalco . L. A. . Weizer . V. G. . Application of the Morse Potential Function to Cubic Metals . Physical Review . 114 . 3 . 1 April 1959 . 0031-899X . 10.1103/physrev.114.687 . 687–690. 1959PhRv..114..687G . 10338.dmlcz/103074 . free .
23. Feuston . B. P. . Garofalini . S. H. . Empirical three-body potential for vitreous silica . The Journal of Chemical Physics . 89 . 9 . 1988 . 0021-9606 . 10.1063/1.455531 . 5818–5824. 1988JChPh..89.5818F .
24. J. F. Ziegler, J. P. Biersack, and U. Littmark. The Stopping and Range of Ions in Matter. Pergamon, New York, 1985.
25. Nordlund . K. . Runeberg . N. . Sundholm . D. . Repulsive interatomic potentials calculated using Hartree-Fock and density-functional theory methods . Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms . 132 . 1 . 1997 . 0168-583X . 10.1016/s0168-583x(97)00447-3 . 45–54. 1997NIMPB.132...45N .
26. Stillinger FH, Weber TA . Computer simulation of local order in condensed phases of silicon . Physical Review B . 31 . 8 . 5262–5271 . April 1985 . 9936488 . 10.1103/physrevb.31.5262 . 1985PhRvB..31.5262S .
27. Ichimura . M. . Stillinger-Weber potentials for III–V compound semiconductors and their application to the critical thickness calculation for InAs/GaAs . Physica Status Solidi A . 153 . 2 . 16 February 1996 . 0031-8965 . 10.1002/pssa.2211530217 . 431–437. 1996PSSAR.153..431I .
28. Ohta . H. . Hamaguchi . S. . 2001 . Classical interatomic potentials for si-o-f and si-o-cl systems . Journal of Chemical Physics . 115 . 14. 6679–90 . 10.1063/1.1400789 . 2001JChPh.115.6679O . 2433/50272 . free .
29. Bazant. M. Z.. Kaxiras. E.. Justo. J. F.. 17860100. Environment-dependent interatomic potential for bulk silicon. Phys. Rev. B. 1997. 56. 14. 8542. 10.1103/PhysRevB.56.8542. cond-mat/9704137. 1997PhRvB..56.8542B.
30. Justo . João F. . Bazant . Martin Z. . Kaxiras . Efthimios . Bulatov . V. V. . Yip . Sidney . Interatomic potential for silicon defects and disordered phases . Physical Review B . 58 . 5 . 1 July 1998 . 0163-1829 . 10.1103/physrevb.58.2539 . 2539–2550. cond-mat/9712058 . 1998PhRvB..58.2539J . 14585375 .
31. Murray S. Daw. Stephen M. Foiles. Michael Baskes . Foiles SM, Baskes MI, Daw MS . Embedded-atom-method functions for the fcc metals Cu, Ag, Au, Ni, Pd, Pt, and their alloys . Physical Review B . 33 . 12 . 7983–7991 . June 1986 . 9938188 . 10.1103/physrevb.33.7983 . 1986PhRvB..33.7983F .
32. Foiles . S. M. . Baskes . M. I. . Daw . M. S. . Erratum: Embedded-atom-method functions for the fcc metals Cu, Ag, Au, Ni, Pd, Pt, and their alloys . Physical Review B . 37 . 17 . 15 June 1988 . 0163-1829 . 10.1103/physrevb.37.10378 . 10378. free .
33. Puska . M. J. . Nieminen . R. M. . Manninen . M. . Atoms embedded in an electron gas: Immersion energies . Physical Review B . 24 . 6 . 15 September 1981 . 0163-1829 . 10.1103/physrevb.24.3037 . 3037–3047. 1981PhRvB..24.3037P .
34. Finnis . M. W. . Sinclair . J. E. . A simple empirical N-body potential for transition metals . Philosophical Magazine A . 50 . 1 . 1984 . 0141-8610 . 10.1080/01418618408244210 . 45–55. 1984PMagA..50...45F .
35. Erratum . Philosophical Magazine A . 53 . 1 . 1986 . 0141-8610 . 10.1080/01418618608242815 . 161. 1986PMagA..53..161. . free .
36. Cleri F, Rosato V . Tight-binding potentials for transition metals and alloys . Physical Review B . 48 . 1 . 22–33 . July 1993 . 10006745 . 10.1103/physrevb.48.22 . 1993PhRvB..48...22C .
37. Kelchner . Cynthia L. . Halstead . David M. . Perkins . Leslie S. . Wallace . Nora M. . DePristo . Andrew E. . Construction and evaluation of embedding functions . Surface Science . 310 . 1–3 . 1994 . 0039-6028 . 10.1016/0039-6028(94)91405-2 . 425–435. 1994SurSc.310..425K .
38. Dudarev . S L . Derlet . P M . A 'magnetic' interatomic potential for molecular dynamics simulations . Journal of Physics: Condensed Matter . 17 . 44 . 17 October 2005 . 0953-8984 . 10.1088/0953-8984/17/44/003 . 7097–7118. 2005JPCM...17.7097D . 123141962 .
39. Olsson . Pär . Wallenius . Janne . Domain . Christophe . Nordlund . Kai . Malerba . Lorenzo . 16118006 . Two-band modeling of α-prime phase formation in Fe-Cr . Physical Review B . 72 . 21 . 21 December 2005 . 1098-0121 . 10.1103/physrevb.72.214119 . 214119. 2005PhRvB..72u4119O .
40. Tersoff J . New empirical approach for the structure and energy of covalent systems . Physical Review B . 37 . 12 . 6991–7000 . April 1988 . 9943969 . 10.1103/PhysRevB.37.6991 . 1988PhRvB..37.6991T .
41. Brenner DW . Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films . Physical Review B . 42 . 15 . 9458–9471 . November 1990 . 9995183 . 10.1103/PhysRevB.42.9458 . 1990PhRvB..42.9458B .
42. Brenner DW . Relationship between the embedded-atom method and Tersoff potentials . Physical Review Letters . 63 . 9 . 1022 . August 1989 . 10041250 . 10.1103/PhysRevLett.63.1022 . 1989PhRvL..63.1022B .
43. Albe. Karsten. Nordlund. Kai. Averback. Robert S.. Modeling the metal-semiconductor interaction: Analytical bond-order potential for platinum-carbon. Physical Review B. 65. 19. 195124. 2002. 0163-1829. 10.1103/PhysRevB.65.195124. 2002PhRvB..65s5124A.
44. de Brito Mota. F.. Justo. J. F.. Fazzio. A.. Structural properties of amorphous silicon nitride. Phys. Rev. B. 1998. 58. 13. 8323. 10.1103/PhysRevB.58.8323. 1998PhRvB..58.8323D.
45. Juslin . N. . Erhart . P. . Träskelin . P. . Nord . J. . Henriksson . K. O. E. . Nordlund . K. . Salonen . E. . Albe . K. . 8090449 . Analytical interatomic potential for modeling nonequilibrium processes in the W–C–H system . Journal of Applied Physics . 98 . 12 . 15 December 2005 . 123520–123520–12 . 0021-8979 . 10.1063/1.2149492 . 2005JAP....98l3520J .
46. Erhart . Paul . Juslin . Niklas . Goy . Oliver . Nordlund . Kai . Müller . Ralf . Albe . Karsten . Analytic bond-order potential for atomistic simulations of zinc oxide . Journal of Physics: Condensed Matter . 18 . 29 . 30 June 2006 . 0953-8984 . 10.1088/0953-8984/18/29/003 . 6585–6605. 2006JPCM...18.6585E . 38072718 .
47. Baskes MI . Application of the embedded-atom method to covalent materials: A semiempirical potential for silicon . Physical Review Letters . 59 . 23 . 2666–2669 . December 1987 . 10035617 . 10.1103/PhysRevLett.59.2666 . 1987PhRvL..59.2666B .
48. Baskes MI . Modified embedded-atom potentials for cubic materials and impurities . Physical Review B . 46 . 5 . 2727–2742 . August 1992 . 10003959 . 10.1103/PhysRevB.46.2727 . 1992PhRvB..46.2727B .
49. Lee. Byeong-Joo. Baskes. M. I.. 2000-10-01. Second nearest-neighbor modified embedded-atom-method potential. Physical Review B. 62. 13. 8564–8567. 10.1103/PhysRevB.62.8564. 2000PhRvB..62.8564L.
50. Heinz H, Lin TJ, Mishra RK, Emami FS. February 2013. Thermodynamically consistent force fields for the assembly of inorganic, organic, and biological nanostructures: the INTERFACE force field. Langmuir. 29. 6. 1754–65. 10.1021/la3038846. 23276161.
51. Heinz H, Ramezani-Dakhel H. January 2016. Simulations of inorganic-bioorganic interfaces to discover new materials: insights, comparisons to experiment, challenges, and opportunities. Chemical Society Reviews. 45. 2. 412–48. 10.1039/c5cs00890e. 26750724.
52. Mishra. Ratan K.. Mohamed. Aslam Kunhi. Geissbühler. David. Manzano. Hegoi. Jamil. Tariq. Shahsavari. Rouzbeh. Kalinichev. Andrey G.. Galmarini. Sandra. Tao. Lei. Heinz. Hendrik. Pellenq. Roland. December 2017. A force field database for cementitious materials including validations, applications and opportunities. Cement and Concrete Research. en. 102. 68–89. 10.1016/j.cemconres.2017.09.003. free.
53. Wang J, Wolf RM, Caldwell JW, Kollman PA, Case DA . Development and testing of a general amber force field . Journal of Computational Chemistry . 25 . 9 . 1157–74 . July 2004 . 15116359 . 10.1002/jcc.20035 . 18734898 . free .
54. Huang J, MacKerell AD . CHARMM36 all-atom additive protein force field: validation based on comparison to NMR data . Journal of Computational Chemistry . 34 . 25 . 2135–45 . September 2013 . 23832629 . 10.1002/jcc.23354 . 3800559 .
55. Bartók. Albert P.. Kondor. Risi. Csányi. Gábor. 118375156. 2013-05-28. On representing chemical environments. Physical Review B. en. 87. 18. 184115. 10.1103/PhysRevB.87.184115. 1098-0121. 1209.3140. 2013PhRvB..87r4115B.
56. Deringer. Volker L.. Csányi. Gábor. 55190594. 2017-03-03. Machine learning based interatomic potential for amorphous carbon. Physical Review B. en. 95. 9. 094203. 10.1103/PhysRevB.95.094203. 2469-9950. 1611.03277. 2017PhRvB..95i4203D.
57. Behler J, Parrinello M . Generalized neural-network representation of high-dimensional potential-energy surfaces . Physical Review Letters . 98 . 14 . 146401 . April 2007 . 17501293 . 10.1103/PhysRevLett.98.146401 . 2007PhRvL..98n6401B .
58. Bartók AP, Payne MC, Kondor R, Csányi G . 15918457 . Gaussian approximation potentials: the accuracy of quantum mechanics, without the electrons . Physical Review Letters . 104 . 13 . 136403 . April 2010 . 20481899 . 10.1103/PhysRevLett.104.136403 . 0910.1019 . 2010PhRvL.104m6403B .
59. Dragoni. Daniele. Daff. Thomas D.. Csányi. Gábor. Marzari. Nicola. 2018-01-30. Achieving DFT accuracy with a machine-learning interatomic potential: Thermomechanics and defects in bcc ferromagnetic iron. Physical Review Materials. 2. 1. 013808. 10.1103/PhysRevMaterials.2.013808. 1706.10229 . 2018PhRvM...2a3808D . 10281/231112. 119252567 . free.
60. Thompson. A.P.. Swiler. L.P.. Trott. C.R.. Foiles. S.M.. Tucker. G.J.. 2015-03-15. Spectral neighbor analysis method for automated generation of quantum-accurate interatomic potentials. Journal of Computational Physics. en. 285. 316–330. 10.1016/j.jcp.2014.12.018. 1409.3880. 2015JCoPh.285..316T. free.
61. Byggmästar. J.. Hamedani. A.. Nordlund. K.. Djurabekova. F.. 201106123. 2019-10-17. Machine-learning interatomic potential for radiation damage and defects in tungsten. Physical Review B. 100. 14. 144105. 10.1103/PhysRevB.100.144105. 1908.07330. 2019PhRvB.100n4105B. 10138/306660. free.
62. Pun GP, Batra R, Ramprasad R, Mishin Y . Physically informed artificial neural networks for atomistic modeling of materials . Nature Communications . 10 . 1 . 2339 . May 2019 . 31138813 . 6538760 . 10.1038/s41467-019-10343-5 . 2019NatCo..10.2339P .
63. Heinz. Hendrik. Vaia. R. A.. Farmer. B. L.. Naik. R. R.. 2008-10-09. Accurate Simulation of Surfaces and Interfaces of Face-Centered Cubic Metals Using 12−6 and 9−6 Lennard-Jones Potentials. The Journal of Physical Chemistry C. 112. 44. 17281–17290. 10.1021/jp801931d. 1932-7447.
64. Liu. Juan. Tennessen. Emrys. Miao. Jianwei. Huang. Yu. Rondinelli. James M.. Heinz. Hendrik. 2018-05-31. Understanding Chemical Bonding in Alloys and the Representation in Atomistic Simulations. The Journal of Physical Chemistry C. 122. 26. 14996–15009. 10.1021/acs.jpcc.8b01891. 51855788 . 1932-7447.
65. Nathanson M, Kanhaiya K, Pryor A, Miao J, Heinz H . Atomic-Scale Structure and Stress Release Mechanism in Core-Shell Nanoparticles . ACS Nano . 12 . 12 . 12296–12304 . December 2018 . 30457827 . 10.1021/acsnano.8b06118 . 53764446 .
66. Ruiz. Victor G.. Liu. Wei. Tkatchenko. Alexandre. 2016-01-15. Density-functional theory with screened van der Waals interactions applied to atomic and molecular adsorbates on close-packed and non-close-packed surfaces. Physical Review B. 93. 3. 035118. 10.1103/physrevb.93.035118. 2016PhRvB..93c5118R. 11858/00-001M-0000-0029-3035-8. 2469-9950. free.
67. Ruiz VG, Liu W, Zojer E, Scheffler M, Tkatchenko A . Density-functional theory with screened van der Waals interactions for the modeling of hybrid inorganic-organic systems . Physical Review Letters . 108 . 14 . 146103 . April 2012 . 22540809 . 10.1103/physrevlett.108.146103 . 2012PhRvL.108n6103R . free . 11858/00-001M-0000-000F-C6EA-3 . free .
68. Ercolessi . F . Adams . J. B . 18043298 . Interatomic Potentials from First-Principles Calculations: The Force-Matching Method . Europhysics Letters (EPL) . 26 . 8 . 10 June 1994 . 0295-5075 . 10.1209/0295-5075/26/8/005 . 583–588. cond-mat/9306054 . 1994EL.....26..583E .
69. Mishin . Y. . Mehl . M. J. . Papaconstantopoulos . D. A. . Embedded-atom potential forB2−NiAl . Physical Review B . 65 . 22 . 12 June 2002 . 0163-1829 . 10.1103/physrevb.65.224114 . 224114. 2002PhRvB..65v4114M .
70. Beardmore . Keith . Smith . Roger . Empirical potentials for C-Si-H systems with application to C₆₀ interactions with Si crystal surfaces . Philosophical Magazine A . 74 . 6 . 1996 . 0141-8610 . 10.1080/01418619608240734 . 1439–1466. 1996PMagA..74.1439B .
71. Mishra. Ratan K.. Flatt. Robert J.. Heinz. Hendrik. 2013-04-19. Force Field for Tricalcium Silicate and Insight into Nanoscale Properties: Cleavage, Initial Hydration, and Adsorption of Organic Molecules. The Journal of Physical Chemistry C. 117. 20. 10417–10432. 10.1021/jp312815g. 1932-7447.
72. Ramezani-Dakhel. Hadi. Ruan. Lingyan. Huang. Yu. Heinz. Hendrik. 2015-01-21. Molecular Mechanism of Specific Recognition of Cubic Pt Nanocrystals by Peptides and of the Concentration-Dependent Formation from Seed Crystals. Advanced Functional Materials. 25. 9. 1374–1384. 10.1002/adfm.201404136. 94001655 . 1616-301X.
73. Chen J, Zhu E, Liu J, Zhang S, Lin Z, Duan X, Heinz H, Huang Y, De Yoreo JJ . 54456982 . 6 . Building two-dimensional materials one row at a time: Avoiding the nucleation barrier . Science . 362 . 6419 . 1135–1139 . December 2018 . 30523105 . 10.1126/science.aau4146 . 2018Sci...362.1135C . free .
74. Swamy . Varghese . Gale . Julian D. . Transferable variable-charge interatomic potential for atomistic simulation of titanium oxides . Physical Review B . 62 . 9 . 1 August 2000 . 0163-1829 . 10.1103/physrevb.62.5406 . 5406–5412. 2000PhRvB..62.5406S .
75. Aguado . Andrés . Bernasconi . Leonardo . Madden . Paul A. . A transferable interatomic potential for MgO from ab initio molecular dynamics . Chemical Physics Letters . 356 . 5–6 . 2002 . 0009-2614 . 10.1016/s0009-2614(02)00326-3 . 437–444. 2002CPL...356..437A .
76. Web site: Interatomic Potentials Repository Project. U.S. Department of Commerce, National Institute of Standards and. Technology. www.ctcms.nist.gov.
77. Web site: Open Knowledgebase of Interatomic Models (OpenKIM).
78. Behler . Jörg . Csányi . Gábor . 2021-07-19 . Machine learning potentials for extended systems: a perspective . The European Physical Journal B . en . 94 . 7 . 142 . 10.1140/epjb/s10051-021-00156-1 . 2021EPJB...94..142B . 1434-6036. free .
79. Rosenbrock . Conrad W. . Gubaev . Konstantin . Shapeev . Alexander V. . Pártay . Livia B. . Bernstein . Noam . Csányi . Gábor . Hart . Gus L. W. . 2021-01-29 . Machine-learned interatomic potentials for alloys and alloy phase diagrams . npj Computational Materials . en . 7 . 1 . 24 . 10.1038/s41524-020-00477-2 . 2021npjCM...7...24R . 2057-3960. 1906.07816 .
80. Kocer. Ko. Behler. Emir. Tsz Wai. Jorg. Annual Review of Physical Chemistry. Neural Network Potentials: A Concise Overview of Methods. 2022. 73. 163–86. 10.1146/annurev-physchem-082720-034254 . 34982580 . 2107.03727 . 2022ARPC...73..163K . 235765258 .
81. Blank. TB. Brown. SD. Calhoun. Doren. DJ. AW. 1995. Neural network models of potential energy surfaces. The Journal of Chemical Physics. 103. 10. 4129–37. 10.1063/1.469597 . 1995JChPh.103.4129B .
82. Rosenbrock . Conrad W. . Gubaev . Konstantin . Shapeev . Alexander V. . Pártay . Livia B. . Bernstein . Noam . Csányi . Gábor . Hart . Gus L. W. . 2021-01-29 . Machine-learned interatomic potentials for alloys and alloy phase diagrams . npj Computational Materials . en . 7 . 1 . 24 . 10.1038/s41524-020-00477-2 . 2021npjCM...7...24R . 2057-3960. 1906.07816 .
83. Behler. J. Parrinello. M. Generalized neural-network representation of high-dimensional potential-energy surfaces. 2007. Physical Review Letters. 148. 14. 146401. 10.1103/PhysRevLett.98.146401. 17501293. 2007PhRvL..98n6401B.
84. Schutt. KT. Arbabzadah. F. Chmiela. S. Muller. KR. Tkatchenko. A. 2017. Quantum-chemical insights from deep tensor neural networks. Nature Communications. 8 . 13890 . 10.1038/ncomms13890 . 28067221 . 5228054 . 1609.08259 . 2017NatCo...813890S .
85. Nature Communications. Towards universal neural network potential for material discovery applicable to arbitrary combinations of 45 elements. Takamoto. So. Shinagawa. Chikashi. Motoki. Daisuke. Nakago. Kosuke. 13. May 30, 2022. 1 . 2991 . 10.1038/s41467-022-30687-9 . 35637178 . 9151783 . 2106.14583 . 2022NatCo..13.2991T .
86. Web site: Matlantis.
87. Anonymous . 2010-04-05 . Modeling sans electrons . Physics . en . 3 . 13 . s48 . 10.1103/PhysRevLett.104.136403. 20481899 . 2010PhRvL.104m6403B . 0910.1019 .
88. Bartók . Albert P. . Payne . Mike C. . Kondor . Risi . Csányi . Gábor . 2010-04-01 . Gaussian Approximation Potentials: The Accuracy of Quantum Mechanics, without the Electrons . Physical Review Letters . 104 . 13 . 136403 . 10.1103/PhysRevLett.104.136403. 20481899 . 2010PhRvL.104m6403B . 0910.1019 .
89. Bartók . Albert P. . De . Sandip . Poelking . Carl . Bernstein . Noam . Kermode . James R. . Csányi . Gábor . Ceriotti . Michele . December 2017 . Machine learning unifies the modeling of materials and molecules . Science Advances . en . 3 . 12 . e1701816 . 10.1126/sciadv.1701816 . 2375-2548 . 5729016 . 29242828. 1706.00179 . 2017SciA....3E1816B .
90. Bartók . Albert P. . Kondor . Risi . Csányi . Gábor . 2013-05-28 . On representing chemical environments . Physical Review B . 87 . 18 . 184115 . 10.1103/PhysRevB.87.184115. 1209.3140 . 2013PhRvB..87r4115B .
91. Book: Rasmussen . Carl Edward . Gaussian processes for machine learning . Williams . Christopher K. I. . 2008 . MIT Press . 978-0-262-18253-9 . 3. print . Adaptive computation and machine learning . Cambridge, Mass..
92. Deringer . Volker L. . Csányi . Gábor . 2017-03-03 . Machine learning based interatomic potential for amorphous carbon . Physical Review B . 95 . 9 . 094203 . 10.1103/PhysRevB.95.094203. 1611.03277 . 2017PhRvB..95i4203D .
93. Bartók . Albert P. . Kermode . James . Bernstein . Noam . Csányi . Gábor . 2018-12-14 . Machine Learning a General-Purpose Interatomic Potential for Silicon . Physical Review X . 8 . 4 . 041048 . 10.1103/PhysRevX.8.041048. 1805.01568 . 2018PhRvX...8d1048B .
94. Szlachta . Wojciech J. . Bartók . Albert P. . Csányi . Gábor . 2014-09-24 . Accuracy and transferability of Gaussian approximation potential models for tungsten . Physical Review B . 90 . 10 . 104108 . 10.1103/PhysRevB.90.104108. 2014PhRvB..90j4108S .
95. Mocanu . Felix C. . Konstantinou . Konstantinos . Lee . Tae Hoon . Bernstein . Noam . Deringer . Volker L. . Csányi . Gábor . Elliott . Stephen R. . 2018-09-27 . Modeling the Phase-Change Memory Material, Ge 2 Sb 2 Te 5, with a Machine-Learned Interatomic Potential . The Journal of Physical Chemistry B . en . 122 . 38 . 8998–9006 . 10.1021/acs.jpcb.8b06476 . 30173522 . 1520-6106.
96. Shenoy . Lakshmi . Woodgate . Christopher D. . Staunton . Julie B. . Bartók . Albert P. . Becquart . Charlotte S. . Domain . Christophe . Kermode . James R. . 2024-03-22 . Collinear-spin machine learned interatomic potential for $__\mathrm$ alloy . Physical Review Materials . 8 . 3 . 033804 . 10.1103/PhysRevMaterials.8.033804. 2309.08689 . }
97. Acevedo O, Jorgensen WL . Advances in quantum and molecular mechanical (QM/MM) simulations for organic and enzymatic reactions . Accounts of Chemical Research . 43 . 1 . 142–51 . January 2010 . 19728702 . 10.1021/ar900171c . 2880334 .