# Bond order potential

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alt=Illustration of how the value of the bond order in a Tersoff-type potential shifts the potential energy minimum towards weaker bond energies and longer bond distances.|thumb|350x350px|Potential energy per bond, illustrating of how the value of the bond order in a Tersoff-type potential shifts the potential energy minimum.
'''Bond order potential''' is a class of empirical (analytical) [interatomic potential](/source/interatomic_potential)s which is used in [molecular dynamics](/source/molecular_dynamics) and [molecular](/source/molecule) statics simulations. Examples include the [Tersoff](/source/Jerry_Tersoff) [potential](/source/Tersoff_potential),<ref name="Tersoff88">{{cite journal
 | first = J.
 | last = Tersoff
 | year = 1988
 | title = New empirical approach for the structure and energy of covalent systems
 | journal = Phys. Rev. B
 | volume = 37
 | issue = 12
 | pages = 6991–7000
 | doi=10.1103/PhysRevB.37.6991|bibcode = 1988PhRvB..37.6991T | pmid = 9943969 }}</ref> the EDIP potential, the Brenner potential,<ref>{{cite journal
 | first = D. W.
 | last = Brenner
 | year = 1990
 | title = Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films
 | journal = Phys. Rev. B
 | volume = 42
 | issue = 15
 | pages = 9458–9471
 | url =http://www.dtic.mil/get-tr-doc/pdf?AD=ADA230023 | archive-url =https://web.archive.org/web/20170922092328/http://www.dtic.mil/get-tr-doc/pdf?AD=ADA230023 | archive-date =September 22, 2017 | doi=10.1103/PhysRevB.42.9458 | bibcode=1990PhRvB..42.9458B | pmid = 9995183| url-access = subscription
 }}</ref> the Finnis–Sinclair potentials,
<ref>{{cite journal
 | first = M. W.
 | last = Finnis
 | year = 1984
 | title = A simple empirical N-body potential for transition metals
 | journal = Philos. Mag. A
 | volume = 50
 | issue = 1
 | pages = 45–55
 | doi = 10.1080/01418618408244210
|bibcode = 1984PMagA..50...45F }}</ref> ReaxFF,<ref>ReaxFF:  A Reactive Force Field for Hydrocarbons, Adri C. T. van Duin, Siddharth Dasgupta, Francois Lorant, and William A. Goddard III, J. Phys. Chem. A, 2001, 105 (41), pp 9396–9409</ref> 
and the second-moment tight-binding potentials.<ref>{{cite journal
 | first = F.
 | last = Cleri
 |author2=V. Rosato
 | year = 1993
 | title = Tight-binding potentials for transition metals and alloys
 | journal = Phys. Rev. B
 | volume = 48
 | issue = 1
 | pages = 22–33
 | doi = 10.1103/PhysRevB.48.22 | bibcode=1993PhRvB..48...22C | pmid = 10006745
}}</ref>
They have the advantage over conventional [molecular mechanics](/source/molecular_mechanics) [force fields](/source/Force_field_(chemistry)) in that they can, with the same parameters, describe several different bonding states of an [atom](/source/atom), and thus to some extent may be able to describe [chemical reaction](/source/chemical_reaction)s correctly. The potentials were developed partly independently of each other, but share the common idea that the strength of a chemical bond depends on the bonding environment, including the number of bonds and possibly also [angles](/source/molecular_geometry) and [bond lengths](/source/bond_length). It is based on the [Linus Pauling](/source/Linus_Pauling) [bond order](/source/bond_order) concept 
<ref name="Tersoff88"/>
<ref>{{cite journal
 | first = G. C.
 | last = Abell
 | year = 1985
 | title = Empirical chemical pseudopotential theory of molecular and metallic bonding
 | journal = Phys. Rev. B
 | volume = 31
 | issue = 10
 | pages = 6184–6196
 |bibcode = 1985PhRvB..31.6184A |doi = 10.1103/PhysRevB.31.6184 | pmid = 9935490
 }}</ref>
and can be written in the form

:<math>
V_{ij}(r_{ij}) = V_\mathrm{repulsive}(r_{ij}) + b_{ijk} V_\mathrm{attractive}(r_{ij}) 
</math>

This means that the potential is written as a simple pair potential depending on the distance between two atoms <math>r_{ij}</math>, but the [strength](/source/bond_strength) of this bond is modified by the environment of the atom <math>i</math> via the bond order <math>b_{ijk}</math>. <math>b_{ijk}</math> is a function that in Tersoff-type potentials depends inversely on the number of bonds to the atom <math>i</math>, the bond angles  between sets of three atoms <math>ijk</math>, and optionally on the relative bond lengths <math>r_{ij}</math>, <math>r_{ik}</math>.<ref name="Tersoff88" /> In case of only one atomic bond (like in a [diatomic molecule](/source/diatomic_molecule)), <math>b_{ijk} = 1</math> which corresponds to the strongest and shortest possible bond. The other limiting case, for increasingly many number of bonds within some interaction range, <math>b_{ijk} \to 0</math> and the potential turns completely repulsive (as illustrated in the figure to the right).

Alternatively, the potential [energy](/source/energy) can be written in the [embedded atom model](/source/embedded_atom_model) form

:<math>
V_{ij}(r_{ij}) = V_\mathrm{pair}(r_{ij}) - D \sqrt{\rho_i}
</math>

where <math>\rho_i</math> is the [electron density](/source/electron_density) at the location of atom <math>i</math>. These two forms for the energy can be shown to be equivalent (in the special case that the bond-order function <math>b_{ijk}</math> contains no angular dependence).<ref>{{cite journal
 | first = D.
 | last = Brenner
 | year = 1989
 | title = Relationship between the embedded-atom method and Tersoff potentials
 | journal = Phys. Rev. Lett.
 | volume = 63
 | issue = 9
 | page = 1022
 |bibcode = 1989PhRvL..63.1022B |doi = 10.1103/PhysRevLett.63.1022 | pmid=10041250}}</ref>

A more detailed summary of how the bond order concept can be motivated by the second-moment approximation of tight binding and both of these functional forms derived from it can be found in.<ref>{{cite journal
 | first = K.
 | last = Albe
 |author2=K. Nordlund
 | year = 2002
 | title = Modeling the metal-semiconductor interaction: Analytical bond-order potential for platinum-carbon
 | journal = Phys. Rev. B
 | volume = 65
 | issue = 19
 | article-number = 195124
 | doi = 10.1103/physrevb.65.195124 |bibcode = 2002PhRvB..65s5124A }}</ref>

The original bond order potential concept has been developed further to include distinct bond orders for [sigma bonds](/source/sigma_bonds) and [pi bonds](/source/pi_bonds) in the so-called BOP potentials.<ref>{{cite journal
 | first = D. G.
 | last = Pettifor
 |author2=I. I. Oleinik
 | year = 1999
 | title = Analytic bond-order potentials beyond Tersoff-Brenner. I. Theory
 | journal = Phys. Rev. B
 | volume = 59
 | issue = 13
 | pages = 8487–8499
 | doi = 10.1103/PhysRevB.59.8487
|bibcode = 1999PhRvB..59.8487P }}</ref>

Extending the analytical expression for the bond order of the [sigma bonds](/source/sigma_bonds) to include fourth moments of the exact tight binding bond order reveals contributions from both sigma- and pi- bond integrals between neighboring atoms. These pi-bond contributions to the sigma bond order are responsible to stabilize the asymmetric before the symmetric (2x1) dimerized reconstruction of the Si(100) surface.<ref name="Kuhlmann07">{{cite journal
 | first = V.
 | last = Kuhlmann
 |author2=K. Scheerschmidt
 | year = 2007
 | title = σ-bond expression for an analytic bond-order potential: Including π and on-site terms in the fourth moment
 | journal = Phys. Rev. B
 | volume = 76
 | issue = 1
 | article-number = 014306
 | doi = 10.1103/PhysRevB.76.014306
|bibcode = 2007PhRvB..76a4306K }}</ref>

Also the [ReaxFF](/source/ReaxFF) potential can be considered a bond order potential, although the motivation of its bond order terms is different from that described here.

== References ==
<references/>

{{DEFAULTSORT:Bond Order Potential}}
Category:Computational chemistry
Category:Computational physics

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Adapted from the Wikipedia article [Bond order potential](https://en.wikipedia.org/wiki/Bond_order_potential) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Bond_order_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.