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 $\backslash textstyle\; V\_\backslash mathrm\{\}$ can

be written as

::$$

V_\mathrm{} = \sum_{i = 1}^N V_1(\vec r_i) + \sum_{i,j = 1}^N V_2(\vec r_i,\vec r_j) + \sum_{i,j,k = 1}^N V_3(\vec r_i,\vec r_j,\vec r_k) + \cdots

Here $\backslash textstyle\; V\_1$ is the one-body term, $\backslash textstyle\; V\_2$ the two-body term, $\backslash textstyle\; V\_3$ the

three body term, $\backslash textstyle\; N$ the number of atoms in the system,

$\backslash vec\; r\_i$ 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

and similarly for the three-body term

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 $\backslash textstyle\; r\_\{ij\}\; =\; |\backslash vec\; r\_i-\backslash vec\; r\_j|$

and angles between the bonds

(vectors to neighbours) $\backslash textstyle\; \backslash theta\_\{ijk\}$.

Then, in the absence of external forces, the general

form becomes

::$$

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 $\backslash textstyle\; V\_3$ the

interatomic distance $\backslash textstyle\; r\_\{jk\}$ is not needed

since the three terms $\backslash textstyle\; r\_\{ij\},r\_\{ik\},\backslash theta\_\{ijk\}$

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 $\backslash textstyle\; V(r)\; \backslash equiv\; 0$ above

some cutoff distance $\backslash textstyle\; r\_\backslash mathrm\{cut\}$,

the summing can be restricted to atoms within the cutoff

distance of each other. By also using a cellular method

for finding the neighbours, 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 $i$ one should take the three-dimensional

derivative (gradient) of the potential $V\_\backslash text\{tot\}$ with respect to the position of atom $i$:

::$$

\vec{F}_i = -\nabla_{\vec {r}_i} V_\mathrm{TOT}

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

$\backslash textstyle\; r\_\{ij\}$. However, for many-body

potentials (three-body, four-body, etc.) the differentiation

becomes considerably more complex

{{cite journal | last1=Beardmore | first1=Keith M. | last2=Grønbech-Jensen | first2=Niels | title=Direct simulation of ion-beam-induced stressing and amorphization of silicon | journal=Physical Review B | volume=60 | issue=18 | date=1 October 1999 | issn=0163-1829 | doi=10.1103/physrevb.60.12610 | pages=12610–12616|arxiv=cond-mat/9901319v2| bibcode=1999PhRvB..6012610B | s2cid=15494648 }}

{{cite journal | last1=Albe | first1=Karsten | last2=Nord | first2=J. | last3=Nordlund | first3=K. | title=Dynamic charge-transfer bond-order potential for gallium nitride | journal=Philosophical Magazine | volume=89 | issue=34–36 | year=2009 | issn=1478-6435 | doi=10.1080/14786430903313708 | pages=3477–3497| bibcode=2009PMag...89.3477A | s2cid=56072359 }}

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 $\backslash textstyle\; \backslash vec\{r\}\_\{i\}$

because of angular and other many-body terms, and hence contribute to the gradient

$\backslash textstyle\; \backslash nabla\_\{\backslash vec\{r\}\_\{k\}\}$.

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.{{cite journal | vauthors = Balamane H, Halicioglu T, Tiller WA | title = Comparative study of silicon empirical interatomic potentials | journal = Physical Review B | volume = 46 | issue = 4 | pages = 2250–2279 | date = July 1992 | pmid = 10003901 | doi = 10.1103/physrevb.46.2250 | bibcode = 1992PhRvB..46.2250B }}

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.{{cite journal | vauthors = Plimpton SJ, Thompson AP | title = Computational aspects of many-body potentials | journal = MRS Bull. | volume = 37 | pages = 513–521 | year = 2012| issue = 5 | doi = 10.1557/mrs.2012.96 | bibcode = 2012MRSBu..37..513P | s2cid = 138567968 }} 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.{{Cite journal | last=Shapeev | first=Alexander V. | s2cid=28970251|date=2016-09-13 | title=Moment Tensor Potentials: A Class of Systematically Improvable Interatomic Potentials|journal=Multiscale Modeling & Simulation | language=en | volume=14 | issue=3 | pages=1153–1173 | doi=10.1137/15M1054183 | issn=1540-3459 | arxiv=1512.06054}} 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.

The arguably simplest widely used interatomic interaction model is the Lennard-Jones potential

{{cite journal |last=Lennard-Jones |first=J. E. |year=1924 |title=On the Determination of Molecular Fields |journal=Proc. R. Soc. Lond. A |volume=106 |issue=738 |pages=463–477 |doi=10.1098/rspa.1924.0082 |bibcode = 1924RSPSA.106..463J |doi-access=free }}.{{Cite journal |last1=Lenhard |first1=Johannes |last2=Stephan |first2=Simon |last3=Hasse |first3=Hans |date=June 2024 |title=On the History of the Lennard-Jones Potential |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.202400115 |journal=Annalen der Physik |language=en |volume=536 |issue=6 |doi=10.1002/andp.202400115 |issn=0003-3804|url-access=subscription |doi-access=free }}

::$$

V_\mathrm{LJ}(r) = 4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6} \right]

where $\backslash textstyle\; \backslash varepsilon$ is the depth of the potential well

and $\backslash textstyle\; \backslash sigma$ is the distance at which the potential crosses zero.

The attractive term proportional to $\backslash textstyle\; 1/r^6$ in the potential comes from the scaling of van der Waals forces, while the $\backslash textstyle\; 1/r^\{12\}$ 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,{{Cite journal |last1=Stephan |first1=Simon |last2=Thol |first2=Monika |last3=Vrabec |first3=Jadran |last4=Hasse |first4=Hans |date=2019-10-28 |title=Thermophysical Properties of the Lennard-Jones Fluid: Database and Data Assessment |url=https://pubs.acs.org/doi/10.1021/acs.jcim.9b00620 |journal=Journal of Chemical Information and Modeling |language=en |volume=59 |issue=10 |pages=4248–4265 |doi=10.1021/acs.jcim.9b00620 |pmid=31609113 |s2cid=204545481 |issn=1549-9596}} 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.{{Cite journal |last1=Stephan |first1=Simon |last2=Horsch |first2=Martin T. |last3=Vrabec |first3=Jadran |last4=Hasse |first4=Hans |date=2019-07-03 |title=MolMod – an open access database of force fields for molecular simulations of fluids |url=https://www.tandfonline.com/doi/full/10.1080/08927022.2019.1601191 |journal=Molecular Simulation |language=en |volume=45 |issue=10 |pages=806–814 |doi=10.1080/08927022.2019.1601191 |s2cid=119199372 |issn=0892-7022|arxiv=1904.05206 }}

Another simple and widely used pair potential is the

Morse potential, which consists simply of a sum of two exponentials.

::$V\_\backslash mathrm\{M\}(r)\; =\; D\_e\; (\; e^\{-2a(r-r\_e)\}-2e^\{-a(r-r\_e)\}\; )$

Here $\backslash textstyle\; D\_e$ is the equilibrium bond energy and

$\backslash textstyle\; r\_e$ the bond distance. The Morse

potential has been applied to studies of molecular vibrations and solids,{{cite journal | last1=Girifalco | first1=L. A. | last2=Weizer | first2=V. G. | title=Application of the Morse Potential Function to Cubic Metals | journal=Physical Review | volume=114 | issue=3 | date=1 April 1959 | issn=0031-899X | doi=10.1103/physrev.114.687 | pages=687–690| bibcode=1959PhRv..114..687G | hdl=10338.dmlcz/103074 | hdl-access=free }} 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

.{{cite journal | last1=Feuston | first1=B. P. | last2=Garofalini | first2=S. H. | title=Empirical three-body potential for vitreous silica | journal=The Journal of Chemical Physics | volume=89 | issue=9 | year=1988 | issn=0021-9606 | doi=10.1063/1.455531 | pages=5818–5824| bibcode=1988JChPh..89.5818F }}

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.

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

: $V(r\_\{ij\})\; =\; \{\; 1\; \backslash over\; 4\; \backslash pi\; \backslash varepsilon\_0\}\; \{Z\_1Z\_2\; e^2\; \backslash over\; r\_\{ij\}\}\; \backslash varphi(r/a)$

Here, $\backslash varphi(r)\; \backslash to\; 1$ when $r\; \backslash to\; 0$. $Z\_1$ 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.J. F. Ziegler, J. P. Biersack, and U. Littmark. The Stopping and Range of Ions in Matter. Pergamon, New York, 1985.

and more accurate ones can be obtained from all-electron quantum chemistry calculations

{{cite journal | last1=Nordlund | first1=K. | last2=Runeberg | first2=N. | last3=Sundholm | first3=D. | title=Repulsive interatomic potentials calculated using Hartree-Fock and density-functional theory methods | journal=Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms | volume=132 | issue=1 | year=1997 | issn=0168-583X | doi=10.1016/s0168-583x(97)00447-3 | pages=45–54| bibcode=1997NIMPB.132...45N }}{{Cite journal |last=Nordlund |first=Kai |last2=Lehtola |first2=Susi |last3=Hobler |first3=Gerhard |date=2025-03-26 |title=Repulsive interatomic potentials calculated at three levels of theory |url=https://journals.aps.org/pra/abstract/10.1103/PhysRevA.111.032818 |journal=Physical Review A |volume=111 |issue=3 |pages=032818 |doi=10.1103/PhysRevA.111.032818|doi-access=free |arxiv=2501.06617 }}

In a comparative study of several quantum chemistry methods, it was shown that pair-specific "NLH" repulsive potentials with a simple three-exponential screening function are accurate to within ~2% above 30 eV, while the universal ZBL potential differs by ~5%–10% from the quantum chemical calculations above 100 eV. In binary collision approximation simulations this kind of potential can be used

to describe the nuclear stopping power.

The Stillinger-Weber potential{{cite journal | vauthors = Stillinger FH, Weber TA | title = Computer simulation of local order in condensed phases of silicon | journal = Physical Review B | volume = 31 | issue = 8 | pages = 5262–5271 | date = April 1985 | pmid = 9936488 | doi = 10.1103/physrevb.31.5262 | bibcode = 1985PhRvB..31.5262S }} is a potential that has a

two-body and three-body terms of the standard form

::$$

V_\mathrm{TOT} = \sum_{i,j}^N V_2(r_{ij}) + \sum_{i,j,k}^N 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 other

elements and compounds

{{cite journal | last=Ichimura | first=M. | title=Stillinger-Weber potentials for III–V compound semiconductors and their application to the critical thickness calculation for InAs/GaAs | journal=Physica Status Solidi A | volume=153 | issue=2 | date=16 February 1996 | issn=0031-8965 | doi=10.1002/pssa.2211530217 | pages=431–437| bibcode=1996PSSAR.153..431I }}

{{cite journal | last1 = Ohta | first1 = H. | last2 = Hamaguchi | first2 = S. | year = 2001 | title = Classical interatomic potentials for si-o-f and si-o-cl systems | journal = Journal of Chemical Physics | volume = 115 | issue = 14| pages = 6679–90 | doi = 10.1063/1.1400789 | bibcode = 2001JChPh.115.6679O | hdl = 2433/50272 | url = https://ir.library.osaka-u.ac.jp/repo/ouka/all/78492/JChemPhys_115_14_6679.pdf | hdl-access = free }}

and also formed the basis for other Si potentials.{{cite journal|last1=Bazant|first1=M. Z.|last2=Kaxiras|first2=E.|last3=Justo|first3=J. F.|s2cid=17860100|title=Environment-dependent interatomic potential for bulk silicon|journal=Phys. Rev. B|date=1997|volume=56|issue=14|page=8542|doi=10.1103/PhysRevB.56.8542|arxiv=cond-mat/9704137|bibcode=1997PhRvB..56.8542B}}

{{cite journal | last1=Justo | first1=João F. | last2=Bazant | first2=Martin Z. | last3=Kaxiras | first3=Efthimios | last4=Bulatov | first4=V. V. | last5=Yip | first5=Sidney | title=Interatomic potential for silicon defects and disordered phases | journal=Physical Review B | volume=58 | issue=5 | date=1 July 1998 | issn=0163-1829 | doi=10.1103/physrevb.58.2539 | pages=2539–2550| arxiv=cond-mat/9712058 | bibcode=1998PhRvB..58.2539J | s2cid=14585375 }}

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

::$V\_\backslash mathrm\{TOT\}\; =\; \backslash sum\_i^N\; F\_i\; \backslash left(\backslash sum\_\{j\}\; \backslash rho\; (r\_\{ij\})\; \backslash right)\; +\; \backslash frac\{1\}\{2\}\; \backslash sum\_\{i,\; j\}^N\; V\_2(r\_\{ij\})$

where $\backslash textstyle\; F\_i$ is a so-called embedding function

(not to be confused with the force $\backslash textstyle\; \backslash vec\; F\_i$) that is a function of the sum of the so-called electron density

$\backslash textstyle\; \backslash rho\; (r\_\{ij\})$. $\backslash textstyle\; V\_2$

is a pair potential that usually is purely repulsive. In the original

formulation {{cite journal |author3-link=Murray S. Daw|author1-link=Stephen M. Foiles|author2-link=Michael Baskes | vauthors = Foiles SM, Baskes MI, Daw MS | title = Embedded-atom-method functions for the fcc metals Cu, Ag, Au, Ni, Pd, Pt, and their alloys | journal = Physical Review B | volume = 33 | issue = 12 | pages = 7983–7991 | date = June 1986 | pmid = 9938188 | doi = 10.1103/physrevb.33.7983 | bibcode = 1986PhRvB..33.7983F }}{{cite journal | last1=Foiles | first1=S. M. | last2=Baskes | first2=M. I. | last3=Daw | first3=M. S. | title=Erratum: Embedded-atom-method functions for the fcc metals Cu, Ag, Au, Ni, Pd, Pt, and their alloys | journal=Physical Review B | volume=37 | issue=17 | date=15 June 1988 | issn=0163-1829 | doi=10.1103/physrevb.37.10378 | pages=10378| doi-access=free }} the electron

density function $\backslash textstyle\; \backslash rho\; (r\_\{ij\})$ 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.

.{{cite journal | last1=Puska | first1=M. J. | last2=Nieminen | first2=R. M. | last3=Manninen | first3=M. | title=Atoms embedded in an electron gas: Immersion energies | journal=Physical Review B | volume=24 | issue=6 | date=15 September 1981 | issn=0163-1829 | doi=10.1103/physrevb.24.3037 | pages=3037–3047| bibcode=1981PhRvB..24.3037P | url=https://aaltodoc.aalto.fi/handle/123456789/30795 }}

However, many other potentials used for metals share the same functional

form but motivate the terms differently, e.g. based

on tight-binding theory

{{cite journal | last1=Finnis | first1=M. W. | last2=Sinclair | first2=J. E. | title=A simple empirical N-body potential for transition metals | journal=Philosophical Magazine A | volume=50 | issue=1 | year=1984 | issn=0141-8610 | doi=10.1080/01418618408244210 | pages=45–55| bibcode=1984PMagA..50...45F }}{{cite journal | title=Erratum | journal=Philosophical Magazine A | volume=53 | issue=1 | year=1986 | issn=0141-8610 | doi=10.1080/01418618608242815 | pages=161| bibcode=1986PMagA..53..161. | doi-access=free }}

{{cite journal | vauthors = Cleri F, Rosato V | title = Tight-binding potentials for transition metals and alloys | journal = Physical Review B | volume = 48 | issue = 1 | pages = 22–33 | date = July 1993 | pmid = 10006745 | doi = 10.1103/physrevb.48.22 | bibcode = 1993PhRvB..48...22C }}

or other motivations

{{cite journal | last1=Kelchner | first1=Cynthia L. | last2=Halstead | first2=David M. | last3=Perkins | first3=Leslie S. | last4=Wallace | first4=Nora M. | last5=DePristo | first5=Andrew E. | title=Construction and evaluation of embedding functions | journal=Surface Science | volume=310 | issue=1–3 | year=1994 | issn=0039-6028 | doi=10.1016/0039-6028(94)91405-2 | pages=425–435| bibcode=1994SurSc.310..425K }}

{{cite journal | last1=Dudarev | first1=S L | last2=Derlet | first2=P M | title=A 'magnetic' interatomic potential for molecular dynamics simulations | journal=Journal of Physics: Condensed Matter | volume=17 | issue=44 | date=17 October 2005 | issn=0953-8984 | doi=10.1088/0953-8984/17/44/003 | pages=7097–7118| bibcode=2005JPCM...17.7097D | s2cid=123141962 }}

.{{cite journal | last1=Olsson | first1=Pär | last2=Wallenius | first2=Janne | last3=Domain | first3=Christophe | last4=Nordlund | first4=Kai | last5=Malerba | first5=Lorenzo | s2cid=16118006 | title=Two-band modeling of α-prime phase formation in Fe-Cr | journal=Physical Review B | volume=72 | issue=21 | date=21 December 2005 | issn=1098-0121 | doi=10.1103/physrevb.72.214119 | page=214119| bibcode=2005PhRvB..72u4119O }}

EAM-like potentials are usually implemented as numerical tables.

A collection of tables is available at the interatomic

potential repository at NIST [http://www.ctcms.nist.gov/potentials/]

Covalently bonded materials are often described by

bond order potentials, sometimes also called

Tersoff-like or Brenner-like potentials.

{{cite journal | vauthors = Tersoff J | title = New empirical approach for the structure and energy of covalent systems | journal = Physical Review B | volume = 37 | issue = 12 | pages = 6991–7000 | date = April 1988 | pmid = 9943969 | doi = 10.1103/PhysRevB.37.6991 | bibcode = 1988PhRvB..37.6991T }}

{{cite journal | vauthors = Brenner DW | title = Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films | journal = Physical Review B | volume = 42 | issue = 15 | pages = 9458–9471 | date = November 1990 | pmid = 9995183 | doi = 10.1103/PhysRevB.42.9458 | bibcode = 1990PhRvB..42.9458B }}

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

::$$

V_{ij}(r_{ij}) = V_\mathrm{repulsive}(r_{ij}) + b_{ijk} V_\mathrm{attractive}(r_{ij})

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 $b\_\{ijk\}$ term. If implemented without

an explicit angular dependence, these potentials

can be shown to be mathematically equivalent to

some varieties of EAM-like potentials

{{cite journal | vauthors = Brenner DW | title = Relationship between the embedded-atom method and Tersoff potentials | journal = Physical Review Letters | volume = 63 | issue = 9 | pages = 1022 | date = August 1989 | pmid = 10041250 | doi = 10.1103/PhysRevLett.63.1022 | bibcode = 1989PhRvL..63.1022B }}

{{cite journal|last1=Albe|first1=Karsten|last2=Nordlund|first2=Kai|last3=Averback|first3=Robert S.|title=Modeling the metal-semiconductor interaction: Analytical bond-order potential for platinum-carbon|journal=Physical Review B|volume=65|issue=19|pages=195124|year=2002|issn=0163-1829|doi=10.1103/PhysRevB.65.195124|bibcode=2002PhRvB..65s5124A}}

Thanks to this equivalence, the bond-order potential formalism has been implemented also for many metal-covalent mixed materials.{{cite journal|last1=de Brito Mota|first1=F.|last2=Justo|first2=J. F.|last3=Fazzio|first3=A.|title=Structural properties of amorphous silicon nitride|journal=Phys. Rev. B|date=1998|volume=58|issue=13|page=8323|doi=10.1103/PhysRevB.58.8323|bibcode=1998PhRvB..58.8323D}}

{{cite journal | last1=Juslin | first1=N. | last2=Erhart | first2=P. | last3=Träskelin | first3=P. | last4=Nord | first4=J. | last5=Henriksson | first5=K. O. E. | last6=Nordlund | first6=K. | last7=Salonen | first7=E. | last8=Albe | first8=K. | s2cid=8090449 | title=Analytical interatomic potential for modeling nonequilibrium processes in the W–C–H system | journal=Journal of Applied Physics | volume=98 | issue=12 | date=15 December 2005 | pages=123520–123520–12 | issn=0021-8979 | doi=10.1063/1.2149492 | bibcode=2005JAP....98l3520J }}

{{cite journal | last1=Erhart | first1=Paul | last2=Juslin | first2=Niklas | last3=Goy | first3=Oliver | last4=Nordlund | first4=Kai | last5=Müller | first5=Ralf | last6=Albe | first6=Karsten | title=Analytic bond-order potential for atomistic simulations of zinc oxide | journal=Journal of Physics: Condensed Matter | volume=18 | issue=29 | date=30 June 2006 | issn=0953-8984 | doi=10.1088/0953-8984/18/29/003 | pages=6585–6605| bibcode=2006JPCM...18.6585E | s2cid=38072718 }}

EAM potentials have also been extended to describe covalent bonding by adding angular-dependent terms to the electron density function $\backslash rho$, in what is called the modified embedded atom method (MEAM).{{cite journal | vauthors = Baskes MI | title = Application of the embedded-atom method to covalent materials: A semiempirical potential for silicon | journal = Physical Review Letters | volume = 59 | issue = 23 | pages = 2666–2669 | date = December 1987 | pmid = 10035617 | doi = 10.1103/PhysRevLett.59.2666 | bibcode = 1987PhRvL..59.2666B }}{{cite journal | vauthors = Baskes MI | title = Modified embedded-atom potentials for cubic materials and impurities | journal = Physical Review B | volume = 46 | issue = 5 | pages = 2727–2742 | date = August 1992 | pmid = 10003959 | doi = 10.1103/PhysRevB.46.2727 | bibcode = 1992PhRvB..46.2727B }}{{Cite journal|last1=Lee|first1=Byeong-Joo|last2=Baskes|first2=M. I.|date=2000-10-01|title=Second nearest-neighbor modified embedded-atom-method potential|journal=Physical Review B|volume=62|issue=13|pages=8564–8567|doi=10.1103/PhysRevB.62.8564|bibcode=2000PhRvB..62.8564L}}

{{main|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.{{cite journal|vauthors=Heinz H, Lin TJ, Mishra RK, Emami FS|date=February 2013|title=Thermodynamically consistent force fields for the assembly of inorganic, organic, and biological nanostructures: the INTERFACE force field|journal=Langmuir|volume=29|issue=6|pages=1754–65|doi=10.1021/la3038846|pmid=23276161}} 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,{{cite journal|vauthors=Heinz H, Ramezani-Dakhel H|date=January 2016|title=Simulations of inorganic-bioorganic interfaces to discover new materials: insights, comparisons to experiment, challenges, and opportunities|journal=Chemical Society Reviews|volume=45|issue=2|pages=412–48|doi=10.1039/c5cs00890e|pmid=26750724}}{{Cite journal|last1=Mishra|first1=Ratan K.|last2=Mohamed|first2=Aslam Kunhi|last3=Geissbühler|first3=David|last4=Manzano|first4=Hegoi|last5=Jamil|first5=Tariq|last6=Shahsavari|first6=Rouzbeh|last7=Kalinichev|first7=Andrey G.|last8=Galmarini|first8=Sandra|last9=Tao|first9=Lei|last10=Heinz|first10=Hendrik|last11=Pellenq|first11=Roland|date=December 2017|title=A force field database for cementitious materials including validations, applications and opportunities|journal=Cement and Concrete Research|language=en|volume=102|pages=68–89|doi=10.1016/j.cemconres.2017.09.003|url=https://www.dora.lib4ri.ch/empa/islandora/object/empa%3A15322|doi-access=free}} molecular fluids, and for biomacromolecules,{{cite journal | vauthors = Wang J, Wolf RM, Caldwell JW, Kollman PA, Case DA | title = Development and testing of a general amber force field | journal = Journal of Computational Chemistry | volume = 25 | issue = 9 | pages = 1157–74 | date = July 2004 | pmid = 15116359 | doi = 10.1002/jcc.20035 | s2cid = 18734898 | doi-access = free }} 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,{{cite journal | vauthors = Huang J, MacKerell AD | title = CHARMM36 all-atom additive protein force field: validation based on comparison to NMR data | journal = Journal of Computational Chemistry | volume = 34 | issue = 25 | pages = 2135–45 | date = September 2013 | pmid = 23832629 | doi = 10.1002/jcc.23354 | pmc = 3800559 }} and COMPASS) to explicitly reactive models with many adjustable fit parameters (e.g. ReaxFF) and machine learning models.

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$$V\_\backslash mathrm\{TOT\}\; =\; \backslash sum\_i^N\; E(\backslash mathbf\{q\}\_i)$$where $\backslash mathbf\{q\}\_i$ is a mathematical representation of the atomic environment surrounding the atom $i$, known as the descriptor.{{Cite journal|last1=Bartók|first1=Albert P.|last2=Kondor|first2=Risi|last3=Csányi|first3=Gábor|s2cid=118375156|date=2013-05-28|title=On representing chemical environments|journal=Physical Review B|language=en|volume=87|issue=18|pages=184115|doi=10.1103/PhysRevB.87.184115|issn=1098-0121|arxiv=1209.3140|bibcode=2013PhRvB..87r4115B}} $E$ 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.{{Cite journal|last1=Deringer|first1=Volker L.|last2=Csányi|first2=Gábor|s2cid=55190594|date=2017-03-03|title=Machine learning based interatomic potential for amorphous carbon|journal=Physical Review B|language=en|volume=95|issue=9|pages=094203|doi=10.1103/PhysRevB.95.094203|issn=2469-9950|arxiv=1611.03277|bibcode=2017PhRvB..95i4203D}} Potentials have been constructed using a variety of machine-learning methods, descriptors, and mappings, including neural networks,{{cite journal | vauthors = Behler J, Parrinello M | title = Generalized neural-network representation of high-dimensional potential-energy surfaces | journal = Physical Review Letters | volume = 98 | issue = 14 | pages = 146401 | date = April 2007 | pmid = 17501293 | doi = 10.1103/PhysRevLett.98.146401 | bibcode = 2007PhRvL..98n6401B }} Gaussian process regression,{{cite journal | vauthors = Bartók AP, Payne MC, Kondor R, Csányi G | s2cid = 15918457 | title = Gaussian approximation potentials: the accuracy of quantum mechanics, without the electrons | journal = Physical Review Letters | volume = 104 | issue = 13 | pages = 136403 | date = April 2010 | pmid = 20481899 | doi = 10.1103/PhysRevLett.104.136403 | arxiv = 0910.1019 | bibcode = 2010PhRvL.104m6403B }}{{Cite journal|last1=Dragoni|first1=Daniele|last2=Daff|first2=Thomas D.|last3=Csányi|first3=Gábor|last4=Marzari|first4=Nicola|date=2018-01-30|title=Achieving DFT accuracy with a machine-learning interatomic potential: Thermomechanics and defects in bcc ferromagnetic iron|url=https://link.aps.org/doi/10.1103/PhysRevMaterials.2.013808|journal=Physical Review Materials|volume=2|issue=1|pages=013808|doi=10.1103/PhysRevMaterials.2.013808|arxiv=1706.10229 |bibcode=2018PhRvM...2a3808D |hdl=10281/231112|s2cid=119252567 |hdl-access=free}} and linear regression.{{Cite journal|last1=Thompson|first1=A.P.|last2=Swiler|first2=L.P.|last3=Trott|first3=C.R.|last4=Foiles|first4=S.M.|last5=Tucker|first5=G.J.|date=2015-03-15|title=Spectral neighbor analysis method for automated generation of quantum-accurate interatomic potentials|journal=Journal of Computational Physics|language=en|volume=285|pages=316–330|doi=10.1016/j.jcp.2014.12.018|arxiv=1409.3880|bibcode=2015JCoPh.285..316T|doi-access=free}}

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,{{Cite journal|last1=Byggmästar|first1=J.|last2=Hamedani|first2=A.|last3=Nordlund|first3=K.|last4=Djurabekova|first4=F.|s2cid=201106123|date=2019-10-17|title=Machine-learning interatomic potential for radiation damage and defects in tungsten|journal=Physical Review B|volume=100|issue=14|pages=144105|doi=10.1103/PhysRevB.100.144105|arxiv=1908.07330|bibcode=2019PhRvB.100n4105B|hdl=10138/306660|hdl-access=free}} or to impose physical constraints on the predictions.{{cite journal | vauthors = Pun GP, Batra R, Ramprasad R, Mishin Y | title = Physically informed artificial neural networks for atomistic modeling of materials | journal = Nature Communications | volume = 10 | issue = 1 | pages = 2339 | date = May 2019 | pmid = 31138813 | pmc = 6538760 | doi = 10.1038/s41467-019-10343-5 | bibcode = 2019NatCo..10.2339P }}

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

.{{Cite journal|last1=Heinz|first1=Hendrik|last2=Vaia|first2=R. A.|last3=Farmer|first3=B. L.|last4=Naik|first4=R. R.|date=2008-10-09|title=Accurate Simulation of Surfaces and Interfaces of Face-Centered Cubic Metals Using 12−6 and 9−6 Lennard-Jones Potentials|journal=The Journal of Physical Chemistry C|volume=112|issue=44|pages=17281–17290|doi=10.1021/jp801931d|issn=1932-7447}}{{Cite journal|last1=Liu|first1=Juan|last2=Tennessen|first2=Emrys|last3=Miao|first3=Jianwei|last4=Huang|first4=Yu|last5=Rondinelli|first5=James M.|last6=Heinz|first6=Hendrik|date=2018-05-31|title=Understanding Chemical Bonding in Alloys and the Representation in Atomistic Simulations|journal=The Journal of Physical Chemistry C|volume=122|issue=26|pages=14996–15009|doi=10.1021/acs.jpcc.8b01891|s2cid=51855788 |issn=1932-7447}} Lennard-Jones potential can typically describe the lattice parameters, surface energies, and approximate mechanical properties.{{cite journal | vauthors = Nathanson M, Kanhaiya K, Pryor A, Miao J, Heinz H | title = Atomic-Scale Structure and Stress Release Mechanism in Core-Shell Nanoparticles | journal = ACS Nano | volume = 12 | issue = 12 | pages = 12296–12304 | date = December 2018 | pmid = 30457827 | doi = 10.1021/acsnano.8b06118 | s2cid = 53764446 }} 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.{{Cite journal|last1=Ruiz|first1=Victor G.|last2=Liu|first2=Wei|last3=Tkatchenko|first3=Alexandre|date=2016-01-15|title=Density-functional theory with screened van der Waals interactions applied to atomic and molecular adsorbates on close-packed and non-close-packed surfaces|journal=Physical Review B|volume=93|issue=3|page=035118|doi=10.1103/physrevb.93.035118|bibcode=2016PhRvB..93c5118R|hdl=11858/00-001M-0000-0029-3035-8|issn=2469-9950|hdl-access=free}}{{cite journal | vauthors = Ruiz VG, Liu W, Zojer E, Scheffler M, Tkatchenko A | title = Density-functional theory with screened van der Waals interactions for the modeling of hybrid inorganic-organic systems | journal = Physical Review Letters | volume = 108 | issue = 14 | pages = 146103 | date = April 2012 | pmid = 22540809 | doi = 10.1103/physrevlett.108.146103 | bibcode = 2012PhRvL.108n6103R | doi-access = free | hdl = 11858/00-001M-0000-000F-C6EA-3 | hdl-access = free }} 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%.

{{cite journal | last1=Ercolessi | first1=F | last2=Adams | first2=J. B | s2cid=18043298 | title=Interatomic Potentials from First-Principles Calculations: The Force-Matching Method | journal=Europhysics Letters | volume=26 | issue=8 | date=10 June 1994 | issn=0295-5075 | doi=10.1209/0295-5075/26/8/005 | pages=583–588| arxiv=cond-mat/9306054 | bibcode=1994EL.....26..583E }}

{{cite journal | last1=Mishin | first1=Y. | last2=Mehl | first2=M. J. | last3=Papaconstantopoulos | first3=D. A. | title=Embedded-atom potential forB2−NiAl | journal=Physical Review B | volume=65 | issue=22 | date=12 June 2002 | issn=0163-1829 | doi=10.1103/physrevb.65.224114 | page=224114| bibcode=2002PhRvB..65v4114M }}

{{cite journal | last1=Beardmore | first1=Keith | last2=Smith | first2=Roger | title=Empirical potentials for C-Si-H systems with application to C₆₀ interactions with Si crystal surfaces | journal=Philosophical Magazine A | volume=74 | issue=6 | year=1996 | issn=0141-8610 | doi=10.1080/01418619608240734 | pages=1439–1466| bibcode=1996PMagA..74.1439B }}

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 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.{{Cite journal|last1=Mishra|first1=Ratan K.|last2=Flatt|first2=Robert J.|last3=Heinz|first3=Hendrik|date=2013-04-19|title=Force Field for Tricalcium Silicate and Insight into Nanoscale Properties: Cleavage, Initial Hydration, and Adsorption of Organic Molecules|journal=The Journal of Physical Chemistry C|volume=117|issue=20|pages=10417–10432|doi=10.1021/jp312815g|issn=1932-7447}}{{Cite journal|last1=Ramezani-Dakhel|first1=Hadi|last2=Ruan|first2=Lingyan|last3=Huang|first3=Yu|last4=Heinz|first4=Hendrik|date=2015-01-21|title=Molecular Mechanism of Specific Recognition of Cubic Pt Nanocrystals by Peptides and of the Concentration-Dependent Formation from Seed Crystals|journal=Advanced Functional Materials|volume=25|issue=9|pages=1374–1384|doi=10.1002/adfm.201404136|s2cid=94001655 |issn=1616-301X}}{{cite journal | vauthors = Chen J, Zhu E, Liu J, Zhang S, Lin Z, Duan X, Heinz H, Huang Y, De Yoreo JJ | s2cid = 54456982 | display-authors = 6 | title = Building two-dimensional materials one row at a time: Avoiding the nucleation barrier | journal = Science | volume = 362 | issue = 6419 | pages = 1135–1139 | date = December 2018 | pmid = 30523105 | doi = 10.1126/science.aau4146 | bibcode = 2018Sci...362.1135C | doi-access = free }}{{cite journal | last1=Swamy | first1=Varghese | last2=Gale | first2=Julian D. | title=Transferable variable-charge interatomic potential for atomistic simulation of titanium oxides | journal=Physical Review B | volume=62 | issue=9 | date=1 August 2000 | issn=0163-1829 | doi=10.1103/physrevb.62.5406 | pages=5406–5412| bibcode=2000PhRvB..62.5406S }}{{cite journal | last1=Aguado | first1=Andrés | last2=Bernasconi | first2=Leonardo | last3=Madden | first3=Paul A. | title=A transferable interatomic potential for MgO from ab initio molecular dynamics | journal=Chemical Physics Letters | volume=356 | issue=5–6 | year=2002 | issn=0009-2614 | doi=10.1016/s0009-2614(02)00326-3 | pages=437–444| bibcode=2002CPL...356..437A }}). 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.{{cite web|url=http://www.ctcms.nist.gov/potentials/|title=Interatomic Potentials Repository Project|first=U.S. Department of Commerce, National Institute of Standards and|last=Technology|website=www.ctcms.nist.gov}} The OpenKIM {{cite web|url=https://openkim.org/|title=Open Knowledgebase of Interatomic Models (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.

{{Main|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){{Cite journal |last1=Behler |first1=Jörg |last2=Csányi |first2=Gábor |date=2021-07-19 |title=Machine learning potentials for extended systems: a perspective |journal=The European Physical Journal B |language=en |volume=94 |issue=7 |pages=142 |doi=10.1140/epjb/s10051-021-00156-1 |bibcode=2021EPJB...94..142B |issn=1434-6036|doi-access=free }} or as 'machine-learned interatomic potentials' (MLIPs).{{Cite journal |last1=Rosenbrock |first1=Conrad W. |last2=Gubaev |first2=Konstantin |last3=Shapeev |first3=Alexander V. |last4=Pártay |first4=Livia B. |last5=Bernstein |first5=Noam |last6=Csányi |first6=Gábor |last7=Hart |first7=Gus L. W. |date=2021-01-29 |title=Machine-learned interatomic potentials for alloys and alloy phase diagrams |url=https://www.nature.com/articles/s41524-020-00477-2 |journal=npj Computational Materials |language=en |volume=7 |issue=1 |page=24 |doi=10.1038/s41524-020-00477-2 |bibcode=2021npjCM...7...24R |issn=2057-3960|arxiv=1906.07816 }} 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, increasing the use of machine learning in the field.{{cite journal|last1=Kocer|last2=Ko|last3=Behler|first1=Emir|first2=Tsz Wai|first3=Jorg|journal=Annual Review of Physical Chemistry|title=Neural Network Potentials: A Concise Overview of Methods|date=2022|volume=73|pages=163–86|doi=10.1146/annurev-physchem-082720-034254 |pmid=34982580 |arxiv=2107.03727 |bibcode=2022ARPC...73..163K |s2cid=235765258 }}{{cite journal|last1=Blank|first1=TB|last2=Brown|first2=SD|last3=Calhoun|last4=Doren|first4=DJ|first3=AW|date=1995|title=Neural network models of potential energy surfaces|journal=The Journal of Chemical Physics|volume=103|number=10|pages=4129–37|doi=10.1063/1.469597 |bibcode=1995JChPh.103.4129B }}

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.{{cite journal|last1=Behler|first1=J|last2=Parrinello|first2=M|title=Generalized neural-network representation of high-dimensional potential-energy surfaces|date=2007|journal=Physical Review Letters|volume=148|issue=14|page=146401|doi=10.1103/PhysRevLett.98.146401|pmid=17501293|bibcode=2007PhRvL..98n6401B}}

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.{{cite journal|last1=Schutt|first1=KT|last2=Arbabzadah|first2=F|last3=Chmiela|first3=S|last4=Muller|first4=KR|last5=Tkatchenko|first5=A|date=2017|title=Quantum-chemical insights from deep tensor neural networks|journal=Nature Communications|volume=8 |page=13890 |doi=10.1038/ncomms13890 |pmid=28067221 |pmc=5228054 |arxiv=1609.08259 |bibcode=2017NatCo...813890S }}{{cite journal|journal=Nature Communications|title=Towards universal neural network potential for material discovery applicable to arbitrary combinations of 45 elements|last1=Takamoto|first1=So|last2=Shinagawa|first2=Chikashi|last3=Motoki|first3=Daisuke|last4=Nakago|first4=Kosuke|volume=13|date=May 30, 2022|issue=1 |page=2991 |doi=10.1038/s41467-022-30687-9 |pmid=35637178 |pmc=9151783 |arxiv=2106.14583 |bibcode=2022NatCo..13.2991T }}

Another class of machine-learned interatomic potential is the Gaussian approximation potential (GAP),{{Cite journal |last=Anonymous |date=2010-04-05 |title=Modeling sans electrons |url=https://physics.aps.org/articles/v3/s48 |journal=Physics |language=en |volume=3 |issue=13 |pages=s48 |doi=10.1103/PhysRevLett.104.136403|pmid=20481899 |bibcode=2010PhRvL.104m6403B |arxiv=0910.1019 }}{{Cite journal |last1=Bartók |first1=Albert P. |last2=Payne |first2=Mike C. |last3=Kondor |first3=Risi |last4=Csányi |first4=Gábor |date=2010-04-01 |title=Gaussian Approximation Potentials: The Accuracy of Quantum Mechanics, without the Electrons |url=https://link.aps.org/doi/10.1103/PhysRevLett.104.136403 |journal=Physical Review Letters |volume=104 |issue=13 |pages=136403 |doi=10.1103/PhysRevLett.104.136403|pmid=20481899 |bibcode=2010PhRvL.104m6403B |arxiv=0910.1019 }}{{Cite journal |last1=Bartók |first1=Albert P. |last2=De |first2=Sandip |last3=Poelking |first3=Carl |last4=Bernstein |first4=Noam |last5=Kermode |first5=James R. |last6=Csányi |first6=Gábor |last7=Ceriotti |first7=Michele |date=December 2017 |title=Machine learning unifies the modeling of materials and molecules |journal=Science Advances |language=en |volume=3 |issue=12 |pages=e1701816 |doi=10.1126/sciadv.1701816 |issn=2375-2548 |pmc=5729016 |pmid=29242828|arxiv=1706.00179 |bibcode=2017SciA....3E1816B }} which combines compact descriptors of local atomic environments{{Cite journal |last1=Bartók |first1=Albert P. |last2=Kondor |first2=Risi |last3=Csányi |first3=Gábor |date=2013-05-28 |title=On representing chemical environments |url=https://link.aps.org/doi/10.1103/PhysRevB.87.184115 |journal=Physical Review B |volume=87 |issue=18 |pages=184115 |doi=10.1103/PhysRevB.87.184115|arxiv=1209.3140 |bibcode=2013PhRvB..87r4115B }} with Gaussian process regression{{Cite book |last1=Rasmussen |first1=Carl Edward |title=Gaussian processes for machine learning |last2=Williams |first2=Christopher K. I. |date=2008 |publisher=MIT Press |isbn=978-0-262-18253-9 |edition=3. print |series=Adaptive computation and machine learning |location=Cambridge, Mass.}} 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{{cite journal | last1=Rowe | first1=Patrick | last2=Deringer | first2=Volker L. | last3=Gasparotto | first3=Piero | last4=Csányi | first4=Gábor | last5=Michaelides | first5=Angelos | title=An accurate and transferable machine learning potential for carbon | journal=The Journal of Chemical Physics | volume=153 | issue=3 | date=2020-07-21 | page=034702 | issn=0021-9606 | doi=10.1063/5.0005084 | doi-access=free | pmid=32716159 | arxiv=2006.13655 }}{{Cite journal |last1=Deringer |first1=Volker L. |last2=Csányi |first2=Gábor |date=2017-03-03 |title=Machine learning based interatomic potential for amorphous carbon |url=https://link.aps.org/doi/10.1103/PhysRevB.95.094203 |journal=Physical Review B |volume=95 |issue=9 |pages=094203 |doi=10.1103/PhysRevB.95.094203|arxiv=1611.03277 |bibcode=2017PhRvB..95i4203D }} Silicon,{{Cite journal |last1=Bartók |first1=Albert P. |last2=Kermode |first2=James |last3=Bernstein |first3=Noam |last4=Csányi |first4=Gábor |date=2018-12-14 |title=Machine Learning a General-Purpose Interatomic Potential for Silicon |url=https://link.aps.org/doi/10.1103/PhysRevX.8.041048 |journal=Physical Review X |volume=8 |issue=4 |pages=041048 |doi=10.1103/PhysRevX.8.041048|arxiv=1805.01568 |bibcode=2018PhRvX...8d1048B }} and Tungsten,{{Cite journal |last1=Szlachta |first1=Wojciech J. |last2=Bartók |first2=Albert P. |last3=Csányi |first3=Gábor |date=2014-09-24 |title=Accuracy and transferability of Gaussian approximation potential models for tungsten |url=https://link.aps.org/doi/10.1103/PhysRevB.90.104108 |journal=Physical Review B |volume=90 |issue=10 |pages=104108 |doi=10.1103/PhysRevB.90.104108|bibcode=2014PhRvB..90j4108S }} as well as for multicomponent systems such as Ge₂Sb₂Te₅{{Cite journal |last1=Mocanu |first1=Felix C. |last2=Konstantinou |first2=Konstantinos |last3=Lee |first3=Tae Hoon |last4=Bernstein |first4=Noam |last5=Deringer |first5=Volker L. |last6=Csányi |first6=Gábor |last7=Elliott |first7=Stephen R. |date=2018-09-27 |title=Modeling the Phase-Change Memory Material, Ge 2 Sb 2 Te 5 , with a Machine-Learned Interatomic Potential |url=https://pubs.acs.org/doi/10.1021/acs.jpcb.8b06476 |journal=The Journal of Physical Chemistry B |language=en |volume=122 |issue=38 |pages=8998–9006 |doi=10.1021/acs.jpcb.8b06476 |pmid=30173522 |issn=1520-6106|url-access=subscription }} and austenitic stainless steel, Fe₇Cr₂Ni.{{Cite journal |last1=Shenoy |first1=Lakshmi |last2=Woodgate |first2=Christopher D. |last3=Staunton |first3=Julie B. |last4=Bartók |first4=Albert P. |last5=Becquart |first5=Charlotte S. |last6=Domain |first6=Christophe |last7=Kermode |first7=James R. |date=2024-03-22 |title=Collinear-spin machine learned interatomic potential for ${\mathrm{Fe}}_{7}{\mathrm{Cr}}_{2}\mathrm{Ni}$ alloy |url=https://link.aps.org/doi/10.1103/PhysRevMaterials.8.033804 |journal=Physical Review Materials |volume=8 |issue=3 |pages=033804 |doi=10.1103/PhysRevMaterials.8.033804|arxiv=2309.08689 }}

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.{{cite journal | vauthors = Acevedo O, Jorgensen WL | title = Advances in quantum and molecular mechanical (QM/MM) simulations for organic and enzymatic reactions | journal = Accounts of Chemical Research | volume = 43 | issue = 1 | pages = 142–51 | date = January 2010 | pmid = 19728702 | doi = 10.1021/ar900171c | pmc = 2880334 }}

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)

{{Reflist|colwidth=30em}}

• [http://www.ctcms.nist.gov/potentials/ NIST interatomic potential repository]
• [https://www.ctcms.nist.gov/~knc6/periodic.html NIST JARVIS-FF]
• [https://openkim.org/ Open Knowledgebase of Interatomic Models (OpenKIM)]

Category:Condensed matter physics

Category:Computational physics

Category:Materials science

Category:Quantum mechanical potentials