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<title>Pair potential</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Pair potential</span></span>

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<p>In <a href="Physics" title="Physics">physics</a>, a <b>pair potential</b> is a function that describes the <a href="Potential_energy" title="Potential energy">potential energy</a> of two interacting objects solely as a function of the distance between them.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>Some interactions, like <a href="Coulomb's_law" title="Coulomb's law">Coulomb's law</a> in <a href="Classical_electromagnetism" title="Classical electromagnetism">electrodynamics</a> or <a href="Newton's_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newton's law of universal gravitation</a> in <a href="Mechanics" title="Mechanics">mechanics</a> naturally have this form for simple spherical objects.

For other types of more complex interactions or objects it is useful and common to approximate the interaction by a pair potential, for example <a href="Interatomic_potential" title="Interatomic potential">interatomic potentials</a> in physics and <a href="Computational_chemistry" title="Computational chemistry">computational chemistry</a> that use approximations like the <a href="Lennard-Jones_potential" title="Lennard-Jones potential">Lennard-Jones</a> and <a href="Morse_potential" title="Morse potential">Morse</a> potentials.

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

<p>The total energy of a system of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.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 N}" loading="lazy"></span> objects at positions <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {R}}_{i}}">

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

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</math></span><img src="./34f37147f1e98252cb1a70455b261eaa49c904fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.564ex; height:3.176ex;" alt="{\displaystyle {\vec {R}}_{i}}" loading="lazy"></span>, that interact through pair potential <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"></span> is given by

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E={\frac {1}{2}}\sum _{i=1}^{N}\sum _{j\neq i}^{N}v\left(\left|{\vec {R}}_{i}-{\vec {R}}_{j}\right|\right)\ .}">

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<annotation encoding="application/x-tex">{\displaystyle E={\frac {1}{2}}\sum _{i=1}^{N}\sum _{j\neq i}^{N}v\left(\left|{\vec {R}}_{i}-{\vec {R}}_{j}\right|\right)\ .}</annotation>

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</p><p>Equivalently, this can be expressed as

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=\sum _{i=1}^{N}\sum _{j=i+1}^{N}v\left(\left|{\vec {R}}_{i}-{\vec {R}}_{j}\right|\right)\ .}">

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<annotation encoding="application/x-tex">{\displaystyle E=\sum _{i=1}^{N}\sum _{j=i+1}^{N}v\left(\left|{\vec {R}}_{i}-{\vec {R}}_{j}\right|\right)\ .}</annotation>

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</p><p>This expression uses the fact that interaction is symmetric between particles <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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

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</math></span><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"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

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</math></span><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"></span>.

It also avoids self-interaction by not including the case where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=j}">

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

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</math></span><img src="./706e0928b2bf0f24076b0c90bb20616ff2068343.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 i=j}" loading="lazy"></span>.

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<div class="mw-heading mw-heading2"><h2 id="Potential_range">Potential range</h2></div>

<p>A fundamental property of a pair potential is its range.

It is expected that pair potentials go to zero for infinite distance as particles that are too far apart do not interact.

In some cases the potential goes quickly to zero and the interaction for particles that are beyond a certain distance can be assumed to be zero, these are said to be short-range potentials.

Other potentials, like the Coulomb or gravitational potential, are long range: they go slowly to zero and the contribution of particles at long distances still contributes to the total energy.

</p>

<div class="mw-heading mw-heading2"><h2 id="Computational_cost">Computational cost</h2></div>

<p>The total energy expression for pair potentials is quite simple to use for analytical and computational work.

It has some limitations however, as the <a href="Computational_cost" class="mw-redirect" title="Computational cost">computational cost</a> is proportional to the square of number of particles.

This might be prohibitively expensive when the interaction between large groups of objects needs to be calculated.

</p><p>For short-range potentials the sum can be restricted only to include particles that are close, reducing the cost to linearly proportional to the number of particles.

</p>

<div class="mw-heading mw-heading2"><h2 id="Infinitely_periodic_systems">Infinitely periodic systems</h2></div>

<p>In some cases it is necessary to calculate the interaction between an infinite number of particles arranged in a periodic pattern.

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<div class="mw-heading mw-heading2"><h2 id="Beyond_pair_potentials">Beyond pair potentials</h2></div>

<p>Pair potentials are very common in physics and computational chemistry and biology; exceptions are very rare. An example of a potential energy function that is <i>not</i> a pair potential is the three-body <a href="Axilrod-Teller_potential" class="mw-redirect" title="Axilrod-Teller potential">Axilrod-Teller potential</a>. Another example is the Stillinger-Weber potential for <a href="Silicon" title="Silicon">silicon</a>, which includes the angle in a triangle of silicon atoms as an input parameter.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Common_pair_potentials">Common pair potentials</h2></div>

<p>Some commonly used pair potentials are listed below.

</p>

<ul><li><a href="Hard_spheres" title="Hard spheres">Hard Sphere</a> potential</li>

<li>Sutherland potential</li>

<li><a href="Buckingham_potential" title="Buckingham potential">Buckingham (or exp-6) potential</a></li>

<li><a href="Mie_potential" title="Mie potential">Mie potential</a></li>

<li><a href="Lennard-Jones_potential" title="Lennard-Jones potential">Lennard-Jones (12-6) potential</a></li>

<li><a href="Stockmayer_potential" title="Stockmayer potential">Stockmayer potential</a></li>

<li><a href="Coulomb's_law" title="Coulomb's law">Coloumb potential</a></li>

<li><a href="Yukawa_potential" title="Yukawa potential">Yukawa potential</a></li>

<li><a href="Morse_potential" title="Morse potential">Morse potential</a></li></ul>

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

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</style><cite id="CITEREFPeiSongMerz_Jr.2020" class="citation journal cs1">Pei, Jun; Song, Lin Frank; Merz Jr., Kenneth M. (June 19, 2020). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://pubs.acs.org/doi/10.1021/acs.jctc.9b01246">"Pair Potentials as Machine Learning Features"</a></span>. <i>J. Chem. Theory Comput</i>. <b>16</b> (8): <span class="nowrap">5385–</span>5400. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Facs.jctc.9b01246">10.1021/acs.jctc.9b01246</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/32559380">32559380</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:219947826">219947826</a><span class="reference-accessdate">. Retrieved <span class="nowrap">26 July</span> 2022</span>.</cite></span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFStillingerWeber1985" class="citation journal cs1">Stillinger, Frank H.; Weber, Thomas A. (15 April 1985). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://journals.aps.org/prb/abstract/10.1103/PhysRevB.31.5262">"Computer simulation of local order in condensed phases of silicon"</a></span>. <i>Physical Review B</i>. <b>31</b> (8): <span class="nowrap">5262–</span>5271. <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/1985PhRvB..31.5262S">1985PhRvB..31.5262S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevB.31.5262">10.1103/PhysRevB.31.5262</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/9936488">9936488</a><span class="reference-accessdate">. Retrieved <span class="nowrap">26 July</span> 2022</span>.</cite></span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFStillingerWeber1986" class="citation journal cs1">Stillinger, Frank H.; Weber, Thomas A. (15 January 1986). <a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevB.33.1451">"Erratum: Computer simulation of local order in condensed phases of silicon [Phys. Rev. B 31, 5262 (1985)]"</a>. <i>Physical Review B</i>. <b>33</b> (2): 1451. <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/1986PhRvB..33.1451S">1986PhRvB..33.1451S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevB.33.1451">10.1103/PhysRevB.33.1451</a></span>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/9938428">9938428</a>.</cite></span>

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</style><div id="Statistical_mechanics157" style="font-size:114%;margin:0 4em"><a href="Statistical_mechanics" title="Statistical mechanics">Statistical mechanics</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theory</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Principle_of_maximum_entropy" title="Principle of maximum entropy">Principle of maximum entropy</a></li>

<li><a href="Ergodic_theory" title="Ergodic theory">ergodic theory</a></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="7" style="width:1px;padding:0 0 0 2px"><div><span class="skin-invert-image" typeof="mw:File"><span></span></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Statistical_thermodynamics" class="mw-redirect" title="Statistical thermodynamics">Statistical thermodynamics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Statistical_ensemble_(mathematical_physics)" class="mw-redirect" title="Statistical ensemble (mathematical physics)">Ensembles</a></li>

<li><a href="Partition_function_(statistical_mechanics)" title="Partition function (statistical mechanics)">partition functions</a></li>

<li><a href="Equation_of_state" title="Equation of state">equations of state</a></li>

<li><a href="Thermodynamic_potential" title="Thermodynamic potential">thermodynamic potential</a>:

<li><a href="Maxwell_relations" title="Maxwell relations">Maxwell relations</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Statistical_model" title="Statistical model">Models</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Spin_model" title="Spin model">Ferromagnetism models</a>

<ul><li><a href="Ising_model" title="Ising model">Ising</a></li>

<li><a href="Potts_model" title="Potts model">Potts</a></li>

<li><a href="Heisenberg_model_(quantum)" class="mw-redirect" title="Heisenberg model (quantum)">Heisenberg</a></li>

<li><a href="Percolation_theory" title="Percolation theory">percolation</a></li></ul></li>

<li>Particles with <a href="Force_field_(chemistry)" title="Force field (chemistry)">force field</a>

<ul><li><a href="Depletion_force" title="Depletion force">depletion force</a></li>

<li><a href="Lennard-Jones_potential" title="Lennard-Jones potential">Lennard-Jones potential</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mathematical approaches</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Boltzmann_equation" title="Boltzmann equation">Boltzmann equation</a></li>

<li><a href="H-theorem" title="H-theorem">H-theorem</a></li>

<li><a href="Vlasov_equation" title="Vlasov equation">Vlasov equation</a></li>

<li><a href="BBGKY_hierarchy" title="BBGKY hierarchy">BBGKY hierarchy</a></li>

<li><a href="Stochastic_process" title="Stochastic process">stochastic process</a></li>

<li><a href="Mean-field_theory" title="Mean-field theory">mean-field theory</a> and <a href="Conformal_field_theory" title="Conformal field theory">conformal field theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Critical_phenomena" title="Critical phenomena">Critical phenomena</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Phase_transition" title="Phase transition">Phase transition</a></li>

<li><a href="Critical_exponent" title="Critical exponent">Critical exponents</a>

<ul><li><a href="Correlation_function" title="Correlation function">correlation length</a></li>

<li><a href="Scaling_(geometry)" title="Scaling (geometry)">size scaling</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Entropy" title="Entropy">Entropy</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Boltzmann's_entropy_formula" title="Boltzmann's entropy formula">Boltzmann</a></li>

<li><a href="Shannon_entropy" class="mw-redirect" title="Shannon entropy">Shannon</a></li>

<li><a href="Tsallis_entropy" title="Tsallis entropy">Tsallis </a></li>

<li><a href="R%C3%A9nyi_entropy" title="Rényi entropy">Rényi</a></li>

<li><a href="Von_Neumann_entropy" title="Von Neumann entropy">von Neumann</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Statistical_field_theory" title="Statistical field theory">Statistical field theory</a>

<ul><li><a href="Elementary_particle" title="Elementary particle">elementary particle</a></li>

<li><a href="Superfluidity" title="Superfluidity">superfluidity</a></li></ul></li>

<li><a href="Condensed_matter_physics" title="Condensed matter physics">Condensed matter physics</a></li>

<li><a href="Complex_system" title="Complex system">Complex system</a>

<ul><li><a href="Chaos_theory" title="Chaos theory">chaos</a></li>

<li><a href="Information_theory" title="Information theory">information theory</a></li>

<li><a href="Boltzmann_machine" title="Boltzmann machine">Boltzmann machine</a></li></ul></li></ul>

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This article is issued from <a class="external text" title="Last edited on 2025-05-26" href="https://en.wikipedia.org/wiki/?title=Pair_potential&oldid=1292360491">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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