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Buckingham potential

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In <a href="Theoretical_chemistry" title="Theoretical chemistry">theoretical chemistry</a>, the Buckingham potential is a <a href="Mathematical_formula" class="mw-redirect" title="Mathematical formula">formula</a> proposed by <a href="Richard_Buckingham" title="Richard Buckingham">Richard Buckingham</a> which describes the <a href="Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a> and <a href="Van_der_Waals_force" title="Van der Waals force">van der Waals energy</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{12}(r)}">

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<annotation encoding="application/x-tex">{\displaystyle \Phi _{12}(r)}</annotation>

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</math><img src="./f75f0b695833c0cb16736048f33cf23050dffa7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.412ex; height:2.843ex;" alt="{\displaystyle \Phi _{12}(r)}" loading="lazy"> for the interaction of two atoms that are not directly bonded as a function of the <a href="Atomic_spacing" title="Atomic spacing">interatomic distance</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

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

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</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy">. It is a variety of <a href="Interatomic_potential" title="Interatomic potential">interatomic potentials</a>.

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<annotation encoding="application/x-tex">{\displaystyle \Phi _{12}(r)=A\exp \left(-Br\right)-{\frac {C}{r^{6}}}}</annotation>

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</math><img src="./39d5ac6758bb1c621090bed49a8a9e20f6f39435.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:27.403ex; height:5.676ex;" alt="{\displaystyle \Phi _{12}(r)=A\exp \left(-Br\right)-{\frac {C}{r^{6}}}}" loading="lazy"></dd></dl>

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

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

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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

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</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

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

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</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> are constants. The two terms on the right-hand side constitute a repulsion and an attraction, because their first <a href="Derivative" title="Derivative">derivatives</a> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

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

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</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> are negative and positive, respectively.

Buckingham proposed this as a simplification of the <a href="Lennard-Jones_potential" title="Lennard-Jones potential">Lennard-Jones potential</a>, in a theoretical study of the <a href="Equation_of_state" title="Equation of state">equation of state</a> for <a href="Gas" title="Gas">gaseous</a> <a href="Helium" title="Helium">helium</a>, <a href="Neon" title="Neon">neon</a> and <a href="Argon" title="Argon">argon</a>.<a href="#cite_note-1">[1]</a>

As explained in Buckingham's original paper and, e.g., in section 2.2.5 of Jensen's text,<a href="#cite_note-jensen-2">[2]</a> the repulsion is due to the interpenetration of the closed <a href="Electron_shell" title="Electron shell">electron shells</a>. "There is therefore some justification for choosing the repulsive part (of the potential) as an <a href="Exponential_function" title="Exponential function">exponential function</a>". The Buckingham potential has been used extensively in simulations of <a href="Molecular_dynamics" title="Molecular dynamics">molecular dynamics</a>.

Because the exponential term converges to a constant as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

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</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy">→<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

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</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">, while the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r^{-6}}">

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

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</math><img src="./5562d12471ec1ca2053ab569f0e5c1451f315d62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.381ex; height:2.676ex;" alt="{\displaystyle r^{-6}}" loading="lazy"> term diverges, the Buckingham potential becomes attractive as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

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

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</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> becomes small. This may be problematic when dealing with a structure with very short interatomic distances, as any nuclei that cross a certain threshold will become strongly (and unphysically) bound to one another at a distance of zero.<a href="#cite_note-jensen-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Modified_Buckingham_(Exp-Six)_potential">Modified Buckingham (Exp-Six) potential</h2></div>

The modified Buckingham potential, also called the "exp-six" potential, is used to calculate the interatomic forces for gases based on Chapman and Cowling collision theory.<a href="#cite_note-3">[3]</a> The potential has the form

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<annotation encoding="application/x-tex">{\displaystyle \Phi _{12}(r)={\frac {\epsilon }{1-6/\alpha }}\left[{\frac {6}{\alpha }}\exp \left[\alpha \left(1-{\frac {r}{r_{min}}}\right)\right]-\left({\frac {r_{min}}{r}}\right)^{6}\right]}</annotation>

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</math><img src="./80e3fc27abeca3b8e89135ee679c5c10bdedc9d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:55.918ex; height:6.343ex;" alt="{\displaystyle \Phi _{12}(r)={\frac {\epsilon }{1-6/\alpha }}\left[{\frac {6}{\alpha }}\exp \left[\alpha \left(1-{\frac {r}{r_{min}}}\right)\right]-\left({\frac {r_{min}}{r}}\right)^{6}\right]}" loading="lazy">

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{12}(r)}">

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</math><img src="./f75f0b695833c0cb16736048f33cf23050dffa7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.412ex; height:2.843ex;" alt="{\displaystyle \Phi _{12}(r)}" loading="lazy"> is the interatomic potential between atom i and atom j, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }">

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

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</math><img src="./c3837cad72483d97bcdde49c85d3b7b859fb3fd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }" loading="lazy"> is the minimum potential energy, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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

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</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> is the measurement of the repulsive energy steepness which is the ratio <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma /r_{min}}">

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

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</math><img src="./ef5f8966d93bdf4441f4595ef34b1b8e063ae033.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.769ex; height:2.843ex;" alt="{\displaystyle \sigma /r_{min}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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

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</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.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 \sigma }" loading="lazy"> is the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

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

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</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{12}(r)}">

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<annotation encoding="application/x-tex">{\displaystyle \Phi _{12}(r)}</annotation>

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</math><img src="./f75f0b695833c0cb16736048f33cf23050dffa7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.412ex; height:2.843ex;" alt="{\displaystyle \Phi _{12}(r)}" loading="lazy"> is zero, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{min}}">

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

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</math><img src="./2d0918727647f5f1a2ccd076d9545df985ab8bc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.277ex; height:2.009ex;" alt="{\displaystyle r_{min}}" loading="lazy"> is the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

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</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> which can achieve the minimum interatomic potential <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }">

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

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</math><img src="./c3837cad72483d97bcdde49c85d3b7b859fb3fd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }" loading="lazy">. This potential function is only valid when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r>r_{max}}">

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

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</math><img src="./49e24adb2e81cba258ac85da7f6a0b784c5acdb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.681ex; height:2.176ex;" alt="{\displaystyle r>r_{max}}" loading="lazy">, as the potential will decay towards <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }">

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

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</math><img src="./ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\rightarrow 0}">

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</math><img src="./47fe923b35572a4c03c177f7cd857e4ebc9b77b6.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\rightarrow 0}" loading="lazy">. This is corrected by identifying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{max}}">

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

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</math><img src="./06da37fcef1872f3f1f9ecfa4410bc7b5f28e58a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.534ex; height:2.009ex;" alt="{\displaystyle r_{max}}" loading="lazy">, which is the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

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</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> at which the potential is maximized; when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\leq {r_{max}}}">

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</math><img src="./24da9ff90083be810a0629ce10bb9f600e6be750.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.681ex; height:2.343ex;" alt="{\displaystyle r\leq {r_{max}}}" loading="lazy">, the potential is set to infinity.

<div class="mw-heading mw-heading2"><h2 id="Coulomb–Buckingham_potential">Coulomb–Buckingham potential</h2></div>

The Coulomb–Buckingham potential is an extension of the Buckingham potential for application to ionic systems (e.g. <a href="Ceramic" title="Ceramic">ceramic</a> materials). The formula for the interaction is

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<annotation encoding="application/x-tex">{\displaystyle \Phi _{12}(r)=A\exp \left(-Br\right)-{\frac {C}{r^{6}}}+{\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}r}}}</annotation>

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</math><img src="./b4a02b51a79bcd09e27f3309cc7d6291d7286950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:36.76ex; height:5.843ex;" alt="{\displaystyle \Phi _{12}(r)=A\exp \left(-Br\right)-{\frac {C}{r^{6}}}+{\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}r}}}" loading="lazy"></dd></dl>

where A, B, and C are suitable constants and the additional term is the <a href="Electrostatic_potential_energy" class="mw-redirect" title="Electrostatic potential energy">electrostatic potential energy</a>.

The above equation may be written in its alternate form as

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<annotation encoding="application/x-tex">{\displaystyle \Phi (r)=\varepsilon \left\{{\frac {6}{\alpha -6}}\exp \left(\alpha \left[1-{\frac {r}{r_{0}}}\right]\right)-{\frac {\alpha }{\alpha -6}}\left({\frac {r_{0}}{r}}\right)^{6}\right\}+{\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}r}}}</annotation>

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</math><img src="./9272eda4ce5c807ddfc2819e6a256f5c01c05959.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:62.843ex; height:6.176ex;" alt="{\displaystyle \Phi (r)=\varepsilon \left\{{\frac {6}{\alpha -6}}\exp \left(\alpha \left[1-{\frac {r}{r_{0}}}\right]\right)-{\frac {\alpha }{\alpha -6}}\left({\frac {r_{0}}{r}}\right)^{6}\right\}+{\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}r}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{0}}">

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</math><img src="./fb12fcfddb65e3d1e6a044215f6e833f0cd4337b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{0}}" loading="lazy"> is the minimum energy distance, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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

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</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> is a free dimensionless parameter and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

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

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</math><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.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 \varepsilon }" loading="lazy"> is the depth of the minimum energy.

<div class="mw-heading mw-heading2"><h2 id="Beest_Kramer_van_Santen_(BKS)_potential">Beest Kramer van Santen (BKS) potential</h2></div>

The BKS potential is a <a href="Force_field_(chemistry)" title="Force field (chemistry)">force field</a> that may be used to simulate the <a href="Interatomic_potential" title="Interatomic potential">interatomic potential</a> between <a href="Silica_glass" class="mw-redirect" title="Silica glass">Silica glass</a> atoms.<a href="#cite_note-4">[4]</a> Rather than relying only on experimental data, the BKS potential is derived by combining <a href="Ab_initio_quantum_chemistry_methods" title="Ab initio quantum chemistry methods">ab initio quantum chemistry methods</a> on small silica clusters to describe accurate interaction between nearest-neighbors, which is the function of accurate <a href="Force_field_(chemistry)" title="Force field (chemistry)">force field</a>. The experimental data is applied to fit larger scale force information beyond nearest neighbors. By combining the <a href="Microscopic_scale" title="Microscopic scale">microscopic</a> and <a href="Macroscopic_scale" title="Macroscopic scale">macroscopic</a> information, the applicability of the BKS potential has been extended to both the silica polymorphs and other tetrahedral network oxides systems that have same cluster structure, such as aluminophosphates, <a href="Carbon" title="Carbon">carbon</a> and <a href="Silicon" title="Silicon">silicon</a>.

The form of this interatomic potential is the usual Buckingham form, with the addition of a <a href="Coulomb's_law" title="Coulomb's law">Coulomb force</a> term. The formula for the BKS potential is expressed as

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<annotation encoding="application/x-tex">{\displaystyle \Phi _{12}(r)=\left[A_{12}\exp \left(-B_{12}r_{12}\right)-{\frac {C_{12}}{r_{12}^{6}}}\right]+{\frac {q_{1}q_{2}}{r_{12}}}}</annotation>

</semantics>

</math><img src="./9ef0dcfdf53df8c8d7bdb7131cd506a1b7ed4d5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:45.036ex; height:7.509ex;" alt="{\displaystyle \Phi _{12}(r)=\left[A_{12}\exp \left(-B_{12}r_{12}\right)-{\frac {C_{12}}{r_{12}^{6}}}\right]+{\frac {q_{1}q_{2}}{r_{12}}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{12}(r)}">

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<annotation encoding="application/x-tex">{\displaystyle \Phi _{12}(r)}</annotation>

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</math><img src="./f75f0b695833c0cb16736048f33cf23050dffa7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.412ex; height:2.843ex;" alt="{\displaystyle \Phi _{12}(r)}" loading="lazy"> is the interatomic potential between atom i and atom j, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{1}}">

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

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</math><img src="./9daa41f6e8f78ea6bb5711d7ac97901ce564b94e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.091ex; height:2.009ex;" alt="{\displaystyle q_{1}}" loading="lazy">and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{2}}">

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

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</math><img src="./fd2d05084feb02b8ba29b0673440fb673b102589.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.091ex; height:2.009ex;" alt="{\displaystyle q_{2}}" loading="lazy"> are the charges magnitudes, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{12}}">

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<annotation encoding="application/x-tex">{\displaystyle r_{12}}</annotation>

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</math><img src="./2342fe122ce8e4694f81037e3d07eec0c4454715.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.925ex; height:2.009ex;" alt="{\displaystyle r_{12}}" loading="lazy"> is the distance between atoms, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{ij}}">

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

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</math><img src="./8272b28f5aae6dbb8d6f829d58bab353b21bde20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.22ex; height:2.843ex;" alt="{\displaystyle A_{ij}}" loading="lazy">,<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{ij}}">

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

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</math><img src="./57c134504644caba8b83f479c9f27dbfa990b79a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.241ex; height:2.843ex;" alt="{\displaystyle B_{ij}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{ij}}">

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

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</math><img src="./70b3f8ea18a9b9a0dbe09dae212a623820b217d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.139ex; height:2.843ex;" alt="{\displaystyle C_{ij}}" loading="lazy"> are constant parameters based on the type of atoms.<a href="#cite_note-:0-5">[5]</a>

The BKS potential parameters for common atoms are shown below:<a href="#cite_note-:0-5">[5]</a>

<caption>BKS parameters

</caption>

<th>i-j

</th>

<th>Aij(eV)

</th>

<th>Bij(Å−1)

</th>

<th>Cij(eV•Å6)

</th></tr>

<tr>

<td>O - O

</td>

<td>1388.7730

</td>

<td>2.76000

</td>

<td>175.0000

</td></tr>

<tr>

<td>O - Si

</td>

<td>18,003.757

</td>

<td>4.87318

</td>

<td>133.5381

</td></tr>

<tr>

<td>Si - Si

</td>

<td>0

</td>

<td>0

</td>

<td>0

</td></tr>

<tr>

<td>Al - O

</td>

<td>16,008.5345

</td>

<td>4.79667

</td>

<td>130.5659

</td></tr>

<tr>

<td>Al - Al

</td>

<td>0

</td>

<td>0

</td>

<td>0

</td></tr>

<tr>

<td>P - O

</td>

<td>9,034.2080

</td>

<td>5.19098

</td>

<td>19.8793

</td></tr>

<tr>

<td>P - P

</td>

<td>0

</td>

<td>0

</td>

<td>0

</td></tr></tbody></table>

An updated version of the BKS potential introduced a new repulsive term to prevent atom overlapping.<a href="#cite_note-6">[6]</a> The modified potential is taken as

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<annotation encoding="application/x-tex">{\displaystyle \Phi _{12}(r)=\left[A_{12}\exp \left(-B_{12}r_{12}\right)-{\frac {C_{12}}{r_{12}^{6}}}\right]+{\frac {q_{1}q_{2}}{r_{12}}}+{\frac {D_{12}}{r_{12}^{24}}}}</annotation>

</semantics>

</math><img src="./fdffe093812bb5c0c4a208500dd533b23dcbd93f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:52.513ex; height:7.509ex;" alt="{\displaystyle \Phi _{12}(r)=\left[A_{12}\exp \left(-B_{12}r_{12}\right)-{\frac {C_{12}}{r_{12}^{6}}}\right]+{\frac {q_{1}q_{2}}{r_{12}}}+{\frac {D_{12}}{r_{12}^{24}}}}" loading="lazy">

where the constant parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{ij}}">

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

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</math><img src="./88badd2360f6d55865a1e8f1b4e3994451bcc075.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.401ex; height:2.843ex;" alt="{\displaystyle D_{ij}}" loading="lazy"> were chosen to have the following values for Silica glass:

<caption>Parameter value for Silica glass

</caption>

<th>

</th>

<th>Si - Si

</th>

<th>Si - O

</th>

<th>O - O

</th></tr>

<tr>

<th>Dij(eV•Å24)

</th>

<td>3423200

</td>

<td>29

</td>

<td>113

</td></tr></tbody></table>

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

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

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</style><cite id="CITEREFBuckingham1938" class="citation journal cs1">Buckingham, R. A. (1938). "The Classical Equation of State of Gaseous Helium, Neon and Argon". Proceedings of the Royal Society A. 168 (933): 264–283. <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/1938RSPSA.168..264B">1938RSPSA.168..264B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1938.0173">10.1098/rspa.1938.0173</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/97239">97239</a>.</cite>

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<li id="cite_note-jensen-2">^ <a href="#cite_ref-jensen_2-0">a</a> <a href="#cite_ref-jensen_2-1">b</a> F. Jensen, Introduction to Computational Chemistry, 2nd ed., Wiley, 2007,

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<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFMason2004" class="citation journal cs1">Mason, Edward A. (2004-12-29). <a rel="nofollow" class="external text" href="https://aip.scitation.org/doi/abs/10.1063/1.1740026">"Transport Properties of Gases Obeying a Modified Buckingham (Exp-Six) Potential"</a>. The Journal of Chemical Physics. 22 (2): 169–186. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1740026">10.1063/1.1740026</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/0021-9606">0021-9606</a>.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFvan_BeestKramervan_Santen1990" class="citation journal cs1">van Beest, B. W. H.; Kramer, G. J.; van Santen, R. A. (1990-04-16). <a rel="nofollow" class="external text" href="https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.64.1955">"Force fields for silicas and aluminophosphates based onab initiocalculations"</a>. Physical Review Letters. 64 (16): 1955–1958. <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/1990PhRvL..64.1955V">1990PhRvL..64.1955V</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%2Fphysrevlett.64.1955">10.1103/physrevlett.64.1955</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/0031-9007">0031-9007</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/10041537">10041537</a>.</cite>

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<li id="cite_note-:0-5">^ <a href="#cite_ref-:0_5-0">a</a> <a href="#cite_ref-:0_5-1">b</a> <cite id="CITEREFKramerFarraghervan_Beestvan_Santen1991" class="citation journal cs1">Kramer, G. J.; Farragher, N. P.; van Beest, B. W. H.; van Santen, R. A. (1991-02-15). <a rel="nofollow" class="external text" href="https://journals.aps.org/prb/pdf/10.1103/PhysRevB.43.5068">"Interatomic force fields for silicas, aluminophosphates, and zeolites: Derivation based onab initiocalculations"</a>. Physical Review B. 43 (6): 5068–5080. <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/1991PhRvB..43.5068K">1991PhRvB..43.5068K</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.43.5068">10.1103/physrevb.43.5068</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/0163-1829">0163-1829</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/9997885">9997885</a>.</cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFCarréIspasHorbachKob2016" class="citation journal cs1">Carré, Antoine; Ispas, Simona; Horbach, Jürgen; Kob, Walter (2016-11-01). <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/pii/S0927025616303688">"Developing empirical potentials from ab initio simulations: The case of amorphous silica"</a>. Computational Materials Science. 124: 323–334. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.commatsci.2016.07.041">10.1016/j.commatsci.2016.07.041</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/0927-0256">0927-0256</a>.</cite>

</li>

</ol></div></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.sklogwiki.org/SklogWiki/index.php/Buckingham_potential">Buckingham potential</a> on <a rel="nofollow" class="external text" href="http://www.sklogwiki.org/SklogWiki/index.php/Main_Page">SklogWiki</a></li></ul></div><div><div class="zim-footer">

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