# Buckingham potential

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{{Short description|Physical model of intermolecular interactions}}

In [theoretical chemistry](/source/theoretical_chemistry), the '''Buckingham potential''' is a model of [intermolecular interaction](/source/intermolecular_interaction)s based on [pair potential](/source/pair_potential)s developed by [Richard Buckingham](/source/Richard_Buckingham). The model describes repulsion by the [Pauli exclusion principle](/source/Pauli_exclusion_principle) and attraction by [van der Waals force](/source/van_der_Waals_force)s between all atom pairs that are not directly bonded as a function of the [interatomic distance](/source/Atomic_spacing) <math>r</math>.

The [interatomic potential](/source/interatomic_potential),
:<math>\Phi_{12}(r) = A \exp \left(-Br\right) - \frac{C}{r^6}</math>
is given by two terms that represent the attraction and the repulsion, respectively. The constants, <math>A</math>, <math>B</math>, and <math>C</math> are parametrizations of the model tuned to the specific type of each atom pair.

Buckingham proposed this as a simplification of the [Lennard-Jones potential](/source/Lennard-Jones_potential), in a theoretical study of the [equation of state](/source/equation_of_state) for [gas](/source/gas)eous [helium](/source/helium), [neon](/source/neon) and [argon](/source/argon).<ref>{{cite journal|pages= 264–283|doi= 10.1098/rspa.1938.0173|jstor=97239|bibcode=1938RSPSA.168..264B|title= The Classical Equation of State of Gaseous Helium, Neon and Argon|journal= Proceedings of the Royal Society A|volume= 168|issue= 933|year= 1938|last1= Buckingham|first1= R. A.|doi-access= }}</ref>

As explained in Buckingham's original paper and, e.g., in section 2.2.5 of Jensen's text,<ref name=jensen>F. Jensen, ''Introduction to Computational Chemistry'', 2nd ed., Wiley, 2007,</ref> the repulsion is due to the interpenetration of the closed [electron shell](/source/electron_shell)s. "There is therefore some justification for choosing the repulsive part (of the potential) as an [exponential function](/source/exponential_function)". The Buckingham potential has been used extensively in simulations of [molecular dynamics](/source/molecular_dynamics).

Because the exponential term converges to a constant with decreasing distance, while the <math>r^{-6}</math> term diverges, the Buckingham potential becomes attractive as <math>r</math> 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.<ref name=jensen/>

== Modified Buckingham (Exp-Six) potential ==
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.<ref>{{Cite journal|last=Mason|first=Edward A.|date=2004-12-29|title=Transport Properties of Gases Obeying a Modified Buckingham (Exp-Six) Potential|url=https://aip.scitation.org/doi/abs/10.1063/1.1740026|journal=The Journal of Chemical Physics|language=en|volume=22|issue=2|pages=169–186|doi=10.1063/1.1740026|issn=0021-9606|url-access=subscription}}</ref> The potential has the form

<math>\Phi_{12}(r) = \frac{\epsilon}{1-6/\alpha}\left[ \frac6\alpha \exp \left[\alpha\left(1-\frac{r}{r_{min}}\right)\right] - \left(\frac{r_{min}}{r}\right)^6\right] </math>

where <math>\Phi_{12}(r) </math> is the interatomic potential between atom i and atom j, <math>\epsilon </math> is the minimum potential energy, <math>\alpha </math> is the measurement of the repulsive energy steepness which is the ratio <math>\sigma/r_{min} </math>, <math>\sigma
 </math> is the value of <math> r </math> where <math>\Phi_{12}(r) </math> is zero, and <math>r_{min} </math> is the value of <math>r </math> which can achieve the minimum interatomic potential <math>\epsilon </math>. This potential function is only valid when <math>r>r_{max} </math>, as the potential will decay towards <math>-\infty</math> as <math>r \rightarrow 0</math>. This is corrected by identifying <math>r_{max} </math>, which is the value of <math>r </math> at which the potential is maximized; when <math>r\leq{r_{max}} </math>, the potential is set to infinity.

==Coulomb–Buckingham potential==

thumb|Example Coulomb–Buckingham potential curve.

The Coulomb–Buckingham potential is an extension of the Buckingham potential for application to ionic systems (e.g. [ceramic](/source/ceramic) materials). The formula for the interaction is

:<math>\Phi_{12}(r) = A \exp \left(-Br\right) - \frac{C}{r^6} + \frac{q_1q_2}{4\pi\varepsilon_0 r} </math>

where ''A'', ''B'', and ''C'' are suitable constants and the additional term is the [electrostatic potential energy](/source/electrostatic_potential_energy).

The above equation may be written in its alternate form as

:<math>\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_1q_2}{4\pi\varepsilon_0 r} </math>

where <math> r_0</math> is the minimum [energy distance](/source/energy_distance), <math> \alpha</math> is a free dimensionless parameter and <math> \varepsilon </math> is the depth of the minimum energy.

== Beest Kramer van Santen (BKS) potential ==
The BKS potential is a [force field](/source/Force_field_(chemistry)) that may be used to simulate the [interatomic potential](/source/interatomic_potential) between [Silica glass](/source/Silica_glass) atoms.<ref>{{Cite journal|last1=van Beest|first1=B. W. H.|last2=Kramer|first2=G. J.|last3=van Santen|first3=R. A.|date=1990-04-16|title=Force fields for silicas and aluminophosphates based onab initiocalculations|url=https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.64.1955|journal=Physical Review Letters|volume=64|issue=16|pages=1955–1958|doi=10.1103/physrevlett.64.1955|pmid=10041537 |bibcode=1990PhRvL..64.1955V |issn=0031-9007}}</ref> Rather than relying only on experimental data, the BKS potential is derived by combining [''ab initio'' quantum chemistry methods](/source/Ab_initio_quantum_chemistry_methods) on small silica clusters to describe accurate interaction between nearest-neighbors, which is the function of accurate [force field](/source/Force_field_(chemistry)). The experimental data is applied to fit larger scale force information beyond nearest neighbors. By combining the [microscopic](/source/Microscopic_scale) and [macroscopic](/source/Macroscopic_scale) 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, [carbon](/source/carbon) and [silicon](/source/silicon).

The form of this interatomic potential is the usual Buckingham form, with the addition of a [Coulomb force](/source/Coulomb's_law) term. The formula for the BKS potential is expressed as

: <math>\Phi_{ij}(r) = \left[ A_{ij} \exp \left(-B_{ij}r_{ij}\right) - \frac{C_{ij}}{r_{ij}^6}\right] + \frac{q_iq_j}{r_{ij}} </math>

where <math>\Phi_{ij}(r) </math> is the interatomic potential between atom <math>i</math> and atom <math>j</math>, <math>q_i</math> and <math>q_j</math> are the charges magnitudes, <math>r_{ij} </math> is the distance between atoms, and <math>A_{ij} </math>,<math>B_{ij} </math> and <math>C_{ij} </math> are constant parameters based on the type of atoms.<ref name=":0">{{Cite journal|last1=Kramer|first1=G. J.|last2=Farragher|first2=N. P.|last3=van Beest|first3=B. W. H.|last4=van Santen|first4=R. A.|date=1991-02-15|title=Interatomic force fields for silicas, aluminophosphates, and zeolites: Derivation based onab initiocalculations|url=https://journals.aps.org/prb/pdf/10.1103/PhysRevB.43.5068|journal=Physical Review B|volume=43|issue=6|pages=5068–5080|doi=10.1103/physrevb.43.5068|pmid=9997885 |bibcode=1991PhRvB..43.5068K |issn=0163-1829}}</ref>

The BKS potential parameters for common atoms are shown below:<ref name=":0" />
{| class="wikitable"
|+BKS parameters
!i-j
!A<sub>ij</sub>(eV)
!B<sub>ij</sub>(Å<sup>−1</sup>)
!C<sub>ij</sub>(eV•Å<sup>6</sup>)
|-
|O - O
|1388.7730
|2.76000
|175.0000
|-
|O - Si
|18,003.757
|4.87318
|133.5381
|-
|Si - Si
|0
|0
|0
|-
|Al - O
|16,008.5345
|4.79667
|130.5659
|-
|Al - Al
|0
|0
|0
|-
|P - O
|9,034.2080
|5.19098
|19.8793
|-
|P - P
|0
|0
|0
|}
An updated version of the BKS potential introduced a new repulsive term to prevent atom overlapping.<ref>{{Cite journal|last1=Carré|first1=Antoine|last2=Ispas|first2=Simona|last3=Horbach|first3=Jürgen|last4=Kob|first4=Walter|date=2016-11-01|title=Developing empirical potentials from ab initio simulations: The case of amorphous silica|url=https://www.sciencedirect.com/science/article/pii/S0927025616303688|journal=Computational Materials Science|language=en|volume=124|pages=323–334|doi=10.1016/j.commatsci.2016.07.041|issn=0927-0256|url-access=subscription}}</ref> The modified potential is taken as

<math>\Phi_{12}(r) = \left[ A_{12} \exp \left(-B_{12}r_{12}\right) - \frac{C_{12}}{r_{12}^6}\right] + \frac{q_1q_2}{r_{12}} + \frac{D_{12}}{r_{12}^{24}} </math>

where the constant parameters <math>D_{ij}</math> were chosen to have the following values for Silica glass:
{| class="wikitable"
|+Parameter value for Silica glass
!
!Si - Si
!Si - O
!O - O
|-
!D<sub>ij</sub>(eV•Å<sup>24</sup>)
|3423200
|29
|113
|}

==References==
{{Reflist}}

==External links==
*[http://www.sklogwiki.org/SklogWiki/index.php/Buckingham_potential Buckingham potential] on [http://www.sklogwiki.org/SklogWiki/index.php/Main_Page SklogWiki]

{{DEFAULTSORT:Buckingham potential}}
Category:Theoretical chemistry
Category:Computational chemistry
Category:Thermodynamics
Category:Chemical bonding
Category:Intermolecular forces
Category:Quantum mechanical potentials

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Adapted from the Wikipedia article [Buckingham potential](https://en.wikipedia.org/wiki/Buckingham_potential) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Buckingham_potential?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.