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Interatomic potential

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Interatomic potentials are mathematical <a href="Function_(mathematics)" title="Function (mathematics)">functions</a> to calculate the <a href="Potential_energy" title="Potential energy">potential energy</a> of a system of <a href="Atom" title="Atom">atoms</a> with given positions in space.<a href="#cite_note-AllenTildesley-1">[1]</a><a href="#cite_note-2">[2]</a><a href="#cite_note-Lesar-3">[3]</a><a href="#cite_note-Brenner2000-4">[4]</a> Interatomic potentials are widely used as the physical basis of <a href="Molecular_mechanics" title="Molecular mechanics">molecular mechanics</a> and <a href="Molecular_dynamics" title="Molecular dynamics">molecular dynamics</a> simulations in <a href="Computational_chemistry" title="Computational chemistry">computational chemistry</a>, <a href="Computational_physics" title="Computational physics">computational physics</a> and <a href="Computational_materials_science" title="Computational materials science">computational materials science</a> 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, <a href="Adsorption" title="Adsorption">adsorption</a>, <a href="Cohesion_(chemistry)" title="Cohesion (chemistry)">cohesion</a>, <a href="Thermal_expansion" title="Thermal expansion">thermal expansion</a>, and <a href="Elasticity_theory" class="mw-redirect" title="Elasticity theory">elastic</a> and <a href="Plasticity_(physics)" title="Plasticity (physics)">plastic</a> material behavior, as well as <a href="Chemical_reaction" title="Chemical reaction">chemical reactions</a>.<a href="#cite_note-AshcroftMermin-5">[5]</a><a href="#cite_note-Kittel-6">[6]</a><a href="#cite_note-Daw93-7">[7]</a><a href="#cite_note-Tersoff_pp._6991–7000-8">[8]</a><a href="#cite_note-FINNIS_2007_pp._133–153-9">[9]</a><a href="#cite_note-Sin12-10">[10]</a><a href="#cite_note-ReferenceA-11">[11]</a>

<div class="mw-heading mw-heading2"><h2 id="Functional_form">Functional form</h2></div>

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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle V_{\mathrm {} }}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle V_{\mathrm {} }}</annotation>

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</math><img src="./7715bef0e59e22997dfa8671c45f8ad3c906f26c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle \textstyle V_{\mathrm {} }}" loading="lazy"> can

be written as <a href="#cite_note-Lesar-3">[3]</a>

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<annotation encoding="application/x-tex">{\displaystyle 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 }</annotation>

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</math><img src="./08828c70390d2afbe15e30d05b7632f797f509fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:57.97ex; height:7.676ex;" alt="{\displaystyle 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 }" loading="lazy"></dd></dl></dd></dl>

Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle V_{1}}">

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</math><img src="./36388f7d4841ad03b7e6edd475fa7233d52e7b8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle \textstyle V_{1}}" loading="lazy"> is the one-body term, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle V_{2}}">

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</math><img src="./b6edd4f7d71777667a670c1d4efbcb59983a36dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle \textstyle V_{2}}" loading="lazy"> the two-body term, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle V_{3}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle V_{3}}</annotation>

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</math><img src="./ed398c6cdd2dd4ebc3f01bc7bdbebc52a223afe9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle \textstyle V_{3}}" loading="lazy"> the

three body term, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle N}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle N}</annotation>

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</math><img src="./a01b816d7d3e71a63798ca1ec640fac754da49dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle \textstyle N}" loading="lazy"> the number of atoms in the system,

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</math><img src="./de93eb4c8bca39012a94e9809c45d7fd677bf975.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.023ex; height:2.676ex;" alt="{\displaystyle {\vec {r}}_{i}}" loading="lazy"> the position of atom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">, etc. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

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</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> 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.<a href="#cite_note-Lesar-3">[3]</a> Alternatively,

the summation of the pair term can be restricted to cases <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle i<j}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle i<j}</annotation>

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</math><img src="./d6e1d1673f74aa85b4a938b3a7138b54c47ecce2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.859ex; height:2.509ex;" alt="{\displaystyle \textstyle i<j}" loading="lazy">

and similarly for the three-body term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle i<j<k}">

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</math><img src="./ad7ae47f7c7125a3c9bf44e253d2b4e5fa344898.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.169ex; height:2.509ex;" alt="{\displaystyle \textstyle i<j<k}" loading="lazy">, if

the potential form is such that it is symmetric with respect to exchange

of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

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</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle r_{ij}=|{\vec {r}}_{i}-{\vec {r}}_{j}|}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle r_{ij}=|{\vec {r}}_{i}-{\vec {r}}_{j}|}</annotation>

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</math><img src="./0dd98f46d07ed43c072f54daeecfcaa47dd267e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.914ex; height:3.009ex;" alt="{\displaystyle \textstyle r_{ij}=|{\vec {r}}_{i}-{\vec {r}}_{j}|}" loading="lazy">

and angles between the bonds

(vectors to neighbours) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \theta _{ijk}}">

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</math><img src="./3c9d721fef93e9f60d7b99f28dfb7db5d8ad8658.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.424ex; height:2.843ex;" alt="{\displaystyle \textstyle \theta _{ijk}}" loading="lazy">.

Then, in the absence of external forces, the general

form becomes

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<annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {TOT} }=\sum _{i,j}^{N}V_{2}(r_{ij})+\sum _{i,j,k}^{N}V_{3}(r_{ij},r_{ik},\theta _{ijk})+\cdots }</annotation>

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</math><img src="./590216589bc559f922b638a21c87f69ece65cd67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:45.912ex; height:7.676ex;" alt="{\displaystyle V_{\mathrm {TOT} }=\sum _{i,j}^{N}V_{2}(r_{ij})+\sum _{i,j,k}^{N}V_{3}(r_{ij},r_{ik},\theta _{ijk})+\cdots }" loading="lazy"></dd></dl></dd></dl>

In the three-body term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle V_{3}}">

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interatomic distance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle r_{jk}}">

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since the three terms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle r_{ij},r_{ik},\theta _{ijk}}">

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are sufficient to give the relative positions of three atoms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j,k}">

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</math><img src="./9550e66fe7e601c4f58bbc9c19ba226301149cde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.04ex; height:2.509ex;" alt="{\displaystyle i,j,k}" loading="lazy"> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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However, if the range of the interatomic potential is finite,

i.e. the potentials <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle V(r)\equiv 0}">

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</math><img src="./23fae8e130d728b128cd20cce8dbe729478b689b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.906ex; height:2.843ex;" alt="{\displaystyle \textstyle V(r)\equiv 0}" loading="lazy"> above

some cutoff distance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle r_{\mathrm {cut} }}">

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the summing can be restricted to atoms within the cutoff

distance of each other. By also using a cellular method

for finding the neighbours,<a href="#cite_note-AllenTildesley-1">[1]</a> the MD algorithm can be

an <a href="Big_O_notation" title="Big O notation">O(N)</a> algorithm. Potentials with an infinite

range can be summed up efficiently by <a href="Ewald_summation" title="Ewald summation">Ewald summation</a>

and its further developments.

<div class="mw-heading mw-heading2"><h2 id="Force_calculation">Force calculation</h2></div>

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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> one should take the three-dimensional

derivative (gradient) of the potential <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\text{tot}}}">

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</math><img src="./a3479cd7add66e4a6062b23315d1e5aae9d6d4d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.689ex; height:2.509ex;" alt="{\displaystyle V_{\text{tot}}}" loading="lazy"> with respect to the position of atom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle {\vec {F}}_{i}=-\nabla _{{\vec {r}}_{i}}V_{\mathrm {TOT} }}</annotation>

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</math><img src="./eddd4ed3b49f837f63de8096f9867bede1143a98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:16.412ex; height:3.843ex;" alt="{\displaystyle {\vec {F}}_{i}=-\nabla _{{\vec {r}}_{i}}V_{\mathrm {TOT} }}" loading="lazy"></dd></dl></dd></dl>

For two-body potentials this gradient reduces, thanks to the

symmetry with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ij}">

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<annotation encoding="application/x-tex">{\displaystyle ij}</annotation>

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</math><img src="./53fcc7b57da64979c370eb150eb5a61a625a08e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.761ex; height:2.509ex;" alt="{\displaystyle ij}" loading="lazy"> in the potential form, to straightforward

differentiation with respect to the interatomic distances

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle r_{ij}}</annotation>

</semantics>

</math><img src="./b12dc7130270f9fe8100538cba685f1e7d4e151f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.526ex; height:2.343ex;" alt="{\displaystyle \textstyle r_{ij}}" loading="lazy">. However, for many-body

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

becomes considerably more complex

since the potential may not be any longer symmetric with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ij}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ij}</annotation>

</semantics>

</math><img src="./53fcc7b57da64979c370eb150eb5a61a625a08e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.761ex; height:2.509ex;" alt="{\displaystyle ij}" loading="lazy"> exchange.

In other words, also the energy

of atoms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> that are not direct neighbours of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> can depend on the position <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\vec {r}}_{i}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle {\vec {r}}_{i}}</annotation>

</semantics>

</math><img src="./1c8ded929337e924e705b7ccc1b512cd961d5ab8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.023ex; height:2.676ex;" alt="{\displaystyle \textstyle {\vec {r}}_{i}}" loading="lazy">

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

<msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \nabla _{{\vec {r}}_{k}}}</annotation>

</semantics>

</math><img src="./b2c0873c00254ce0116f42587960aa119a45a470.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.93ex; height:3.176ex;" alt="{\displaystyle \textstyle \nabla _{{\vec {r}}_{k}}}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Classes_of_interatomic_potentials">Classes of interatomic potentials</h2></div>

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.<a href="#cite_note-Bal92-14">[14]</a>

The true interatomic interactions

are <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanical</a> in nature, and there is no known

way in which the true interactions described by

the <a href="Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> or <a href="Dirac_equation" title="Dirac equation">Dirac equation</a> for

all electrons and nuclei could be cast into an analytical

functional form. Hence all analytical interatomic

potentials are by necessity <a href="Approximation" title="Approximation">approximations</a>.

Over time interatomic potentials have largely grown more complex and more accurate, although this is not strictly true.<a href="#cite_note-Plimpton2012-15">[15]</a> 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.<a href="#cite_note-Shapeev2016-16">[16]</a> 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.

<div class="mw-heading mw-heading3"><h3 id="Parametric_potentials">Parametric potentials</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Pair_potentials"><a href="Pair_potential" title="Pair potential">Pair potentials</a></h4></div>

The arguably simplest widely used interatomic interaction model is the <a href="Lennard-Jones_potential" title="Lennard-Jones potential">Lennard-Jones potential</a>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {LJ} }(r)=4\varepsilon \left[\left({\frac {\sigma }{r}}\right)^{12}-\left({\frac {\sigma }{r}}\right)^{6}\right]}</annotation>

</semantics>

</math><img src="./e90b275eab3b2b359da8e002162965157b0de494.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.158ex; height:6.176ex;" alt="{\displaystyle V_{\mathrm {LJ} }(r)=4\varepsilon \left[\left({\frac {\sigma }{r}}\right)^{12}-\left({\frac {\sigma }{r}}\right)^{6}\right]}" loading="lazy"></dd></dl></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \varepsilon }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \varepsilon }</annotation>

</semantics>

</math><img src="./7133ce73ec025b79174c7655031030387cc34b06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \textstyle \varepsilon }" loading="lazy"> is the depth of the <a href="Potential_well" title="Potential well">potential well</a>

and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \sigma }</annotation>

</semantics>

</math><img src="./9a5e2f2768ed33d5a464e9171f7915ed32f9dab5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \textstyle \sigma }" loading="lazy"> is the distance at which the potential crosses zero.

The attractive term proportional to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle 1/r^{6}}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle 1/r^{6}}</annotation>

</semantics>

</math><img src="./a3553378b1c8cb1113d3ccc213a2f6c005b81bc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.428ex; height:3.009ex;" alt="{\displaystyle \textstyle 1/r^{6}}" loading="lazy"> in the potential comes from the scaling of <a href="Van_der_Waals_forces" class="mw-redirect" title="Van der Waals forces">van der Waals forces</a>, while the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle 1/r^{12}}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle 1/r^{12}}</annotation>

</semantics>

</math><img src="./d2ef21636d314109244948bb08cd0c276179bbd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.25ex; height:3.009ex;" alt="{\displaystyle \textstyle 1/r^{12}}" loading="lazy"> repulsive term is much more approximate (conveniently the square of the attractive term).<a href="#cite_note-Kittel-6">[6]</a> On its own, this potential is quantitatively accurate only for noble gases and has been extensively studied in the past decades,<a href="#cite_note-19">[19]</a> but is also widely used for qualitative studies and in systems where dipole interactions are significant, particularly in <a href="Force_field_(chemistry)" title="Force field (chemistry)">chemistry force fields</a> to describe intermolecular interactions - especially in fluids.<a href="#cite_note-tandfonline.com-20">[20]</a>

Another simple and widely used pair potential is the

<a href="Morse_potential" title="Morse potential">Morse potential</a>, which consists simply of a sum of two exponentials.

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {M} }(r)=D_{e}(e^{-2a(r-r_{e})}-2e^{-a(r-r_{e})})}</annotation>

</semantics>

</math><img src="./f6758a438d41e4482325b5e1e256c42bf3f5532d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.19ex; height:3.343ex;" alt="{\displaystyle V_{\mathrm {M} }(r)=D_{e}(e^{-2a(r-r_{e})}-2e^{-a(r-r_{e})})}" loading="lazy"></dd></dl></dd></dl>

Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle D_{e}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle D_{e}}</annotation>

</semantics>

</math><img src="./3f4a4d660dcd7f920106b186dcfaeadebb1a232d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.923ex; height:2.509ex;" alt="{\displaystyle \textstyle D_{e}}" loading="lazy"> is the equilibrium bond energy and

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle r_{e}}</annotation>

</semantics>

</math><img src="./09611d01a31b52bab0d2248c822aa57570571a54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.047ex; height:2.009ex;" alt="{\displaystyle \textstyle r_{e}}" loading="lazy"> the bond distance. The Morse

potential has been applied to studies of molecular vibrations and solids,<a href="#cite_note-Girifalco_Weizer_pp._687–690-21">[21]</a> 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

<a href="Buckingham_potential" title="Buckingham potential">Buckingham pair potential</a>, and a long-range <a href="Coulomb_potential" class="mw-redirect" title="Coulomb potential">Coulomb potential</a>

giving the ionic interactions between the ions forming the material. The short-range

term for ionic materials can also be of many-body character

.<a href="#cite_note-Feuston_Garofalini_1988_pp._5818–5824-22">[22]</a>

Pair potentials have some inherent limitations, such as the inability

to describe all 3 <a href="Elasticity_(physics)" title="Elasticity (physics)">elastic constants</a> of

cubic metals or correctly describe both cohesive energy and vacancy formation energy.<a href="#cite_note-Daw93-7">[7]</a> Therefore, quantitative <a href="Molecular_dynamics" title="Molecular dynamics">molecular dynamics</a> simulations

are carried out with various of many-body potentials.

<div class="mw-heading mw-heading5"><h5 id="Repulsive_potentials">Repulsive potentials</h5></div>

For very short interatomic separations, important in <a href="Radiation_material_science" title="Radiation material science">radiation material science</a>,

the interactions can be described quite accurately with screened <a href="Coulomb_potential" class="mw-redirect" title="Coulomb potential">Coulomb potentials</a> which have the general form

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V(r_{ij})={1 \over 4\pi \varepsilon _{0}}{Z_{1}Z_{2}e^{2} \over r_{ij}}\varphi (r/a)}</annotation>

</semantics>

</math><img src="./3e529ce71fcd815013526bcf2b394c10cdb130a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.717ex; height:6.343ex;" alt="{\displaystyle V(r_{ij})={1 \over 4\pi \varepsilon _{0}}{Z_{1}Z_{2}e^{2} \over r_{ij}}\varphi (r/a)}" loading="lazy"></dd></dl>

Here, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (r)\to 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (r)\to 1}</annotation>

</semantics>

</math><img src="./fe6a11342decd1667c460f9d4ad0fd6f83211169.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.155ex; height:2.843ex;" alt="{\displaystyle \varphi (r)\to 1}" loading="lazy"> when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\to 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r\to 0}</annotation>

</semantics>

</math><img src="./779a77ed3f68acc5bee34346099daacfcb0ceee7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.825ex; height:2.176ex;" alt="{\displaystyle r\to 0}" loading="lazy">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z_{1}}</annotation>

</semantics>

</math><img src="./cea9e950915c77b3dcf9d4d54101820f538bc077.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle Z_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z_{2}}</annotation>

</semantics>

</math><img src="./c98d433ae289ecb2b88f895b407538b0e4183b28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle Z_{2}}" loading="lazy"> are the charges of the interacting nuclei, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> is the so-called screening parameter.

A widely used popular screening function is the "Universal ZBL" one.<a href="#cite_note-23">[23]</a>

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

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.<a href="#cite_note-:4-25">[25]</a> In <a href="Binary_collision_approximation" title="Binary collision approximation">binary collision approximation</a> simulations this kind of potential can be used

to describe the <a href="Stopping_power_(particle_radiation)" title="Stopping power (particle radiation)">nuclear stopping power</a>.

<div class="mw-heading mw-heading4"><h4 id="Many-body_potentials">Many-body potentials</h4></div>

The Stillinger-Weber potential<a href="#cite_note-Stillinger_Weber_pp._5262–5271-26">[26]</a> is a potential that has a

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

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {TOT} }=\sum _{i,j}^{N}V_{2}(r_{ij})+\sum _{i,j,k}^{N}V_{3}(r_{ij},r_{ik},\theta _{ijk})}</annotation>

</semantics>

</math><img src="./3f66cbe717035baceec2aa273d0a309de7c3c4c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:40.348ex; height:7.676ex;" alt="{\displaystyle V_{\mathrm {TOT} }=\sum _{i,j}^{N}V_{2}(r_{ij})+\sum _{i,j,k}^{N}V_{3}(r_{ij},r_{ik},\theta _{ijk})}" loading="lazy"></dd></dl></dd></dl>

where the three-body term describes how the potential energy changes with bond bending.

It was originally developed for pure Si, but has been extended to many other

elements and compounds

and also formed the basis for other Si potentials.<a href="#cite_note-29">[29]</a>

Metals are very commonly described with what can be called

"EAM-like" potentials, i.e. potentials that share

the same functional form as the <a href="Embedded_atom_model" title="Embedded atom model">embedded atom model</a>.

In these potentials, the total potential energy is written

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {TOT} }=\sum _{i}^{N}F_{i}\left(\sum _{j}\rho (r_{ij})\right)+{\frac {1}{2}}\sum _{i,j}^{N}V_{2}(r_{ij})}</annotation>

</semantics>

</math><img src="./b06e026e40ef92c91d0a473ca5cff41e2b86fe67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:43.434ex; height:7.676ex;" alt="{\displaystyle V_{\mathrm {TOT} }=\sum _{i}^{N}F_{i}\left(\sum _{j}\rho (r_{ij})\right)+{\frac {1}{2}}\sum _{i,j}^{N}V_{2}(r_{ij})}" loading="lazy"></dd></dl></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle F_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle F_{i}}</annotation>

</semantics>

</math><img src="./b2615c70665cac2cd2fe3b481ab49c7f6219eaf4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.294ex; height:2.509ex;" alt="{\displaystyle \textstyle F_{i}}" loading="lazy"> is a so-called embedding function

(not to be confused with the force <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\vec {F}}_{i}}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle {\vec {F}}_{i}}</annotation>

</semantics>

</math><img src="./688c2f407125bc2efa27b8e7950c2b4aa0c7cab0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.571ex; height:3.176ex;" alt="{\displaystyle \textstyle {\vec {F}}_{i}}" loading="lazy">) that is a function of the sum of the so-called electron density

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \rho (r_{ij})}</annotation>

</semantics>

</math><img src="./fd003959c818d8fa262769c87051e1cb502b5a1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.537ex; height:3.009ex;" alt="{\displaystyle \textstyle \rho (r_{ij})}" loading="lazy">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle V_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle V_{2}}</annotation>

</semantics>

</math><img src="./b6edd4f7d71777667a670c1d4efbcb59983a36dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle \textstyle V_{2}}" loading="lazy">

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

formulation <a href="#cite_note-Foiles_Baskes_Daw_pp._7983–7991-31">[31]</a><a href="#cite_note-Foiles_Baskes_Daw_pp._10378–10378-32">[32]</a> the electron

density function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \rho (r_{ij})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \rho (r_{ij})}</annotation>

</semantics>

</math><img src="./fd003959c818d8fa262769c87051e1cb502b5a1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.537ex; height:3.009ex;" alt="{\displaystyle \textstyle \rho (r_{ij})}" loading="lazy"> was obtained

from true atomic electron densities, and the embedding function

was motivated from <a href="Density-functional_theory" class="mw-redirect" title="Density-functional theory">density-functional theory</a> as the energy needed

to 'embed' an atom into the electron density.

.<a href="#cite_note-Puska_Nieminen_Manninen_pp._3037–3047-33">[33]</a>

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

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

on <a href="Tight-binding_model" class="mw-redirect" title="Tight-binding model">tight-binding theory</a>

or other motivations

.<a href="#cite_note-Olsson_Wallenius_Domain_Nordlund_p.-39">[39]</a>

EAM-like potentials are usually implemented as numerical tables.

A collection of tables is available at the interatomic

potential repository at NIST <a rel="nofollow" class="external autonumber" href="http://www.ctcms.nist.gov/potentials/">[1]</a>

Covalently bonded materials are often described by

<a href="Bond_order_potential" title="Bond order potential">bond order potentials</a>, sometimes also called

Tersoff-like or Brenner-like potentials.

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

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<annotation encoding="application/x-tex">{\displaystyle V_{ij}(r_{ij})=V_{\mathrm {repulsive} }(r_{ij})+b_{ijk}V_{\mathrm {attractive} }(r_{ij})}</annotation>

</semantics>

</math><img src="./cc6c2a012440bd8ff62f0bf228e15efac4d7d1d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:41.639ex; height:3.009ex;" alt="{\displaystyle V_{ij}(r_{ij})=V_{\mathrm {repulsive} }(r_{ij})+b_{ijk}V_{\mathrm {attractive} }(r_{ij})}" loading="lazy"></dd></dl></dd></dl>

where the repulsive and attractive part are simple exponential

functions similar to those in the Morse potential.

However, the <a href="Bond_strength" class="mw-redirect" title="Bond strength">strength</a> is modified by the environment of the atom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

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</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> via the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{ijk}}">

<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{ijk}}</annotation>

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</math><img src="./06cd002fe0d1c37ecaf235548cd04472b239face.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.331ex; height:2.843ex;" alt="{\displaystyle b_{ijk}}" loading="lazy"> term. If implemented without

an explicit angular dependence, these potentials

can be shown to be mathematically equivalent to

some varieties of EAM-like potentials

Thanks to this equivalence, the bond-order potential formalism has been implemented also for many metal-covalent mixed materials.<a href="#cite_note-Alb02-43">[43]</a><a href="#cite_note-44">[44]</a>

EAM potentials have also been extended to describe covalent bonding by adding angular-dependent terms to the electron density function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

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</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy">, in what is called the modified embedded atom method (MEAM).<a href="#cite_note-47">[47]</a><a href="#cite_note-48">[48]</a><a href="#cite_note-49">[49]</a>

<div class="mw-heading mw-heading5"><h5 id="Force_fields">Force fields</h5></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Force_field_(chemistry)" title="Force field (chemistry)">Force field (chemistry)</a></div>

A <a href="Force_field_(chemistry)" title="Force field (chemistry)">force field</a> 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 <a href="Computational_chemistry" title="Computational chemistry">computational chemistry</a> community.<a href="#cite_note-:1-50">[50]</a> 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,<a href="#cite_note-:0-51">[51]</a><a href="#cite_note-52">[52]</a> molecular fluids,<a href="#cite_note-tandfonline.com-20">[20]</a> and for biomacromolecules,<a href="#cite_note-53">[53]</a> 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,<a href="#cite_note-:1-50">[50]</a> <a href="CHARMM" title="CHARMM">CHARMM</a>,<a href="#cite_note-54">[54]</a> and COMPASS) to explicitly reactive models with many adjustable fit parameters (e.g. <a href="ReaxFF" title="ReaxFF">ReaxFF</a>) and machine learning models.

<div class="mw-heading mw-heading3"><h3 id="Non-parametric_potentials">Non-parametric potentials</h3></div>

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 <a href="Machine_learning" title="Machine learning">machine learning</a> methods. The total energy is then written<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathrm {TOT} }=\sum _{i}^{N}E(\mathbf {q} _{i})}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {TOT} }=\sum _{i}^{N}E(\mathbf {q} _{i})}</annotation>

</semantics>

</math>where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {q} _{i}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {q} _{i}}</annotation>

</semantics>

</math><img src="./c8ce6be9c9335b20681f1b784557c574a69f28e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.211ex; height:2.176ex;" alt="{\displaystyle \mathbf {q} _{i}}" loading="lazy"> is a mathematical representation of the atomic environment surrounding the atom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">, known as the <a href="Molecular_descriptor" title="Molecular descriptor">descriptor</a>.<a href="#cite_note-:3-55">[55]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> is a machine-learning model that provides a prediction for the energy of atom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> to its neighbours, yielding a machine-learned pair potential. However, more complex many-body descriptors are needed to produce highly accurate potentials.<a href="#cite_note-:3-55">[55]</a> It is also possible to use a linear combination of multiple descriptors with associated machine-learning models.<a href="#cite_note-56">[56]</a> Potentials have been constructed using a variety of machine-learning methods, descriptors, and mappings, including <a href="Artificial_neural_network" class="mw-redirect" title="Artificial neural network">neural networks</a>,<a href="#cite_note-57">[57]</a> <a href="Gaussian_process_regression" class="mw-redirect" title="Gaussian process regression">Gaussian process regression</a>,<a href="#cite_note-58">[58]</a><a href="#cite_note-59">[59]</a> and <a href="Linear_regression" title="Linear regression">linear regression</a>.<a href="#cite_note-60">[60]</a><a href="#cite_note-Shapeev2016-16">[16]</a>

A non-parametric potential is most often trained to total energies, forces, and/or stresses obtained from quantum-level calculations, such as <a href="Density_functional_theory" title="Density functional theory">density functional theory</a>, 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,<a href="#cite_note-61">[61]</a> or to impose physical constraints on the predictions.<a href="#cite_note-62">[62]</a>

<div class="mw-heading mw-heading2"><h2 id="Potential_fitting">Potential fitting</h2></div>

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 <a href="Surface_energy" title="Surface energy">surface energy</a> of a solid

.<a href="#cite_note-63">[63]</a><a href="#cite_note-64">[64]</a> Lennard-Jones potential can typically describe the lattice parameters, surface energies, and approximate mechanical properties.<a href="#cite_note-:2-65">[65]</a> 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 <a href="Density-functional_theory" class="mw-redirect" title="Density-functional theory">density-functional theory</a>.<a href="#cite_note-66">[66]</a><a href="#cite_note-67">[67]</a> For solids, a many-body potential

can often describe the <a href="Lattice_constant" title="Lattice constant">lattice constant</a> of the equilibrium crystal structure, the <a href="Cohesion_(chemistry)" title="Cohesion (chemistry)">cohesive energy</a>, and <a href="Elasticity_(physics)" title="Elasticity (physics)">linear elastic constants</a>, as well as basic <a href="Point_defect" class="mw-redirect" title="Point defect">point defect</a> properties of all the elements and stable compounds well, although deviations in surface energies often exceed 50%.

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.<a href="#cite_note-71">[71]</a><a href="#cite_note-72">[72]</a><a href="#cite_note-73">[73]</a><a href="#cite_note-Swamy_Gale_pp._5406–5412-74">[74]</a><a href="#cite_note-Aguado_Bernasconi_Madden_2002_pp._437–444-75">[75]</a>). Key aspects here are the correct representation of chemical bonding, validation of structures and energies, as well as interpretability of all parameters.<a href="#cite_note-:0-51">[51]</a> Full transferability and interpretability is reached with the Interface force field (IFF).<a href="#cite_note-:1-50">[50]</a> 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.<a href="#cite_note-Bal92-14">[14]</a>

The NIST interatomic potential repository provides a collection of fitted interatomic potentials, either as fitted parameter values or numerical

tables of the potential functions.<a href="#cite_note-76">[76]</a> The OpenKIM <a href="#cite_note-77">[77]</a> 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.

<div class="mw-heading mw-heading2"><h2 id="Machine-learned_interatomic_potentials">Machine-learned interatomic potentials</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Machine_learning_potential" class="mw-redirect" title="Machine learning potential">Machine learning potential</a></div>

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)<a href="#cite_note-78">[78]</a> or as 'machine-learned interatomic potentials' (MLIPs).<a href="#cite_note-nature.com-79">[79]</a> Such machine learning potentials help fill the gap between highly accurate but computationally intensive simulations like <a href="Density_functional_theory" title="Density functional theory">density functional theory</a> 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.<a href="#cite_note-ML-80">[80]</a><a href="#cite_note-81">[81]</a><a href="#cite_note-nature.com-79">[79]</a>

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.<a href="#cite_note-ML-80">[80]</a><a href="#cite_note-82">[82]</a>

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 <a href="Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graphs</a> 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.<a href="#cite_note-83">[83]</a><a href="#cite_note-ML-80">[80]</a><a href="#cite_note-84">[84]</a>

Another class of machine-learned interatomic potential is the Gaussian approximation potential (GAP),<a href="#cite_note-85">[85]</a><a href="#cite_note-86">[86]</a><a href="#cite_note-87">[87]</a> which combines compact descriptors of local atomic environments<a href="#cite_note-88">[88]</a> with Gaussian process regression<a href="#cite_note-89">[89]</a> to machine learn the <a href="Potential_energy_surface" title="Potential energy surface">potential energy surface</a> 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<a href="#cite_note-90">[90]</a><a href="#cite_note-91">[91]</a> Silicon,<a href="#cite_note-92">[92]</a> and Tungsten,<a href="#cite_note-93">[93]</a> as well as for multicomponent systems such as Ge2Sb2Te5<a href="#cite_note-94">[94]</a> and austenitic <a href="Stainless_steel" title="Stainless steel">stainless steel</a>, Fe7Cr2Ni.<a href="#cite_note-95">[95]</a>

<div class="mw-heading mw-heading2"><h2 id="Reliability_of_interatomic_potentials">Reliability of interatomic potentials</h2></div>

Classical interatomic potentials often exceed the accuracy of simplified quantum mechanical methods such as <a href="Density_functional_theory" title="Density functional theory">density functional theory</a> at a million times lower computational cost.<a href="#cite_note-:0-51">[51]</a> 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 <a href="Quantum_chemistry" title="Quantum chemistry">quantum chemistry</a> methods can be locally used.<a href="#cite_note-96">[96]</a>

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.

<ul><li><a href="Computational_chemistry" title="Computational chemistry">Computational chemistry</a></li>

<li><a href="Computational_materials_science" title="Computational materials science">Computational materials science</a></li>

<li><a href="Molecular_dynamics" title="Molecular dynamics">Molecular dynamics</a></li>

<li><a href="Force_field_(chemistry)" title="Force field (chemistry)">Force field (chemistry)</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-AllenTildesley-1">^ <a href="#cite_ref-AllenTildesley_1-0">a</a> <a href="#cite_ref-AllenTildesley_1-1">b</a> M. P. Allen and D. J. Tildesley. Computer Simulation of Liquids. Oxford University Press, Oxford, England, 1989.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> Daan Frenkel and Berend Smit. Understanding molecular simulation: from algorithms to applications. Academic Press, San Diego, second edition, 2002.

</li>

<li id="cite_note-Lesar-3">^ <a href="#cite_ref-Lesar_3-0">a</a> <a href="#cite_ref-Lesar_3-1">b</a> <a href="#cite_ref-Lesar_3-2">c</a> R. Lesar. Introduction to Computational Materials Science. Cambridge University Press, 2013.

</li>

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</style><cite id="CITEREFBrenner2000" class="citation journal cs1">Brenner, D.W. (2000). "The Art and Science of an Analytic Potential". Physica Status Solidi B. 217 (1): 23–40. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2000PSSBR.217...23B">2000PSSBR.217...23B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F%28SICI%291521-3951%28200001%29217%3A1%3C23%3A%3AAID-PSSB23%3E3.0.CO%3B2-N">10.1002/(SICI)1521-3951(200001)217:1<23::AID-PSSB23>3.0.CO;2-N</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0370-1972">0370-1972</a>.</cite>

</li>

<li id="cite_note-AshcroftMermin-5"><a href="#cite_ref-AshcroftMermin_5-0">^</a> N. W. Ashcroft and N. D. Mermin. Solid State Physics.Saunders College, Philadelphia, 1976.

</li>

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</ol></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.ctcms.nist.gov/potentials/">NIST interatomic potential repository</a></li>

<li><a rel="nofollow" class="external text" href="https://www.ctcms.nist.gov/~knc6/periodic.html">NIST JARVIS-FF</a></li>

<li><a rel="nofollow" class="external text" href="https://openkim.org/">Open Knowledgebase of Interatomic Models (OpenKIM)</a></li></ul></div><div><div class="zim-footer">

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